exploring polymer dynamics with single …...exploring polymer dynamics with single dna molecules a...
TRANSCRIPT
EXPLORING POLYMER DYNAMICS WITH
SINGLE DNA MOLECULES
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF PHYSICS
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Thomas Tupper Perkins
July 1997
ii
© Copyrigth 1997 by Thomas Tupper Perkins
All Rights Reserved
iii
I certify that I have read this dissertation and that in my opinion it is fully adequate,
in scope and in quality, as a dissertation for the degree of Doctor of Philosophy.
_______________________________________
Steven Chu Principal Advisor
I certify that I have read this dissertation and that in my opinion it is fully adequate,
in scope and in quality, as a dissertation for the degree of Doctor of Philosophy.
_______________________________________
Gerald Fuller
I certify that I have read this dissertation and that in my opinion it is fully adequate,
in scope and in quality, as a dissertation for the degree of Doctor of Philosophy.
_______________________________________
Robert Pecora
Approved for the University Committee on Graduate Studies
_______________________________________
iv
Abstract
We have developed physical and biochemical techniques for manipulating and
visualizing single DNA molecules enabling direct tests of polymer physics at the single
molecule level. These methods are applied to isolated polymers, to polymers in flows
and to concentrated polymer solutions.
In concentrated solutions, we stained a fluorescently labeled test chain in a
background of unstained chains and observed tube-like motion. This result confirms the
central assumption of de Gennes’ reptation theory.
We measured the entropic relaxation of single molecules and the scaling as a function of
length. These results demonstrate the importance of hydrodynamic coupling within the
chain even for extended molecules. We also measured the stretching of single tethered
polymers in a uniform flow. These detailed results agree with a bead-spring model
developed by R. Larson and establishes a quantitative agreement between single DNA
molecule experiments and classical polymer theory.
We imaged single DNA molecules in a homogeneous elongational flow in the
vicinity of a stagnation point where two channels cross. We measured the dynamic
unwinding of individual chains, their steady-state behavior and the time evolution of the
probability distribution of molecular extension as a function of strain rate. By analyzing
those molecules that reach steady-state, we report the first direct observation of the coil-
stretch transition as evidenced by a rapid non-linear increase in the steady-state extension
at a critical strain rate. However, the dynamics were complex. There was not a simple
and simultaneous unwinding as soon as the strain rate exceeded a critical strain rate. The
rate of stretching depended strongly on the molecular conformation. Configurations
having a dumbbell shape (similar to Ryskin’s yo-yo model) stretched most rapidly while
molecules with folded configurations stretched the slowest.
v
Acknowledgements
I want to thank my advisor Steve Chu for allowing me to work in his lab. He has
a clear and incisive understanding of experimental physics as well as a gift for presenting
specialized science to a more general audience. I have grown as a person and a scientist
from my interactions with him. His lab has been a stimulating and challenging
environment.
Next, I want to thank my lab-mate Doug Smith who is a co-author on all of our
papers. We have spent countless hours working together over the last 5 years and his help
has been invaluable. Most of all, it is a pleasure to work so long and so closely with a
friend.
Jim Spudich has been very supportive of my work. A collaboration has allowed
me to work in his lab and this collaboration has been crucial to the timely and successful
nature of my graduate school career. The members of the biochemistry department and,
in particular, the members of the Spudich lab have been friendly and unselfishly generous
with their time and guidance. I am particularly indebted to Hans Warwick, Holly
Goodson, Glen Nuckolls and Jeff Finer.
I want to thank Steve Quake who I collaborated with on the first experiment. He
initiated our work into polymer physics and is responsible for kindling my interests in
polymer dynamics.
I want to thank Ron Larson for innumerable long phone and email conversations,
which were crucial for my developing understanding of polymer physics. I have also
enjoyed my valuable interactions with Bruno Zimm at UCSD. I appreciate the time and
advice Robert Pecora and Gerry Fuller have contributed. They have been particularly
helpful making sure we were aware of and presenting our results within the broader
context of polymer physics literature.
I would like to thank all of the members of the Chu lab for many conversations
and in particular Heun Jin Lee, Achim Peters, and Brent Young. I also want to thank my
vi
original housemates Heun Jin Lee, Tony Loeser and John Uglum for some great times
and wonderful conversations.
I especially want to thank Marcia Keating who has been wonderfully helpful. Her
enthusiasm and warmth has made the physics department a more enjoyable community.
Outside of science, I have had two longstanding friends in the bay area, Thomas
Gewecke and Keao Caindec. I have enjoyed our many wide ranging conversations and
their unwavering friendship over the last six years. I am also grateful to my friends Frank
Huerta, Daniele Schecter, Brian Trelstad, Ben Eiref, and Jacob Farmer.
Finally, I want to thank my family for their love and support. My brother John,
my sister-in-law Marla and my sister Katherine have helped me through the tough times
and celebrated with me when times were good. My parents have been particularly
generous with their unwavering support both emotionally and financially.
vii
Contents
CHAPTER 1 INTRODUCTION..................................................................................... 1
1.1 DIRECT IMAGING OF SINGLE MOLECULES AS A MEASUREMENT TECHNIQUE ............. 1
1.2 DNA AS A MODEL POLYMER.................................................................................... 2
1.3 OVERVIEW OF THESIS............................................................................................... 4
CHAPTER 2 EXPERIMENTAL TECHNIQUE........................................................... 6
2.1 SAMPLE PREPARATION ............................................................................................. 6
2.1.1 Attaching microspheres to λ-DNA.................................................................. 6
2.1.2 Making dimers of λ-DNA.............................................................................. 18
2.1.3 Making long DNA ......................................................................................... 21
2.2 STAINING AND SOLUTIONS ..................................................................................... 23
2.2.1 Fluorescent stains ......................................................................................... 23
2.2.2 Solution ......................................................................................................... 26
2.2.3 Staining ......................................................................................................... 29
2.3 EXPERIMENTAL EQUIPMENT................................................................................... 29
2.3.1 Microscope and Imaging .............................................................................. 30
2.3.2 Motors ........................................................................................................... 33
2.3.3 Pumps and Plumbing .................................................................................... 34
2.3.4 Micro-fabricated flow cells........................................................................... 35
CHAPTER 3 RELAXATION OF SINGLE POLYMER CHAINS ........................... 45
3.1 INTRODUCTION ...................................................................................................... 45
3.2 MANIPULATION AND VISUALIZATION OF SINGLE MOLECULES............................... 46
3.3 RELAXATION MEASUREMENTS .............................................................................. 48
3.4 ANALYSIS .............................................................................................................. 50
3.4.1 Inverse Laplace transform ............................................................................ 50
viii
3.4.2 Rescaling relaxation curves: Data collapse ................................................. 53
3.5 DISCUSSION ........................................................................................................... 54
3.6 FUTURE PROSPECTS ............................................................................................... 55
CHAPTER 4 CONCENTRATED POLYMER DYNAMICS: REPTATION .......... 57
4.1 INTRODUCTION ...................................................................................................... 57
4.2 MATERIALS AND METHODS ................................................................................... 58
4.3 MEASUREMENT OF TUBELIKE MOTION................................................................... 60
4.4 EFFECT OF THE MICROSPHERE ON THE BACKGROUND CHAIN.................................. 61
4.5 OBSERVATION OF TUBELIKE MOTION .................................................................... 64
4.6 CONSTRAINT RELEASES, EXCESS CHAIN SEGMENT DIFFUSION AND TUBE
DEFORMATION ............................................................................................................... 64
4.7 CONCLUSIONS........................................................................................................ 66
CHAPTER 5 SINGLE POLYMERS IN A UNIFORM FLOW ................................. 70
5.1 INTRODUCTION ...................................................................................................... 70
5.2 APPLYING UNIFORM FLOW TO DNA....................................................................... 71
5.3 INITIAL MEASUREMENTS (22µM < L < 84 µM) ....................................................... 73
5.4 LONGER CHAINS (L < 150 µM)............................................................................... 82
5.5 CONCLUSIONS........................................................................................................ 85
CHAPTER 6 SINGLE POLYMERS IN A ELONGATIONAL FLOW ................... 86
6.1 INTRODUCTION ...................................................................................................... 86
6.2 PREVIOUS EXPERIMENTAL WORK ........................................................................... 87
6.3 EXPERIMENTAL TECHNIQUE .................................................................................. 90
6.3.1 Measurement of τ1......................................................................................... 91
6.4 EXPERIMENTAL RESULTS....................................................................................... 94
6.4.1 Data reduction .............................................................................................. 94
6.4.2 Extension versus residency time ................................................................... 97
6.4.3 Average and steady-state properties........................................................... 101
6.4.4 Steady-state measurements fit by the dumbbell model ............................... 104
6.4.5 Conformational dependent dynamics at highest strain rates ..................... 105
ix
6.4.6 Master curves.............................................................................................. 108
6.4.7 Affine deformation ...................................................................................... 112
6.4.8 Dynamic data, intrinsic viscosity and the dumbbell model ........................ 114
6.4.9 Comparison to previous experimental work ............................................... 117
6.4.10 Limitation of applicability........................................................................... 127
6.5 SUMMARY............................................................................................................ 127
6.6 FUTURE PROSPECTS.............................................................................................. 128
CHAPTER 7 FUTURE PROSPECTS........................................................................ 131
7.1 DILUTE SOLUTIONS .............................................................................................. 131
7.1.1 Relaxation ................................................................................................... 131
7.1.2 Stretching in a uniform flow ....................................................................... 131
7.1.3 Polymers in shear flow................................................................................ 132
7.1.4 Mixed flows ................................................................................................. 133
7.1.5 Dynamic force measurements ..................................................................... 133
7.2 CONCENTRATED SOLUTIONS ................................................................................ 134
7.2.1 Constrain release ........................................................................................ 134
7.2.2 Step-strain experiments............................................................................... 135
7.2.3 Flows of concentrated solutions ................................................................. 135
7.2.4 Star molecules............................................................................................. 135
BIBLIOGRAPHY:........................................................................................................ 136
x
List of Tables
Table 1: Sequences. ........................................................................................................... 9
Table 2: Ratio of hydrodynamic drag .............................................................................. 83
xi
List of Figures
Figure 1: Oligonucleotides ............................................................................................... 11
Figure 2: Coupling of polystyrene microspheres to DNA molecules............................... 11
Figure 3: Making dimers of λ-DNA. ................................................................................ 19
Figure 4: Concatemers of λ-DNA..................................................................................... 21
Figure 5: Photostability of DNA....................................................................................... 25
Figure 6: Schematic of optical layout for fluorescence imaging. ..................................... 29
Figure 7: Diagram of pump and tubing the for elongational flow experiment................. 32
Figure 8: Calibration of elongational flowcell.................................................................. 33
Figure 9: Flow cell for elongational flow experiment. ..................................................... 38
Figure 10: Velocity as a function of depth in the inlet channel. ....................................... 39
Figure 11: Velocity along the outlet channel.................................................................... 40
Figure 12: Velocity along the inlet channel...................................................................... 41
Figure 13: The center of mass motion of an individual DNA .......................................... 43
Figure 14: Schematic of the relaxation apparatus............................................................. 47
Figure 15: Relaxation of a single molecule of DNA ........................................................ 48
Figure 16: Relaxation of several different length molecules ............................................ 49
Figure 17: Spectrum of decaying exponentials ................................................................ 52
Figure 18: Data collapse of relaxation curves .................................................................. 53
Figure 19: Schematic representation of the reptation experiment. ................................... 59
Figure 20: A series of tiled images showing the tube-like motion ................................... 60
Figure 21: Loop formed in viscous Newtonian ............................................................... 61
Figure 22: Loop formed in concentrated polymer ............................................................ 62
Figure 23: Relaxation in dilute and concentrated polymer solutions ............................... 65
Figure 24: Stretching in a uniform flow ........................................................................... 72
Figure 25: Extension versus the viscosity times velocity ................................................ 74
Figure 26: Extension compared to worm-like chain force law......................................... 75
Figure 27: Extension versus velocity for different lengths............................................... 77
xii
Figure 28: Fluctuations in extension................................................................................. 78
Figure 29: Time-averaged images .................................................................................... 80
Figure 30: Average chain conformation ........................................................................... 81
Figure 31: Rescaling of extension versus veloctity .......................................................... 84
Figure 32: Dumbbell model .............................................................................................. 87
Figure 33: Crossed slot .................................................................................................... 88
Figure 34: Schematic of elongational flow experiment. ................................................... 89
Figure 35: Relaxation of 14 individual DNA molecules averaged together..................... 93
Figure 36: Illustrations of the seven main configurations observed. ................................ 94
Figure 37: Extension versus accumulated fluid strain ε tres ............................................. 95
Figure 38: Normalized probability distributions .............................................................. 96
Figure 39: Two molecules stretching at the same time..................................................... 98
Figure 40: Free energy curves for a simple dumbbell model ........................................... 99
Figure 41: Asymmetric probability distribution. ............................................................ 100
Figure 42: Summary of elongational flow data. ............................................................. 102
Figure 43: Molecules stretching to steady-state.............................................................. 103
Figure 44: Dynamic unwinding of different conformations ........................................... 106
Figure 45: Pie charts showing the distribution of conformations at two strain rates...... 107
Figure 46: Average extension versus tres for different conformations ............................ 108
Figure 47: Comparison between dumbbell and folded configurations........................... 109
Figure 48: Individual traces of extension versus tres . ..................................................... 110
Figure 49: Dynamics of molecules that best-typify dumbbell molecules....................... 111
Figure 50: Comparison of master curves between the dominant ................................... 111
Figure 51: Rates of deformation and normalized rates of deformation for all ε . .......... 112
Figure 52: Rates of molecular and fluid deformation..................................................... 113
Figure 53: Dynamics of dumbbell model compared to measured dynamics.................. 116
Figure 54: Steady state measurement versus birefringence data for λ-DNA ............... 118
Figure 55: Radius of gyration ......................................................................................... 119
Figure 56: Single DNA molecule in a shear with 1τγ ≅ 8 at η = 41 cp. ........................ 132
1
Chapter 1 Introduction
Polymers, a long chain of simple molecular subunits, are central to modern
chemical industries. In everyday life, we tend to think of polymers as synthetic material
such as plastics, rubbers and adhesives. But, DNA is also an important example of a
polymer. Polymers are in our cells not only as the genetic code (DNA) but also as the tiny
protein ropes by which our muscles contract (actin). Indeed all proteins are polymers
whose initial step in folding from a highly extended step into a partially extended state is
often modeled as an entropic collapse.
Understanding the physical properties of dilute and concentrated solutions of
polymers has long been a major goal of basic and applied research in physics, chemistry,
chemical engineering, and material science [1-5]. However, it has been hard to test
polymer theories directly against experiments. While theorists have modeled the motion
of individual polymer chains, tradition polymer measurements must average over a huge
number of molecules (109-1023). This thesis focuses on the development of techniques to
visualize and to manipulate single DNA molecules. These techniques then provide a
method to test the basic postulates and predictions of polymer dynamics in both dilute
and concentrated solutions.
Imaging single molecules eliminates five important experimental limitations of
classical techniques. (i) Direct imaging yields the full conformation of individual
polymers. (ii) We can observe the full time evolution of individual polymers by tracking
single polymers. (iii) For dilute solutions, we use single, isolated molecules which
eliminates polymer-polymer interactions as well as polymer induced alterations of the
flow field. (iv) With optical tweezers, polymers can be prepared in a variety of initial
states including states far from equilibrium. (v) The inherent uniformity in size of
lambda bacteriophage DNA (λ-DNA, Lstained ≅ 22 µm) eliminates the complications due
to polydispersity and enables accurate calculation of ensemble averages.
1.1 Direct imaging of single molecules as a measurement technique
As outlined above, direct observation of individual molecules provides a simple
and unique method for testing models of polymer dynamics. Visualization of single
2
molecules is generally done with video enhanced fluorescence microscopy. Such
measurements require an intensified camera and a high numerical aperture (NA > 1.2)
microscope objective.
In this chapter, we focus on the deformation of polymers in flow. But in general,
optical tweezers [6] are used to manipulate micron-sized microspheres [7-14] although
some work is done with magnetic microspheres [15] or electric fields [16-20]. The
susceptibility of the DNA to the applied electromagnetic field from the laser is too small
to be directly manipulated using optical tweezers. Therefore polystyrene microspheres
(~1 µm) are attached to one or both the ends of the molecules. These spheres are the
handles by which the molecule is manipulated.
Direct observation and manipulate of single molecules is not limited to
fluorescently labeled DNA or to polymers. These techniques are making important
contributions in systems as diverse as colloidal suspension [11, 21] and molecular motors
[12-14]. For instance, Crocker and Grier measured the interaction potential between
isolated pairs of charge colloidal spheres [11] while Finer et al [12] measured the force
and step size of a single myosin molecule on an actin filament, the fundamental
interaction that leads to muscle contraction.
1.2 DNA as a model polymer
There has been a considerable recent work based around visualizing and
manipulating single DNA molecules [8-10, 14-20, 22-32]. DNA acts as a scaled up
version of the smaller, more flexible synthetic polymers. Locally, it is quite stiff with a
persistence length p of 50 nm [33] and has a hydrodynamic diameter of ~2 nm [34, 35].
But, for sufficiently long chains, polymers act like a flexible chain and the entropic and
hydrodynamic effects dominate. In this limit, many properties of polymers can be
described by three variables: the contour length of the chain, the persistence length, and
the quality of the solvent, a description of monomer-solvent interaction [1, 2]. For the
results presented in this work, L/p ranges from 400 to 3,000 and measurements indicate
that aqueous solutions are a good solvent for DNA [30].
Initially, the interpretation of single DNA molecule research relied upon this
universality of polymer dynamics when L/p >> 1 [1, 2, 8, 9]. However, the interpretation
3
of single DNA molecule data as a representation of polymers in general is now grounded
in quantitative agreement between experiment and theory [22, 24, 36]. For instance,
Smith et al measured non-linear elasticity of single DNA molecules [15] while
Vologodoski [32] and Marko and Siggia [22, 25] showed that this data corresponded to
the elasticity of a worm-like chain. Marko and Siggia [25] provided a simple analytical
approximation to this non-linear elasticity that is a function only of the fractional
extension and persistence length:
⎭⎬⎫
⎩⎨⎧ −−+•= −
41)1(
41)()( 2
Lx
Lx
pTk
LxF b
MS Eq. 1
where x is the end-to-end extension. Furthermore, in collaboration with R. Larson, we
have established a quantitative agreement between a bead-spring model with self-
consistent hydrodynamics [24] and our previous measurements of a tethered polymer
stretching in a uniform flow [10]. Zimm has also developed a model that describes the
stretching of a tethered polymer in a uniform flow [10]
DNA molecules have three unique properties over standard synthetic molecules:
length, uniformity and ease of manipulation.
First and foremost is length and, importantly, size relative to the wavelength of
visible light. DNA molecules of up to 100 µm can be routinely manipulated. We use λ-
bacteriophage DNA which consists of 48,502 base pairs giving a length of L = 16.3 µm
for an unstained molecule or concatemers of λ-DNA. Stained λ-DNA is ~22 µm long
due to the intercalation of the dye molecules in to the double helix structure [10]. With
polymers this long, both its extension and general features of its internal conformation are
observable using standard video enhanced microscopy [9, 28].
Second, DNA is inherently monodisperse. Enzymatic replication guarantees that
each copy of the λ-DNA is exactly the same size. For longer molecules, concatemers of
λ-DNA can be made or alternative bacteriophages (T2, T4, T5) can be used. However,
for the experiments described here, we have found commercial sources of T2, T4 or T5
DNA were not adequately monodisperse. Presumably, this polydispersity was caused by
shearing and we did not have an independent method to measure the length of each
4
molecule. So, given the diversity of the dynamics that will be discussed in chapter 6, we
choose to limit our study to the dynamics of λ-DNA. Shorter lengths of DNA can be
generated using restriction enzymes. Flexibility is perhaps DNA biggest advantage for
studying polymer dynamics over another important bio-polymer, actin. Nevertheless,
actin’s offers the advantage that is very stiff (p = 8-20 µm) and its polymer properties
have been extensively studied at the single molecule level [37].
In addition, we utilize a number of advances in molecular biology. DNA
molecules can be “cut and pasted” together and purified from contaminating protein and
DNA on a routine basis. DNA can be stained with highly efficient fluorescent dyes.
Such bright dyes enable single molecules to be visualized [26, 38]. Furthermore, these
dyes allow for the observation of a single, stained chain in a concentrated solution of
unstained DNA molecules enabling tests of concentrated polymer solutions. Finally, the
ends of DNA can be functionalized to facilitate the coupling of DNA molecules to
polystyrene beads. Typically this coupling is done via a protein-ligand bond.
1.3 Overview of thesis
In chapter 2, we give a detailed report on the experimental technique, including
protocols for attaching DNA molecules to polystyrene microspheres, methods for making
microfabricated flow cells, and techniques for calibrating the fluid flow. We also discuss
the design criterion for the flow cells as well as the benefits and limitations of different
fluorescent dyes.
In chapter 3, we demonstrate our technique by visualizing the relaxation of single
polymers from near full extension [9]. We determined the longest relaxation times as a
function of length and showed that this relaxation time obeyed a simple scaling law
(τ~L1.66). This scaling exponent agreed more closely with the scaling exponent predicted
by the Zimm model [39], which includes hydrodynamic interaction within the chain, than
the Rouse model [40], which neglects such intramolecular interaction. This is in
disagreement with the expectation that the Rouse model is a better description than the
Zimm model for extended chains [41].
In chapter 4, we test the central assumption of the de Gennes reptation theory for
concentrated polymer solutions. de Gennes postulated that polymers would be confined
5
to a tube due to topical constraints of the surrounding polymers. Any one polymer moves
by a one-dimensional diffusion within the imaginary tube. From this assumption, he
predicted that the characteristic time separating the elastic and viscous behavior of these
solution would scale with length as τ ~ L3. However, experiments measure τ ~ L3.4 [42].
This difference in the scaling exponent has been used to argue that the fundamental
conjecture of tube-like motion in concentrated solution is invalid [42]. By visualizing
and manipulating single, fluorescently-stained DNA molecule in a background of
unstained DNA molecules, we demonstrated the molecule undergo tube-like motion and
confirmed the central assumption of the reptation theory [8].
In chapter 5, we continue our exploration of dilute polymer dynamics. We
extended tethered polymers in a uniform flow and measured their extension as a function
of applied velocity and length. These results further emphasize the importance of
hydrodynamic interaction within the chain and provide experimental results in a highly
simplified geometry against which theories can be tested.
In chapter 6, we study the dynamics of single, isolated polymers in an
elongational flow, a model for turbulent drag reduction. Our results reveal that the
concept of a discrete and abrupt transition from a coiled to an extend state is limited to
the steady-state. Further, our results reveal several distinct conformations with differing
dynamics. And, our results show that previous light scattering and birefringence
measurements at the stagnation point do not measure steady-state properties of the
polymer.
In chapter 7, we cover some future prospects for using single DNA molecules to
explore polymer dynamics, including both dilute and concentrated polymer solutions.
6
Chapter 2 Experimental Technique
In this chapter, we present the experimental techniques used in the visualization
and manipulation of single DNA molecules. The first section focuses on sample
preparation while the second section focus on the experimental equipment.
2.1 Sample preparation
In this section, we discuss the technical aspects of preparing samples. In section
2.1.1, we discuss attaching polystyrene microspheres to one or both ends of the DNA.
These microspheres act as the handles by which we manipulate the DNA because the
optical forces by a focus laser beam is too small to allow it to be directly manipulated. In
section 2.1.3, we present an efficient method for making dimers of λ-DNA. Longer DNA
molecules are desirable because their dynamics are slower and allows the dynamics to be
studied as a function of length. In section 2.1.4, we give a method for creating very long
concatemers of λ-DNA. DNA molecules of up to 200 µm have been constructed and
attached to microspheres.
For the DNA, we exclusively used lambda-bacteriophage DNA (λ-DNA). λ-
DNA is 48,502 bp long, which leads to an unstained contour length L of 16.3 µm. λ-
DNA serves as an excellent substrate for several reasons: (i) Its contour length L is much
longer than its persistence length p (50 nm). In this regime of L >> p, λ-DNA’s
dynamics are governed by statistical mechanics and hydrodynamics not the details of its
local chemical structure. (ii) The ends of λ-DNA are differentiated, which allows us to
selectively attach a microsphere at each end. DNA from T2, T4, T7 is blunt ended. (iii)
λ-DNA is the longest commercially available DNA that is monodisperse – the molecules
are all the same length.
2.1.1 Attaching microspheres to λ-DNA
2.1.1.1 Choice of microspheres
We used polystyrene microspheres. They are commercially available,
monodisperse, and come with a variety of functional groups on the surface. These
microspheres are stabilized in solution by a large surface charge density. If the salt
7
concentration is too high, clumping of the microspheres into large aggregates occurs. To
help prevent clumping, we add 0.1% Tween-20. This also helps prevent the microspheres
from sticking to the glass slides and coverslips.
One major drawback to polystyrene microspheres is their absorption of
fluorescent dye. When viewed in fluorescence, these beads become bright balls of
fluorescence next to the dimmer DNA. An ideal microsphere would not absorb dye, be
monodisperse, and attach easily to DNA. Silica microsphere absorb significantly less dye
that polystyrene microspheres but, in preleminary trials, we were not able to reliably
couple DNA to these microspheres nor did we find an adequate supplier of monodisperse
silica microspheres.
An alternative to polystyrene microspheres, one can couple the DNA to gold
microspheres. These microspheres typically have diameters < 40 nm and come coated
with or without streptavidin. Preliminary work showed DNA was successfully attached
to both types of microspheres but we did not use them for the trapping experiments due to
inability to trap gold microspheres of this size.
Thus, polystyrene microspheres offered aquick, reliable bonding technique.
However, the use of these polystyrene microspheres still left the problem of absorption of
the fluorescent dye by the microsphere. Smaller microspheres have less surface area,
which leads to less fluorescence. But, decreasing the size of the microspheres leads to
weaker trapping.
In the beginning, we used 1.0-µm microspheres. But, after our initial proof-of-
principle work [8, 9], we found that this sphere of fluorescence near the DNA was
unacceptably bright. Consequentially, we investigated smaller microspheres. The
smallest microspheres that we were able to trap with the Zeiss 63×, 1.4 objective were
0.15 µm. However, these 0.15-µm microspheres trapped too weakly to be of general use.
Therefore, we adopted 0.3-µm microspheres as the minimum standard size for
manipulating λ-DNA in aqueous solutions. For manipulating longer DNA or
manipulating DNA in concentrated solution, we used 0.5-0.6 µm microspheres. An
unexpected benefit of smaller microspheres is their reduced probability of binding
multiple DNA molecules.
8
2.1.1.2 Possible coupling techniques
There are several possible different methods to attach DNA to microspheres. We
currently couple the DNA to microspheres via a streptavidin-biotin bond. Streptavidin is
a protein which binds its receptor, biotin, with a binding constant of KD ≅ 10 -15 . This
coupling technology is mature and widely used. It yields a fast, efficient and almost
permanent bond. The preparation of short synthetic oligonucleotides is also a mature
technology and oligonucleotides can be purchased with a wide variety of chemical groups
at the end including primary amines (-NH2), thiols (-SH), and biotin as well as
fluorescent dyes.
Ideally, the permanency of covalent coupling is desirable. Initially, we
investigated several covalent coupling methods. Maleimides couple a thiol group to a
primary amine and carbodimides (EDC) couple a carboxylic acid (-C00H) group to a
phosphate (P04). While the coupling of the DNA to the microspheres was verified by
radioactive assays, the microspheres clumped into large aggregates that were
unacceptable. This chemistry has not been revisited since the early trials in the spring of
1991.
Another possible covalent coupling technique that has not been actively pursued
is the coupling of DNA to activated microspheres. Activated microspheres come with
highly reactive groups on the surface. These microspheres simply need to be mixed with
the target molecules. For instance, we couple streptavidin to aldehyde (-C0H)
microspheres via the primary amines on the surface. This reaction is effective and stable
as long as the solution is not too basic (pH < 9). Presumably, the primary amines on
oligonucleotides should couple to these microspheres efficiently. The length of the
flexible carbon–carbon spacer arm or the number of non-hybridizing base pairs may need
to be optimized. There are also epoxy microspheres as well as the proprietary coupling
chemistry developed by Duke Scientific. Note, when using any of these methods, the
chemical structure of all components of the solution needs to be known. For instance, if
one tried to couple streptavidin to aldehyde-coated microspheres in a Tris-HCl buffer, it
would not work because Tris has a primary amine and its molar concentration is several
orders of magnitude higher than the protein. Note that if the synthetic oligonucleotides
are not part of double stranded DNA, the bases adenine, guanine, and cytosine have
9
amine groups. Therefore, any spacer arm away from the double stranded DNA should be
thymine. Our current oligonucleotides do not use thymine as a spacer but instead use
adenine.
2.1.1.3 Streptavidin-biotin coupling
Streptavidin is a multimeric protein consisting of four identical domains, each of
which binds a biotin molecule. It is a remarkably stable protein and can be subjected to a
wide variety of pH, salt concentrations, and temperatures without denaturing. We have
found that the streptavidin-biotin bond is stable at 4 °C for several months. At elevated
temperatures, it is stable for at least several days but we have not tested its long term
stability.
Recent work with atomic force microscopes have measured the rupture strength
of streptavidin to be 160 pico-Newtons (pN), which is significantly smaller than forces
exerted on the DNA (< 20 pN) by our optical trap.
2.1.1.4 λ-DNA and oligonucleotides
Name Sequence (5’ - to – 3’)
λ-end-A GGGCTGCGACCT
λ-end-B AGGTCGCCGCCC
BA Biotin-AAAACTGTGCTACGACGGCT
SA Phosphate-AGGTCGCCGCCCAGCCGTCGTAGCACAG
BB Biotin-AAAATCTGCTACTGTGAAGG
SB Phosphate-GGGCGGCGACCTCCTTCACAGTAGCAGA
Table 1: Sequences for the cohesive ends of λ-DNA, the linking oligonucleotides (SA,
SB), and the biotinylated oligonucleotides (BA, BB).
λ-DNA has complimentary, 12 bp overhangs at each end. Therefore, it can
circularize to form concatemers depending on the DNA’s concentration and the ionic
condition of the solution. These 12 bp overhangs are distinctly different than the smaller
10
overhangs generated from restriction enzymes; they are not palindromic. Thus, with
suitably constructed oligonucleotides, the ends can be differentiated and circulization
prevented. In Table 1, we list the sequences of the λ-DNA’s cohesive ends, the
oligonucleotides used to differentiate the ends, and the biotinylated oligonucleotides.
These sequences were designed, using OLIGO 4.0, to minimize any intra-molecular or
inter-molecular base pairing common to single-stranded DNA.
We purchase our λ-DNA from either Gibco/BRL or NEB. The reported stock
concentration of the DNA is ~500 µg/ml. This concentration was verified by measuring
the optical density at 260 nm and 280 nm. For reference, 500 µg/ml of λ-DNA
corresponds to a concentration of 17 nM assuming 650 daltons per double-stranded
basepair. The effective intra-molecular concentration of one cohesive end to the other is
0.6 nM (~ 3−GR ).
Note, when handling DNA or DNA containing solutions we use cut or wide-bore
pipette tips to minimize shearing of the DNA. DNA is more susceptible to shearing at
dilute concentrations so keeping the concentration high reduces shearing. In general, λ-
DNA does not shear with reasonable handling. However, it is a good idea to routinely
run gels to and, occasionally, look at the DNA under a microscope just to make sure.
Concentrated λ-DNA should not be heated over 65°C. Several times at ~70°C
we found that the DNA had become hopelessly entangled. Since this temperature is
below the temperature for complete melting of the two strands, we postulate that local
melting occurred followed by hybridization between chains. This entanglement is easily
diagnosed by staining the stock λ-DNA with picogreen or ethidium bromide and
inspecting the sample by eye with an optical microscope. If the sample is uniformly
distributed, the DNA is fine. However, if there are clusters analogous to a pom-pom then
the DNA should be discarded.
2.1.2 Basic scheme The basic scheme to attach DNA to microspheres is to first ligate on
oligonucleotides (SA,SB) to λ-DNA to differentiate each end. Next, biotinylated
oligonucleotides are attached to streptavidin microspheres (Figure 1). Finally, these
microspheres are mixed with the modified λ-DNA, which leads to a hybirdization
11
between the oligonucleotides on the microspheres and the modified ends of the λ-DNA
(Figure 2).
λ-DNA
SA
SBBA
BB
Figure 1: Oligonucleotides for creating microsphere-DNA-microsphere complexes. A 16 bp overhang is used to attach the modified λ-DNA to the biotinylated oligonucleotides attached to the streptavidin coated microspheres.
Streptavidin
Aldehyde-sulfateLatex bead
B
Biotinylated oligonucleotide
λ-DNA
Figure 2: Coupling of polystyrene microspheres to DNA molecules. Single DNA molecules are attached to the microspheres via a streptavidin-biotin bond.
2.1.2.1 Protocol
A detailed protocol for attaching microspheres to λ-DNA is given. The relevant
buffers are listed at the end of this section.
Ligating oligonucleotides to λ-DNA -- Be careful with the DNA and do not heat
the DNA higher than 65°C. Occasionally, overheating leads to huge “starbursts” of
DNA stuck together (see sect. 2.1.1.4). If you suspect this, stain the DNA and put it
under the microscope with perhaps 1,000 molecules per field of view. The DNA
solution should be uniform. If the DNA is entangled, it is best to start with fresh DNA.
0) Pre-heat aluminum heating block to 65 °C.
12
1) Add:
270 µl ddH20
40 µl 10X ligase buffer (see below)
40 µl 20 µM 5’ phosphoralated oligonucleotides
In an eppendorf tube. Vortex.
2) Add:
50 µl λ-DNA (GIBCO-BRL cat# 25250-028, 500 µg/ml, 17 nM)
Gently pipette whole solution three times slowly with a cut pipette tip.
3) Heat at 65 °C for 5 min, remove heating block from heater and let cool to room
temperature. Generally, we let this solution sit at room temperature for several hours (3-
6) ostensibly to allow the oligonucleotides to attach to the DNA for a longer time.
We typically start the ligation immediately before going home for the evening. We do
not worry about the small amount of liquid that condenses on the top of the eppendorf
tube.
4) Add:
2 µl Ligase (NEB)
4 µl 0.1 M ATP
Mix gently with pipette until the index of refraction mismatch disappears (3-5
times). Put at 16°C overnight. There are lots of temperatures and durations one can use
for ligations. This is the combination that we use.
5) Heat to 65°C for 10 min. Let cool by putting eppendorf tubes in a standard
rack at room temperature. This kills the ligase, and hopefully, unhybradizes any non-
ligated oligonucleotides, which are removed in the next step.
Remove unbound oligonucleotides -- The idea here is that the oligonucleotides
(28 bp) are much smaller and diffuse much more rapidly than the DNA. So, we need to
quantitatively remove them from solution or, otherwise, they will block attachment of the
modified λ-DNA to the microspheres. There are several possible methods including spin
columns, dialysis, and/or gel recovery. We use a laborious method of
microconcentrators (Centricon-100, Amicon) but it is very reliable and also works well
for large concatemers of λ-DNA. This protocol is conservative but we want to be very
13
sure that all the excess oligonucleotides are removed. Amicon also makes
microconcentrators for standard bench-top microcentrifuges that work equally as well.
1) Set centrifuge for 1000g @ 25°C (2700 rpm in a JA20 or SS-34 rotor).
Centrifuging at room temperature significantly speeds up the spins. Do not wait for the
centrifuge to come up to temperature, just start and within an hour or so it will be up to
19°C. It takes a very long-time (3 hr) to come up to 25°C.
2) To prevent non-specific sticking of the DNA to the centricons, we pre-spin the
centricons with TE and BSA. Add 1 ml. TE to each centricon. Then add, pre-mixed 1ml
TE and 1 µl 100X BSA (10 mg/ml). You can probably get away with less BSA or just
pre-rinsing the centricons. The only potential pitfall is that if you do not purify enough of
the BSA away it can precipitate when you heat the retentate (DNA) to 65°C. This does
not harm anything but it is unnerving to see.
3) Spin 3x for about 15 min. each time filling each centricon up to 2 ml with TE.
There should be about 40-100 µl of liquid in the retenate of the centricon.
4) Fill centricon with 1 ml TE, add ligation solution and then fill up to 2 ml. with
TE. Spin 7-10 X for 10-12 min repeatedly filling to 2 ml with TE. This does not spin
down to 40 µl but allows for more rapid spins. When adding TE to the centricon, hold
the centricon at about 80 degrees from vertical so the TE is gently added to the retentate.
The basic test for how successfully you have removed oligonucleotides is if the DNA will
attaches to the microspheres. If they do not, it is possible that you did not remove
enough. Since this takes a couple hours to figure out and test, we error on the side of
caution.
5) Add the cup to the top of the centricons, flip them over and spin at 1000 rpm
for 5-10 min. Pipette of the retentate in to an eppendorf tube. Repeat the process for
ligating on to the other end if necessary. Or, if we are attaching the DNA to one
microsphere, we dilute the final volume down into a total volume of 400-1000 µl
depending on the diameter of the microspheres -- the microspheres are at a fixed
concentration ( 1%). Thus, the large the microspheres the more dilute the DNA can be
due to the lower number density of the microspheres.
6) Optionally, heat the DNA to 65°C for 5 min to make sure any non-ligated
oligonucleoties are not hybradizing. This all depends on your application.
14
7) Run a gel to verify that you have DNA and that it is monodisperse. Generally,
80 -90% of the original DNA is recovered.
Attaching streptavidin to microspheres -- This is very simple. The only thing that
can go wrong is that the microspheres clump together. Typically about 1 microsphere in
40 or so will be a clump. To check, just look at then with a microscope in brightfield. If
the clumping is worse than 1 in 10 then try again. Aldehydes attach to amines on the
protein so this works great unless you have other amines around. For instance, Tris has
an amine so that is why the reaction is done in water. You can also do it in phosphate
buffer. We store the streptavidin in water and it does fine for about a 6 months. If any
precipitate forms, spin at full speed in a microfuge for 15-20 min and save the
supernatant.
1) Add:
750 µl ddH20
40 µl streptavidin ( ∼20 mg/ml, Prozyme)
Vortex. Then add:
250 µl 4% 0.6 µm aldehyde-sulfate microspheres (Interfacial Dynamics)
And vortex. Leave at 4°C for 5 hrs. This is excess protein but it works fine.
2) Divide into two eppendorf tubes, add 500 µl SB-0 to each. Vortex, and spin
down at 8,000 rpm until pelleted at room temperature in a standard microfuge (about 3-4
min)
3) Repeat two more time re-suspending each time in to 200 µl SB-0. Pipette
vigorously to break up the pellet but without introducing lots of bubbles. Vortex briefly.
4) After the last spin re-suspend into 500 µl SB-50. Check under the microscope
that no clumping has occurred. Oligonucleotides are reasonably expensive so ther is no
reason to proceed to the next step of the microspheres are already clumped.
Attaching biotinylated oligonucleotides to streptavidin coated microspheres -- For
microspheres of different diameters, we always work with 1% microspheres and scale the
amount of oligos as the total surface area. We remove the excess oligonucleotides
immediately before using the microspheres.
1) Add:
500 µl 1%, 0.6 µm streptavidin, coated microspheres in SB-50
15
40 µl 2 mM biotinylated oligonucleotides
Vortex and let sit overnight at room temperature or 4°C.
Attaching DNA to “hairy” microspheres --Remember that these oligonucleotides
diffuse much faster than those attached to the microspheres and it is important to have
any excess ones removed. As mentioned before, excess oligonucleotides block the
attachment of the modified λ-DNA to the microsphere. The off-rate from streptavidin is
of order 1 month and there are about 100,000 streptavidins on a microspheres. So,
without any reattachment, there could be 3,000 oligonucleotides falling off per day. Of
course, this is not what really happens but it is an important point to remember. We
always wash the microspheres before each use. We store them in SB-50 at 4 C.
Attaching microspheres to DNA will vary from batch to batch of DNA and also
depends on the Na+ concentration. We typically have no problems working at 50 -200
mM Na+. If problems develop, the Tween-20 concentration can be raised to 0.2%.
Divalent ions (Mg2+) easily clump microspheres so avoid them. Higher salt
concentrations lead to the faster kinetics of DNA-microsphere attachment. Generally, we
get increasing attachment with time for the first 12-24 hrs, after which there is no
discernable improvement. We use ∼5-50 DNA molecules per microsphere and incubate
the reaction for 4 hrs at 37°C, though room temperature works fine as well. With the best
batches of DNA and microspheres, the DNA-to-microsphere ratio can be reduced down
to 1-2 DNA per microsphere. Clearly, working at higher concentration speeds things up
but it also wastes DNA and can lead to multiple DNA molecules per microsphere.
Multiple attachment is significantly less of a problem for 0.3 µm microspheres in
comparison to 1 µm microspheres. The first time any particular batch of DNA is used,
we monitor it every 2 hrs. The best results are about 70% of the microspheres having a
single DNA and routinely work with one at 1 in 2 and 1 in 3. Typical results are about
20-50%.
1) Wash the microspheres 5x in 200 µl SB-50 (the first time we use a batch of
microspheres we them wash 7x). Remember to keep track of the volume of microspheres
used so the concentration of microspheres stays constant. Keep an eye of the whiteness
of the microsphere solution. If it goes from a whole milk to a skim milk color, the
microsphere concentration is seriously changing.
16
2) Add:
2-5 µl DNA
40 µl SB-50, 100, or 200
Mix gently and put on a rotator (LabQuake). If we have time, we usually let
these sit of an hour or even overnight to let the DNA diffuse. Trying to mix concentrated
DNA by pipetting can lead to shearing. This is more of a problem with long DNA. It
seems to be helpful to have the DNA well dispersed before adding the microspheres
because it presents the microspheres with a more uniform concentration and makes the
removal of excess DNA easier.
3) Add:
5 µ 1% microspheres
Put on rotator at 25°C or 37°C. Higher temperatures lead to faster kinetics.
Presumably, the optimum temperature is 50°C but we use 37°C because the biochemistry
department has a 37°C room. Do not use a simple heating block because the liquid is
slowly transferred to the top of the eppendorf tube by evaporation.
4) Sample at various times ( 2 hrs., 6 hrs., overnight) to see how the reaction is
proceeding. We image by eye under a standard fluorescence microscope using a 63X,
1.4 NA objective. In an eppendorf tube add 20 µl (TE+ 50 mM NaCl) + 1 µl sample, mix
gently and then spin down at 4,000 rpm for 4 min. Gently resuspend in the same buffer
— this removes the TWEEN-20 allowing the microspheres to stick to the microscope
slides. Then, the slight flows from evaporation will stretch out the DNA. There are
many different stains for visualizing DNA. Generally, we use YOYO-1 but for looking
at DNA by eye without having to adding oxygen scavenging enzymes, picogreen is a
great dye. Make up a 50 µl 1:10 dilution of stock PG in ( TE + 50 mM NaCl + 10 % β-
mercaptoethanol). Take 2 µl sample + 4 µl of the diluted dye solution. mix in an
eppendorf tube, and then deposit on to a slide covering with a 22 mm #1 coverslip.
Remember, always pipette with a cut pipette tip. When the sample is about 1 DNA
attached for every 2 microspheres, stop the reaction. You can check the initial ratio of
DNA to microspheres by not spinning down in the first step. Sometimes the DNA will
stick to the surface regardless of whether or not a microspheres is attached. In a SIT or
other intensified camera, the ball of stained DNA at the attachment site will misleadingly
17
look like a microsphere. For this reason, we always do the initial sample inspection by
eye.
5) To remove the unbound DNA from the microsphere-DNA-complexes, take the
sample, add 40 µl SB-50. Let sit on rotator until the microspheres are uniformly
dispersed (20 - 60 min). Spin down at 2,000 rpm until pelleted (10-20 min). Remove the
supernatant, being careful not to pull the meniscus across the pellet. Add 40 µl SB-50.
Do not pipette to mix. Instead, put on a rotator for 2 hours. If it has not completely
dispersed, gently draw the pellet up once or twice and put back on to the rotator until the
microspheres are uniformly dispersed.
6) Store on a rotator at 4°C to keep the microspheres from pelleting and to
decrease any thermal activation for unbinding of the streptavidin-biotin bond. When
stored this way, microsphere-DNA solutions are good for several months or more without
any serious degradation.
Buffers:
SB-50 Buffer (Standard microsphere buffer: 50 refers to the NaCl concentration. ( i.e.
SB-0 has no added NaCl whereas SB-200 has 200 mM add NaCl)
100 ml TE
1 ml 5 M NaCl
100 µl TWEEN-20 (Biorad)
TE Buffer
10 mM Tris HCl pH 8
2 mM EDTA
Autoclaved
10 x Ligase Buffer (many varieties but this is what we use.)
500 µl 1M Tris HCl pH 8
20.4 µl 4.9 M MgCl2
100 µl 1M DTT
50 µl BSA (100x from NEB, 10 mg/ml)
329 µl ddH20
Add together in an eppendorf tube, vortex, and aliquot in 50 µl units and freeze.
18
2.1.2.2 Notes
To create a microsphere-DNA-microsphere complex, the second oligonucleotide
needs to be attached. After the “remove unbound oligonucleotides” step, simply repeat
the ligation but adjusting the amount of H20 to accommodate the variable volume
recovered from the centricon. The “remove unbound oligonucleotides” step needs to be
repeated as well. We add the second microspheres at a one-to-one ratio after purifying
away the excess DNA. To check the preferred order, we set up a reaction with type-A
and type-B microspheres. Generally, one type of microsphere will yield better
attachment. This microsphere is added second. The yield to microsphere-DNA-
microsphere complex is much lower, typically around 2-10% but has been as high as
50% for 0.3 µm microspheres.
Presumably, attaching the microspheres to the DNA at higher temperatures will
lead to a higher yield and a higher rate of attachment. We did not investigate
temperatures higher than 37°C. One easy way to do this would be to put a small rotator
in the oven with the power for the rotator coming in though the hole for the temperature
probe.
There is considerable variation in yield to microsphere-DNA complexes. This
variation appears to be more correlated with the DNA preparations than with the
microsphere preparations. The origin of the variation is unknown.
2.1.3 Making dimers of λ-DNA
2.1.3.1 Basic scheme
DNA molecules twice as long as λ-DNA can be made efficiently (~80%) and
simply by concatenation two λ-DNA molecules together. A schematic of the order in
which these reactions are done is shown in Figure 3. However, the method in which the
products of reaction #1 and reaction #2 are mixed is critical to the success of reaction #3.
Simple kinetics suggests that higher concentrations lead to higher rates. However, this is
not true for λ-DNA due to the very slow, reptative-type diffusion which drastically
reduces the rate of diffusion. Therefore, even if the two different λ-DNAs are mixed at
high concentration, the different types of DNA will simply not inter-mix and reaction #3
19
does not proceed. To overcome this limitation, the products of reaction #1 and reaction
#2 must be first diluted and then mixed. Finally, this dilute solution containing both types
of modified λ-DNA is concentrated and then ligated.
Reaction #1 Reaction #2
SA
λ -DNA λ -DNA SB
Reaction #3SA
SB
Figure 3: Making dimers of λ-DNA. Reactions #1 and #2 are done simultaneously. Ligation of SA and SB prevent multimers or circles from being formed. The ends of λ-DNA are complimentary so the free end from reaction #1 and reaction #2 will hybrodize. The final reaction yields dimers but its efficiency depends on the method of mixing.
2.1.3.2 Recipe
When we initially tested this protocol, it was a background experiment.
Therefore, some of the mixing times are longer than are needed but reflect the
approximate times used.
1) Follow the protocol for “Ligating oligonucleotides to λ-DNA” but setup up two
parallel for the SA and SB oligonucleotides.
2) Next, purify the excess oligonucleotides from the λ-DNA given by “Remove
unbound oligonucleotides”
3) Now take the modified λ-DNA from reaction #1 and reaction #2 and separately
dilute them up to 1 ml in TE. Mix gently several times over the next day.
4) Mix the two 1 ml samples and let sit for another day.
20
5) Concentrate the 2 ml sample down to ~200 µl using a centricon-100
microconcentrator following the general guide lines of the protocol “Remove unbound
oligonucleotides.”
6) Add:
200 µl DNA from centricon
40µl 10X ligase buffer
150 µl ddH20
Mix gently and then place in a 65°C heating block for 10 min. and let cool to
room temperature. We next let this hybridization reaction sit for four days to attempt to
compensate for the slower, reptative dynamics at these higher concentrations. Unlike
reaction #1 and reaction #2 where one of the reactants was a small molecule, both of the
reactants in reaction #3 are large molecules and must diffuse in a concentrated polymer
solution.
7) Add:
5 µl ligase
4 µl 0.1 mM ATP
Mix gently until the gradient in the index of refraction disappears and then put in
at 16°C overnight.
8) Heat to 65°C for 10 min. to inactivate the ligase.
9) Add TE up to 2 ml and concentrate down to 400 µl in a microconcentrator
three times. This process is done to transfer the DNA from the ligase buffer, which
contains Mg2+ into TE buffer, which does not. Mg2+ can lead to clumping of the
microspheres as well as digestion of the DNA by contaminating enzymes. The DNA is
now ready for general use and can be attached to microspheres as detailed in “Attaching
DNA to microspheres”. The DNA should not be concentrated below 400 µl to prevent
entanglement effects that happen at very high DNA concentrations.
2.1.3.3 Notes
Since reaction #3 takes such a long time, we always verify that reaction #1 and
reaction #2 proceeded efficiently by attaching them to their respective type of
microspheres before starting reaction #3. Typically, when attaching this dimer-DNA to
21
microspheres, the yield to dimers can be estimated by counting the extended DNA
molecules by eye. An alternative method is to run a gel. However, the difference
between a monomer of λ-DNA and a dimer of λ-DNA is not well resolved by a standard
gel and we used the pulse field gel apparatus in the lab of Prof. Ron Davis. The standard
pulse field gel protocol was followed.
2.1.4 Making long DNA
2.1.4.1 Basic scheme
We make long DNA by using concatemers of λ-DNA (Figure 4). To control the
distribution of the products, the relative concentration of λ-DNA to oligonucleotide is
changed. However, the control of this distribution in limited. For instance, one can not
get a narrow distribution of around concatemers of 5-7 λ. Additionally, the number
density of the smaller products generally exceeds the longer products. Therefore, we
combine the production of concatemers shown in Figure 4 with the sequential ligation
shown in Figure 3 to yield a distribution of lengths biased towards larger concatemers.
1λ
2λ
4λ3λ
Figure 4: Concatemers of λ-DNA.
2.1.4.2 Recipe
To generate long concatemers, two of the following reactions must be setup with
SA and SB oligonucleotides.
1) Add:
40 µl 10X ligase buffer
2 µl 5’ phosphoralated oligonucleotides
125 µl dd H20
Vortex.
22
2) Add:
50 µl λ-DNA (~500 µg/ml)
Mix gently but thoroughly. Heat to 65°C for 10 min. and then let cool to room
temperature. Let sit for at least 3 days at room temperature. Again, as with the dimers of
λ-DNA, the hybridization of λ-DNA into concatemers is very slow. Experimentally,
overnight was not a long enough time for the reaction to finish.
3) Next, excess oligonucleotides are removed. This process gets more laborious
with the longer DNA because the DNA can not be concentrated too much. Again, if the
DNA is too highly concentrated, especially with long DNA, it becomes hopelessly
entangled. Therefore, we proceed by the protocol “Remove unbound oligonucleotides”
except the volume is never allowed to drop below 200 µl. Note that the amount of time
need to spin concentrate the DNA will be longer for the higher concentrations of and
longer lengths of DNA used here. The lengths of the resulting DNA should be analyzed
by pulsed field gel electrophoresis. The ability of the DNA to attach to microspheres
should also be checked.
4) Each sample of DNA should be diluted up to 1 ml in TE and mixed with a wide
bore pipette tip gently over several days to allow the DNA to become uniformly
dispersed. Note that this dispersal is not rapid. An entangled ball of semi-dilute
polymers does not immediately disentangle. Schematically, we think of it as peeling
away successive layers of an onion. The outer polymer layer must diffuse into the
diluting solution before the underlying layer can diffuse out of the entangled ball. This
process is slower for longer polymers because the reptation time scales as L3.3. This
scaling limits the length of DNA concatemers that can effectively be used. If the
distribution of lengths is skewed towards longer and longer molecules, this
disentanglement processes becomes too long.
5) The separate, diluted samples of concatemers are gently mixed and let sit for 1+
days at room temperature while occasionally being mixed by pipetting.
6) The samples are concentrated down to 400 µl by spinning in a centricon-100
micoconcentrator (see “remove unbound oligonucleotides” for details).
7) Add:
60 µl 10 X ligase buffer
23
124 µl dd H20
Vortex and then add:
400 µl DNA.
Mix gently but thoroughly. Place in a 65°C heating block for 10 min. Remove
the heating block from heater and let cool to room temperature. Let sit at room
temperature for 4 days.
8) Add:
8 µl ligase
6 µl 0.1 M ATP
Mix thoroughly and place at 16°C overnight.
9) Dilute up to 2 ml in TE and concentrate four times in a micrconcentrator but
never letting the volume decrease below 600 µl. This step can be replace by dialysis.
The DNA is now ready to use.
2.1.4.3 Notes
2.2 Staining and solutions
Imaging single DNA molecules require fluorescent dyes with a high quantum
yield and a low background. However, fluorescent dyes rapidly photobleach. Thus to
study the physical properties of polymers one molecule at a time, we needed a highly
efficient dye with a low background and a solution that minimized photobleaching of the
dye molecules.
2.2.1 Fluorescent stains
2.2.1.1 YOYO/TOTO
YOYO and TOTO are a family of dyes sold by Molecular Probes, Inc. Different
wavelengths are available depending which specific dye is used. YOYO-1 is efficiently
excited by the 488 nm line of an argon ion laser with a peak emission at 509 nm. YOYO-
2 and YOYO-3 are excited and emit further into the red than YOYO-1. YOYO and
TOTO are excellent nucleic dyes with two very useful properties. First, the dye exhibits
a 100 to 1000 fold increase in its quantum yield upon binding to DNA, giving a strong
24
image signal against a low background [43]. Second, for YOYO at least, the binding is
so tight that one chain can be stained in a concentrated solution of unstained DNA [8].
This high binding constant is achieved through both an electrostatic interaction and a
hydrophobic interaction. The electrostatic interaction is between a positively charged
portion of the dye molecule and the negatively charged phosphates on the DNA
backbone. The hydrophobic interaction is between the planar portion of the dye molecule
and the planar base pairs. These dyes intercalate or slide in between the base pairs of the
DNA. This intercalation alters the local structure of the DNA and increases the contour
length of the DNA. At our staining levels, the contour length increased from 16.3 µm to
21.1 µm.
YOYO and TOTO do not work efficiently in solutions containing Mg2+ ions.
Fluorescence disappears at Mg2+ concentrations greater than 2.5 mM. Presumably, Mg2+
ions are displacing the dye molecules from the DNA. Alternatively, the Mg2+ ions could
be quenching the fluorescence. To determine which process is taking place, the length of
the DNA needs to be measured by stretching the DNA between two optical traps. The
inability of DNA to be stained by YOYO or TOTO in the presence of Mg2+means that
YOYO is unsuitable for investigating DNA-enzymatic reactions that require Mg2+. For
instance, the digestion of DNA by exonuclease could not be observed.
Molecular Probes states that YOYO-1 exhibits a fast binding followed by a slow
binding mode. Thus, slow binding mode happens within approximately 1 hour.
Therefore, DNA should be stained for at least 1 hour to assure that it is in a reproducible
state.
YOYO and TOTO are stored frozen in DMSO, a non-aqueous solvent. Note that
DMSO allows chemicals to pass through both latex gloves and human skin so suitable
gloves should be used. While in DMSO, YOYO and TOTO can be repeatedly thawed
with no apparent degredation. However, to limit our exposure to this potential problem,
we aliquot the stock dye solution in 2 µl aliquots. Once diluted into an aqueous buffer,
the dye can not be refrozen.
Occasionally, stock samples of YOYO and TOTO will deteriorate. This
deterioration manifest itself by clumps of fluorescence along the DNA that should
otherwise be uniform. These clumps can be eliminated with strong forces but they will
25
reappear once the DNA is allowed to relax. If this problem appears, we purchase new
dye.
YOYO and TOTO are extremely photolabile and will photobleach and fragment
the DNA in 1-3 seconds when no precautions are taken (see sect 2.2.2.1).
2.2.1.2 PicoGreen
YOYO
Time (s)
0 1000 2000 3000 4000
Frac
tion
of s
tart
ing
exte
nsio
n
0.7
0.8
0.9
1.0Sytox
Time (s)
0 1000 2000 3000 4000
Figure 5: Stability of extension of as a function of time at a constant strain rate to determine if the contour length is a function of time. The extension was measured by trapping a microsphere at the stagnation point in an elongational flow. The flow rate was slightly above the critical strain rate for the transition from a coiled to the extended state. (A) YOYO-1stained DNA: the illuminating laser power was at the minimum intensity to observe a molecule for the first 2,000 seconds after which it was increased approximately 10 fold and the measured extension decrease linearly with time. (B) Sytox stained DNA: the laser power was increased at 0 and 2,000 seconds. The extension changed but came into a new equilibrium value in approximately 500 seconds. Both measurements were taken with excess dye in the solution.
Picogreen is also a bright fluorescent nucleic acid dye manufactured by Molecular
Probes, Inc. The background is much higher, however images with good contrast are
achievable. Picogreen’s virture is that it is not as sensitive to Mg2+ ions. However, in
experiments with exonuclease, picogreen lowered or inhibited enzymatic activity.
Picogreen is not as photolabile as YOYO and is much brighter than ethidium-bromide.
26
Thus, picogreen is an excellent dye for surveying the efficiency of coupling DNA to
microspheres.
2.2.1.3 Sytox Sytox is another a bright fluorescent nucleic acid dye manufactured by Molecular
Probes, Inc. It has a higher background than YOYO but can yield excellent images. The
main difference between YOYO and Sytox is that Sytox is in fast equilibrium with the
surrounding dye molecules while YOYO is not (Figure 5). This fast equilibrium has two
beneficial properties. First, the physical properties of the stained DNA can be constant
for an extended period. Second, constant replacement of the dye molecules on the DNA
by the surrounding dye led to an increase lifetime of the DNA before the photobleaching
of the dye cleaved the DNA. With this technique, individual molecules have been
imaged for up to 6.5 hours. However, this technique requires a flow to act as a reservoir
of unexposed dye molecules.
2.2.2 Solution The persistence length p of unstained DNA is a function of the salt concentration
[33]. For monovalent ions such as Na+, p is approximately constant at 50 nm for [Na+]
>10 mM. Therefore, all our experiments in dilute solutions were done at 10 mM NaCl.
The “reptation” experiment was done at 2 mM NaCl.
The variation of p for unstained DNA with the concentration of Na+ should not be
naively extended to stained DNA. This variation for native unstained DNA arises from
the electrostatic interaction along the highly-charged, phosphoester backbone of DNA.
The dye molecules are also highly charged. For instance, YOYO has four positive
charges. These positive charges, from quatenary nitrogens, interact with the negatively
charged phosphoester backbone and this electrostatic interaction leads to YOYO’s
remarkably tight binding. This electrostatic interaction must screen the highly charged
backbone in the same manner as the free ions in solution. Therefore, the electrostatic
contribution to DNA’s persistence length is most likely automatically screened for
stained DNA. However, to our knowledge, this has not been directly measured.
The general solution consisted of 10 mM Tris HCl pH 8, 2 mM EDTA, 10 mM
NaCl, 4% β-mercaptoethanol, ∼50 µg/ml glucose oxidase (Behringer-Mannheim) and
27
∼10 µg/ml catalase (Behringer-Mannheim) plus glycerol, sucrose or glucose to enhance
the viscosity. Tris HCl is a buffering agent to keep the pH constant. EDTA binds traces
quantities of Mg2+ to prevent digestion of the DNA by contaminating enzymes. NaCl
screens the highly charged phosphates on the DNA backbone so the persistence length is
not a sensitive function of the ionic concentration. To remove any contaminates,
solutions were filtered through a 0.4 µm filter.
2.2.2.1 Preventing photobleaching
The main requirement of the solution was to reduce photobleaching of the
fluorescent dye. When dye molecules photobleach, two detrimental processes happen.
First, the contour length of the DNA shortens. Second, the photobleaching of the dye
molecules can fragment the DNA. Without any precautions, YOYO-1 stained DNA will
fragment in 1-3 seconds.
The dominant cause of photobleaching is oxygen radicals (O-). Therefore, we
used two standard techniques to eliminate oxygen radicals. First, we added 4% (v/v) β-
mercaptoethanol, a reducing agent that directly attacks the oxygen radicals. Second, we
enzymatically scavenged the oxygen from solution. In this process, two enzymes,
glucose oxidase and catalase, are added and the net reaction is to transfer dissolved
oxygen from the solution onto a glucose molecule. The effectiveness of this enzymatic
process was much higher than simple physical purging of the oxygen by bubbling dry
nitrogen through the solution. Using these techniques, single DNA molecules could be
visualized for up to 6 hours with Sytox at 1 cP. For YOYO-1, single molecule could be
observed for 2 hours at 1 cP. At higher viscosities, the observation time was reduced
presumably by the reduced efficiency of the enzymatic oxygen scavenging system.
Note glucose oxidase and catalase sold by Sigma contains a large amount of
sheared DNA. This sheared, unwanted DNA will stain along with the intended DNA and
dramatically increase the background. The purity of the Behringer-Mannheim enzymes
were much higher.
28
2.2.2.2 High viscosity solutions
The solution viscosity was increased to slow down the dynamics of the molecules
or to increase the hydrodynamic force exerted on the molecules. The dynamics and
scaling were the same in water:glycerol as in water:sucrose solutions [9]. This similarity
implies that the solvent quality did not change. However, we found that the rate of
photobleaching was lower in sucrose solution than in glycerol solution. Viscosity
measurements were done using a Brookfield cone-plate viscometer. For the 41 cP
measurements, the viscometer was temperature stabilized.
2.2.2.2.1 41 cP
To get good images of the DNA in an elongational flow, we performed our
experiments in a high viscosity aqueous solution. Polymers begin to stretch when
.ετ 1 05≥ or, alternatively, when .ε τ≥ 05 1 . In water where the τ1 ≅ 0.1 s, this requires a
strain rate of at least 5 s-1 which necessitates a flow rate of 5 µm/s at 1 µm from the
stagnation point and 10 µm/s at 2 µm. At any appreciable distance from the stagnation
point, the velocity of the molecule is too great to image with video microscopy ( 301 s per
video frame). To overcome this limitation, we increased the viscosity η of the solution
from 1 cP to 41 cP using a combination of sucrose and glucose. This increase in η lead
to a corresponding increase in τ1 and reduction in ε , assuming a constant ετ 1 . Thus, an
increase in η slows down the bulk and internal motion of the molecules such that rate of
motion is amenable to our imaging techniques.
To achieve a high viscosity (η = 41 cP) buffer with minimal photobleaching, the
solution consisted of 10 mM Tris HCl pH 8, 2 mM EDTA, 10 mM NaCl, 4% β-
mercaptoethanol, ∼50 µg/ml glucose oxidase (Behringer-Mannheim) and ∼10 µg/ml
catalase (Behringer-Mannheim), ~18% (w/w) glucose and ~40% (w/w) sucrose.
Note that the viscosity of such a solution is temperature sensitive and we
measured the viscosity of each solution in a temperature stabilized viscometer and
adjusted it as needed.
When mixing the DNA into the high viscosity solution, great care was taken to
avoid shearing (breaking) of the DNA molecules. First, the stained DNA sample was
29
gently pipetted with a wide-bore pipette tip into the high viscosity solution. Next, the
solution was slowly mixed with a helix-shaped plastic rod for 10 minutes at ~0.25
revolutions/sec. Finally, we visually inspected each solution using fluorescence
microscopy to verify that the molecules were homogeneously distributed in the solution
and un-sheared. The viscosity of each DNA containing solution was measured and the
concentration of solvent was slightly adjusted to produce the desired viscosity.
2.2.3 Staining
We typically stained the DNA in 10-7 M YOYO-1 at a ratio of 1 dye molecule per
4 base pairs. For DNA molecules attached to microspheres, this ratio is difficult to asses
exacly because the microspheres bind a majority of the dye. In this case, the ratio of
number of microspheres to dye was kept constant to assure that each sample had the same
degree of staining. Staining was done at room temperature in the dark for at least 1 hour.
2.3 Experimental equipment
LASER
SHUTTER
FLOW CELL
DICHROICBEAMSPLITTER
VIDEOCAMERA
OBJECTIVE
EMISSIONFILTER
Gre
en
Figure 6: Schematic of optical layout for fluorescence imaging.
30
2.3.1 Microscope and Imaging
We imaged single DNA molecules using video enhanced fluorescence
microscopy (Figure 6). We used a Zeiss Axioplan microscope with either a Zeiss Plan-
Neofluor 63X NA 1.4, a Zeiss a c-Apochromat 60X NA 1.2 or a Nikon Plan Apo 60 X
NA 1.2 objective. The fluorescent dye emited in the green portion of the spectrum when
excited by 488 nm light from an argon-ion laser.
Excitation with a laser leads to speckle in the image plane due to the coherence of
the laser light. This coherence was greatly reduced by focusing the light on to a spinning,
ground glass disk. To efficiently couple the light into the microscope, we used the
normal illumination path except the spinning disk took the place of excitation lamp.
Images were recorded by a silicon intensified target (S.I.T.) camera (Hamamatsu
C2400-08), processed by a image processor (Hamamatsu Argus 10), and the recorded on
to S-VHS video tape or directly digitized by the computer.
Due to lag in the phosphor of a S.I.T. camera, rapidly moving objects appear to
have comet-like tails. The decay time is about 3 video frames (33 ms per video frame).
To overcome this limitation for the elongational flow experiment, we stroboscopically
illuminated the DNA using a mechanical shutter (Uniblitz) to block the laser. The shutter
was synchronized to the camera so it opened 4 ms before the start of a frame. A synch
pulse was also recorded on to the audio track of the videotape. By using the timing
information stored on the audio portion of the videotape, only the fully illuminated
images were transferred to an optical disk recorder (Panasonic, TQ-2028F). This video
data was then digitized by a Data Translation Quick Capture board in sequences of
approximately 7500 frames.
2.3.1.1 Synchronizing the shutter with the camera
To get good, evenly illuminated images while stroboscopically illuminating the
DNA, we need to synchronize the shutter to our camera. For this synchronization, we
triggered an SRS timer with the vertical synchronization pulse from the camera. Our
S.I.T. camera progressively reads out all the odd rows (field #1) and then all the even
rows (field #2). These two fields make up one standard video frame, which is 33 ms in
duration. If the shutter is turned on halfway through a field, the illumination intensity is
31
uneven for that field. If the illumination is turned on exactly at the start of a field, the
first rows have accumulated essentially no exposure before they were read out. The best
images were achieved by opening the shutter 4 ms before the start of field #1. A CCD
accumulates free charge in each well and all pixels are exposed simultaneously and for
the same duration. Therefore, these issues are specific to our camera. Note, our S.I.T.
camera puts out a vertical synchronization pulse every field so to make sure the phase is
correct, the video is recorded on to a s-VHS videotape. A VCR, on pause, shows only
field #2 by copying the contents of field #2 into field #1. So, if there is a pre-flash shown
on the bottom half of a still image, the shutter is opening during the field #2 as desired.
2.3.1.2 Trapping as a function of depth
Trapping strength decreases as a function of depth. This was a particularly
severe problem for the standard Zeiss Plan Apo Chromat 63X, NA 1.4 objective. The
maximum depth varied depending on the objective and microsphere size but was limited
to less than 40 µm. For short DNA molecules and for concentrated solutions, the physics
is not altered. However, for long DNA molecules (~150 µm), the perturbation in the
hydrodynamic interaction by a solid plane 15 µm away starts to become non-negligble.
Therefore, it is desirable to increase the maximum trapping depth.
We achieved strong trapping down to 180 µm by changing objectives to the
Nikon Plan Apo 60 X NA 1.2. This new objective design was optimized for confocal
microscopy deep into aqueous solutions by minimizing spherical aberration. Besides
leading to better trapping, the image quality is significantly improved. The Nikon lens is
not infinity corrected and has a different thread pattern than standard Zeiss objectives.
Zeiss makes an adapter for Nikon objectives but does not always acknowledge that it
exists. For trapping, we found that the Nikon confocal objective was significantly better
than the corresponding Zeiss objective, at least at our wavelength (1.06 µm). However,
the Zeiss objective had better image quality, particularly the evenness of illumination in
the imaging plane.
An alternative method for trapping microspheres deep in solution was developed
by Steve Smith and Carlos Bustamante [31]. They developed a counter propagating
optical trap with two low NA (0.8) objectives. This trapping geometry has the added
32
benefit that it leads to significantly higher trapping forces and the forces can be directly
measured by the deflection of one of the trapping beams. However, this geometry
requires optical access from both sides of the sample, does not lead to good imaging
because of the low numerical aperture used. Further, it requires the two traps to be
precisely aligned. Presumably, a double optical trap based on the confocal objective
would lead to the best of both methods but would be very expensive.
2.3.1.3 Filters
Laser illumination eliminates the need for an excitation filter. The dichroic and
emission filters were optimized for the 488 nm argon-ion line. We used a long pass
emission filter, rather than a more typically notch filter, because more light is collected.
A notch filter is only needed if multiple fluorophores are being imaged.
MainPump
(100ml)
Flow cellSamplePump
(1ml)
Sample injectionvalve
Sample loop
Shunt
Figure 7: Diagram of pump and tubing the for elongational flow experiment
Our trapping beam passes through the 488 nm dichroic. This dichoic puts
interference fringes on one polarization of the trapping beam. The fringes reduce the
trapping efficiency and shift the trapping plan down ~2 µm from the image plane. This
33
shift in the trapping plane relative to the image plan can be compensated for by placing a
2m concave lens in the appropriate beam path.
2.3.2 Motors
The stage motors were used to generate a uniform flow around the trapped
stationary microsphere. For our application, we needed a constant velocity and we
purchased our stage motors from Oriel because they specified a RMS variation of 1% for
velocities over 5 µm. The motors can be refurbished by Oriel. Note that some new
motors are not smooth to 1% and the stability should be checked for each new motor as it
arrives. A crude check is to run the motor at 200 µm/s and listen for variation in pitch.
Also, the range of motion should be checked.
Pump rate (ml/min)0 200 400 600 800 1000 1200
Stra
in ra
te (s
-1)
0.0
0.2
0.4
0.6
0.8
1.0
Figure 8: Calibration of elongational flowcell showing the strain rate is linear with pump rate.
34
2.3.3 Pumps and Plumbing
In designing the flow system for the elongation flow experiment, the main design
consideration was to minimize fluctuations in the flow rate. However, the highest flow
rates through the flow cell were 10 µl/min -- too slow for even the best syringe pumps to
provide truly pulse free flow. To increase the pumps operating flow rate while keeping
the flow rate in the flow cell low, we created a 100:1 bypass shunt for the fluid (Figure
7). The ratio of shunting was constant over the range of pump rates used in the
experiment (Figure 8).
The main pump was an Isco Model 100 D syringe pump temperature stabilized at
22.7 ºC with a Lauda RM6 circulator. The main pump provided fluid flows from 10
µl/min to 25,000 µl/min. The maximum flow rate used in this experiment was 1,000
µl/min. The sample pump was a Harvard Apparatus 55 syringe pump loaded with a 1 ml
syringe (Unimetrics). The tubing (1.0 mm i.d.), connectors, filters, valves and tees were
purchased from Upchurch Scientific. The sample injection valve allows the introduction
of the DNA solution without the introduction of bubbles into the system. To minimize
contaminates, we loaded the fluid into the main pump through a 2 µm filter.
We attached the tubing to the flow cell using a 125 µm thick, epoxy-fiber glass
adhesive film (Ablefilm, Ablestick, Rancho Dominguez, Ca). Donut-shaped pieces were
stamped out and placed around the holes on the backside of the wafer. The wafer was
then briefly heated to 65 ºC. A PEEK ferrule and nut were placed at the end of 1.0 mm
i.d., stainless steel tubing which had previously been attached to a stainless steel tee. The
tubing was then screwed partway in to a 1/2” thick copper block, which provided the
mechanical support to hold the tubing properly aligned to holes in the wafer. The silicon
flow cell was then centered over the tubing and was gently clamped down. The nuts
were then tightened until the ferrules just made contact with the Ablefilm. The epoxy
was cured by baking the whole assembly for 2 hours at 120 ºC.
Note that silicon wafers are single crystals that are 500 µm thick. They
can fracture along a crystal axis under a minimal amount of applied stress. To prevent
excess stress, the copper block was smoothed with grinding powder (∼10 µm) to generate
a bump-free surface. To prevent a metal-silicon-water contact, the aluminum frame used
35
to clamp the wafer to the copper block was coated with 1/16′′ thick Teflon to cushion the
applied stress. The affixed tubing should be handled gently and not bumped since it is
directly coupled to the silicon or, otherwise, the flowcell will fracture.
Fluctuations in the flow can also be reduced using a trapped air bubble. The
bubble contracts when the pressure increases and expands if the pressure drops. We
introduced such a bubble using a tee and a partially filled syringe. The diameter of the
tubing down stream of the bubble and the diameter of the syringe define a time constant
over which the fluctuations in pressure are suppressed. This system is directly analogous
to RC filter of a voltage.
2.3.4 Micro-fabricated flow cells
Flow cells were etched out of silicon wafers to provide a well defined,
homogeneous elongational flow.
2.3.4.1 Flow cell design
A homogeneous elongational flow is defined by a linear velocity gradient along
the direction of flow such that v yy = ε . The strain rateε is constant but the residency
time tres of the polymers in these flows is limited. To increase tres, we used a standard,
cross-slot flow geometry. This generated a planar homogenous extensional flow where
ε x = -ε , ε y = ε and ε z = 0. Note, this is slightly different than an axial extensional flow
generated by an opposing jet flow where ε x = - ε 2 , ε y = ε and ε z = -ε 2 .
The main design consideration was to ensure that we studied dynamics of
polymers unwinding from equilibrium. That is, we wanted to avoid any pre-deformation
of the polymer prior to its entering the elongational flow. To eliminate pre-deformation
of the molecules caused by velocity gradient at the entrance to the flow cell, we used long
channels (25 mm) on the inlet. This increased the transit time for a molecule down the
inlet channel and allowed any molecule deformed upon entering the flow cell sufficient
time (> 8τ1) to relax back to an equilibrium configuration.
In a preliminary experiment with a 50-µm deep channel, we imaged polymers in
the center of the flow as they moved down the inlet channel. Some polymers were in
highly extended states (~15 µm) along the flow direction ( x ). To clarify the origin of
36
this deformation, we pulsed the pump and imaged only molecules that started within the
inlet channel. We still found some polymers in highly extended states.
We hypothesized that the shear rate, ∂ ∂v zx , is strong enough in a 50 µm deep
flow cell to cause significant pre-deformation for molecules at the center of the parabolic
fluid flow. A characterization of the strength of shear rate is given by a dimensionless
shear rate or a Weissenberg number (Wi = γ τ 1 ) where γ ∂ ∂= v zx -- the velocity
gradient perpendicular to the flow direction. Polymers start to deform when γτ 1 ≥1. To
estimate the dimensionless shear rate, we assume the velocity profile as a function of
depth is parabolic at low Reynolds numbers (~10-4). The shear rate in the flow is then
∂∂
γv zz
vz
z zx inlet
centercenter
( ) ( )= = −2 2 Eq. 2
where vinlet is the velocity at the center of flow in the x direction and zcenter is the distance
from the top surface to the middle of the flow. Although a naive interpretation of this
formula would indicate that γ = 0 at z = zcenter, DNA molecules have a finite size (RG =
0.7 µm) which leads to some shearing even for molecules at z = zcenter. Further, the
microscope has a small but finite depth of field (∼1µm). Thus, an in-focus molecule
nominally at the center of the flow will experience a shear rate approximately equal to a
molecule displaced by 1 µm from zcenter. Thus, we estimate γτ 1 = 8 for our preliminary
experiment, which supports our hypothesis that shearing was the origin of the observed
pre-deformation. Further experimentation at Wi = 1 showed little to no pre-deformation.
In order to prevent pre-deformations was due to shearing, the flow cell was
redesigned to minimize the shear experienced by the polymers along the inlet channel.
Inspection of Eq. 2 shows this can be achieved by increasing zcenter or decreasing vinlet. By
increasing the depth of the flow cell, zcenter is increased. This is limited by the working
distance (170 µm) of the high numerical aperture objectives. It is important to be able to
image several 10’s µm past zcenter to be able to accurately measure zcenter (see 2.3.4.3). By
narrowing the width of the flow cell, vinlet is reduced. A narrower channel leads to a
smaller region over which there is an elongational flow. Thus, a smaller vinlet is needed to
37
achieve the same strain rate. This is limited by shear flow arising from flow gradients in
the y direction which, until now, has been neglected.
Our goal was to measure polymers in an elongational flow up to ετ 1 = 3.5 while
keeping γτ 1 < 0.5. To achieve this goal within these above constraints, we choose the
depth of the flow cell to be 220 µm and the width of the inlet channel to be 650 µm
(Figure 9). The corners of the crossed slot were rounded with a radius of curvature of
325 µm.
2.3.4.2 Flow cell manufacture
Following the example of Volkmuth and Austin [18, 19], the flow cells were
made by etching channels into silicon wafers and then anionically bonding [44] Pyrex
coverslips to the silicon to seal the top surface of the channels. We generated the crossed
channel on the front side and holes for plumbing connections on the back side of the
wafers by performing two successive silicon etches in KOH. Typically KOH etching of
(100) wafers yields walls that are angled 54º from vertical. But, by rotating the pattern by
45º to the crystal axis [45, 46], vertical side walls are achieved along the inlet and outlet
channels.
Nitride is not etched by KOH and serves as the protective layer during the KOH
etch. Therefore, a 800-Å nitride layer was grown on 500-µm thick, 4” dia., L-Prime
wafers. To pattern the nitride layer, we used 1-µm thick photoresist as the protective
layer and then exposed the sections of the photoresist using a lithographic mask.
Subsequent development of photoresist removed the exposed photoresist leaving areas of
un-coated nitride and a plasma etch removed this unprotected nitride. After an acid wash
in 9:1 H2SO4:H202 to remove the photoresist, we placed the wafers in 30% KOH at 80 ºC
for 3.5 hours which etched the sections of silicon unprotected by the nitride. This first
etching cycle created 200 µm deep features on the back side of the wafer. The wafers are
washed in 5:1:1 H20:H202:HCl to remove any residual KOH. After an alignment between
a second lithographic mask and the pattern on the back side of the wafer, we patterned
and etched the front surfaces by the same process. The front surface was etched to a
depth of 170-250 µm deep depending on the duration of the etch. During the second etch,
the holes on the backside of the wafer were further etched and made connections from the
38
backside to the newly created channels on the front side. Next, the wafers were taken
through a second KOH decontamination and then plasma etched on the front side of the
wafers to remove the remaining nitride. A final acid wash in 9:1 H2SO4:H202 cleaned the
wafers. For some unknown reason, some of the wafers were not optically flat but had
large (300-500 µm), shallow (1-3 µm) depressions and these wafers were not used.
Silicon
Pyrex
1000 µm i.d.
51 mm
11 mm
650 µmr = 325 µm
A
B240 µm
Figure 9: Flow cell for elongational flow experiment.
To seal the flow cell, we anionically bonded Pyrex coverslips to the silicon wafer.
First, the Pyrex was cleaned immediately prior to bonding. Next, the Pyrex was placed
on the silicon wafer and heated on a hot plate at 400 º C. Upon reaching this temperature,
we then applied -400 V to the coverslip relative to the wafer, which sealed the Pyrex to
the silicon. Pyrex is used instead of another glass because its thermal coefficient of
expansion matched that of the silicon. Although it is possible to bond to Pyrex to the
nitride layer, we found bonding directly to the silicon yielded significantly cleaner bonds.
39
The sealed flow cells were cut out of the wafer using a wafer saw to a final dimension of
3″ × 1″ to facilitate mounting.
2.3.4.3 Flow cell calibration
Depth (µm)0 50 100 150 200
Velo
city
(µm
/s)
0
10
20
30
40
50
60
70
Pixels shifted60 80 100
Cor
rela
tion
valu
e
0
4
8
12
16
Figure 10: Velocity as a function of depth in the inlet channel.
The flow cell was calibrated in a three-step process using fluorescent beads. We
found that the magnitude of the strain rate is uniform to within 2%. In addition, we
determined that the strain rate turns on abruptly ( 02.0≤∆tε ). Or, in other words, the
polymers experienced an approximate step function increase in the strain rate as they
moved from the uniform-flow, strain-rate free motion in the inlet channel to the
elongational flow in the crossed-slot region as opposed to a gradual increase in the strain
40
rate. This calibrated region is the 22 µm on either side of the center- line of the flow cell
and the imaging area (100 x 94 µm).
Position (µm)-50 -25 0 25 50
Velo
city
( µm
/s)
-20
-10
0
10
20
Figure 11: Velocity along the outlet channel as a function of distance from the stagnation point.
Although the depth of the flow cell was measured to be ~220 µm deep with a 10X
NA 0.3 air objective, it is important to precisely determine the center of the flow to
minimize the shear on the DNA molecules (see 2.3.1.1). The index of refraction ne of
the immersion fluid (ne = 1.33) is not the same as the index of refraction of the
glucose/sucrose solution (ne ≈1.44). Thus, a vertical movement of the objective by 100
µm does not correspond to a change in the focal plane of 100 µm.
41
Position (µm)0 250 500 750 1000
Velo
city
( µm
/ s)
0
25
50
75
100
Figure 12: Velocity along the inlet channel as a function of distance from the stagnation point.
Therefore, the vertical center of the flow zcenter was found by measuring the
parabolic velocity profile in the inlet channel as a function of depth. To do this, we
injected a high density of microspheres (0.8-µm dia., Polysciences) in to the flow cell
such that there were ~30 microspheres in focus on the screen. The flow was turned on
and images were digitized every 0.22 seconds. An autocorrelation between successive
images determined the average displacement (Figure 10). This autocorreletation was
repeated for a hundred images and the resulting data was averaged to determine the
velocity for that depth. Successive sets of data were taken as a function of depth from 20
µm to 160µm (Figure 10). From fitting this data to a parabolic curve, we determined the
42
center of the flow to be at 102 µm. We repeated the measurement and, again, determined
zcenter = 102 µm.
The second calibration measured ε over the imaging area (100 µm X 96 µm)
around the stagnation point. The stagnation point is centered and images of fluorescent
microspheres are digitized. The y-coordinate of at least 20 microspheres is then tracked
between successive images. When plotted as vy vs. y, this data showed a linear
relationship verifying a uniform strain rateε ∂ ∂≡ v yy within the field of view (Figure
11). A linear fit to this data determined the strain rate for a given pump rate. The process
was repeated for different pump rates. The resulting linear relationship between strain
rate and pump rate shows the shunting ratio is constant over these pump rates (Figure 8).
For this experiment, a flow rate of 1,000 µl/min corresponded to a ε of 0.85 s-1
The third calibration measured the velocity gradient along the inlet channel. This
is important because it located the onset of the elongation flow. Since tres = 0 at the onset,
this calibration allowed for an accurate determination of the accumulated fluid strain
(ε ε= tres ) of the polymers. For this measurement, the pump was set to a 100 µl/min,
which corresponded to a velocity in the inlet (vinlet) of 85 µm/s at the center of the flow.
The whole flow cell, which is mounted on a motorized x-y stage, was moved at a
constant velocity in relation to the objective. If the stage velocity matched the velocity
of the fluorescent microspheres, the microspheres are stationary with respect to the
imaging optics and do not move on the video image. This measurement can be done
along the inlet channel by starting with the image area located 1500 µm upstream from
the stagnation point. Next, the velocity of the stage was set to values from 10 % to 90%
of vinlet. When the fluorescent microspheres were stationary in the video image, the stage
was stopped and the location of the stage as determined by the motor controller, was
recorded. The resulting measured velocity gradient along the inlet ∂ ∂v xx was linear
(Figure 12). A fit to this data yields a strain rate of -0.86 s-1 in excellent agreement with
calibration of the strain rate 0.86 s-1 as determined by the second method. By
extrapolating to vinlet, the onset of the elongational flow xonset was determined to be 960
µm. This value of xonset agrees well with the geometry of the flow cell which would
43
predict 975 µm since the channels are 650 µm wide and there is a 325 µm radius of
curvature at the cross.
ycm (µm)0 10 20 30 40
v y cm ( µ
m)
0
5
10
15
20
25
Figure 13: The center of mass motion of an individual DNA molecule. The measured strain rate from the DNA molecules (circles) agrees within 2% of the calibrated strain rate (line) determined by tracking fluorscent beads.
Taken together, these three calibrations show the elongational flow field is
uniform. Since the fluid is incompressible, we have
44
∇ • = + +vvx
vy
vz
x y z∂∂
∂∂
∂∂
= 0 Eq. 3
From the geometry of the planar flow cell, we have ∂ ∂v zz = 0. With this information
and the calibration ofε along the inlet (∂ ∂v xx = -ε ) and at the stagnation point
(∂ ∂v yy =ε ), we know the onset of the elongation flow field is sudden and the polymers
experience a constant strain rate. The uniformity of the strain rate is also confirmed by
showing the trajectories of the center of mass of indivdual fluorescent microspheres and
individual DNA molecules were hyperbolic. Furthermore, the velocity of the center of
mass motion of the individual DNA molecules increased with distance and the strain rate
calculated from this motion agreed within 2% of the strain rate calculated from the
motion of fluorescent beads (Figure 13).
2.3.4.4 Washing the flow cell
The flow cell system is cleaned to reduce the background fluorescence. DNA,
protein and microspheres bind to the glass and silicon, even in the presence of 0.1%
Tween-20. First, the system is washed in a high salt buffer (SB-500) to overwhelm any
electrostatic interactions. Next, the system was cleaned with an enzymatic detergent
(Tergazyme) to attack any protein on the surface. The system was then flushed with 50
ppm chlorine to attach any dye molecules in the system and to inactivate any enzymes
still present from the preceeding step. The system was finally flushed with SB-50 + 0.1%
β-mercaptoethanol to reduce the chlorine.
45
Chapter 3 Relaxation of Single Polymer Chains
3.1 Introduction
An understanding of the static and dynamic properties of a single isolated chain
forms the foundation of polymer physics [1, 2, 47]. One can then ask questions about the
behavior of dilute solutions, and eventually consider more concentrated solutions, melts,
and gels in which interactions between chains become important. Theoretical approaches
have involved many techniques: thermodynamic analysis, field theory, scaling,
renormalization group theory, and computer simulation [1, 2, 47].
Many of the observable static properties of polymers in dilute solutions are well
described by scaling relations. If A is an observable quantity, A ~ Mν ~ Lν, where M and
L are the molecular weight and length of the polymer and ν is the scaling exponent. The
value of ν is independent of the local molecular structure of the polymer but does depend
on temperature and monomer-solvent interactions [2]. It has been proposed by de
Gennes [48] that these scaling laws can be generalized to dynamical properties such as
relaxation and elongation rates, but the experimental verification of this idea has been on
less firm ground.
Measurements of rheology, light scattering, and birefringence of bulk samples
have been used to study the dynamics of polymer chains [49-57]. For example, Keller et
al used birefringence measurements to determine the bulk orientational order in regions
of dilute polymer solutions [49]. A sharp increase in the birefringence is observed in an
extensional flow at a critical strain rate cε , and a relaxation time τ was extracted by
assuming that cε τ ~1. In these experiments, it was found that τ ∼ ηL1.5/kBT where η is
the solvent viscosity and the scaling exponent is 3ν = 1.5, in accordance with the Zimm
model for a Θ-solvent [39]. Surprisingly, this result was found to be independent of the
"quality" of the solvent as it was varied from a Θ-solvent (3ν = 1.5), where attractive
monomer-monomer interactions are canceled by swelling due to monomer-solvent
interactions, to a "good" solvent (3ν = 1.8) where there is net swelling. This apparent
suppression of excluded volume effects is in contradiction with the theoretically
46
predicted scaling exponent of 3ν = 1.8 [1] which has been observed in intrinsic viscosity
measurements [58]. However, the relationship between birefringence and extension is
not entirely clear since it requires independent knowledge of the polarizability and chain
conformation. In fact, light scattering experiments seem to show that the chains in such
flows are only extended by a factor of order 2 from their equilibrium size [50].
Light scattering has been used to study the fluctuations of polymers from
equilibrium. Measurements of time correlations in scattered light intensities yield a
structure factor, which is theoretically predicted to scale as a power law with the
exponent ν [3]. For good solvents, the measured exponent is about 0.55 [54-57], which is
slightly smaller than the theoretical value of 0.6.
Crothers and Zimm measured the intrinsic viscosity of DNA as a function of
molecular weight and determined the scaling exponent to be 0.66 [59]. Within the Zimm
model, the intrinsic viscosity scales as [η]~ L3ν-1 [1], and this gives an experimental value
of 3ν = 1.65 for chains with molecular weights up to 1.3 ×108 (62.7 µm long).
While these traditional methods of experimentation have given much insight, they
have an inherent disadvantage: the relaxation properties of a single chain must be inferred
from indirect measurements averaged over a macroscopically large number of chains. In
addition, the polymer solutions used are not always monodisperse. Finally, in most
hydrodynamic flow experiments, the time during which a measurement may be made on
a particular volume of sample is short because the fluid element is rapidly carried away
by the flow [49]. In the experiment reported here, these problems are avoided because
the full relaxation of a single polymer is directly observed.
3.2 Manipulation and Visualization of Single molecules
A number of technical developments have allowed us to observe and manipulate
single polymers. DNA can be used as a model polymer [33, 34] and because of its large
size (10's µm), detailed observation in an optical microscope is possible. Although the
molecule can not be directly manipulated with optical tweezers [6], a microsphere on the
end of the DNA can be controlled quite easily. Molecular biology techniques facilitated
the attachment of the DNA to a micron-sized microsphere and highly-efficient,
47
fluorescent dyes allowed us to directly visualize the molecules in an optical microscope.
DNA has also been manipulated using magnetic microspheres [15, 60].
Microscope Objective
Oil
∼20 µm30-40 µm
DNA
Infrared Laser Beam
Bead
Cover Slip
Slide
Buffer
Figure 14: Schematic of the apparatus: a single latex microsphere is optically trapped approximately halfway between the slide and coverslip in a sample with a thickness of 30-40 µm. A single molecule of DNA attached to the microsphere is stretched to its full extension in a fluid flow, and then allowed to relax. The stained DNA was imaged using video enhanced fluorescence microscopy. Reprinted with permission from Science 264, 819, 1994. Copyright 1994 American Association for the Advancement of Science.
Our experiment was performed using DNA from 4 to 43 µm long, where
molecules larger than 16 µm were formed by linking several lambda-phage DNA (1.9 ×
106 g/mol/µm). The variation in DNA length used in our experiment was probably due
to shearing of the DNA attached to microspheres during pipetting. Single DNA
molecules were attached to a 1.0 µm polystyrene sphere via a streptavidin-biotin bond.
The molecules were observed in an aqueous buffer with a viscosity of about 15 cP
(measured by a Brookfield cone-plate viscometer) consisting of either a sucrose or a
glycerol solution with 10 mM Tris-HCl pH 8, 1 mM EDTA and 10 mM NaCl. At this
ionic strength, DNA has a persistence length, a measure of polymer stiffness, of about 50
nm [33].
48
The optical tweezer was made by focusing a 100 mW Nd:YAG laser beam
through a Zeiss 63×, 1.4 numerical aperture microscope objective. The DNA was stained
with YOYO-1 (Molecular Probes, Inc.), a fluorescent dye that emits in the green portion
of the spectrum when excited by 488 nm light from an argon-ion laser. To stretch the
DNA, a feedback stabilized motor (Oriel) moved the microscope stage at a constant
velocity and generated a fluid flow around the trapped, stationary microsphere, as shown
in Figure 14. A flow velocity of 20 µm/s was chosen to almost fully extend the DNA.
The fluorescent images were recorded by a S.I.T. camera (Hamamatsu), processed by a
image processor (Hamamatsu Argus 10), and digitized by a Data Translation Quick
Capture board at a maximum rate of 15 frames/s.
3.3 Relaxation Measurements
Figure 15: Relaxation of a single molecule of DNA. Initially, the DNA is stretched to its 39 µm, which is 85% of its full extension. Each subsequent frame was spaced at 4.5 s intervals. An initial rapid recoil of the DNA is evident. Reprinted with permission from Science 264, 819, 1994. Copyright 1994 American Association for the Advancement of Science.
The length (L) of each stained DNA molecule was measured by recording 5 video
frames before the motor was turned off and the relaxation time of the fluid flow (~ 0.1 s)
was measured by observing the motion of a trapped microsphere after the motor was
49
stopped. Later measurements have shown that this value of the length was 85% of the
full contour length [10]. After the flow stopped, video frames were digitized until the
chain was near equilibrium. A typical time sequence of video images is shown in Figure
15. Relaxation measurements were repeated 3-5 times for each molecule. As seen in
Figure 15, there was a rapid recoil of the free end to about 70% of its full length. In part,
such a fast initial relaxation might have been expected from static force measurements of
elongated DNA which showed a highly nonlinear dependence of the force on length at
>75% percent extensions [15].
Time (s)0 10 20 30 40 50
Visu
al le
ngth
(µm
)
0
10
20
30
40
Figure 16: Relaxation of long (39.1 µm), medium (21.6 µm), and short (7.7 µm) molecules. The visual length of the DNA for every frame was measured and each data set shown consists of five individual relaxation measurements. The solid lines are fits to a continuous spectrum of decaying exponentials as determined by an inverse Laplace transformation of the data in the limit of small statistical regularization [61]. Reprinted
50
with permission from Science 264, 819, 1994. Copyright 1994 American Association for the Advancement of Science.
It is interesting to note that the relaxing end of the DNA sometimes formed a
compact ball of about 0.5 to 1 µm in diameter. Similarly, as the chain was stretched in a
flow, a coiled section usually appeared to unravel from the stretching end. This behavior
may be qualitatively similar to that of the yo-yo model proposed by G. Ryskin [62, 63].
In addition, possible evidence for knots within a single molecule of DNA has been
observed by a spot of increased fluorescence intensity along the DNA which moved as if
it were in a fixed location on the relaxing chain.
Several plots of the visible length along the flow direction versus time are shown
in Figure 16. Each length was measured on the video screen using an overlaid computer-
generated cursor and two model-independent methods of analyzing the data were
performed.
3.4 Analysis
We used two model independent methods to analyze the data.
3.4.1 Inverse Laplace transform
The relaxation data were analyzed for a spectrum of exponential decays, as
shown in Figure 17A, using an inverse Laplace transform algorithm developed by S. W.
Provencher [61]. The inverse Laplace transforms determines a fit using a weighted least
squares estimator with an added quadratic form as a statistical regularizer. The
amplitudes of the exponentials were constrained to be non-negative. The inversion of
noisy data is generally not unique and using statistical regularization one finds a family of
acceptable solutions as a function of the regularization parameter. As the regularization
parameter α is increased, the fit is biased toward smoother spectra and the algorithm
searches for the simplest solution consistent with the data but without over-interpretation.
For many of our data sets the number and position of peaks did not change as α was
increased, they merely broadened, indicating that these decay components were robust.
In such cases, the simplest solutions would be the ones for small α. However, in about a
third of our data sets, some pairs of peaks merged as α was increased, leaving us with a
51
higher degree of uncertainty in the decay times. The sharp peaks are the spectra in the
limit where α is small (α = 1.3×10-8 for the 38.3 µm chain and 7.9×10-7 for the 12.8 µm
chain) and the broader curves are the chosen solutions (with α = 4.3×10-4 for the 38.3 µm
chain and 8.4×10-4
for the 12.8 µm chain).
For our data, discrete peaks were always present in the spectra and in the limit of
small regularization they were very sharp, essentially corresponding to a discrete sum of
exponentials (Figure 17A). As shown in Figure 17b, the longest relaxation times
followed a scaling law with chain length τ ~ L3ν. The scaling exponent 3ν was measured
to be 1.66 ± 0.10 in limit of small regularization and 1.60 ± 0.10 for the broader spectra
chosen by Provencher's algorithm. These results fall in between the values of 1.5 and 1.8
predicted theoretically for theta and good solvents in the Zimm model and are slightly
higher than the value of 1.5 found in birefringence measurements.
The classical theories of chain dynamics are based on the notion of normal modes
of relaxation which arise as solutions to linearized equations of motion [1, 39, 40]. As
pointed out by de Gennes [2], it is not clear that the concept of modes would remain valid
without the simplifying assumptions in these models. While the mode picture predicts
sharp peaks in the decay spectrum, the actual spectrum for dilute chains has been difficult
to obtain experimentally because of the weak signals involved in standard experiments.
de Gennes hypothesized that nonlinearities from the hydrodynamic interaction, from the
excluded volume interaction, and possibly from knot formation would tend to broaden
peaks in the decay spectrum. For these reasons, he was not certain that distinct peaks
would even appear the in spectrum [2].
While our analysis does show distinct peaks in the spectrum, the structure of the
spectrum was not in agreement with the mode structure of the Zimm model.
Experimentally, the ratio of the decay times for the first to the second peak in the
spectrum was 8.3 ± 3.2 and the ratio of their amplitudes was 2.0 ± 0.8. These decay
times are consistent with the Zimm prediction of τp ∼ τ1/p3ν for p = 1,3,5 . . .. Note that
the symmetry of the tethered chain yields only the odd modes with the Zimm and Rouse
models and the amplitudes for the Zimm model are approximate and are give by Zimm et
al. [64]. The ratio of the amplitudes is in clear disagreement with the predicted value of 9
52
from the dependence of cp ~ c1/p2 for both a Θ-solvent and a good solvent [1, 64]. Note
that the theoretical ratio of cp is found by projecting the initial starting configuration of a
straight line for the end-to-end distance on to the normal modes of the Zimm model with
only the odd modes having a non-zero amplitude.
Decay time (s)
0.1 1 10 100
Am
plitu
de ( µ
m)
0
10
20
30
40
A
log[Length(µm)]
0.5 1.0 1.5 2.0lo
g[D
ecay
tim
s (s
-1)
-0.5
0.0
0.5
1.0
1.5
2.0
B
Figure 17: The spectrum of decaying exponentials describing each data set was determined by an inverse Laplace transformation of the relaxation data. (A) Spectra for chain lengths of 38.3 µm and 12.8 µm in the 15 cP glycerol solution. The spectrum for the 38.3 µm chain was shifted up by 10 µm for clarity. (B) Scaling of the longest decay times (τ) with length for the chosen solutions. The black points are data taken in a sucrose solution and the white ones in a glycerol solution. The solid line is a linear fit to all of the data and yielded a slope of 1.60 ± 0.10. A similar analysis in limit of small α gives a slope of 1.66 ± 0.10. These fits demonstrate dynamical scaling of τ with length and the slopes correspond to scaling exponents. Reprinted with permission from Science 264, 819, 1994. Copyright 1994 American Association for the Advancement of Science.
On theoretical grounds, this is not surprising since the Zimm model describes
polymers fluctuating near equilibrium and its approximations break down for large chain
extensions. The relaxation of a chain is also a dynamical process. Recent experimental
measurements in which a single DNA molecule is held extended between two optical
traps have shown a normal mode basis does describe the equilibrium fluctuation of an
extended molecule [28]. Further, it was determined that τp ∼ τ1/pα where α = 1.65-1.70.
53
The given experimental ratios were in the limit of small regularization. The ratios
for the solution chosen by Provencher's algorithm were 10.1 ± 4.0 for the times and 2.3 ±
0.9 for the amplitudes
Time (s)0 5 10 15 20
Visu
al le
ngth
( µm
)
0
5
10
15
20
25
Time (s)0 5 10 15 20
Res
idua
ls ( µ
m)
-1
0
1
log[Length (µm)]0.5 1.0 1.5 2.0
log[
λ t]
-2.5
-1.5
-0.5
0.5A B
Figure 18: Data collapse of relaxation curves (A) The results of one such rescaling of a template data from a 42 µm chain to a data set for a 21 µm chain yielded a time rescaling parameter (λt ) of 0.31. The rescaled curves lie on top of each other quite well. The inset shows the difference between the two curves. (B) Dependence of the time rescaling parameter (λt ) on length. The black points are data taken in the sucrose solution and the white ones in the glycerol solution. The solid line is a linear fit to all of the data and an yielded a scaling exponent of 1.79 ± 0.08. For template sizes of 11 µm and 23 µm, the scaling exponents were 1.52 ± 0.05 and 1.63 ± 0.05 respectively. The linear behavior of the data shows dynamical scaling in agreement with the results shown in Figure 17B. Reprinted with permission from Science 264, 819, 1994. Copyright 1994 American Association for the Advancement of Science.
3.4.2 Rescaling relaxation curves: Data collapse
Data collapse is another method of measuring scaling relations without reference
to a specific model and is based on the similarity between the curves of the data as shown
in Figure 18. In the data collapse method, the relaxation data is rescaled in time and
length as well as a baseline offset until it lies on top of a template curve. The time
rescaling parameter (λt ) then indicates how the relaxation time scales with length as the
template is collapsed on to each of the other data sets.
54
Three templates of varying length were arbitrarily chosen and collapsed onto the
rest of the data. A typical collapse is shown in Figure 18A, with residual errors shown in
the inset. The time rescaling parameter for the longest template (L = 41µm) is plotted
versus length in Figure 18B and fits a scaling law λt ~ L3v with an exponent 3ν = 1.79 ±
0.08. The medium length template (L = 20µm) gave 3ν = 1.68 ± 0.05 and the shortest
template (L = 11µm) gave 3ν = 1.53 ± 0.05. The longer templates may be more indicative
of the true value of the scaling exponent because the data for longer pieces of DNA
contained more information, though the value reported in the abstract is the average value
of the three templates. These scaling exponent values are consistent with the results
obtained from the inverse Laplace transforms and within the theoretical values discussed
above. The systematic dependence of the exponent on template length may suggest that
dynamical scaling might not be perfect. However, this dependence may simply be due to
the floating baseline in the relaxation data sets (especially for the shorter lengths of
DNA), causing the fitting program to trade off time rescaling for baseline rescaling.
3.5 Discussion
The relaxation of a tethered chain is not the same as that of a free chain. In fact
the tethered relaxation is that of a free chain of twice the length, considering symmetry
about the center of mass and neglecting hydrodynamic interactions between the two
halves. We have not measured the end-to-end distance of the polymer chain, rather its
visual length within the depth of field of the microscope (~ 1 µm). From an experimental
point of view, the difference between this relaxation and that of the end-to-end separation
is probably not very significant, for we see that the transverse excursions are almost
always less than 10% of the chain length. Also, a computer simulation of the Zimm
model showed that the maximum displacement (visual length) under traction was within
about 2% of the end-to-end displacement, for displacements above the radius of gyration.
In a recent experiment, Manneville et al measured the relaxation of single DNA
molecule extended in a shearing flow [65]. They analyzed the initial rapid recoil and
determined it followed the scaling law L0-x(t) ∼ t 0.51 where L0 is the “full” extension of
the molecule. This scaling was determined for λ-DNA and fragments of λ-DNA that
were 50% and 75% the length of λ-DNA. These results agrees with the predictions of
55
Brochard-Wyart [66]. However, in their analysis, this group used L0 = 16.3 µm, the
crystallographic length for unstained DNA, instead of the true contour length of the
molecule, which is presumably 21-22 µm due to the staining of the DNA. We performed
a similar analysis on our relaxation data and found that the scaling exponent depended on
the choice of L0. Nonetheless, this experiment presents an excellent and straight forward
method to extend DNA without attaching it to microspheres. The scaling of the initial,
rapid relaxation is an interesting problem and can be further investigated with the long
DNA we are now able to manipulate.
In summary, we have measured the relaxation of a single polymer chain from near
full extension. Although a highly extended chain will be a different regime of dynamics
from most theories of polymer dynamics that treat small fluctuations about equilibrium,
our data agree with dynamical scaling when both the longest relaxation time τ and the
rescaling parameter λt are plotted on a log-log plot versus length. Our two average values
of the scaling exponent agree even though the inverse Laplace transform analysis
determined the scaling for the longest decay time, which described the approximately the
last third of the relaxation curve and the data collapse method, which rescaled the full
relaxation curve. Our measured values 1.66 ± 0.12 also agrees quantitatively with the
scaling exponent measured by intrinsic viscosity (3ν = 1.65) [59] and dynamic light
scattering (3ν = 1.65) [54-57]. The scaling exponent determined from an extended chain
is closer to that predicted by the Zimm model, where hydrodynamics coupling within the
chain is included, than that predicted by the Rouse model, where such intrachain
hydrodynamic coupling is neglected. This is in disagreement with the expectation that
the Rouse model is a better description than the Zimm model for extended chains [41].
3.6 Future Prospects
The data for the results presented here were taken in the Fall of 1993. We have
since developed to the ability to manipulate and measure the relaxation of DNA
molecules up 150 µm long. The scaling of τ has not been investigated at these new
longer lengths. Theoretically, molecules of the length of 4 –40 µm do not have a large
variation in the hydrodynamic drag [24]. However, for longer molecules, the variation
becomes more significant and lead to a change in the scaling for molecules extended in a
56
uniform flow [10, 24]. Thus, it will be of interest to investigate whether this change in
scaling is observed in the relaxation measurements.
57
Chapter 4 Concentrated polymer dynamics: Reptation
4.1 Introduction
Understanding the behavior of a concentrated solution of polymers is a
fascinating and challenging problem in condensed matter physics. Interactions between
entangled polymer chains lead to a rich variety of unusual properties. On long time
scales these materials flow like a viscous liquid, on shorter time scales they have the
elastic response of rubber, and for still shorter times they display glass-like behavior [1,
2, 47]. These viscoelastic materials play an important role in many biological systems
and have very broad commercial applications.
The most popular and successful theoretical model for a concentrated polymer
solution is the reptation model of de Gennes, Edwards and Doi [1, 2, 48, 67, 68]. The
key assumption of the model is that an entangled polymer chain is confined to a tube and
moves in snake-like fashion through the tube. The notion of a tube, originally due to
Edwards [67], is based on the idea that a chain is topologically constrained by its
neighbors from undergoing transverse displacements. As a consequence, the chain
behaves as if it were trapped in a small tube that follows its own contour. The motion of
an individual chain is governed by one-dimensional diffusion within the tube. de Gennes
originally modeled the motion of the polymer in a fixed tube as diffusion of a gas of non-
interacting defects carrying stored length along the chain. The chain ends can diffuse or
"reptate" out of the original tube creating new portions of tube. He conjectured that the
results might be extended to the case of concentrated or molten polymers in which the
tubes would be effectively frozen on time scales less than the characteristic time
separating the viscous from the elastic domain. de Gennes [48] and Doi and Edwards [1]
extended the basic reptation model to include tube deformation and constraint release,
which would allow the tube to undergo limited transverse motion on longer time scales.
Tube deformation is predicted to occur when a strained polymer relaxes its strain by
deforming the surrounding polymers. Constraint release, referring to the release of
topological constraints due to reptation of the surrounding chains, is also expected to
occur on long time scales. The great theoretical advantage of the reptation model is that
58
it allows the simplification of a many-body problem to a single-body problem moving in
an effective mean field.
While experimental studies of the bulk properties of polymers have tested the
predictions of reptation [1, 42], the assumptions of the reptation model must be tested by
probing the dynamics of individual chains. Neutron scattering measurements [69, 70]
using deuterated polymers as test chains have shown the dynamics are consistent with
tube-like constraints on time scales faster than 10-8 s, but were unable to investigate
longer time behavior. Recently, Russell et al [71] have shown that ends of a polymer at
the interface of two layers diffuse more rapidly than the central portion of the chain by
differentially labeling chains with deuterium and probing the motion across a boundary
with secondary-ion mass spectroscopy. While these methods of experimentation yield
results consistent with reptation, they have an inherent disadvantage: the detailed motion
of a single chain cannot be measured and its dynamics must be inferred from indirect
measurements averaged over a macroscopically large number of chains.
We present direct molecular observation of tube-like motion of a single polymer
chain using fluorescence microscopy. This method of observing reptation by
fluorescently staining a single DNA molecule in a background of unstained DNA was
first suggested by Chu [7].
4.2 Materials and Methods
Solutions of DNA are an excellent system for studying polymer dynamics [7, 9,
33]. Modern molecular biology techniques offer precise control in the preparation and
manipulation of DNA molecules, facilitating the preparation of monodisperse polymer
solutions. Single molecules of DNA are of sufficient size (10's µm) that they can be
directly visualized using fluorescence microscopy using efficient, tightly-bound
fluorescent dyes. We used YOYO-1 from Molecular Probes Inc., which allows for one
molecule to be selectively stained in a background of unstained DNA molecules. For
technical reference, see H. S. Rye et al [43]. Several groups have observed individual
DNA molecules undergoing gel electrophoresis [16, 17].
For the test chain, single DNA molecules of up to 100 µm in length were formed
by linking several λ-phage DNA (Lstained ≅ 22 µm) molecules together. One end of the
59
DNA molecule was biotinylated and attached to a streptavidin coated polystyrene
microsphere. The concentrated polymer solution consisted of 600 µg/ml of highly
purified lambda DNA in a buffer solution of 12 mM Tris-HCl pH 8, 1.2 mM EDTA,
0.01% Tween-20 and 2 mM NaCl. The DNA was concentrated from a stock solution
(NEB) using a speed-vac. By weighing the tube before and after applying the vacuum,
the change in volume can be calculated. This density of 600 µg/ml of DNA
corresponded to a concentration of 12 molecules/µm3. Because the radius of gyration of
a single λ-DNA molecule is 0.71 µm in dilute solution [30], there were approximately 18
polymers packed into the space that one would occupy in dilute solution (c = 18 c*) and a
large degree of entanglement was expected. Prior to using the concentrated DNA, the
solution was heated to 70°C for a few minutes to linearize any circular DNA molecules.
The experiment was carried out at room temperature.
BA
Figure 19: Schematic representation of the experiment. (A) A stained DNA molecule in a concentrated solution of unstained λ-phage DNA was manipulated with optical tweezers by means of a 1.0-µm polystyrene microsphere attached to one end of the molecule. (B) The topological constraints of all the background chains collectively act to confine the polymer motion within a tube-like region along its own contour. Reprinted with permission from Science 264, 822, 1994. Copyright 1994 American Association for the Advancement of Science.
In a modified optical microscope using a high power objective (Zeiss NA 1.4,
63x), 1-µm diameter polystyrene microspheres were trapped and manipulated in an
aqueous environment using optical tweezers [6]. An attenuated Nd:YAG laser producing
about 100 mW at the focus formed the tweezer. The fluorescently stained DNA was
60
excited by an argon-ion laser operating at 488 nm and the resulting fluorescent images
were taken by a S.I.T. video camera (Hamamatsu), processed by a image processor
(Hamamatsu Argus 10), and recorded by a VCR.
Figure 20: A series of tiled images showing the tube-like motion of a relaxing DNA molecule in a concentrated (12 molecules/µm3) polymer solution. The time spacing in the first row is 1.9 s and for rows 2-6 it is 5.1 s. The DNA was approximately 80 µm long and the image of the 1-µm microsphere is 1.5 times its true size due to blooming in fluorescence.
4.3 Measurement of Tubelike motion
The fluorescently labeled test polymer chain was dragged through the entangled
solution using optical tweezers at velocities from 5-100 µm/s (Figure 19). This allowed
61
us to stretch the chain into various conformations in the focal plane of the microscope
and produce kinks and loops on length scales smaller than the radius of gyration of the
background polymers or the diameter of the microsphere. As the test chain was pulled or
relaxed through the concentrated polymer solution, it closely followed the path of the
microsphere, allowing for various conformation to be drawn with the test chain (Figure
20). This tube-like motion is strikingly different than the motion of a polymer chain in a
viscous Newtonian fluid (Figure 21). In a Newtonian fluid (~25 cP), the chain moved in
a direction perpendicular to its contour even on the most rapid time scales investigated
(1/30 s).
Figure 21: A series of tiled images showing that a loop formed in viscous Newtonian fluid (25 cP) rapidly disappeared as the DNA chain moved transverse to its contour. The time for the images are 0, 0.5, 1.7, 2.1, 2.9, 3.6, 4.4, 5.1, 5.9, 6.6, 7.4 s. The solution was 65% (w/w) sucrose in an aqueous solution containing 5 mM NaCl, 1 mM Tris HCl pH 8, 0.01% Tween-20 and 0.5 mM EDTA. Reprinted with permission from Science 264, 822, 1994. Copyright 1994 American Association for the Advancement of Science.
4.4 Effect of the microsphere on the background chain In our experiment, a primary concern was to understand the disturbance of the
microsphere on the background polymer solution. In standard polymer experiments, a
62
polymer chain in solution is deformed by local stress forces in the bulk fluid or by
Brownian motion. The dynamics of deformed chains in the presence of other chains
leads to the bulk properties of the solution. In our case, we directly deform a single chain
by pulling on it with a microsphere. We were concerned that a path created in the
entangled polymers would define a tube that was independent of the reptation model.
Furthermore, if the tube collapsed back to molecule dimensions, there might still be a
residual “weakness” in the entanglements that could survive for a long time. We
conducted three experiments to investigate these two possibilities.
Figure 22: A tiled series of images showing tube deformation and stress recovery in a concentrated polymer solution. As the microsphere was rapidly moved away from the loop, tension in the chain increased, applying a stress that decreased the diameter of the loop. After the microsphere stopped, the loop recovered to its original size. The times for the images are 0, 1.3, 1.6, 2.3, and 3.0 s. Reprinted with permission from Science 264, 822, 1994. Copyright 1994 American Association for the Advancement of Science.
We measured the relaxation time for the background chains to interact with the
test chain after the passage of the microsphere by comparing the relaxation of a linearly
stretched-out chain in the polymer solution to that in the solvent alone (buffered solution
with a viscosity of 0.95 cP). If the movement of the microsphere carrying the test chain
formed a hole of pure solvent in the network of background chains, the test chain would
be free of other entangling polymers, and one would expect the test chain to relax as if it
63
were in pure solvent. The relaxation of the test chain stretched at 80 µm/s in the polymer
solution became much slower than the relaxation in the pure solvent in less than 1 s
Figure 23 [9].The relaxation data were analyzed for a spectrum of decaying exponential
with an inverse Laplace transformation [61], which yielded peaks at a time constant of
about 0.6 s in the pure solvent and 0.7 s and 14 s in the concentrated polymer solution.
Hence, after about 1 s, the background chains have become close enough to strongly
interact with the test chain. Furthermore the surrounding chains are about one fifth the
length of the longest test chain and, within the reptation theory, should reptate 125 times
faster and rapidly relieve any induce stress in the background polymers.
We made another estimate of the background polymer relaxation by stretching out
the test chain linearly by moving the microsphere at constant velocity (10 µm/s) and then
suddenly reversing the direction of travel so that the microsphere went back along its
original path. After the change of direction, the trailing polymer chain was pulled back
on itself by the microsphere. For a brief time, the section of chain following the
microsphere was dragged along with the microsphere; but after about 1 s, the chain
caught on an entanglement and became anchored at a particular point, around which it
was pulled like a rope on a pulley. This experiment showed that the test chain was
entangled with the background chains and the entanglement withstood the elastic forces
of the stretched DNA. Thus, any path weakened by the passage of the microsphere
through the polymer solution recovered in about 1 s.
Finally, we directly investigated the disturbance of the moving microsphere
through the background chains by observing a fluorescently labeled chain entangled with
the other unlabeled background chains while rapidly moving the microsphere back and
forth across the labeled molecule. Very little deformation of the entangled chain was
observed, suggesting that the entangled chains are elastically deformed by the
microsphere and relaxed very quickly without disentangling. Rarely, this test chain
would elastically deform up to about 2 µm but would elastically retract back to its
original conformation in less than 1 s.
64
4.5 Observation of Tubelike motion
To test for tube-like motion of the test chain on longer time scales, small loops
that encircled a bundle of background chains were drawn with our joystick controlled
optical tweezers. Encircled chains imposed a topological constraint on the test chain,
which could easily be visualized. As shown in Figure 20, the topological constraint of
such a small loop persisted for at least 120 s. We were unable to observe the topological
constraint at longer times because the test chain, relaxing inside the tube, went around the
loop. We have shown that a stretched polymer relaxes to its equilibrium state in a tube
defined by a configuration created with optical tweezers and that the tube lasts for times
in excess of 120 s. Thus, the tube-like constraints in a concentrated polymer solution
exist and are stable for long times. In future work, this time limit can be overcome by
attaching each end of the DNA to a microsphere, and drawing a loop while holding both
ends to prevent relaxation of the DNA.
4.6 Constraint releases, excess chain segment diffusion and tube
deformation
Whereas the constraints of a small loop can stay fixed for very long times, straight
sections of the tube may drift and fluctuate slightly. In the bottom four rows of Figure
20, one sees that at about a 1 min, the middle section of the tube between the loop and the
microsphere began to sag downwards and to develop small wiggles as excess length
moved into the region. This was most likely due to constraint release, as surrounding
chains reptated in and out of the way of the linear section of the chain. Constraint release
near the more contorted section of the chain forming the loop, which hardly drifted at all,
was evidently much slower.
Evidence for chain segment diffusion is also seen in Figure 20. The chain
segment or monomer density can be inferred by the fluorescent intensity along the chain.
The first row of images in Figure 20 shows the sharp increase in the fluorescent intensity
of the chain near the relaxing end without a corresponding increase in intensity of the
chain close to the microsphere. These excess chain segments only slowly moved through
the entanglements along the test chain. At 9.3 s after the microsphere stopped, the image
data shows last half of the chain is significantly brighter than the half closer to the
65
microsphere. After 24 s, excess chain segment density has diffused along the full length
of the chain as evidenced by the increased fluorescent intensity along the whole length of
the chain. This diffusion was driven by the tension within the chain and slowed by
entanglements and contorted chain confirmations. There was a ball of excess chain
segments at relaxing end until approximately 100 s.
Time (s)0 5 10 15
Leng
th (µ
m)
0
6
12
18
Figure 23: Comparison of the relaxation of a 17 µm molecule linearly stretched to full extension at 80 µm/sec in a concentrated polymer solution of 12 molecules/µm3
(white points) and in pure solvent (black points). The relaxation in the polymer solution became much slower than that in the pure solvent in a time of less than 1 s, indicating that background chains were strongly interacting with relaxing polymer in less than 1 s. Reprinted with permission from Science 264, 822, 1994. Copyright 1994 American Association for the Advancement of Science.
As shown in Figure 22, tube deformation and microscopic elasticity were also
observed. After the loop was drawn, the microsphere was rapidly moved away from the
66
loop pulling the chain taught. This increased tension in the chain squeezed down the loop
deforming the original tube with an applied stress. The entangled polymers, which form
the constraint, were compressed until they balanced the applied stress. After the
microsphere was stopped, the stress applied by the loop decreased as excess chain
segments diffused in from the relaxing free end and relieved the tension. With this
reduction is stress, the loop grew back to its initial size. In contrast, loops formed in
Newtonian fluids always decreased in time. Thus, we have shown the deformation of a
tube with an applied stress and its subsequent recovery when the stress is relieved.
4.7 Conclusions
In summary, tube-like motion of a single polymer in a concentrated
polymer system has been directly observed with a fluorescently labeled test chain
manipulated using optical tweezers. The disturbance due to the motion of the
microsphere relaxed to the molecular dimensions of the polymer in less than 1 s while
topological constraints imposed by the tube persisted for at least 120 s.
Subsequently, in an experiment led by D. Smith, we measured the diffusion
coefficient of fluorescently labeled DNA molecules in an entangled solution of unstained
λ-DNA [29]. The measured lifetime (τconstr > 2 min.) of the tube-like topological
constraints were longer than the self-diffusion time (τself = 20 s), where we have included
corrections for the change in RG with concentration and the change in length between the
stained and unstained DNA. The scaling of D with polymer length and concentration
was consistent with the predictions of reptation for the highest concentrations and longest
chains studied though these values were determined over a very small range in length and
concentration. Thus, self-diffusion measurements further support the assumptions and
predictions of the reptation theory.
Recently, Musti et al [72] performed bulk viscoelastic measurements on semi-
dilute solutions of T2 DNA (164 000 bp) and found behavior consistent with
measurements on synthetic polymers and with the reptation theory for concentrations
greater than Ce = 0.25 mg/ml. The prediction of λ-DNA (48 500 bp) based on their
results (correcting for the length difference between λ and T2 DNA) is Ce = 0.6 mg/ml.
This is approximately the concentration we used in our studies and is consistent with our
67
observation that at 0.63 mg/ml, the diffusion followed but at 0.40 mg/ml it did not. It is
also consistent with the observation that at 0.63 mg/ml λ-DNA followed reptation’s
predictions but shorter fragments did not. Taken together, Musti et al [72] and Smith et
al [30] allow us to understand the time scale of tube-like constraints the we have
observed and quantitatively illustrate the role of reptation.
In other subsequent work, D. Wirtz coupled a magnetic microsphere to a DNA
molecule and reported to measure the transport properties of a single DNA molecule in a
concentrated solution of unstained DNA [60]. While Wirtz’s use of magnetic
microsphere technique developed by Smith et al [15] offers an excellent opportunity to
study the motion of a labeled test chain, there were the numerous errors in the paper and,
in light of these errors, the conclusions were not justified
Wirtz [60] tethered a single DNA molecule to a magnetic bead and observed its
molecular conformation using fluorescence microscopy. For a given magnetic force (F)
applied to the bead, the resulting transport velocity of DNA molecules (v) and extension
(x) of a fluorescently labeled molecule were measured in a “concentrated” DNA solution
of unlabeled molecules. We point out several large discrepancies between the data
presented in ref. [60] and other experimental data on DNA molecules.
It is stated that the experiment [60] was done at a concentration of “18.1 vol %”
of DNA having a length L = 40 µm. Considering the partial specific volume of DNA
(0.55 ml/g) [73], this is an extraordinarily high concentration (≈330 mg/ml DNA). For
comparison, the transition from isotropic to structured anisotropic (cholesteric) liquid
crystalline phases of 50 nm long DNA occurs at about 100 mg/ml in a solution with 0.01
M Na+. Solutions of DNA molecules above this concentration are turbid and opalescent.
Therefore, it is difficult to believe that the Wirtz experiment was done at 18.1 vol %
DNA.
By measuring the friction coefficient (ζ ≡ F/v) of the molecule in the small-
velocity regime when the DNA is essentially unperturbed, Wirtz calculated a value of D
= kBT/ζ = 9.33 ± 0.42 µm2/s for a 28 µm fluorescently labeled DNA in this
“concentrated” solution. For comparison, at the zero concentration limit, we measure the
value of D to be 0.47 ± 0.03 µm2/s for a single, isolated fluorescently labeled λ-phage
DNA (contour length 22 µm) [30]. Our value agrees with the value of 0.54 ± 0.15 µm2/s
68
obtained by Scalettar et al [74] and both of these values are consistent with the
relationship RG = kBT/(6πηD) predicted by the Zimm model [39].
There appears to be at least two errors in the calculation of D given the data in
Fig. 3 of this paper. From the linear portion of Fig 3, the data shows ζ ≅ 2 pN/ µm s-1
not ζ = 4.41 pN/ µm s-1 as stated in the text. Given this value, we calculate D = kBT/ζ ≅
0.002 µm2/s not 9.33 µm2/s. Working backwards from this recalculated value of D using
the reptation model [68] and the data presented in Smith et al [29] (D=6.56 x 10-3 µm2/s
for L=22 µm when at C = 0.63 mg/ml λ-phage DNA), we calculate C ≅ 1 mg/ml or a
0.005 vol% DNA solution.
However, C ≅ 1 mg/ml is inconsistent with Fig 2 of Wirtz’s paper where the
extension of a molecule reported to be 28 µm is shown for 5, 10, 65 µm/s. These
extension measurement are remarkably close to the values we previously reported for the
stretching of a single, 34.6 µm long, isolated DNA molecule in a uniform flow of pure
aqueous solvent (Fig 1 b in Perkins et al [10]). Therefore, it is difficult to reconcile these
extensions vs. velocity results with a nonzero concentration of background chains.
Further, the magnitude of the applied forces seem disproportionate to molecular
deformation given the measured elasticity of DNA. Fig 3 and 2 b show a measured
velocity of 5 µm/s for an applied force of about 4.5 pN and an extension (x) of about 10
µm ( x/L ≅ 0.3 ). This is difficult to reconcile with elasticity measurements of Smith et al
[15, 22, 25], which show x/L ≅ 0.3 when a force of about 0.04 pN is applied across the
molecule’s ends.
Wirtz interpreted the average conformation within a reptation-based model for
concentrated solutions [75]. Previously we measured the lifetime of the tube-like
constraints to be >2 min. in 0.6 mg/ml λ-phage DNA and a longest entropic relaxation
time of ≈20 µm long DNA molecule to be 14 s at this concentration [8]. Given these
values, a molecule moving at 5 µm/s, as shown in Fig 2b in Wirtz’s paper, would not
have an opportunity to adopt any confirmation via reptation except a line with a width of
about the entanglement spacing -- the motion of the free ends could not diffuse fast
enough relative to the translating background chains which form the tube. However, the
image in Fig. 2B of Wirtz’s paper shows a transverse width of 3 µm. To verify our
69
intuition, we observed a 34 µm long chain at 0.43 mg/ml λ-DNA for v= 1-20 µm/s. We
did not see such significant transverse fluctuations at any velocity and we measured x/L =
0.66 at v = 5 µm/s as expected considering the higher frictional drag in a semi-dilute
solution.
The only way we can reconcile the data is by assuming the forces are lower by
about 1/250 and assuming a dilute solution limit. The first assumption corrects the
measured friction coefficient and brings D to a dilute solution value.
In summary, the discrepancies Wirtz’s paper and existing data on DNA make it very
difficult to evaluate the DNA concentration, the scale of the applied forces, and the
relationship of the data to theoretical models.
70
Chapter 5 Single Polymers in a Uniform Flow
5.1 Introduction
The deformation of polymers in hydrodynamic flows is a fundamental and still
incompletely resolved problem in polymer physics [1, 4, 5]. The major difficulty in
theoretical descriptions of polymer chain dynamics is the hydrodynamic coupling within
the chain -- the motion of one part of the chain perturbs the surrounding flow and
modifies the hydrodynamic force exerted on another part. Here, we present results of the
stretching of single, tethered DNA molecules in a uniform fluid flow.
Direct observation and controlled deformation of individual DNA molecules
gives insights into the previously inaccessible regime of single polymer dynamics [7-9,
15]. Earlier, we observed the relaxation of stretched DNA molecules in dilute and
concentrated polymer solutions using optical tweezers [6] and fluorescence microscopy
[7-9]. The present experiment addresses the balance of forces between hydrodynamic
drag on a deformed polymer and its entropic elasticity. In earlier work, Smith e. a.
measured the extension of single DNA molecules stretched by a force applied at the ends.
They attached one end to a surface and exerted a force on a magnetic microsphere at the
other end by a combination of magnetic and hydrodynamic forces [15]. Since the forces
exerted on the microsphere were larger than the hydrodynamic drag due to the DNA,
Smith et al measured primarily the elastic force, a static property of a polymer in
solution. Theoretical calculations [22, 25, 32, 76] show quantitative agreement between
the elasticity data and the force law for a worm-like chain and give an approximate
formula [25] of
( )FA k T x L x LB/ / /= − − +−14
114
2 Eq. 4
where F is the force applied across the ends, A is the persistence length, x is the
extension, L is the length of the polymer and kBT is the thermal energy.
Our measurements are made by optically trapping a microsphere attached to one
end of a DNA molecule while the other end remains free allowing us to investigate the
71
hydrodynamic interaction between the polymer and the fluid. The chain is positioned
away from any surface and elongated in a uniform (non-shearing) flow by translating the
fluid past the trapped, stationary microsphere. By determining the scaling properties of
the system as a function of polymer length and solvent viscosity, we investigated the
effects of hydrodynamic coupling. As Zimm showed, the hydrodynamic coupling within
a polymer near equilibrium causes, in the simplest case, the total hydrodynamic drag Fdrag
to scale as Fdrag ~ L0.5 [39]. If hydrodynamic coupling is negligible, the chain is said to
be “free draining” and the hydrodynamic drag would be Fdrag ~ L. It is often assumed
that a polymer would become free-draining in the limit of large extensions. For example,
de Gennes modeled the hydrodynamic shape of a polymer in an elongational flow as
cylinder into which the flow field penetrated on a finite length scale making the chain
free draining in the limit of large extensions [41, 77].
5.2 Applying uniform flow to DNA
To generate a flow around the trapped, stationary microsphere, the microscope
stage was moved at a constant velocity of 1 to 200 µm/s with a feedback-controlled
motor. The motor was stable to within 1% as determined by a signal analysis of the
pulses from its 0.1 µm/pulse optical encoder. A Fourier transform of 400 extension
measurements had no peaks in the spectrum, showing the fluctuations were not driven by
mechanical resonances.
The microsphere was held stationary against the applied flow using optical
tweezers and suspended 12 µm below the surface of the coverslip. The coverslip and
microscope slide were separated by two 75-µm wires and sealed with epoxy. The optical
tweezers were made by focusing a 175 mW Nd:YAG laser beam with an oil-immersed
microscope objective (Zeiss x63, NA 1.4). The DNA was stained with 0.1 µM TOTO-1
(Molecular Probes), excited by 488 nm light from an argon-ion laser and imaged with an
intensified video camera (Hamamatsu C2400-08). The molecules were observed in an
aqueous solution of 10 mM Tris-HCl (pH 8), 1 mM EDTA, 10 mM NaCl, 0.8%
β-mercaptoethenol, 50 µg/ml glucose oxidase, 0.1% glucose, and 10 µg/ml catalase.
DNA molecules up to about 100 µm long were constructed by ligating together several λ-
phage DNAs (16.3 µm) and linked them to a 0.3 µm or 1 µm latex microsphere by a
72
streptavidin-biotin bond. At this dye concentration, the length of a single λ-DNA was
measured to be 22 µm by attaching each end to a 1 µm microsphere and stretching it
between two optical traps with a peak force of order 10 pN. Variation of lengths away
from multiples of 22 µm were probably due to shearing of the DNA attached to a
microsphere during pipetting.
Figure 24: Images of a fluorescently labeled, 64.6 µm long DNA molecule tethered by one end to a 0.3 µm latex sphere and deformed by constant fluid flows of v = 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 30, 40 and 50 µm/sec with a viscosity of η = 0.95 cP. The fractional extensions are (x/L) = 15, 22, 31, 38, 42, 49, 60, 63, 67, 72, 78, 81 and 83 % respectively. Reprinted with permission from Science 268, 83, 1995. Copyright 1995 American Association for the Advancement of Science.
The extensions were measured by allowing the chain to reach their equilibrium
extension and then digitizing 40 video frames with a delay (0.3 - 1.2 s) between each
frame to allow the fluctuating chain to completely explore the phase space of accessible
conformations. The extension of the molecule along the direction of flow was measured
using a computer-generated cursor and the average extension x and its standard deviation
73
σx were calculated at each velocity from the 40 individual extension measurements. To
minimize the fluid disturbance due to the microsphere, we used a 0.3 µm microsphere at
a viscosity of η = 0.95 cP. To generate larger hydrodynamic forces, we increased the
viscosity to η = 2.9 cP and, to trap at these higher forces, we used 1.0 µm diameter
sphere.
5.3 Initial measurements (22µm < L < 84 µm) In our experiment, we used a broad range of fluid velocities v and two different
viscosities η (Figure 24 & Figure 25). When the extension is plotted versus the product
of the viscosity times the velocity, the data for both viscosities overlap, suggesting that
the hydrodynamic drag on the chain is a function of ηv. Although this would be expected
on theoretical grounds, our measurements do not directly prove it since we measured the
extension, not the drag force. The drag depends on the conformation of the chain. To
investigate the effect of the perturbed flow by the microsphere on the chain deformation
we compared the stretching of a chain of the same length attached to both microsphere
sizes. The measured extension was the same for both microsphere diameters at velocities
greater than 1-2 µm/s, indicating that the 0.3 µm microsphere was, at most, a small
perturbation on the fluid flow over the range of velocities studied (1 - 200 µm/s).
Remarkably, the functional dependence of the extension versus flow velocity is
the same as extension versus force data of Smith et al for fractional extensions of about
20% to 90% (Figure 26). This correspondence is unexpected and non-trivial since the
nature of the force applied to the chain in the two cases is quite different. In our case, the
applied force is zero at the free end in contrast to the experiment of Smith et al in which
the dominant force was applied to the end of the chain via the attached microsphere.
Nevertheless, this functional correspondence allows us to determine an asymptotic length
of partially stretched chains by fitting our data to the force law given by Eq. 4. This
fitted, asymptotic length is robust and changes by 3% as the last data point was decreased
from 183 cP-µm/s to 12 cP-µm/s. Note, however, that our data at the largest extensions
(>90%) is slightly larger than the values predicted by Eq. 4. Similar behavior was found
in the experimental data of Smith et al and it was suggested that this might be due to
deformation of the DNA helix beyond the limit of entropic elasticity. In fact, a transition
74
from normal B-DNA to a stretched form has been observed to occur at a tension of 65 pΝ
[23, 31].
ηv (cP-µm/s)1 10 100
Exte
nsio
n ( µ
m)
1
10
ηv (cP-µm/s)0 200 400 600
Exte
nsio
n ( µ
m)
0
10
20
30
40
Figure 25: A Log-Log plot of the extension versus the product of viscosity (η) and velocity (v) for a 34.6 µm long molecule. To minimize the fluid disturbance due to the microsphere, we used a 0.3 µm microsphere and a viscosity of η = 0.95 cP (black points). To generate larger hydrodynamic forces, we increased the viscosity to η = 2.9 cP and, to trap at these higher forces, we used 1.0 µm diameter microsphere (white points). (Inset) All of the data plotted with linear axis scales. Reprinted with permission from Science 268, 83, 1995. Copyright 1995 American Association for the Advancement of Science.
To look for scaling properties of the system, we measured the extension as a
function of velocity (1-80 µm/s) and chain length (22-84 µm) using a 0.3 µm sphere to
minimize the disturbance to the fluid flow (Figure 27A). These sets of data collapsed
onto a single curve when the fractional extension (x/L) was plotted versus vLα with α =
0.54 ± 0.05 (Figure 27) showing that the fractional extension is a function of ηvL0.54
75
when combined with the functional dependence shown in Figure 25. When x/L is plotted
versus vL, as suggested by the free draining model, the data for different length molecules
do not collapse onto a universal curve, even for the largest extensions investigated
(x/L≈0.85).
ηv ( cP-µm/s) or F (pN/122)1 10 100 1000
Frac
tiona
l ext
ensi
on (x
/L)
0.1
1
V-1/20.1 1
L - x
1
10
100
Figure 26: Comparison of the extension versus flow velocity data (white points) to the predicted form of the force versus extension law for a worm-like chain with a force applied at the ends (solid line) and the elasticity data of Smith et al. (black points). The elasticity data (in picoNewtons) was multiplied by 122 to overlay our experimental data. (Inset) (L-x) plotted versus v-1/2. This shows the deviation of the experimental data from the functional form of dominant term of Eq. 4 at large extensions ( A/(1-x/L)-2) and, within the dumbbell model, the linear portion of the plot shows independence of hydrodynamic drag on conformation. The first four data points (highest ηv) deviate from the linear behavior and were not used in determining an asymptotic length. Reprinted with permission from Science 268, 83, 1995. Copyright 1995 American Association for the Advancement of Science.
76
This functional form of the extension over the range of lengths from 22 µm to 84
µm suggests that the DNA molecule is not free draining. For a free draining chain, the
tension would increase as F(s)=γvs where γ is a constant drag coefficient per unit length
and s is position along the chain contour from the free end (Figure 27, Inset). The tension
at one segment of the chain is due to the hydrodynamic drag of the downstream portion
of the chain. The corresponding projected distance (z) anti-parallel to the flow direction
is computed by integrating the local, coarse-grained stretching of the chain,
( )dz ds f FA k TB= −1 where f −1 is the inverse of a force function of the form
FA k T f x LB/ ( )= . The total deformation is given by
xdzds
ds fFAk T
dss
L
Bs
L= =
⎛
⎝⎜
⎞
⎠⎟=
−
=∫ ∫
01
0.
Eq. 5
Numerical integration of this expression with the force function given by Eq. 4 shows
that x/L for a free draining polymer is a universal function of ηvL, in disagreement with
the observed dependence. This analysis shows that the DNA is not free draining, despite
being extended to almost its full contour length (~85%).
A non-free draining chain interacts with and modifies the surrounding fluid flow
differently than a free draining chain. DNA undergoing electrophoresis in solution does
not separate by molecular weight and is thought to be free draining due to counter-ion
flow [78]. Our measurements highlight the difference between electric and
hydrodynamic forces on DNA. Single molecule experiments could test the assumption
that DNA driven by electric fields is free draining. The interaction of extended polymers
with the surrounding fluid is also responsible for turbulent drag reduction [41]. These
results suggest the interaction is different than previously believed and, thus, may aid in
understanding this effect.
On an empirical level, our data shows that a dumbbell model (with one bead held
fixed, the other bead free, and a spring obeying the force law of Eq. 4) may be used to
predict the steady-state extension of the chain in a uniform flow. The flow extends the
chain as if a force proportional to ηvL0.54 were applied at its end via a fictitious bead. The
77
fictitious bead’s radius (Reff), and thus the chain’s hydrodynamic drag, would be constant
for a given chain length and scale as L0.54. As v→0, the size of the fictitious bead should
scale as the radius of gyration (Rg) since the coil is only slightly deformed. Such
independence of hydrodynamic drag on extension is consistent with calculations of
Larson and Magda for extensional flow [79]. Our measurements give Reff = 0.6 µm for a
34.6 µm chain. This value is consistent with the estimate of the drag force in the
experiment of Smith et al [15]. In calculating the hydrodynamic force applied to the end
of the chain, they increased the radius of the 2.9 µm diameter magnetic microsphere by
0.7 µm to account for the hydrodynamic drag of their 32.6 µm long DNA.
v L0.54 (µm1.54 /s)0 250 500 750
Frac
tiona
l ext
ensi
on (x
/L)
0.0
0.2
0.4
0.6
0.8
1.0
Velocity (µm/s)
0 25 50 75 100
Exte
nsio
n ( µ
m)
0
25
50
75
z
s
ds
v
dz
A B
Figure 27: (A) Plots of the extension versus applied flow velocity for DNA molecules of lengths 22.4, 34.8, 44.0, 53.1, 63.6, and 83.8 µm attached to a 0.3 µm sphere with a viscosity of η=0.95 cP (B) The fractional extension (x/L) plotted versus vL0.54 collapsed the data to a universal curve. (Inset) Diagram used in the computation of the chain deformation (where s is the distance along the contour and z is the projection of s in the -v direction). Reprinted with permission from Science 268, 83, 1995. Copyright 1995 American Association for the Advancement of Science.
78
While a dumbbell model describes the steady-state extension, it should only be
valid for dynamic processes slower than slowest relaxation time of the polymer. For
example, the rate of relaxation predicted by this model (dx/dt = -F/6πηReff) in conjunction
with the measured force law (Eq. 4) does not agree with the measured relaxation of
tethered DNA molecules. Besides the contribution from higher order modes to the
relaxation not modeled by the dumbbell model, the chain may not have sufficient time to
fully explore its phase space and, consequentially, its elasticity may be different from the
steady-state values.
Fractional extension (x/L)0.0 0.2 0.4 0.6 0.8
σ x2
/ L ( µ
m)
10-4
10-3
10-2
10-1
x/L0.0 0.4 0.8
σ x/x
10-2
10-1
Figure 28: Fluctuations, measured by the standard deviation (σx) about the mean extension, are compared to a dumbbell model prediction of thermally driven fluctuations in the end-to-end distance. Standard deviations corresponding to less than 2 pixels were excluded, which limited the fractional extension to lower than 80%. (Inset) Log-linear plot of the relative size of fluctuations in extension versus fractional extension. Reprinted with permission from Science 268, 83, 1995. Copyright 1995 American Association for the Advancement of Science.
79
Fluctuations of the chain conformation are an important part of polymer
dynamics. These fluctuations are due to Brownian motion and, in a flow, may also be
driven by a variation in the hydrodynamic drag as the chain conformation changes.
Present theories do not consider this type of variation in the hydrodynamic drag. To
characterize the observed fluctuations, we computed the standard deviation (σx) in
extension (Figure 28). The relative of size for the fluctuations were well described by σx
/x ≅ 0.42 exp(-4.9 x/L). This decrease in fluctuations at higher extensions is caused by
the non-linear elasticity. As the spring stiffens, the fluctuations decrease. Within the
dumbbell model, the fluctuations in extension can be modeled as thermal fluctuations of a
fictitious bead in the potential well resulting from the balance between the elastic (Eq. 4)
and the hydrodynamic forces. This calculation gives
( )[ ]
σ x
LA
x L
2
3
2
1 2=
− +−
Eq. 6
which has the correct qualitative dependence and is only 35% smaller in magnitude than
the experimental data.
Direct visualization of the chain conformation gives us further insight into the
deformation problem. Since the chain is uniformly labeled with dye molecules and the
imaging system has linear gain, the chain segment distribution may be inferred from
intensity measurements from instantaneous images (Figure 24) and time-averaged images
(Figure 29). The images are a two dimensional projection of an object fluctuating in three
dimensions. To extract information of the chain conformation, we analyzed only those
video frames when the molecule was reasonably well focused (50-70% of the total
number of frames). Although the fluctuations of the chain take place roughly within a
cone-shaped envelope (Figure 30A), the chain segment distribution does not vary
uniformly along the flow direction (Figure 30B). The transverse width 2σ of the chain
versus position along v was determined by fitting the intensity distribution along each
perpendicular line to a Gaussian distribution. To reduce high frequency noise for the
purpose of fitting, the images were smoothed by convoluting each pixel with a two-
dimensional Gaussian distribution (σ ≅ 0.13 µm). Normalized intensity profiles along the
80
flow direction (v ) for each of the images were obtained by summing the total intensity
along each digital line perpendicular to v . These plots show the appearance, on average,
of a ball of excess chain segments at the free end of the chain and the progressive
decrease in size of the ball as the flow velocity is increased and the chain extends. The
zero position is location of the tether point and the values associated with a small, fixed
spot in the images were removed from the analysis. The data was normalized so that the
area under each curve is equal to 1. There is a localized increase in chain segments at free
end of the chain where the tension goes to zero. A successful theoretical description
should account for the observed conformations as well as the dependence of the
extension on velocity and length.
Figure 29: Time-averaged images of a 67.2 µm long tethered DNA molecule deformed by constant fluid flows of v = 2, 4, 6, 8, and 10 µm/sec with a viscosity of η = 0.95 cP. The relative extensions are (x/L) = 19, 34, 44, 50, and 55% respectively. Approximately 300-400 video frames were digitally captured and averaged with a delay of 0.5 s/frame. A background level was subtracted from the image and then the contrast was linearly enhanced. Fluctuations of the molecule out of the depth of field were not included in the average. Reprinted with permission from Science 268, 83, 1995. Copyright 1995 American Association for the Advancement of Science.
81
Position (µm)0 10 20 30 40
0
1
2
3
Wid
th ( µ
m)
0
1
2
32 µm/s
4 µm/s
6 µm/s
8 µm/s
A
Position (µm)0 10 20 30 40
0
1
2
0
1
2
Rel
ativ
e In
tens
ity (x
10-2
)
B 2 µm/s
4 µm/s
6 µm/s
8 µm/s
10 µm/s10 µm/s
Figure 30: (A) Average transverse width versus position from the tether point for different velocities. The dashed line represents the measured resolution limit for the microscope (B) Intensity as a function of distance along the chain, where the intensity was determined by integrating the transverse dimension. Reprinted with permission from Science 268, 83, 1995. Copyright 1995 American Association for the Advancement of Science.
Our results indicate that the Brochard theory [80] does not describe tethered DNA
molecules. For small deformations in the Hookean regime ( x/L < 0.3), the extension
82
increases with velocity as x ~ v0.70 ± 0.08 in disagreement with Brochard’s prediction of
quadratic or exponential growth depending on solvent quality. Although the chains we
studied are 400-1600 persistence lengths, the scaling picture requires that each blob
contain a large enough number of statistical segments to be described by a random walk.
The blobs, especially those near the microsphere, may not be in the scaling regime and
may not entirely exclude the fluid flow. The dynamics of the unwinding process was also
analyzed within the blob model [81].
Zimm, motivated by our measurements, modeled the stretching of a tethered
polymer near equilibrium using a chain composed of multiple beads and Hookean springs
[36]. He replaced the force on each bead (Fbead) by the average over all beads and then
computed the Oseen tensor using the Kirkwood-Riseman approximation [1, 82, 83]. This
leads to Fbead ~ ηv/L1/2 and, by summing the tension along chain, the end-to-end
extension R/L~ ηvL1/2, close to our experimental result of x/L ~ (ηvL0.54)0.7 for small
deformations. As v→0, the scaling of x with v may have an exponent that is smaller than
unity because x is constrained to approach a constant, Rg, at zero velocity rather than zero
while R is not so constrained. While in this model each bead experiences the same force,
it is the hydrodynamic coupling within the chain that leads to the rescaling of Fbead and
thus the observed functional dependence. To investigate the behavior far from
equilibrium, a detailed comparison between theory and experiment can be made with
numerical simulations by including Brownian motion, the worm-like-chain force law, and
by recalculating the hydrodynamic coupling between the beads at each extension [24]. In
future studies, instantaneous forces on the trapped microsphere can be measured [12] and
these techniques can be extended to the dynamic unwinding of a polymer from a coil and
to more complex flows involving velocity gradients [27].
5.4 Longer chains (L < 150 µm) Larson theoretically investigated our original work (sect 5.3) by analytical and
numerical techniques [24]. This analytical work lead to an unexpected result that the
ratio of the hydrodynamic drag of a DNA molecule ζcoil in a coil and in a fully extended
state ζrod was small, ranging from 1.7-2.8. From simple scaling arguments based on the
difference between the scaling of a rod (∼L) and a coil (∼L0.5), this ratio was expected to
83
be much larger [41]. However, the exact formula for an extended rod has a logarithmic
term is given by [1]:
)ln(
2
dLTk
L
B
rodηπζ =
Eq. 7
where d is the diameter of the rod, i.e., the diameter of a DNA molecule, which is about 2
nm [34] and the experimental values for ζcoil can be determined from our diffusion
measurements [30]. The resulting ratios are shown in Table 2. These results suggested
that the original scaling function dependence of extension on vL0.55 maybe caused by a
limited dynamical range.
L (µm) ζcoil = 1/D ( s-µm-2) ζrod ( s-µm-2) ζcoil /ζrod
22.4 2.1 3.5 1.7
44.0 3.6 16.4 1.8
67.2 4.8 9.4 1.9
83.8 5.4 11.5 2.1
151 7.5 19.6 2.6
Table 2: Ratio of hydrodynamic drag of a DNA molecule in a coil to a fully extended
DNA molecule.
To investigate this theoretical prediction, we measured extension versus velocity
data for molecules up to 154 µm long. As with the original data, the shape of the curves
was very accurately described by Eq. 4. But, the complete range of data from 22 µm to
154 µm did not collapse on to a universal curve. To look at scaling dependence on L, we
determined a rescaling parameter λ by scaling each set of extension versus velocity
measurement to Eq. 4. This analysis revealed that there is a transition from the original
scaling exponent of 0.54 for L < 84 µm to a scaling exponent of 0.75 for 44 µm < L<154
µm (Figure 24).
84
log(L)1.2 1.6 2.0 2.4
log
( λ)
-0.6
-0.4
-0.2
0.0
0.2
m = 0.
75(.0
2)
Figure 31: Rescaling of all the x vs. v data yields a scaling parameter λ. The data for molecules from 44 µm to 154 µm (black points) are well described by a scaling exponent of 0.75 but all of the data is not described by a single scaling exponent
Although the range of lengths investigated by in our experiment is small in
comparison to standard polymer experiments, direct visualization yields a wealth of
detailed information including the extension and conformation. In collaboration with R.
Larson, we developed a numerical model based on a multiple bead-spring model to
compare our detailed experimental results. The model contains no free parameters and
was based on the measured elasticity [15], our diffusion data [30], and the drag of rod
(Eq. 7). Also, the model calculated the Oseen tensor at each velocity based in a self-
consistent manner. This numerical model accurately predicted the overall shape and
scaling of our data. Thus, there is a quantitative agreement between a general polymer
85
model and our detailed DNA data. This agreement emphasizes the generality of polymer
physics and suggests that more complicated experiments with single DNA can yield
additional insight into polymer science.
5.5 Conclusions We have presented the first direct, detailed measurements of a polymer extending in
a uniform flow. We measured the extension, fluctuations in extension as well as time-
averaged conformational quantities. Detailed comparison showed that our results, while
initially unexpected, can be understood within the context of classical polymer physics.
86
Chapter 6 Single Polymers in a Elongational Flow
6.1 Introduction
The behavior of dilute polymers in an elongational flow has been an outstanding
problem in polymer science for several decades [3, 5, 41]. In these flows, a velocity
gradient along the direction of flow can stretch polymers far from equilibrium. Extended
polymers exert a force back on the solvent leading to the important, non-Newtonian
properties of dilute polymer solutions such as viscosity enhancement and turbulent drag
reduction.
A homogeneous elongational flow is defined by a linear velocity gradient along
the direction of flow such that v yy = ε where ε ≡ ∂ ∂v yy , the strain rate, is constant.
Theory suggests that the onset of polymer stretching occurs at a critical velocity gradient
or strain rate ε c of
.ετc ≈0 5
1
Eq. 8
where τ1 is the longest relaxation time of the polymer [79]. For ε < ε c , the molecules
are in a “coiled” state. But as ε is increased above ε c , the hydrodynamic force exerted
across the polymer just exceeds the linear portion of the polymer’s entropic elasticity and
the polymer stretches until its non-linear elasticity limits the further extension of this
“stretched” state.
The simplest model that captures this coil-stretch transition is a dumbbell model
with a non-linear spring. In the dumbbell model, two beads of radius Rbead are connect by
a spring (Figure 32). The two beads represent the coupling between the fluid and the
polymer while the spring represents the entropic elasticity of the molecule. A non-linear
spring is necessary to account for the finite extensibility of the polymer since a polymer
can not extend beyond its contour length L. Neglecting Brownian motion and assuming
steady-state, the two forces on each bead are given by
87
xR
vRF
bead
fluidbeadhydro
εηπ
ηπ
3
6
=
=
Eq. 9
)()(
)/(3 LxxOxk
LxfF
spring
spring
<<+≅
=
Eq. 10
where Fspring is approximately linear for small extensions x. Thus, when εηπ beadR3
exceeds kspring, the molecule undergoes a transition from a coiled to a stretched state. de
Gennes predicted that this “coil-stretch transition” would be sharpened by an increase in
the hydrodynamic drag of the stretched state relative to the drag of the coiled state [41].
Fhyd
Fspr
Fhyd
Fspr
Figure 32: Dumbbell model
6.2 Previous experimental work
In many types of elongational flows, such as flow through a pipette tip or a
contraction in a pipe, the time a polymer interacts with the elongational flow field is
limited. Also, such flow fields are elongational but not homogeneous. The classical
technique to simplify the experimental situation is to use a stagnation point flow (Figure
33). Such a flow increases the polymer’s interaction or residency time tres in the velocity
gradient and it also generates a homogeneous velocity gradient. For polymers whose
88
molecular trajectories approaches the stagnation point, tres diverges and measurements are
made on those molecules near of the stagnation point.
Birefringence is a commonly used technique for inferring the degree of polymer
deformation in this geometry [49, 52, 53, 84, 85]. For example, in dilute polystyrene
solutions, Keller and Odell reported a rapid increase in the birefringence for ε above
ε c followed by a saturation [49]. Such saturation was interpreted as an indication that the
polymers had reached equilibrium in a highly extended state.
Stagnationpoint
Figure 33: Crossed slot flow to increase interaction time between the polymers and flow.
In addition to these birefringence experiments, molecular weight analysis shows
that, at sufficiently high ε , chains are fractured in this flow and such fracture occurs
preferentially at the center of the chain [86-88]. This result further supports the
hypothesis that the polymers reached full extension because the tension within the chain
is highest at the center of the polymer only if the chain is extended.
An alternative method to probe the deformation of a polymer is to measure the
radius of gyration RG by light scattering [50, 89, 90]. To make a direct comparison
89
between light scattering and birefringence, Menasveta and Hoagland performed both
types of measurements on the same experimental apparatus [50]. Their birefringence
results showed the same saturation with increasing ε as seen previously. Similarly, their
light scattering measurements of showed increase in RG at the same cε as their
birefringence data, followed by a saturation. However, the saturating value of RG was
only 2 times the equilibrium size implying that the saturation in birefringence did not
represent a highly extended state.
Imaging Area
Velocitygradient
Stagnation point
Path of DNAalong flow line
x
X-axis (µm)
Y- a
xis
(µm
) x (µ
m)
0 25 50 75
0
15
30
45
Time (s)4 8 12
0
8
16
Stagnation point
Top end
Bottom end
Figure 34: Schematic of experiment.
90
Both of these measurement techniques have an inherent disadvantage: the state of
the polymer must be inferred from an indirect measurement of an optical property not
directly related to the polymer’s conformation. Furthermore, the signal is averaged over
molecules having a broad range of tres, inherent in stagnation point flows. And, the
dynamics of individual polymers are hidden within an average over a macroscopic
number of molecules. Finally, recent birefringence experiments show the same
saturation at the stagnation point but when the probe region is moved towards an outlet,
the birefringence increased [51]. Thus, the origin and meaning of the saturation in
birefringence at the stagnation point is uncertain.
Many rheological effects also remain unexplained. In 1980, James and Saringer
measured a pressure drop of a dilute polymer solution in a converging flow that was
significantly higher than predicted by simple models [91]. Recently, Tirtaatmadja and
Srhidar measured extensional viscosities ηE in filament stretching experiments, which
were several thousand times higher than the shear viscosities [92]. At large
deformations, ηE saturated suggesting again that the polymers were fully extended.
However, the measured stress was significantly lower than expected for fully extended
polymers and the plateau in ηE occurred before the polymers could have become fully
extended [93]. Taken together, these results imply that full extension had not actually
been achieved. The stress relaxation after the exponential deformation was stopped
contained both a strain-rate independent, “elastic” and a strain-rate dependent,
“dissipative” component. The molecular origin of the dissipative component was
uncertain [94, 95].
Examples such as these indicate that, even after a tremendous amount of work,
the deformation of polymers in elongational flows is still ill-understood [93, 96].
6.3 Experimental Technique
In this paper, we report the direct visualization of individual polymers in an
elongational flow. We fluorescently labeled DNA molecules and imaged them in a
standard optical microscope using a low light video camera. By positioning the field of
view of the camera at the stagnation point, we can image the unwinding dynamics of
single polymers as illustrated in Figure 34. From this visual data, we measure the
91
conformation and extension x of each molecule as a function of strain rate ε and
residency time tres in the applied velocity gradient.
Imaging single molecules eliminates four important experimental limitations of
classical techniques. First, direct imaging yields the full conformation of individual
polymers whereas birefringence measures average local orientation of the chain
segments. Second, by tracking single polymers near the stagnation point, we know tres of
each polymer, eliminating the distribution of tres normally found in stagnation point
flows. Third, by working with single, isolated molecules, we eliminate polymer-polymer
interactions and polymer induced alterations of the flow field. Finally, the inherent
uniformity in size of lambda bacteriophage DNA (λ-DNA, Lstrained ≅ 21-22 µm)
eliminates the complications due to polydispersity and enables accurate calculation of
ensemble averages.
6.3.1 Measurement of τ1
To compare our measurements to classical experiments, we needed to establish a
relationship between our single molecule results and the classical definition of τ1 from
bulk measurement. Within the dumbbell model, this relation is easily established
between the stress relaxation and the relaxation of a single chain via
)()()( RFtRnt •=σ Eq. 11
where n is the density of polymers, R(t) is the end-to-end distance and F(R) is the tension
with in the polymer. Since for small displacements F(R) is linear in R, we have
)()()( tRtRnt •∝σ Eq. 12
Thus, the classical relaxation time can be directly related to the measurements from
single molecules. The relaxation time reported here is from a fit over the region where
x/L < 0.3 to
92
G1 R2 -) (-t/ exp c = >x(t)x(t)< τ• Eq. 13
where x(t) is the maximum visual extension and τ1, c and RG were free parameters.
We measured the relaxation of 14 individual molecules from a highly extended
(>16 µm) state (Figure 35). The molecules were extended in the elongational flow and
relaxed when the flow was stopped. Two sets of data were taken in the first and final
preparation of the high viscosity buffer. The statistical nature of the relaxation prevented
an accurate calculation of τ1 from a single relaxation. We therefore averaged the data
between relaxations of different molecules. To account for the small differences in
initial extension of the molecule, all data sets were aligned such that t = 0 for x = 15 µm.
We then fit the averaged data over region x/L < 0.3 to Eq. 13. The relaxation time
between the initial and final solution was constant within the statistical error for the
measurement. By averaging all the data together, we determined τ1 = 3.89 ± 0.05 s.
In comparison, the stress relaxation time reported for unstained λ-DNA in water (
η = 1 cP) is τ1 = 0.067 s, where we determined τ1 for λ-DNA by scaling τ1 = 0.046 s
reported by Klotz and Zimm [97] for T7-DNA in a creep recovery experiment and scaled
for the slight difference in length between T7 (L = 13.4 µm) and λ-DNA (L = 16.3 µm)
with a scaling exponent of 1.66 [9]. This value of τ1 = 0.067 s is in agreement, after
scaling for length, with τ1 = 0.058 to 0.068 s from intrinsic viscosity [97], light scattering
[98], birefringence [99], and flow dichroism experiments [100].
To compare our measured τ1 to these previous measurements, we scaled by the
change in length (21.1/16.3)1.66 and for the change in viscosity (41/1). This calculation
yields τ1 = 0.061 s and does not take in to account any possible changes in persistence
length. The present measurements were done in a sucrose-glucose-water solution while
the stress relaxation measurements were done in a glycerol-water solution. But, we have
measured the relaxation of single DNA molecules in both a glycerol and sucrose
enhanced aqueous buffered solution [9]. The relaxation time and scaling behavior were
the same. Thus, there was no change in solvent quality between the glycerol-water and
sucrose-water solutions.
93
Time (s)
0 5 10 15 20
<R(t)
R(t)
> ( µ
m2 )
0
50
100
150
200τrelax= 3.89 (0.05)
Figure 35: Relaxation of 14 individual DNA molecules averaged together.
Hence, we see that there is quantitative agreement between τ1 measured via a
single molecule and the classical definition of τ1 by bulk measurements. Both our single
molecule measurements and the stress birefringence experiments determined τ1 by fitting
the last portion of the relaxation to equilibrium to a single exponential. In this limit, the
contribution of the higher order modes of the polymer’s relaxation are small, and the
measured τ1 is approximately equal to the fundamental or longest relaxation mode of a
polymer [97].
It is important to note that our τ1 reported here is different than the relaxation time
determined from <x(t)> = c exp(t/τ) - RG, which yields τ = 6.17 ± 0.15 s for free λ-
DNA. This is the relaxation time we reported in an earlier experiment where the DNA
was tethered to a bead [9]. Also τ measured for a tethered polymer and a free polymer
94
are not the same but, rather, a tethered polymer relaxation approximately corresponds to
the relaxation of a free polymer twice as long.
6.4 Experimental Results
6.4.1 Data reduction
D
umbb
ell
Hal
f-du
mbb
ell
Kink
edFo
lded
Uni
form
Coi
led
Figure 36: Illustrations of the seven main configurations observed.
To convert the raw image data to numerical data, a computer program was used to
track and to determine the extension of individual molecules as they traversed the field of
view. Molecules were required to be in-focus and their center-of-mass was required to
start within 22 µm of the center-line of the inlet channel at the edge of the screen (83
µm). This eliminated those molecular paths with less than 6 individual images where
ε ∆t was ~ 0.1 between successive frames. Molecules were tracked until a portion of the
molecule went off the screen or the molecules came within 5 µm of another molecule. A
schematic sketch of this process is illustrated in Figure 34. For each individual image,
nine fields of data were taken that described the extension, time, conformation, center of
intensity, total integrated intensity, and positions of the ends of the molecule. The
95
extension of each molecule was checked visually. Further, the conformations were also
determined visually. The conformation was specified as one of seven different
conformations: kinked, folded, dumbbell, half-dumbbell, uniform, coil or extended.
Illustrations of these conformations are shown in Figure 36.
Figure 37: Extension versus accumulated fluid strain ε tres and reduced residency time tres/τ1. (A) ετ 1=0.2, (B) ετ 1 =0.32, (C) ετ 1 =0.47, (D) ετ 1= 0.58, (E) ετ 1 =0.82 (F) ετ 1 = 1.2, (G) ετ 1= 2.0 , (H) ετ 1= 3.3. Graph H reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
Accumulated fluid strain
2.5 3.0 3.5 4.0
Exte
nsio
n ( µ
m)
0
2
4
6
Reduced residency time 12 14 16 18 20
A
Accumulated fluid strain2 4 6 8 10
Exte
nsio
n ( µ
m)
0
2
4
6
8
Reduced residency time 10 20 30
B
Accumulated fluid strain2 4 6 8 10
Exte
nsio
n ( µ
m)
0
3
6
9
125 10 15 20
Reduced residency time
C
Accumulated fluid strain2 4 6 8 10 12
Exte
nsio
n ( µ
m)
0
5
10
15
Reduced residency time5 10 15 20
D
Accumulated fluid strain2 4 6 8
Exte
nsio
n ( µ
m)
0
5
10
15
Reduced residency time3 5 7 9
E
Accumulated fluid strain2 4 6 8
Exte
nsio
n ( µ
m)
0
5
10
15
20
Reduced residency time2 3 4 5 6
F
2 4 6 8
Exte
nsio
n ( µ
m)
0
5
10
15
20
Reduced residency time1 2 3 4
Accumulated fluid strain
G
Reduced residency time 1.0 1.5 2.0
Exte
nsio
n ( µ
m)
0
5
10
15
20
Accumulated fluid strain 3 5 7
H
96
Extension (µm)0 5 10 15 20
B
ε tres = 2.48
Nor
mal
ized
pro
babi
lity
dist
ribut
ion
ε tres = 4.58
ε tres = 3.53
Extension (µm)0 5 10 15 20
C
ε tres = 2.48
Nor
mal
ized
pro
babi
lity
dist
ribut
ion
ε tres = 3.25
ε tres = 4.02
ε tres = 4.78
Extension (µm)0 5 10 15 20
D
ε tres = 2.48
Nor
mal
ized
pro
babi
lity
dist
ribut
ion
ε tres = 4.88
ε tres = 4.09
ε tres = 3.28
Extension (µm)0 5 10 15 20
E
ε tres = 2.48
Nor
mal
ized
pro
babi
lity
dist
ribut
ion
ε tres = 5.18
ε tres = 4.27
ε tres = 3.38
Extension (µm)0 5 10 15 20
F
ε tres = 2.48N
orm
aliz
ed p
roba
bilit
y di
strib
utio
n
ε tres = 4.88
ε tres = 4.07
ε tres = 3.29
Extension (µm)0 5 10 15 20
H
ε tres = 2.48
Nor
mal
ized
pro
babi
lity
dist
ribut
ion
ε tres = 3.42
ε tres = 4.29
ε tres = 5.23
Folded
Extension (µm)0 5 10 15 20
G
ε tres = 2.48
Nor
mal
ized
pro
babi
lity
dist
ribut
ion
ε tres = 3.14
ε tres = 3.86
ε tres = 4.52
Extension (µm)0 5 10 15 20
Nor
mal
ized
pro
babi
lity
dist
ribut
ion
A
ε tres = 2.76
ε tres = 2.48
ε tres = 3.04
Figure 38: Normalized probability distributions of molecular extension at different accumulated fluid strain ε tres . (A) ετ 1 =0.2, (B) ετ 1 =0.32, (C) ετ 1 =0.47, (D) ετ 1= 0.58, (E) ετ 1=0.82 (F) ετ 1= 1.2, (G) ετ 1 = 2.0 , (H) Graph H ετ 1= 3.3 reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
97
The reduced numerical data was then stored in a database. The raw extension
data was smoothed by a weighted average with its nearest neighbors of xi = 0.21 xi-1 +
0.58 xi + 0.21 xi+1.
To calculate tres, the time a polymer interacts or resides in the elongational flow
field, we needed to know the polymer’s residency time in the elongational flow field
tresinitial prior to its appearance in the field of view. This is simple to calculate since the
strain rate is constant and leads to
txxres
initial onset
screen
=1 ln( )ε
Eq. 14
where xonset is the position of the onset of the elongational flow field and xscreen is
position of the edge of the screen. For our flow cell and imaging system, these values
were 960 and 83 µm respectively. Since we do not image the molecules continuously but
between illuminated video frames, the molecules are not imaged exactly at the edge of
the imaging area. The average distance the molecules travel bewteen illuminated frames
is <v∆t>. Thus, our value of xscreen takes into account this motion and reduces the true
location of xscreen by <v∆t>/2. Having calculated tresinitial , we add tres
initial to the time each
molecule is visualized in the imaging area to yield tres.
While tresinitial is dependent on the flow rate, ε tres
initial is dictated by the geometry of
the flow cell and the imaging system. Hence, all molecules experience the same amount
of accumulated strain before they are visualized. To visualize the earlier time evolution,
the position of the objective was moved away from the stagnation point towards the inlet.
6.4.2 Extension versus residency time
We performed the experiment at 8 different flow rates. In our earlier report on
these results [27], we showed the raw data for only the highest strain rate. Here, in we
present all of the individual traces of extension versus time (Figure 37). We also plot the
average extension <x(tres)> as well as highlighting several individual traces. Due to the
nature of a stagnation point flow, the number of molecules observable for a given tres
decreased exponentially. Our analysis started with ~1000 molecules for the five highest
98
strain rates and ~400 molecules for the remaining strain rates. For tres> tresinitial , we
calculated averages from at least 40 individual molecules unless stated otherwise.
One immediately striking feature in all of the data sets is the large heterogeneity
in dynamics from molecule to molecule. Such heterogeneity is not observable in classical
bulk experiments. But from an ensemble of individual measurements, we can not only
measure <x(tres)> we can also measure the time evolution of the full probability
distribution for molecular extension (Figure 38A-H).
Absolute time (s)
30 45 60 75
Exte
nsio
n ( µ
m)
0
2
4
6
8
10
ετ =0.46
∆t = 6.4
Figure 39: Two molecules stretching at the same time. The difference in their tres is 6.4 s.
Clearly, the molecules are not undergoing a simple and simultaneous unwinding
as soon as ε > ε c . Since the molecules experienced the same ε and tres, this
heterogeneity in dynamics must arise from a combination of the polymer’s initial internal
configuration and Brownian motion. This heterogeneity underscores the necessity of a
monodisperse solution. By using λ-DNA, we know that these diverse dynamics are not
caused by a distribution in lengths.
Further, this diversity in dynamics was not caused by pump fluctuations (σv/v =
0.028) but by the dynamics of the molecule. To illustrate this, we plot the extension of
99
two molecules that were imaged at the same time (Figure 39); one molecule is stretching
while the other remains a coil and then while the first molecule is retracting, the second
molecule begins to grow.
Extension (µm)
0 3 6 9 12
Free
ene
rgy
(kBT)
0
1
2
3
4
5ετ1= 0.59ετ1= 0.46ετ1= 0.32ετ1= 0.2
εc = 0.4
Figure 40: Free energy curves for a simple dumbbell model with no Brownian motion illustrating the large fluctuations near the ε c .
We now separately discuss the data associated with each strain rate. For ετ 1 = 0.2
(Figure 37A), the average extension <x(tres)> remains approximately constant but the
fluctuations are very large. Since we have ε = 0.5 ε c , we expect the molecule to
fluctuate about a coiled configurations. Qualitatively, the magnitude of these
fluctuations can be understood as thermal fluctuations within the dumbbell model
(Figure 32 & Figure 40 ). The other striking feature of this data set is the apparent
shortening of the molecules at larger values of ε tres . But, we know that the molecules
are in a constant ε and that the molecules are not shortening due to photobleaching.
We suggest that this is a misleading visual effect due to small number statistics at larger
value ε tres . Since ετ 1 = 0.2 is below ε c , we approximate the true distribution of
molecular extension from a histogram of all of the data (Figure 41). Near ε tres = 2.5,
100
there are approximately 400 individual traces while at ε tres = 3.0 there are approximately
40. Thus, at ε tres = 2.5, there are a sufficient number of molecules to expect a few of
these independent samples to reach ~4 µm. As the number of molecules decreases, it
becomes increasingly unlikely that a molecule will extend to ~4 µm and the data tends
to cluster around peak in the distribution at x = 1.8 µm
Extension (µm)0 1 2 3 4 5 6
Prob
abili
ty d
istr
ibut
ion
0.0
0.1
0.2
0.3
ετ1= 0.2
Figure 41: Asymmetric probability distribution.
For ετ 1 = 0.32, we have ε = 0.8 ε c and the fluctuations become more
pronounced. Such behavior is often seen at phase transitions. In this case, the
fluctuations are increasing because the hydrodynamic force exerted by the elongational
flow almost exactly equals the linear elasticity of the chain. Within the dumbbell model,
this means the free energy of the chain has a shallow minimum and thermal fluctuations
about the mean lead to large changes in x (Figure 40). As shown by the top axis label of
Figure 37, these fluctuations are happening on time scales significantly longer than τ1, the
natural time scale for the polymer to relax to equilibrium. <x(tres)> increases slowly to a
value of ~3.2 µm. To demonstrate that <x(tres)> does not further increase with tres, we
plot the <x(tres)> averaged for 20-40 molecules in the smaller open circles.
101
For ετ 1 = 0.47, we have ε = 1.1ε c and the fluctuations become even more
pronounced. Some molecules reach an extension of 9 µm and then relax back towards 6
µm. Other molecules did not stretch appreciably until tres/τ1 > 17 and then stretched
rapidly. The diversity in the dynamics and the magnitude of the fluctuations underscore
the difficulty a mean-field theory will have describing this data (see 6.4.9.6).
For ετ 1= 0.58, we have ε = 1.5ε c . Once a molecule started to stretch, there was
a general trend towards a highly-extended, steady-state value with smaller fluctuations
about xsteady. But there was a large variation in tres at which such stretching begins. To
describe this, we define tonset as the tres at which significant started began. At this flow
rate, tonset can be greater than 10 tres/τ1 emphasizing how long it takes these polymers to
come into equilibrium with the applied velocity gradient.
For ετ 1 = 0.82, we have ε = 2.1ε c . The large fluctuations seen near equilibrium
were suppressed by the higher tension within the chain associated with larger extensions.
However, there was still a large variation in tonset. Interestingly, some molecules overshot
xsteady and then relaxed back towards xsteady.
For ετ 1 = 1.2, we have ε = 3ε c . The large variation in tonset was still very
pronounced while the thermal fluctuations during the stretching and around were further
suppressed.
For ετ 1 = 2.0, we have ε = 5ε c . The differences in dynamics between the
individual traces and <x(tres)> illustrates how the large variation tonset alters <x(tres)>. One
of the highlighted molecules had significantly slower dynamics. This diversity of
dynamics became more pronounced as ε was increased.
For ετ 1= 3.3, we have ε = 8.3ε c . While the variation in tonset is still present,
there is now a large variation in the rate of stretching as well. The extension of some
molecules, independent of their tonset, increased exponentially with time. Other, slow-
stretching molecules grew linearly in time.
6.4.3 Average and steady-state properties: coil-stretch transition observed
The simplest analysis of the x vs. tres data is to calculate the average extension
<x(tres)> as a function of ε and tres. To compare the data on an equal basis, we plot the
102
data as a function of the accumulated fluid strain or Henky strain (ε ε= tres ) in Figure
42A. This analysis plots the average extension of the polymer in comparison to the
deformation of the underlying fluid element where the deformation of the fluid element is
given by exp(ε). <x(ε)> increased monotonically with ε indicating that the average
extension had not reached steady-state within the measurement time except for the lowest
values of ε where there was no deformation.
Dimensionless strain rate0 1 2 3 4
Steady-state extension xsteady /L
0.0
0.2
0.4
0.6
0.8
1.0
Accumulated fluid strain1 2 3 4 5 6
Ave
rage
ext
ensi
on <
x>/L
0.0
0.2
0.4
0.6
0.8
1.0 A B0.86 s-1
0.51 s-1 0.31 s-1 0.21 s-1 0.15 s-1
0.12 s-1 0.08 s-1
0.05 s-1
ε =
Bulk: avg overall data
Dumbbell model
Figure 42: Summary of data. (A) Average extension as a function of accumulated fluid strain ε ε= tres . (B) Steady-state extension (open symbols) and a bulk spatio-temporal average (closed symbols) determined as a function of the dimensionless strain rate 1τε . A fit of the dumbbell model (solid line) to the steady-state data. Reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
Uniquely, however, we can determine the steady-state extension xsteady. By
analyzing the subset of molecules that reach steady-state (Figure 43), we plot xsteady vs.
1τε (Figure 42B). 1τε is the appropriate dimensionless strain rate or “Deborah number”
which characterizes the rate of deformation of the fluid relative to the relaxation time of
the polymer. Note that xsteady rises sharply at a critical strain rate of ε τc relax ≅ 0.4.
Thus, we report the first direct observation of the coil-stretch transition [41] as
evidenced by the rapid non-linear increase in the steady-state extension at a critical strain
103
rate Figure 42. At ε ≅ 0.9 ε c , there are large fluctuations (σx/x = 0.4). Similar behavior
is seen near phase transitions. The nature of this experiment does not allow us to show
that this transition is a first order phase transition or the presence of hysteresis as
proposed by de Gennes [41]. However, our future single DNA molecule experiments
will be able to resolve this question.
Accumulated Fluid Strain2 4 6 8
Exte
nsio
n ( µ
m)
0
5
10
15
20xsteady
ετ1=3.35
Figure 43: Molecules stretching to steady-state.
A comparison between <x(ε)> and xsteady shows that <x(ε)> had not yet reached
xsteady for up to ε ≅ 5.7. This corresponds to a deformation of the underlying fluid
element by a factor of exp(5.7) or ~300.
In classical bulk measurements by light scattering and birefringence, the response
of individual chains is averaged over the diameter of the probe laser beam. Since the
measurements are done at the stagnation point, the spatial average due to the diameter of
the laser beam leads to an average over a broad distribution in tres as well. To compare
our single molecule results to classical bulk measurements, we plot a spatio-temporal
average xbulk of all of our data (Figure 42B). Our imaging area is similar in size to that
the probed by lasers in previous birefringence experiments [50, 51]. This analysis shows
104
that xbulk is significantly smaller than xsteady. This is not unexpected since <x(ε)> is less
than xsteady up to the largest values of ε measured. Thus, our data suggest that classical
bulk measurements in this finite region around the stagnation point do not measure the
steady-state properties.
In part, this mismatch between xbulk and xsteady arises from the limited tres
associated with measurements in the vicinity of the stagnation point. However, if all the
molecules started to stretch immediately upon enter the elongational flow field, xbulk
would be significantly larger. Thus, visualizing single molecules reveals how the large
and previously unobservable variation in tonset reduced xbulk in comparison to xsteady.
6.4.4 Steady-state measurements fit by the dumbbell model
Theoretical descriptions of dilute polymer rheology are often based on
constituitive equations. Many of these equations are derived from the simplest of
polymer models: a dumbbell model [5]. In this case, the complexity of the complete
dynamics is drastically simplified. The hydrodynamic coupling between the fluid and the
polymer is represented by two beads while the entropic elasticity is represented by a
spring. For flows that generate significant deformation, finitely extensible chains with
non-linear elasticity are used. Fortunately, for modeling DNA with a dumbbell model,
the steady-state elasticity of DNA has been measured [15] and found to agree with the
elasticity of a worm-like chain [25]. Marko and Siggia provided a simple analytic
approximation to this elasticity [25]. Therefore, our dumbbell model is based on two
beads connected by a Marko-Siggia spring. Previously, this simple model has described
the steady-state extension of a tethered polymer (DNA) in a uniform flow [10]. In
collaboration with R. Larson, we developed a molecular understanding of the origin of
this agreement based on simulations [24].
This model does not include Brownian motion, which can lead to a change in
xsteady even when ε < ε c (Figure 40). Further, when trying to fit the data to the model,
the fit is most sensitive to fluctuations at or near ε c where xsteady is rapidly increasing.
Therefore, we fit the model to the data for ε > 0.15 s-1 to
105
x Lb b
b=
− − +⎛
⎝⎜
⎞
⎠⎟
2 1 4 12
Eq. 15A
where b is
bR Lpk T
bead
B= − +4
12πη ε
Eq. 15B
and the two free parameters are Rbead, the radius of each bead, and L, the contour length
of the polymer. As shown in Figure 42B, our steady-state results for free polymers in an
elongational flow is also approximately characterized by this simple model with the
parameters (Rbead = 0.15 µm and L = 21.1 µm). This value for the stained length agrees
closely with previous measurements of the length of stained DNA (L = 22 µm [10]). An
extrapolation of the model to x = 0 gives a critical strain rate of ε τc relax ≅ 0.4, which is
close to the theoretical value of 0.5 calculated from the Zimm model and by the
numerical calculation of Larson and Magda [101]. The mismatch between xsteady and the
model at ε ≅ε c is caused by the increase in xsteady due to Brownian motion, which was
neglected in this simple model.
6.4.5 Conformational dependent dynamics at highest strain rates
The variation in tonset is not the only cause of the slower average dynamics. At
higher ε (Figure 37H), we see a second source: heterogeneity in the rate of stretching.
To help understand the origin of this heterogeneity, we analyzed the conformation of the
molecule. In general, at each instance in time, the molecules could be classified as one of
seven different conformations: kinked, folded, dumbbell, half-dumbbell, uniform,
extended and coiled. Well defined states that were robust to Brownian motion only
existed when the molecules were subject to a ε significantly greater than the inverse
relaxation time (τ relax−1 = 0.26 s-1; τrelax =3.89 s (see 6.3.1)). Only at the highest two strain
rates did we observe folded and kinked conformations. Several examples of each
conformation, except extended, are shown in Figure 36.
106
Figure 44: Dynamic unwinding of different conformations. From top to bottom, the images are classified as dumbbell, kinked, half-dumbbell, and folded. Reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
Presumably, the differences in conformation as well as the variations in tonset arise
directly from the multitude of accessible conformations at equilibrium, where thermal
fluctuations cause instantaneous deviations away from a spherically symmetric
distribution. For instance, some of these accessible conformations of an equilibrium coil
have both ends on the same side of the center of mass. When subject to a 1/1 τε >> ,
these ends will not be able diffuse across the length of the molecule and such types of
initial conformations presumably lead to folded configurations.
While this simple classification could describe each of our individual images of a
polymer’s conformation, we could not uniquely classify the full series of images that
showed a polymer stretching from a coiled to an extended state. For example, a molecule
at x = 5 µm may appear to be in a dumbbell configuration but, by x = 10 µm, one of the
“balls” of fluorescence disappeared (unraveled) and it is now in a half-dumbbell
configuration. Nonetheless, to look for conformation dependent dynamics, we assigned
each molecule a conformation to its entire stretching process by creating a hierarchy of
107
conformations. The order for classification was kinked, folded, dumbbell, half-dumbbell,
uniform, extended and coiled. For example, we classified the molecule shown in the
second row of Figure 44 as kinked. At the highest strain rate of ε = 0.86 s-1, this general
classification of all molecules yielded 5.4% kinked, 24% folded, 20% dumbbell, 35%
half-dumbbell, 8.3% uniform, 5% extended, and 3% coils (Figure 45B). Thus at ε = 0.86
s-1, the dominant conformations at the highest strain rate were half-dumbbell, folded and
dumbbell.
B
Half-dumbbells(35%)
Kinked(5%)
Folded(24%)
Dumbbell(20%)
Uniform(8%)
Extended(5%)
Coils(3%)
Half-dumbbells(33%)
Folded(10%)
Extended (4%)
A
Coils(9%)
Kinked(4%)
Dumbbell(31%)
Uniform(10%)
ετ1 = 3.3ετ1 = 2.0
Figure 45: Pie charts showing the distribution of conformations at two strain rates.
The coiled and extended conformations are the starting and ending configurations
are necessary to describe those molecules whose evolution in extension are not observed
due to the limited range in observation time. A coil is simply a molecule that did not
deform from a ball-shaped configuration during its tres and its tonset is larger than its
maximum tres. At the other extreme, an extended molecule is a molecule that is already
stretched to near its steady-state extension before it was visualized and its conformation
during the stretching process was not observed.
A preliminary analysis of the average dynamics of based on this first, general
classification showed differences between the folded and the dumbbell configuration
(Figure 46). But, these differences were not as large as observed in some individual
traces. However, there was a lot of scatter within each category. The dynamics of a
folded molecule whose initial fold was 30% of its extension stretches differently than a
108
molecule whose initial fraction of fold was 80%. To highlight the differences between
the conformations, we determined those molecules that best typified each conformational
class and then re-analyzed these molecules. For classification purposes, the molecules
that best typified a dumbbell configuration had approximately symmetric “coils” at each
end. For classification as folded, we required the initial percentage of the folded section
to be >75%. This classification yielded 30 dumbbells, 34 folded, and 43 half-dumbbells
out of 992 molecules and these molecules were re-analyzed. This data clearly shows a
distinct difference in the dynamics of molecules in the folded and dumbbell configuration
(Figure 47).
Residency time (s)2 4 6 8
< x(
t res)
>
0
5
10
15
20
AllDumbbellHalf-dumbbellFoldedCoiled
Figure 46: Average extension versus tres for different conformations. Reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
6.4.6 Master curves
To more accurately compare the differences in the unwinding dynamics between
conformations, we wanted to calculate an average rate of unwinding. However, the large
variability in tonset obscures this analysis for the case of a simple time average (Figure 48).
Conceptually, we can eliminate tonset by sliding each curve along the time axis until the
curves superimpose to form a master curve. An alternative method to accomplish the
109
same thing is to calculate to the rate of extension ( )x x as a function of x and then
integrate it to get a “master” curve.
Residency time (s)4 6 8
Exte
nsio
n ( µ
m)
0
5
10
15
20
DumbbellFolded
Figure 47: Comparison between dynamics of dumbbell and folded configurations. Reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
Specifically, we calculate ( )x x by fitting 5 successive data points to a line. The
average dynamics < ( )x x > is calculated by binning the individual ( )x x every 0.5 µm in x
and then averaging them to determine < ( )x x > for each bin. Note, this calculates the
average dynamics by averaging all molecules at a given x not a given tres, so there is not a
distribution about x as there would normally be when calculating such an average.
For those molecule in a dumbbell configuration (Figure 49), this analysis of the
rate of stretching ( )x x as function of x shows that once a molecule starts to stretch, its
dynamics follows a specific time evolution. Up to x/L = 0.6, we observed a linear
increase in x with x at ε = 0.86 s-1. When integrated, this yields an initial exponential
growth of the master curve. The relationship between the individual curves, the master
110
curve and <x(tres)> is shown in Figure 48 and shows how the master curve does a much
better job capturing the averaging unwinding dynamics than <x(tres)>.
Residency time (s)2 4 6 8
Exte
nsio
n ( µ
m)
0
5
10
15
20
Master curve
<x(tres)>
Figure 48: Individual traces of extension versus tres for molecules that best typify the dumbbell configuration. The master curve does a much better job than a simple time average at describing the dynamics because of the large variability in tres.
In Figure 50, we show three such master curves generated from the molecules that
best-typify each of the dominant conformations at ε = 0.86 s-1. Clearly, there is a strong
dependence of the rate of stretching on conformation. For comparison, we show
<x(tres)> in Figure 47 for the full data set as well as for several of the different
conformational classes arising from the first, general classification.
In addition to looking for conformational dependent rate of stretching, we
calculated < ( )x x > for all ε (Figure 51). This analysis averaged over all conformations.
For ε > 0.21 s-1 and at low extensions, < ( )x x > was a linear function of x. This linear
relationship indicates that once the polymers began to deform their deformation increased
exponentially with time for small x. This deformation is similar to and caused by the
exponential deformation of the underlying fluid element.
111
x (µm)0 5 10 15 20
x ( µ
m/s
)
0.0
2.5
5.0
7.5
10.0
Figure 49: Dynamics of molecules that best-typify dumbbell molecules. Reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
Time (s)0 2 4 6
Exte
nsio
n ( µ
m)
0
5
10
15
20DumbbellHalf-dumbbellFolded
Figure 50: Comparison of master curves between the dominant conformations. Reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
112
6.4.7 Affine deformation
A polymer is said to deform “affinely” with the fluid if the molecular deformation
equals the deformation of the surrounding fluid element. It has been postulated that when
11 τε >>> affine deformation becomes an increasingly valid approximation [102-104].
Our observed exponential growth in the master curves suggests that a comparison
between our data and the approximation of affine deformation.
In the simplest analysis, we note that <x(tres)> did not reach xsteady even after an
accumulated fluid strain of ε ε= tres ≅ 5.7, which corresponds to an e5.7 or ∼300 fold
distortion of the fluid element (Figure 42B). For comparison, the required molecular
distortion to fully extend stained λ-DNA is L/RG ≅ 30 where RG, the radius of gyration, is
0.73 µm.
< x >
( µm
/s)
0
2
4
6
8
Extension (µm)0 5 10 15 20
< x >
/ ε (
µm)
0
2
4
6
8
0.51 s-1
0.41 s-1
ε = 0.86 s-1
0.21 s-1
0.15 s-1
0.12 s-1
0.08 s-1
Figure 51: Rates of deformation and normalized rates of deformation for all ε .
113
0.0 0.2 0.4 0.6 0.80.0
0.5
1.0
1.5
ε mol = ε fluid
- ε c
Dumbbell configurationAll configurations
ε fluid
/ (ε flu
id -
εc)
ε mol
( s-1
)
0.0
0.2
0.4
0.6
0.8
εfluid - εc(s-1)
A
B
Dumbbell configurationAll configurations
Figure 52: Comparison of the rate of molecular deformation to rate of deformation of the fluid element minus the necessary fluid slip to overcome the linear elasticity of the polymer. Graph A reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
In part, this lack of affine deformation in <x(tres)> arises from the large variation
in tonset. Notwithstanding this variation which is intrinsically non-affine, we wanted to
know if molecules deform affinely once they start to stretch. To do so, we analyzed the
dynamics of the master curve, which suppresses the variation in tonset by computing
< >( )x x instead of < >( )x tonset . To compare the molecular deformation and the
deformation of the underlying fluid element, we define a molecular strain rate ε mol by
fitting this linear region of < ( )x x > vs. x to < ( )x x > = ε mol x + b. The linear portion of
< ( )x x > vs. x extended up to 5.2, 7.8, 8.8, 10.2, and 11.8 µm in increasing ε .
114
At moderate strain rates, affine deformation is not expected because there must be
some slip between the polymer and the fluid to create the hydrodynamic force necessary
to overcome the native elasticity of the polymer. This required slip is simply the critical
strain rate ε c identified in the analysis of the steady-state extension below which there is
no significant deformation. To account for this threshold, we simple rescale the rate of
deformation of the fluid εε ≡fluid by subtraction off ε c , and plot ε mol versus cfluid εε −
(Figure 52A). At lower ε , the molecules are stretching near the theoretically expected
limit. At higher ε , the data shows a marked departure and it is clear that the affine
deformation approximation breaks down. Furthermore, when plotted as ε mol / ( )ε εfluid c−
vs. ( )ε εfluid c− , the data is decreasing at 0.86 s-1 (Figure 52B).
Thus, the average data shows neither an absolute nor a fractional approach toward
affine deformation at higher ε even after eliminating the large variation in tonset.
Rather, the approximation is becoming increasingly worse at higher ε . This failure
arises from the introduction of intramolecular constraints (primarily folds) which
dramatically slow down the average dynamics. In contrast to the average dynamics, the
subset of molecules in a dumbbell configuration stretched almost as fast as can be
theoretically expected (Figure 52B, open symbol). This rapid stretching of the dumbbell
configuration makes intuitive sense. This rapid stretching of the dumbbell configuration
makes intuitive sense. The “coils” at each end lead to a large hydrodynamic drag force
because they are at the ends of the molecule and therefore experience the full extent of
the velocity gradient across the molecule. Furthermore, the force necessary to stretch
chain segments from the coil is low because the elasticity force is small at low
deformations.
6.4.8 Dynamic data, intrinsic viscosity and the dumbbell model
Having seen that the dumbbell model’s simplified representation of a polymer
described our steady state results, we wanted to see if this model could self-consistently
describe the dynamics. The measured dynamics have a large variation in tonset and this
model does not include any Brownian motion or a distribution of initial starting
configurations. Therefore, we compared the model to the dynamics of the master curve
115
because the master curve suppresses the variation in tonset. by calculating < ( )x x > instead
of <x(tres)> (see 6.4.6). To self consistently calculate the predicted dynamics, we use the
parameters determined from the steady-state results. Specifically, we calculated the force
on each bead as the sum of the entropic elasticity Fspring = FMS (x/L) [25] and
hydrodynamic force Fhydro given by
Fhydro = 62
πηR vx
bead fluidpredicted( )−
Eq. 16
where the fluid velocity at each bead is given by vfluid = 0.5ε x. The net force is set to
zero
Ftotal = Fspring + Fhydro = 0. Eq. 17
We can then solve for the predicted rate of stretching of the total chain at each x to be
( )x R v F Rpredicted bead fluid spring bead= −2 6 6πη πη Eq. 18
Note that x predicted is also calculated a given x not a tres, and the comparison to < ( )x x > is
valid.
As shown in Figure 53, this simple model overestimates the average dynamics.
Therefore, we expect difficulty predicting the rheological properties of dilute polymer
solutions using this simplified model (see 6.4.9.4). However, when x predicted is compared
to the dynamics of molecules in the dumbbell configuration, the disagreement is much
smaller. But, recall that in this analysis the large variation in tres has been suppressed.
Clearly, the average dynamics are decreased from the fastest dynamics by the
introduction of folds. In essence, this is analogous to the breakdown in the affine
deformation approximation at higher ε . But, in that case, the details of the elastic
properties of the chain and its hydrodynamic drag were not explicitly written out as a
function of Rbead and L but incorporated in ε c .
116
x (µm)0 5 10 15 20
<x(x
)> (
µm/s
)
0
2
4
6
8
10Dumbbell modelDumbbell configurationAll data
Figure 53: Comparison between the simplified, measured dynamics of the master curve for all molecules and those in the dumbbell configuration at ε = 0.86 s-1 and the prediction of the dumbbell model using the parameters determined from the a fit to the measurements of the steady state extension. Reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
Previously, “internal viscosity” [2, 3, 5] has been added to the dumbbell model to
explain the delay in shear thinning [5]. Similarly, on might be tempted to add a term
proportional to - x predicted because such a term can approximately compensate for the
slower average dynamics. Such a term is suggested by the measurements of ηE because it
leads a dissipative component of the stress relaxation [94]. In particular, for ε = 0.86 s-1,
the average dynamic data is best fit by multiplying x predicted in Fhydro by (1+α) where α =
0.55 (Figure 53). Note that the trend in Figure 52 away from ε ε εmol fluid c= − shows that
this coefficient α is dependent on ε .
Importantly we stress that the observed slower elongation rates arise from folded
configurations and not from the monomer-monomer friction typically associated with the
assumed internal viscosity. We also note that there are addition terms besides - x predicted
that can lead to dissipative stresses.[62, 63, 105-108]
117
6.4.9 Comparison to previous experimental work
Given our measurements of the dynamic, steady-state and ensemble-averaged
properties of polymers in an elongational flow, we now compare our data to previous
experimental and theoretical results.
6.4.9.1 Birefringence
As mentioned in the introduction, birefringence has been the dominant
experimental technique for studying polymers in elongational flows. These experiments
have shown the birefringence abruptly increases at a critical strain rate followed by
saturation at higher strain rates. The classical interpretation has been that this saturation
implies the chains are highly extended [49]. However, the implication of this saturation
is still debated [51, 109]. To help clarify the relationship between extension and
birefringence, we compare our data to the previous measurements of the birefringence of
λ-DNA as well as synthetic polymers.
Atkins and Taylor have measured the birefringence of dilute solutions of λ-DNA.
This data was also taken in a planar elongational flow. To appropriately compare the
birefringence data for unstained DNA and our steady-state extension data for stained
DNA, we normalized their birefringence measurements by its saturating value and xsteady
by L. For both measurements, the strain rate was scaled by the appropriate τ1 to yield the
dimensionless strain rate 1τε (see 6.3.1 for a complete discussion of τ1). Clearly, as
shown in Figure 54, our ability to select only those molecules that have reached steady-
state extensions reveals a much sharper transition occurring at a lower 1τεc . Since
birefringence averages over a broad range of positions and tres, we also plot xbulk, a spatio-
temporal average of all data at a given ε .
The large offset between xsteady and the birefringence might lead one to suspect an
error in calculating 1τεc for either our stained λ-DNA or their unstained λ-DNA.
However, our measurement of τ1, after scaling for the ∼30% change in length due to
staining, is in good agreement with consensus value of τ1 in the literature (see 6.3.1).
Further, the large offset between xsteady and the birefringence is not limited to DNA. For
very dilute polystyrene solutions, Nguyen et al determined critical values of 1τεc ranging
118
from 3 to 8 polystyrene [51]. Taken together, these results argue that this offset does not
arise from a concentration dependent effect or a property of the polymer’s local chemical
structure.
Dimensionless strain rate
0.0 1.5 3.0 4.5 6.0
Nor
mal
ized
Ext
ensi
on (x
stea
dy/L
)
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Normalized Birefringence (Atkins and Taylor)
Bulk averageover all data
xsteady
Dumbbellmodel
Norm
alized Birefringence
Figure 54: Steady state measurement versus previous birefringence data for λ-DNA by Atkins and Taylor [85]. A bulk average is plotted is a better comparison since it averages all molecules within the field of view which is similar to the average of all molecules within the diameter of the laser beam used to measure the birefringence. Reprinted with permission from Science 276, 2016, 1997. Copyright 1997 American Association for the Advancement of Science.
There is no direct correspondence between either xsteady or xbulk and the
birefringence at the stagnation point. Since birefringence measures orientation rather
than extension, some disagreement would be expected based on the observed
conformational features such as folds. Since folds cause a premature saturation in
birefringence with respect to xsteady , one might have expected the birefringence to
saturate before xsteady but our data shows the birefringence increased after xsteady.
Our results highlight the difficulties in interpreting birefringence and other bulk
measurements. Moreover, we suggest that this difficulty may be even greater for
119
synthetic polymers where the larger ratio of L/RG requires an even larger accumulated
fluid strain than is needed to extend λ-DNA.
Other important differences between our data and previous birefringence
measurements are the uniformity in the strain rate and the absolute monodispersity of λ-
DNA. Our ε is uniform to within 2% over the region of interest. Further, ε turns on
abruptly. Both of these reasons may contribute to the sharper, more sudden rise in xsteady
than is seen in the birefringence.
A
Accumulated fluid strain2 3 4 5 6
Leng
th ( µ
m)
0
5
10
15<x>RG
RG
+
-
<x(tres)> (µm)0 5 10 15
<RG( t r
es)>
/ <x
( t res
)>
0.27
0.28
0.29
0.30 B
Figure 55: (A) The average radius of gyration along in the income (-) and outgoing axes (+) calculated from the conformation of the polymer. (B) The normalized average radius of gyration compared to the average extension.
120
6.4.9.2 Light scattering
Previously, light scattering experiments have shown a small increase (~2X) in RG
[50]. In particular, Menasveta and Hoagland showed that RG increased at the same cε as
determined from birefringence [50]. In contrast to these previous light scattering results
on synthetic polymers, our data shows extensions significantly greater than ~2 RG though
RG. For a uniformly extended molecule, RG is xsteady/4. In general, the large difference
between RG and xsteady/4 is caused by the broad distribution in tres for the population of
molecules measured by light scattering. Hence, xbulk/4, not xsteady/4, should be used for
comparison. In addition, the highly asymmetric mass distribution of the most common
conformations (half-dumbbell) would further reduce RG.
Since the optical response of our S.I.T. camera is linear, a two dimensional
projection of RG can be directly calculated from an image of a polymer’s conformation.
For each image, the background is subtracted from the individual intensity values ix′,y′
where x′ and y′ are the coordinates of the pixel. RG is calculated about both the
incoming ( x -axis) and the outgoing ( y -axis). Since we are already using x to describe
the extension along the outgoing axis, we refer to +GR for the deformation along the
outgoing axis and to −GR for deformation along the incoming axis. We calculate these
values using
∑ −=+
','
2','
2 )''(1)(yx
cmyxtotal
G yyiI
R Eq. 19
∑ −=−
','
2','
2 )''(1)(yx
cmyxtotal
G xxiI
R Eq. 20
where Itotal is the total intensity, and ( x′cm , y′cm) is the center of intensity. < +GR > and
< −GR > are the root mean square data calculated from an ensemble average over all
molecule in the field of view. In Figure 55A, we plot < +GR > and < −
GR > as a function of
the accumulated fluid strain ( restε ) as well as <x> at 1τε = 1.2 s-1. Clearly, < +GR > is
much smaller than <x> and the ratio between these two measurements is almost constant
121
at 0.29 at each value of 1τε (Figure 55B). For an object with a uniformly distributed
mass, this ratio is 0.25. < −GR > does not decrease during the measurement because its
value is bounded by the resolution limit of the microscope (~0.4 µm).
To compare directly to the light scattering data, we compute a spatio-temporal
average over all measurements of +GR at 1τε = 1.2 s-1. This yields bulkGR+ = 2.2 µm which
is ~7 times smaller than xsteady while being ~3 times larger than RG (=0.73 µm). Thus, our
results of single molecule results emphasize +GR is much smaller than xsteady. The limited
but broad distribution in tres and geometric factors lead to a bulkGR+ that is much smaller
than the xsteady and only a small factor larger than the equilibrium value of RG. This
agreement between our measurements of bulkGR+ and Menasveta and Hoagland’s
measurements emphasize the inability of bulk experiments to draw conclusions of steady-
state properties of molecules by making measurements at the stagnation point.
6.4.9.3 Stagnation point flow fracture
At ε >>> 1/τ1, polymers flowing through stagnation point flow fracture at the
center of the chain [86-88, 110]. This mid-point chain fracture implies the chains are
highly extended. This follows because, if it were not true, the center of the chain would
not be a unique point and the chains would fracture into a broad distribution of lengths.
These results have been used to argue that all the chains are highly extended. However,
our results suggest that mid-point chain fracture does not imply all chains are extended.
The large variability in x shown in Figure 37H indicates a few molecules rapidly
reached steady-state. If we extrapolate our results to a 100 times higher ε and if the
heterogeneity in the onset of stretching persists, it is these highly-extended, early-
stretching molecules that will experience a large enough force to fracture at or near their
center. This value of ε is in agreement with the minimum necessary force to rupture
DNA when calculated within the dumbbell model assuming a rupture force of 470 pN
[111] and assuming the DNA did not have time to undergo a transition to a hyper-
extended state [23, 31]. Nonetheless the number of such chains that are rapidly stretching
and start stretching early is relatively small. Thus, due to the limited tres, only a fraction
122
of the total number of chains fracture in agreement with the results of bulk experiments.
The fracture of some chains does not imply that all chains are extended.
6.4.9.4 Rheology
Rheologists often infer molecular deformation from bulk viscoelastic
measurements [3, 5]. For instance, within the dumbbell model, one can derive the
extensional stress σE (Eq. 11) and the extensional viscosity η σ εE E= using the data in
Figure 37, the known elasticity of DNA and classical results in rheology (Eq. 11).
However, because the molecules are in highly non-equilibrium configurations (Figure
44), it is inaccurate to used the steady-state elasticity for molecules at ε >> τ1. Further,
such a simple analysis would fail to predict the stress associated with molecules in the
folded configuration. From this and the lack of a physically significant mean described
in 6.4.9.6, our results suggest difficulty in inferring an average conformation from bulk
rheological measurements.
6.4.9.5 Filament stretching
In filament stretching experiments, dilute quantities of high molecular weight
polymers are suspended in a fluid of low molecular weight polymers [92, 94, 95, 112].
This composite fluid is the placed between two disks. The distance between the disks is
increased exponentially in time while monitoring the force between the polymeric
solution and the disk. Such filament stretching experiments are an excellent method for
determining the extensional stress within the fluid. While they average over a large
number of molecules, most molecules have a well defined tres, unlike stagnation point
flows. Therefore, the average properties are well determined. Filament stretching
devices have measured extensional viscosities several thousand times greater than the
shear viscosity. Also, the stress relaxation from such experiments contained both a
strain-rate dependent, ‘elastic’ component as well as a strain-rate independent,
‘dissipative’ component [94, 95].
We do not directly compare our single molecule results to these filament
stretching experiments due to our inability to accurately calculate the transient elasticity
of the chain in the variety non-equilibrium configurations (Figure 44). Nonetheless, we
123
can draw several qualitative conclusions. The heterogeneity in x(tres) imply that a large
value of resτε (>6) will be needed to for the all of the molecules to reach steady-state.
Further, this value of resτε is likely to be larger for synthetic polymers, which have a
larger ratio of L/RG than for λ-DNA where the ratio is 30. However, the presence of
folded configurations suggests that the stress may plateau before the steady-state
extension is achieved.
As stated in the previously, these filament stretching experiments measure a
strain-rate dependent and a strain-rate independent relaxation after the applied
deformations is stopped. The presence of the different conformations offers a qualitative
explanation to the origin of dissipative or strain-rate dependent stress (see 6.4.9.8). For
example, a molecule folded in half generates stress as the fluid slips past it. A molecule
in a dumbbell configuration also leads to a dissipative-like stress. For a given extension,
a dumbbell configuration has more tension within its extended portion than a molecule
under a uniform tension since a smaller fraction of the molecule accomplishes the same
extension. When the applied flow field is stopped, only a small amount of excess chain
segment density within the coils rapidly needs to diffuse inward from each end because
the elasticity is a highly non-linear at large extension. This relaxation is akin to the
higher order modes in the Zimm and Rouse model [39, 40]. These relaxations are too
fast to be measured in the filament stretching device and therefore lead to an apparent
dissipative stress [105]. So, the measured dissipative stress relaxation most likely arises
from both the slip past an extended object and from the higher order modes of molecular
relaxation.
6.4.9.6 Mean field theories
Mean field theories, such as the Zimm model for dilute solutions as well as the
reptation model for concentrated solution, have been very successful in describing
polymer dynamics [39, 68]. Mean fields are based on the assumption of a well defined
mean behavior. However, our data shows that the probability distribution for molecular
extension is not a narrow distribution about a mean but rather a broad, oddly shaped
distribution (Figure 38). This oddly shaped distribution arises from the presence of
several distinctly different dynamical processes at high ε as well as the broad
124
distribution in tres at all ε (Figure 37). Further, the differences in x and tonset imply a
sensitive dependence on the polymer’s initial conformation when it enters the velocity
gradient.
Specifically, for elongational flows, the Peterlin approximation is often used [113,
114]. In this approximation, x2(tres) is replaced by <x2(tres)> in calculating the elasticity
of the chain in conjunction with the dumbbell model. Such an approximation yields
mathematical closure and enables derivation of constituitive equations from kinetic
theory [3, 5]. These constituitive equations are then used to predict the stress in the fluid
or, alternatively, to determine the molecular configuration from bulk rheological
properties.
Our results highlight previously known problems with the Peterlin approximation.
The broad distribution in tonset and the conformational dependent dynamics show the
evolution of <x(tres)> is inherently different from the dynamics of individual molecules
(Figure 37). Further, even within one conformational class, the dynamics of <x(tres)> is
different from the dynamics of the individual molecules (Figure 48). Thus, our data
shows that the approximation of x2 (tres) by <x2(tres)> is a poor one.
The heterogeneity in our data that leads to the breakdown of the Peterlin
approximation is also seen in Keunings’ stochastic simulations of the finitely extensible
dumbbell model [114]. While this simplified model of polymer dynamics based on
kinetic theory yields histograms that are in semi-quantitative agreement with our data
(Figure 38), simulations using the Peterlin approximation in conjunction with kinetic
theory lead to qualitatively different results.
Thus, any closed form analytical solution describing the observed dynamics
seems doubtful given the crucial role of fluctuations in the initial condition. The best
opportunity for capturing the diversity in dynamics lies in stochastic simulation of multi-
element chains. Such chains are necessary to generate the internal conformations of the
molecule. Simulations also offer the best opportunity to investigate the relationship
between a polymer’s initial conformation when it enters the flow and the resulting
dynamics.
125
6.4.9.7 Comments on dumbbell model
The dumbbell model is the simplest of models to describe a polymer. It can
extend and orient with the flow. This model, when using a Marko-Siggia spring, has been
remarkably successful in describing our current as well as previous steady-state results
[10]. Such steady-state results are in equilibrium and justify the use of the steady-state
elasticity to describe the spring.
However, the dumbbell model can not model the dynamics over a broad range of
ε . In part, the dumbbell model fails due to its inherent simplicity; it can not model folds.
Yet, folds are becoming increasingly prevalent as ε is increased. Specifically, our data
shows that, at lower values ε where there is not conformational dependent rate of
stretching, the dumbbell model approximately described the average unwinding dynamics
of the master curve, where the large variation in tonset was suppressed. But, at the highest
ε investigated, the dumbbell model overestimated even these simplified, average
dynamics. This overestimation occurred, in part, because of the appearance of the folded
configuration, which greatly slowed down the average dynamics. The fraction of
molecules in the folded configuration dramatically increased from 9% to 24% as ε τ1 was
increased from 2 to 3.3. Therefore, our data suggests that the range of applicability of the
dumbbell model to describe dynamics is limited to small values of ε τ1. Simple
modifications to the dumbbell model, such as the introduction of a dissipative term -
x (1+α), are most likely of limited utility since α would have to be a function of
ε (Figure 52). More importantly, the origin of such a dissipative term from internal
viscosity is not physically justified in light of our data.
Another limitation of this model at higher ε is the assumption of adiabaticity.
For ε >> 1/τ1, molecules do not have time to sample the full, accessible configurational
space at each x. Therefore, it is unrealistic to expect the steady-state elasticity, which is
derived from the change in the number of accessible states with x, to describe the
elasticity of a rapidly deforming chain. Thus for processes that occur on time scales
much faster than τ1, the fundamental time scale of the polymer, the polymer is not in
equilibrium with the flow. Figure 44 emphasizes the highly non-equilibrium
configurations observed at ε τ1 = 3.3 and Figure 37 emphasizes how long it can take the
126
polymers to come into equilibrium with the flow. Yet the dumbbell model assumes the
deformation is adiabatic by using the steady-state elasticity to describe the spring.
Therefore, on general terms, we expect difficulty in predicting the dynamics of polymers
at higher ε using the dumbbell model.
While the dumbbell model can not robustly model the average dynamics, it
approximately describes the dynamics of the dumbbell configuration. In this case, there
are not internal constraints to slow down the dynamics. But, the overall usefulness of this
approximate agreement may also be limited since the fractional number of molecules in
the dumbbell configuration decreased from 31% to 20 % as ε τ1 was increased from 2 to
3.3 (Figure 45).
6.4.9.8 Proposed conformations
The dynamics of polymers in elongational flows has been a challenging
theoretical problem for several decades [3, 5, 41]. To simplify the full complexity of the
dynamics, several different theoretical models have been investigated. The most popular
of which is the finitely extensible dumbbell model [3, 5, 41]. But, as noted above, this
model can not describe a polymer’s internal conformation, which can dramatically effect
the dynamics. Therefore, over the last 15 years, several additional models have been
developed [62, 63, 96, 107, 108]. These models make assumptions about the
conformation of the polymer and then derive the consequence of these assumptions.
These models were developed, in part, to account for the excess stress measured by
James and Saringer[91].
Ryskin developed the “yoyo” model which describes a molecule as two coils
connected by a highly taught region. This model is very similar to the observed dumbbell
configuration while the half dumbbell model lacks a second coil at one end [62, 63].
Larson and Hinch developed “kink dynamics” models by assuming the polymer is
compressed into a one-dimensional object and the dynamics of this object are dominated
by its kinked or multiply folded structure [96, 107]. This conformation is similar to our
folded configuration, but these simulations in which there were multiple folds were done
at much values of ε τ1 (~40). King and James proposed a conformation based on
internal entanglements [108]. This configuration may be similar to our kinked
127
conformation. But, within the resolution of our microscope, it is not possible to assert
whether kinked conformations arise from internal constrains or, rather, they are arise
from a variation on a half dumbbell conformation in which a filament of taught DNA
coming from each side of a coil.
No one of these theories describes the complete range of observed conformations.
Rather, the individual conformations assumed in these theories represent one of the
several observed conformations. Notwithstanding, the presence of these observed
conformations provides a qualitative explanation for the dissipative component of stress
found in measurements of ηE.
6.4.10 Limitation of applicability
The results presented here should not be generalized to polymers in a mixed
elongational and shearing flow or to polymers in an elongational flow that were pre-
sheared. Our data indicates that the processes involved in the diverse dynamics arise
from the variation in tonset and from internal configurations (i.e. folds). In mixed flows, a
large fraction of the molecules are partially extended due to shearing and this may
eliminate some of the internal constraints that led to the observed dynamics.
However, the dynamics molecules in mixed flows can easily be studied.
Preventing pre-shearing of the molecules along the inlet was the main design criterion for
the flow cell. To study pre-sheared molecules, the microscope simply needs to be
focused to a different depth. Molecules in a mixed flow can be studied in a similar
manner.
6.5 Summary
We have presented the first results of single DNA molecules in an elongational
flow where the velocity gradient is uniform and the polymers are isolated. The results
reveal complex, diverse dynamics. Such results were previously unobservable in bulk
measurements and current theoretical models only capture a portion of the observed
complexity. Through these methods, we have illustrated the use of DNA as a model
polymer for investigating an outstanding problem in polymer dynamics. In using DNA,
one gains a detailed knowledge of the conformation and internal dynamics of individual
128
polymers. This knowledge eliminates many of the ambiguities associated with classical
bulk experiments that average over a macroscopic large number of molecules.
Unique to this method we were able to identifying those individual molecules that
reach steady state elongations. This analysis has led to the first direct observation of the
coil-stretch transition in the steady-state extension as evidenced by the sudden, nonlinear
increase in the steady-state extension of polymers at a critical strain rate. This behavior is
well characterized by a simple finitely extensible dumbbell model using Marko-Siggia
spring [25] and the critical strain rate agrees with theoretical and numerical calculations
[101].
Our data indicates that the concept of a discrete and abrupt coil-stretch transition
is limited to the steady-state. One can not think of polymers undergoing a simple,
collective and simultaneous unwinding as soon as ε ε> c . The mismatch between
<x(tres)> and xsteady implies that the non-Newtonian properties of dilute polymer solutions
in most practical elongational flows (where ε tres < 5.5) are dominated by the dynamic
and not the steady-state properties.
However, even for molecules of identical length and strain-history, these
dynamics are complex. By visualizing single molecules, we identified distinct
conformational classes with differing dynamics. These conformations provide a
qualitative explanation for the high stress observed by James and Saringer [91] and the
dissipative stress observed in the measurements of ηe [92, 94, 95]. Furthermore, the
variety of conformations and the large variation in tonset imply difficulties with any mean
field description of polymers in an elongational flow.
The data presented here should serve as a guide in developing improved
microscopic theories for the polymer dynamics and the bulk rheological properties of
such solutions.
6.6 Future prospects
Within the context of elongational flows, the appearance of conformational
dependent rate of stretching at 1τε > 2 suggests that these experiments be extended to
higher strain rates and longer chains. At higher strain rates, the fraction of chains in each
conformation can be measured, while longer chains allow for multiple folds.
129
Furthermore, by positioning a molecule at the stagnation point [115] and by pulsing the
pump, the evolution of the conformation from equilibrium can be imaged.
We have confirmed the presence of a sharp coil-stretch transition but de Gennes
postulated that polymers in an elongational flow will undergo a hysteresis due to an
increase in the drag of an extended state relative to the coiled state. Our present
experiment is unable to probe this question. However, the presence of such hysteresis is
being experimentally investigated by holding one end of a long DNA molecule at the
stagnation point. This geometry mimics co-moving with the fluid and allows for an
indefinite residency time.
Our data shows that affine deformation is becoming an increasingly worse
approximation at 1τε ≅ 3.3. However, previous simulations of polymers in elongational
flows at high strain rates ( 1τε ≅40) have shown the polymer deform affinely with the fluid
[107]. This apparent contradiction needs to be investigated both experimentally and
theoretically.
Imaging single DNA molecules is not limited to such a simple system. By
coupling fluorescence microscopy with optical tweezers, one can go beyond passive
observation of molecules undergoing Brownian motion by driving the dynamics with
forces and stresses applied at the level of single chains and then measure the resulting
response. Quantitative measurements of extension [9, 10, 27], internal dynamics [28],
and forces [22] are now possible.
For instance, DNA offers the opportunity to measure the non-equilibrium force
generated in a polymer. DNA can be attached between two microspheres and held in
tension between two optical traps. By monitoring the position of the first trap while the
second trap is rapidly moved, the non-equilibrium tension with the chain can be
measured [116]. Alternatively, the force exerted on the bead can be measured during
the relaxation of a chain from a highly extended state. Such non-equilibrium
measurements will be critical step towards developing an understanding of dynamics of
polymers in at the very high strain rates.
Moreover, the application of this technique is not limited to single, isolated
molecules. For example, we have observed the tube-like motion of individual molecules
in concentrated solutions by staining one DNA molecule in a background of unstained
130
DNA [8]. Future prospects include studying concentrated polymers under flow, DNA-
based polymer brushes, and polymer dynamics in reduced dimensions [18].
131
Chapter 7 Future Prospects In the preceding chapters, we have presented the visualization and manipulation of
single DNA molecules as a new tool in polymer science. These techniques have been
applied to several simple systems to address basic questions in polymer physics. These
techniques can continue to be applied to answer basic question in polymer physics as well
as more practical problems in polymer science. In this chapter, we present several
suggested additional experiments for both dilute and concentrated solutions.
7.1 Dilute solutions
7.1.1 Relaxation
In chapter 3, we measured that the longest relaxation time scale as τ ~ L1.66. The
results were determined from a limited range in length of 4 - 40 µm. We have seen in
chapter 5 that there was a transition in the scaling of stretching of a tethered polymer in a
uniform flow when the length range was extended from 83 µm to 154 µm. It would be
interesting to learn if such a transition occurs in the scaling of the relaxation time.
Perhaps, for the longer molecules, a more Rouse-like scaling might be revealed. Such a
transition would arise due to the increase in the difference in hydrodynamic drag of the
DNA in extended and coiled configurations. With our new Nikon objective, these
experiments can be done at 100 - 150 µm away from any surface to eliminate any
coupling between the polymer and the edge of the fluid cell defined by the cover slip.
Another important relaxation experiment is to determine the effect of
conformation on the relaxation time. In chapter 6, we showed that rate of stretching in an
elongational flow depended strongly on the conformation. For many practical flows, the
residency time of the polymer time is too low to fully extend the polymer. Thus, these
polymers relax from a partially extended state. The question to address is how different
are the relaxation times from the different conformations.
7.1.2 Stretching in a uniform flow
In chapter 5, we observed a transition in the scaling behavior for the stretching of
a polymer in a uniform flow. However, the polymer was only 15 µm from the coverslip
132
even though the polymers were up to 154 µm long. Theoretical estimates suggest that the
effect of this wall was small [24]. Nonetheless, with our new objective, we can
experimentally verify the generality of the earlier results and eliminate any outstanding
uncertainties.
7.1.3 Polymers in shear flow
Time (s)
0 50 100 150 200 250 300
Exte
nsio
n (m
m)
0
5
10
15
Figure 56: Single DNA molecule in a shear with 1τγ ≅ 8 at η = 41 cp.
In chapter 6, we studied the dynamics of polymers in an elongational flow. The
ability of elongational flows to stretch polymers far from equilibrium and the possibility
a coil-stretch phase transition has made it an active area of theoretical and experimental
work [2, 3, 5, 41, 49]. However, the behavior of polymers in a shear flow is also of
great practical importance since flow through a pipe leads to a strong shear due to the
parabolic flow profile.
Shear flow can be broken down in a rotation component and an elongational
component. The elongation component leads to polymer stretching and the rotational
component leads to a reversal of the stretching and compressive axes. Thus, polymers
133
are thought to undergo a cycle of extension and retraction in a shearing flow. However,
the expectation is that polymers do not become highly extended in shearing flows.
The dynamics of polymers in shear flow can easily be measured with our existing
apparatus. To study shear, one simply has to image the polymers in the long inlet
channels away from the center. So, for instance, in a flow cell that was 220 µm deep, the
shear experiment could be done at a depth of 15 µm. At this depth, even though the flow
is parabolic, the shear rate would be constant to 1% per µm. To image single molecules
for an extended period of time, the stage can be moved to match the net velocity of the
polymer.
In preliminary experiments using this technique, we have shown that λ-DNA can
be extended up to ~ 75 % of its contour length at a shear rate of 1τγ = 8. Further, these
preliminary experiments verified that the polymers undergo a series of extensions and
retractions as if tumbling in the fluid flow. However, in this limited data set, we could
not determine a dominant frequency component to this tumbling motion. The main
technical difficulties will be establishing that photobleaching does not adversely effect
the molecule during the long observation time. Stroboscopic illumination should help
alleviate this concern.
7.1.4 Mixed flows
These flows can be investigated with a linear combination of elongational and
shearing flows. Also, polymers in elongational flow that are pre-sheared can be studied
simply by changing the depth at which the polymers are imaged in the elongational flow
cell.
7.1.5 Dynamic force measurements
Polymers at high strain rates are deformed on times scales much shorter than their
longest relaxation time. So, rapidly deforming polymers are not in equilibrium with the
applied flow and will not have a sufficient time to fully explore their accessible phase
space. Hence, it is inaccurate to describe the elasticity of rapidly deforming polymers
using their steady-state elasticity. To experimentally measure this non-equilibrium
134
elasticity, one can use the optical tweezer as a force transducer. The displacement of the
microsphere in the optical trap is directly related to the force applied to it by the DNA.
Using this technique, there are three simple experiments that can be performed.
First, the force exerted on the microsphere during a relaxation can be measured. Second,
the force exerted by a tethered polymer can be correlated with fluctuations in extension.
This will determine if fluctuations in the hydrodynamic interaction are important. Third,
the DNA can be held between two optical traps by attaching a microsphere to each end.
As one of the microspheres is rapidly moved away, the development of tension on the
other microsphere is measured [116].
7.2 Concentrated solutions
Most of our work has focused on developing an understanding the dynamics of
polymers in dilute solution. We have tried to study simple but central questions to
validate our new experimental technique. First, we measured the relaxation of an isolated
chain. Next, we measured the stretching of a tethered polymer in a uniform flow. And
last, we measured the dynamics of polymers in an elongational flow. But, perhaps, the
most exciting future research will be in concentrated polymers systems. Concentrated
polymers solutions are complex systems that can exhibit remarkably universal properties
[2]. We have already shown that polymers undergo tube-like motion [8] and that such
tube-like motion exists for time longer than the self-diffusion time [29].
7.2.1 Constrain release
Our previous measurements of tube-like motion were limited by the relaxation of
our test chain. By attaching microspheres to both ends of the DNA, one can draw loop
and hold each end, preventing the normal relaxation of the chain within the tube. Since
the ends are constrained, the extended, looped test chain relaxes by constrain release.
An alternative geometry would be to draw an arc and measure the rate at which the arc
relaxes in to a straight line.
For a background polymer, each time an entanglement moves out of the way, the
background chain moves in a random direction but, for the test chain, the tension within
the chain biases this random walk towards a relaxed state. Thus, the rate of motion of
135
the test chain may simply be the rate of constrain release multiplied by the average
distance between constrains.
7.2.2 Step-strain experiments.
In concentrated linear polymer solutions, the fast time scale dynamics after a large
step are not well defined. According to Doi, the deformation is thought to be affine but
it has not experimentally verified. We can investigate these dynamics by selective
staining a few chains as test chains. Then, approximately linear steps can be generated
with a cone-plate geometry where the plate is a cover-slip.
7.2.3 Flows of concentrated solutions
Our measurement techniques for dilute polymers solutions in elongational and
shear flows can be extended to concentrated solutions. The slower dynamics of
concentrated solutions should alleviate several of the problems encountered in dilute
solutions. Also, the alteration of the flow pattern by the increase in polymer concentration
can be measured by either the bulk motion of the fluorescently labeled chains or by
adding small fluorescent microspheres as tracer particles.
In particular, it will be interesting to determine how the tumbling motion present
in dilute polymer solutions will be modified in concentrated solutions and how this
change in motion will correlate with the change in the velocity profile of the fluid.
7.2.4 Star molecules
Star molecules in concentrated solutions diffuse much more slowly due to their
additional topological constraints. However, the exact dynamics are still an active area of
research. The dominant process is chain retraction. Such chain retraction can be studied
by stretching polymer in a concentrated solution. Alternatively, star molecules might be
constructed using biotinylated-DNA and streptavidin. Streptavidin has four binding sites
so presumably each streptavidin molecule attaches four different DNA molecules. The
difficulty will lie in asserting how many individual linear DNA molecules are in each star
molecule.
136
Bibliography:
[1] M. Doi, S. E. Edwards, The Theory of Polymer Dynamics (Oxford Press, New York,
1986).
[2] P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press,
Ithica, 1979).
[3] R. B. Bird, C. F. Curtiss, R. C. Armstrong, O. Hassager, Dynamics of Polymeric
Liquids (Wiley, New York, ed. 2, 1987), vol. 2.
[4] J. D. Ferry, Viscoelastic Properties of Polymers (Wiley, New York, ed. 3, 1980).
[5] R. G. Larson, Constitutive Equations for Polymer Melts and Solution (Buttersworths,
New York, 1988).
[6] A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, Optics Letters 11, 288-90
(1986).
[7] S. Chu, Science 253, 861-66 (1991).
[8] T. T. Perkins, D. E. Smith, S. Chu, Science 264, 819-22 (1994).
[9] T. T. Perkins, S. R. Quake, D. E. Smith, S. Chu, Science 264, 822-6 (1994).
[10] T. T. Perkins, D. E. Smith, R. G. Larson, S. Chu, Science 168, 83-87 (1995).
[11] J. C. Crocker, D. G. Grier, Physical Review Letters 73, 352-5 (1994).
[12] J. T. Finer, R. M. Simmons, J. A. Spudich, Nature 368, 113-9 (1994).
[13] K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, Nature , 721-7 (1993).
[14] H. Yin, et al., Science 270, 1653-7 (1995).
[15] S. B. Smith, L. Finzi, C. Bustamante, Science 258, 1122-26 (1992).
[16] S. B. Smith, P. K. Aldridge, J. B. Callis, Science 243, 203-6 (1989).
[17] D. C. Schwartz, M. Koval, Nature 338, 520-2 (1989).
[18] W. D. Volkmuth, R. H. Austin, Nature 358, 600-2 (1992).
[19] W. D. Volkmuth, T. Duke, M. C. Wu, R. H. Austin, Physical Review Letters 72,
2117-20 (1994).
[20] L. Mitnik, C. Heller, J. Prost, J. L. Viovy, Science 267, 219-22 (1995).
137
[21] A. D. Dinsmore, A. G. Yodh, D. J. Pine, Nature 383, 239-42 (1996).
[22] C. Bustamante, J. F. Marko, E. D. Siggia, S. B. Smith, Science 265, 1599-1600
(1994).
[23] P. Cluzel, et al., Science 271, 792-4 (1996).
[24] R. G. Larson, T. T. Perkins, D. E. Smith, S. Chu, Physical Review E 55, 1794-1797
(1997).
[25] J. F. Marko, E. D. Siggia, Macromolecules 28, 8759--70 (1995).
[26] K. Morikawa, M. Yanagida, Journal of Biochemistry 89, 693-696 (1981).
[27] T. T. Perkins, D. E. Smith, S. Chu, Science 276, 2016-21 (1997).
[28] S. R. Quake, H. P. Babcock, S. Chu, Nature (in press).
[29] D. E. Smith, T. T. Perkins, S. Chu, Physical Review Letters 75, 4146-9 (1995).
[30] D. E. Smith, T. T. Perkins, S. Chu, Macromolecules 29, 1372-3 (1996).
[31] S. B. Smith, Y. Cui, C. Bustamante, Science 271, 795-9 (1996).
[32] A. Vologodskii, Macromolecules 27, 5623-5 (1994).
[33] P. J. Hagerman, Annual Review of Biophysics and Biophysical Cemistry 17, 265-86
(1988).
[34] R. Pecora, Science 251, 893--8 (1991).
[35] W. Eimer, R. Pecora, Journal of Chemical Physics 94, 2324--9 (1991).
[36] B. H. Zimm, preprint submitted to biophsical Journal (1997).
[37] J. Kas, et al., Biophysical Journal 70, 609-25 (1996).
[38] S. Matsumoto, K. Morikawa, M. Yanigida, Journal of Molecular Biology 110, 501
(1981).
[39] B. H. Zimm, Journal of Chemical Physics 24, 269-278 (1956).
[40] P. E. Rouse, Journal of Chemical Physics 21, 1272 (1953).
[41] P. G. de Gennes, Journal of Chemical Physics 60, 5030--42 (1974).
[42] T. Lodge, N. Rotstein., S. Prager, Advances in Chemical Physics 79, 1 (1990).
[43] H. S. Rye, J. M. Dabora, M. A. Quesada, R. A. Mathies, A. N. Glazser, Analalytical
Biochemistry 208, 144-50 (1993).
[44] G. Wallis, D. Pomerantz, Journal of Applied Physics 40, 3946-9 (1969).
[45] C. Hu, S. Kim, Applied Physics Letters 29, 582-5 (1976).
[46] K. E. Petersen, Proceedings of the IEEE 70, 420-57 (1982).
138
[47] P. Flory, Statistical Mechanics of Chain Molecules (Interscience, New York, 1969).
[48] P. G. de Gennes, Macromolecules 9, 587 (1976).
[49] A. Keller, J. A. Odell, Colloid & Polymer Science 263, 181-201 (1985).
[50] M. J. Menasveta, D. Hoagland, A., Macromolecules 24, 3427-33 (1991).
[51] T. Q. Nguyen, G. Yu, H.-H. Kausch, Macromolecules 28, 4851-60 (1995).
[52] G. G. Fuller, L. G. Leal, Rheologica Acta 19, 580-600 (1980).
[53] C. A. Cathey, G. G. Fuller, Journal of Non-Newtonian Fluid Mechanics 34, 63-88
(1990).
[54] M. Adam, M. Delsanti, Macromolecules 10, 1229 (1977).
[55] M. Adam, M. Delsanti, J. Phys. Paris 37, 1045 (1976).
[56] Y. Tsunashima, N. Nemoto, M. Kurata, Macromolecules 16, 584 (1983).
[57] Y. Tsunashima, N. Nemoto, M. Kurata, Macromolecules 16, 1184 (1983).
[58] J. Brandrup, E. Immergut, Eds., Polymer Handbook (Wiley, New York, 1989).
[59] D. M. Crothers, B. H. Zimm, Journal of Molecular Biology 12, 525 (1965).
[60] D. Wirtz, Physical Review Letters 75, 2436-9 (1995).
[61] S. W. Provencher, Comput. Phys. Commun. 27, 213 (1982).
[62] G. Ryskin, Physical Review Letters 59, 2059-62 (1987).
[63] G. Ryskin, Journal of Fluid Mechics 178, 423-449 (1987).
[64] B. H. Zimm, G. M. Roe, L. F. Epstein, Journal of Chemical Physics 24, 279 (1956).
[65] S. Manneville, P. Cluzel, J. L. Viovy, D. Chatenay, F. Caron, Europhysics Letters
36, 413-8 (1996).
[66] F. Brochard-Wyart, Europhysics Letters 30, 387-92 (1995).
[67] S. F. Edwards, Proc. Phys. Soc. London 92, 9 (1967).
[68] P. G. de Gennes, Journal of Chemical Physics 55, 572-9 (1971).
[69] D. Richter, Physical Review Letters 64, 1389 (1990).
[70] J. S. Higgins, J. E. Roots, J. Chem. Soc. Faraday. Trans. 81, 757 (1985).
[71] T. Russel, et al., Nature 365, 235 (1993).
[72] R. Musti, J.-L. Sikorav, D. Lairex, G. Jannink, M. Adam, Comptes Rendus De
L'Academie des Sciences Series II 352, 599-605 (1995).
[73] R. L. Rill, T. E. Strzelecka, M. W. Davidson, D. H. Van Winkle, Physica A 176, 87
(1991).
139
[74] B. Scalettar, J. Hearst, M. Klein, Macromolecules 22, 4550 (1989).
[75] A. Ajdari, et al., Physica A 204, 17-39 (1994).
[76] M. Fixman, J. Kovac, Journal of Chemical Physics 58, 1564-8 (1971).
[77] P. Debye, A. Bueche, Journal of Chemical Physics 16, 573 (1948).
[78] B. H. Zimm, S. D. Levene, Quaterly Reviews of Biophysics 25, 171-204 (1992).
[79] R. G. Larson, J. J. Magda, Macromolecules 22, 3004-3010 (1989).
[80] F. Brochard-Wyart, Europhysics Letters 23, 105-11 (1993).
[81] F. Brochard-Wyart, H. Hervet, P. Pincus, Europhysics Letters 27, 511-6 (1994).
[82] J. G. Kirkwood, J. Riseman, Journal of Chemical Physics 16, 565-73 (1948).
[83] J. G. Kirkwood, Journal of Polymer Science 12, 1-14 (1954).
[84] P. N. Dunlap, L. G. Leal, Journal of Non-Newtonian Fluid Mechanics 23, 5-48
(1987).
[85] E. D. T. Atkins, M. A. Taylor, Biopolymers 32, 911-23 (1992).
[86] J. A. Odell, A. Keller, Y. Rabin, Journal of Chemical Physics 88, 4022-8 (1988).
[87] J. A. Odell, A. Keller, A. J. Muller, Colloid & Polymer Science 270, 307-324
(1992).
[88] J. A. Odell, M. A. Taylor, Biopolymers 34, 1483-93 (1994).
[89] K. A. Smith, E. W. Merrill, L. H. Peebles, S. H. Banijamali, Colloq. Int. C.N.R.S.
233, 341 (1975).
[90] J. L. Lumley, Physics of Fluids 20, s64 (1977).
[91] D. F. James, J. H. Saringer, Journal of Fluid Mechics 97, 655-671 (1980).
[92] V. Tirtaatmadja, T. Sridhar, Journal of Rheology 37, 1081-1102 (1993).
[93] D. F. James, T. Sridhar, Journal of Rheology 39, 713-24 (1995).
[94] S. H. Spiegelberg, G. H. McKinely, Journal of Non-Newtonian Fluid Mechanics 67,
49-76 (1996).
[95] N. V. Orr, T. Sridhar, Journal of Non-Newtonian Fluid Mechanics 67, 77-103
(1996).
[96] E. J. Hinch, Journal of Non-Newtonian Fluid Mechanics 54, 209-230 (1994).
[97] L. C. Klotz, B. H. Zimm, Journal of Molecular Biology 72, 779-800 (1972).
[98] K. S. Schmitz, R. Pecora, Biopolymers 14, 521-542 (1975).
[99] D. S. Thompson, S. J. Gill, Journal of Chemical Physics 47, 5008-17 (1967).
140
[100] P. R. Callis, N. Davidson, Biopolymers 8, 379-90 (1969).
[101] J. J. Magda, R. G. Larson, M. E. Mackay, Journal of Chemical Physics 89, 2504-
13 (1988).
[102] S. Daoudi, F. Brochard, Macromolecules 11, 751-8 (1978).
[103] P. G. de Gennes, Physica 140A, 9-25 (1986).
[104] E. J. Hinch, Physics of Fluids 20, 522 (1977).
[105] J. M. Rallison, Journal of Non-Newtonian Fluid Mechanics 68, 61-83 (1997).
[106] P. S. Doyle, E. S. G. Shaqfeh, A. P. Gast, Journal of Fluid Mechics 334, 251-291
(1997).
[107] R. G. Larson, Rheologica Acta 29, 371-384 (1990).
[108] D. H. King, D. F. James, Journal of Chemical Physics 78, 4749-54 (1983).
[109] S. P. Carrington, J. A. Odell, Journal of Non-Newtonian Fluid Mechanics 67, 269-
283 (1996).
[110] Y. Rabin, Journal of Non-Newtonian Fluid Mechanics 30, 119-123 (1988).
[111] D. Bensimon, A. J. Simon, V. Croquette, A. Bensimon, Physical Review Letters
74, 4754-7 (1995).
[112] V. Tirtaatmadja, T. Sridhar, Journal of Rheology 39, 1133-60 (1995).
[113] A. Peterlin, Makromol. Chem. 44 (1961).
[114] R. Keunings, Journal of Non-Newtonian Fluid Mechanics 68, 85-100 (1997).
[115] D. E. Smith, T. T. Perkins, S. Chu, Macromolecules 29, 1372-3 (1995).
[116] U. Seifert, W. Wintz, P. Nelson, Physical Review Letters 77, 5389-92 (1996).