Review of class 4 – using DNA to purify CNT species

1. View CNTs as segment of graphene cut (so C’s areat edge) rolled up and reattached -> n,m classes

2. ssDNA acts like soap to “solubilize” CNTs with basesstacking (flat) against C rings, P’s interacting w/H20

3. DNA binding has little sequence specificity, yetelution of DNA-CNTs from IEX column -> specificpeaks of CNT species w/different DNA oligos. Practically very important, but why does it work?

4. How were different CNT species recognized?semi-conductor CNTs have (n,m)-specific abs and emis peaks due to electron band structure

5. Model of ssDNA bound to graphene: bases on alternate sides of sugar-P backbone;adjacent strands anti-parallel, novel base pairing

6. Imagine cutting this DNA-graphene into stripsand rolling into CNTs:for DNA strands to be at edges of cuts and tobase pair across seam may require specialmatching between sequence and n,m type

7. Hypothesis: only such uniformly “tiled” CNTs elutein IEX chromatography as narrow peak; non-uniformly tiled species stick in column or eluteover broad range of [salt]

8. Nice idea, but not clear what data support hypoth.

Mechanical properties of DNA under stretching and twisting

Why important –

biology: curved/bent DNA important in packing intonuclei and viruses, in regulation of transcription,

various enzymes bend/twist DNA during replication, transcription, recombination

polymer physics: model for understanding basic force-length relationships for well-defined polymer

technology: important for using DNA as tool to pull, twist objects; to study how enzymes that act on DNA work as nano-machines

What we’ll cover: class 5 stretching

at low force, concept of entropic spring freely-jointed chain and worm-like chain modelsat high force – structural change in double helix B->S form, similarity to phase changemethods used to study hydrodynamic drag paramagnetic beads laser traps

class 6 twisting at low torque, twisting compliance supercoils, topological changesat high torque – structural changes (P, L forms)coupling between twisting and stretching

Linear polymers and Hooke’s Law

Freely jointed chain (FJC) model n segments of length b joined

at freely rotating joints Brownian (thermal)

motion randomizes fi

applied force pulls out chain fixed at one end ( ) contour length L = nb (b also called “Kuhn” length)

<x>/L = coth(Fb/kBT) – kBT/FB in 3-d

= tanh(Fb/kBT) in 1-d (see Nelson, Biol. Phys. ch 9.2, for derivation)

f1

Fb

x

Where (in the world) do these formulas come from?

In 1-d, x = bSsi where si =+1

p({si})=Z-1exp[-(-FbSsi)/kBT]

<x> = Sall states p({si}) x

= Ss1=+1 …Ssn=+1 Z-1 {exp[-(-FbSsi)/kBT]} bSsi

= kBT (d/dF) ln {Ss1=+1 …Ssn=+1 exp[-(-FbSsi)/kBT]}

= kBT (d/dF) ln [exp(Fb/kBT) + exp(-Fb/kBT)]n

= nb tanh(Fb/kBT) = L tanh(Fb/kBT)

f1

Fb

x

<x>/L = tanh(Fb/kBT) tanh(z) = (ez – e-z)/(ez + e-z) -> z for z<<1-> 1 for z>>1

Low force regime F << kBT/b, F -> <x> (kBT/Lb) => ksp = kBT/Lb

the longer L, the more compliantthe higher T, the less compliant

empirically, b ~ 100nm so “low” F < 0.04pN

At F = 0, equipartition theorem => ksp<x2> = kBT <x2>1/2 = xrms = (Lb)1/2

note xrms independent of T at F=0

High force regime: F>>kBT/b, <x> -> L

Several groups tried to measure b by pulling on DNA

Bustamante (Science 258:1122 (1992)

dimer of dsDNA from l phage, 48kb x2, L ~30mm, one end attached to glass, other to r ~ 1mm paramagnetic bead

att. pt. determined by varyingflow and magnetic field; forflow v, Fflow= 6phrv(1+9r/16d)

Ftotal = Fflow/cos qmeasure <x> as function of Ftotal

<x>

<x> (mm)

F (pN)

Problem – poor fit to 3-d FJC model no matter what L or brest of paper = complicated disc. of possible explanations

b = 50nm 100 200

Next paper – WLC model fits same data beautifully!

Worm-like chain model - randomly oriented smooth chain with “stiffness” defined by:

Persistence length p = length over which orientational correlation falls to characteristic value

<x(F)>/L does not have analytic solution, but in high and low force limits, Fp/kBT = ¼ (1-<x>/L)-2 – ¼ + <x>/L

<cosq(s)>

sp

1

st1

t2^^q DNA

x

FJC model

WLC model

WLC modelfits force-extension data much better than FJC

Bustamante, Science 265:1599 (1994)

Fp/kBT = ¼ (1-<x>/L)-2 – ¼ + <x>/L

At low force, <x>/L<<1, Fp/kBT -> 3<x>/2L

F -> (3kBT/2pL) <x> => ksp= 3kBT/2pL

p L/unit empirically, dsDNA 50nm 0.34nm/bp

ssDNA ~1nm ~0.6nm/b

ssDNA is “tighter” entropic spring, because more flexible!

At high force, Fp/kBT -> ¼ (1-<x>/L)-2 =>

<x>/L -> 1 – (kBT/4pF)1/2

What use are these formulae?

Can estimate: Average end-end distance of a DNA of given L

Average length of ss or ds DNA under given F

For immobilized enzyme pulling on DNA (or RNA), the maximum force it can exert, given <x>/L

For enzyme unwinding a dsDNA held at given F, the length unwound, given <x>/L

Next paper – what happens at higher force?

Slight discrepancywith WLC modelat 10-60pN ->estimate of Young’s(elastic) modulus E:F/A=EDx/x

->E = 3x108N/m2

cf ~ 109 for nylon 1012 for CNT

Dx

At F ~ 65pN, DNA suddenly begins to stretch

Further pullinglengthens DNA >Lw/ little increaseF until new, fullystretched state is reached (~1.7 L)

Smith et al Science 271:795 (1996)

Stretched “S”-form of DNA probably has base-stacking interactions disrupted -> change in helix pitch

3.4nm/10bp 5.8nm/10bp

“Cooperativity” of transition (occurrence over just 2-3 pN)suggests S-form segment spreads along DNA; similar to phase change (e.g. ice->water, adding heat energy doesn’t change temperature until all ice melted: adding energy via stretching doesn’t change tension until all DNA converted to S-form); can be used to clamp F at 65pN

>65pN

Nicks in single DNA strands may -> partially unravelingon further stretching -> mix of ss and ds regions withintermediate behavior

<x> mm

F (p

N)

ds

ss

mix

Stretching experiments used laser trapNobel prize to generate high force (only get few pN w/magnets)

Highly focused laser pulls object with higher index of refraction towards brightest part of laser beam (x=0); small displacement x -> restoring force ~ -kspx. Given trap strength

ksp, observing x, one can infer F and vice versa

Mechanism – light E-field polarizes object with diff.dielectric constant -> attractive dipole force

--> -- ++E

Newman and Block, Rev Sci Instr 75:2787 (2004)

Alternative explanation – photons carry momentum; bending ray changes photon momentum; momentum conserva- tion => object feels opposing force; if beam asymmetric, force from brightest region dominates

Their set up: apposedlasers with | polarization trap bead; pol. beam splitter (PBS) sends light from left laser to right position detector;bead displacement altersposition of laser light ondetector; flow andFdrag = 6phrv can be usedto calibrate position detector signal to force

PBS

DNA attached to streptavidinbeads via biotin tags at DNA ends

Example of DNAstretched betweenbead held on pipetand bead held inlaser trap: segment of biotinylated DNAin center binds smallSA bead (from nextweek’s paper)

DNA madeby pcr andrestr. enz.cutting +ligation;ss nickallowsbottompart to swivel

apply torque

keep const. tension

observe ~ w t

This kind of set-up can also be used to investigateeffect of twisting forces on DNA – next week’s topic

Summary

DNA extension with force best fits WLC model

Fp/kBT = ¼ (1-<x>/L)-2 – ¼ + <x>/L for high and low Fp = 50nm for dsDNA, ~1nm for ssDNAL = 0.34nm/bp for dsDNA, ~0.6nm/b for ssDNAWhen F<<kBT/p, F ~ ksp<x> where ksp = 3kBT/2pLWhen F=0, ksp<x2> = kBT

When F > 65pN, B-form DNA (3.4nm/10bp) convertsto S-form (1.7x longer); might be able to usemixed B-S form DNA as force clamp