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1.021, 3.021, 10.333, 22.00 Introduction to Modeling and SimulationSpring 2011
Part I – Continuum and particle methods
Markus J. BuehlerLaboratory for Atomistic and Molecular MechanicsDepartment of Civil and Environmental EngineeringMassachusetts Institute of Technology
Applications to biophysics and bionanomechanics Lecture 10
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Content overview
I. Particle and continuum methods1. Atoms, molecules, chemistry2. Continuum modeling approaches and solution approaches 3. Statistical mechanics4. Molecular dynamics, Monte Carlo5. Visualization and data analysis 6. Mechanical properties – application: how things fail (and
how to prevent it)7. Multi-scale modeling paradigm8. Biological systems (simulation in biophysics) – how
proteins work and how to model them
II. Quantum mechanical methods1. It’s A Quantum World: The Theory of Quantum Mechanics2. Quantum Mechanics: Practice Makes Perfect3. The Many-Body Problem: From Many-Body to Single-
Particle4. Quantum modeling of materials5. From Atoms to Solids6. Basic properties of materials7. Advanced properties of materials8. What else can we do?
Lectures 1-13
Lectures 14-26
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Overview: Material covered so far…Lecture 1: Broad introduction to IM/S
Lecture 2: Introduction to atomistic and continuum modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion)
Lecture 3: Basic statistical mechanics – property calculation I (property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function)
Lecture 4: Property calculation II (Monte Carlo, advanced property calculation, introduction to chemical interactions)
Lecture 5: How to model chemical interactions I (example: movie of copper deformation/dislocations, etc.)
Lecture 6: How to model chemical interactions II (EAM, a bit of ReaxFF—chemical reactions)
Lecture 7: Application to modeling brittle materials I
Lecture 8: Application to modeling brittle materials II
Lecture 9: Application – Applications to materials failure
Lecture 10: Applications to biophysics and bionanomechanics
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Lecture 10: Applications to biophysics and bionanomechanics
Outline:1. Protein force fields2. Single molecule mechanics3. Fracture of protein domains – Bell model
Goal of today’s lecture: Force fields for organic materials, and specifically proteinsBasic introduction into modeling of biological materials Fracture model for protein domains
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1. Force fields for organic chemistry -how to model proteins
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Significance of proteins
Proteins are basic building blocks of life
Define tissues, organs, cells
Provide a variety of functions and properties, such as mechanical stability (strength), elasticity, catalytic activity (enzyme), electrochemical properties, optical properties, energy conversion
Molecular simulation is an important tool in the analysis of protein structures and protein materials
Goal here: To train you in the fundamentals of modeling techniques for proteins, to enable you to carry out protein simulations
Explain the significance of proteins (application)
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Human body: Composed of diverse array of protein materials
Eye’s cornea (collagen material)
Skin (complex composite of collagen, elastin)
Cells (complex material/system based on proteins)
Muscle tissue (motor proteins)
Nerve cells
Blood vessels
Tendon(links bone, muscles)
Cartilage (reducefriction in joints)
Bone (structuralstability)
http://www.humanbody3d.com/ and http://publications.nigms.nih.gov/insidethecell/images/ch1_cellscolor.jpg
Image removed due to copyright restrictions.
Human Body 3D View™ image of whole bodies.
Image courtesy of NIH.
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Cellular structure: Protein networks
Cell nucleus
Actin network
Microtubulus(e.g. cargo)
Vimentin(extensible, flexible, providestrength)
= cytoskeleton Image courtesy of NIH.
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Protein structures define the cellular architecture
Intermediate filaments
Image removed due to copyright restrictions; see image now: http://www.nanowerk.com/spotlight/id2878_1.jpg. Source: Fig. 2.17 in Buehler, Markus J. Atomistic Modelingof Materials Failure. Springer, 2008.
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How protein materials are made – the genetic code
Proteins: Encoded by DNA (three “letters”), utilize 20 basic building blocks (amino acids) to form polypeptides
Polypeptides arrange in complex folded 3D structures with specific properties1D structure transforms into complex 3D folded configuration
ACGT
Four letter code “DNA”
Combinationof 3 DNA letters equals a amino acid
E.g.: Proline –
CCT, CCC, CCA, CCG
Sequence of amino acids“polypeptide” (1D structure)
Transcription/translation
Folding (3D structure)
.. - Proline - Serine –Proline - Alanine - ..
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Chemical structure of peptides/proteins
R = side chain, one of the 20 natural amino acids
20 natural amino acids differ in their side chain chemistry
Peptide bond
Typically short sequence
of amino acids
Longer sequence of amino acids, often complex 3D structure
…
…
side chains
© source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.
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There are 20 natural amino acids
Difference in side chain, R
Forms peptide bond
Image by MIT OpenCourseWare.
H
H
C COO-H3N+
CH3
H
C COO-H3N+
CH
CHCH3 CH3
H
C COO-H3N+
CH2
CH3 CH3
H
C COO-H3N+
CH2
CHCH3
CH3
H
C COO-H3N+
Glycine (Gly) G6.0
Alanine (Ala) A6.0
NE
CH2
H
C COO-H3N+
Phenylalanine (Phe) F5.5E
CH2
CH3OH
H
C COO-H3N+
Serine (Ser) S5.7NE
HCOH
H
C COO-H3N+
Threonine (Thr) T5.6E
CH2
SH
H
C COO-H3N+
Cysteine (Cys) C5.1NE
CH2
NH2
CO
H
C COO-H3N+
C
Asparagine (Asn) N5.4NE
CH2
H
OH
C COO-H3N+
Tyrosine (Tyr) Y5.7NE
CH3
CH3
CH2
S
H
C COO-H3N+
Methionine (Met) M5.7E
CH2
H
H
C COO-H3N+
Tryptophan (Trp) W5.9E
CH2CH2
CH2
H
C COO-H2N+
Proline (Pro) P6.3NE
NE
Valine (Val) V6.0E
Leucine (Leu) L6.0E
Isoleucine (lle) l6.0E
CH2
H
C COO-H3N+
Histidine (His) H7.6E
N
CH2
O-O-
CO
H
C COO-H3N+
Aspartic acid (Asp) D2.8NE
CH2
CH2
NH2O
H
C COO-H3N+
Glutamine (Gln) Q5.7NE
C
CH2
CH2
O
H
C COO-H3N+
Glutamic acid (Glu) E3.2NE
CH2
CH2
CH2
CH2
NH3
H
C COO-H3N+
Lysine (Lys) K9.7E
NHHN
+
+
CH2
CH2
CH2
C
NH
NH2
H
C COO-H3N+
Arginine (Arg) R10.8E
NH2
+
Nonpolar Amino Acids
Polar Amino Acids (Neutral)
Basic Amino AcidsAcidic Amino Acids
δ-δ+
charges
R
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Chemistry, structure and properties are linked
Cartoon
Chemical structure
• Covalent bonds (C-C, C-O, C-H, C-N..)• Electrostatic interactions (charged amino acid side chains)• H-bonds (e.g. between H and O)• vdW interactions (uncharged parts of molecules)
Presence of various chemical bonds:
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Concept: split energy contributions
Covalent bond described as
1. Bond stretching part (energy penalty for bond stretching)2. Bending part (energy penalty for bending three atoms)3. Rotation part (energy penalty for bond rotation, N ≥ 4)
Consider ethane molecule as “elastic structure”
bondHvdWMetallicCovalentElec −++++= UUUUUUtotal
rotatebendstretchCovalent UUUU ++=
=0 for proteins
EthaneC2H6
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Force fields for organics: Basic approach
bondHvdWMetallicCovalentElec −++++= UUUUUUtotal
rotbendstretchCovalent UUUU ++=
∑=
−=
pairsstretchstretch
20stretchstretch )(
21
φ
φ
U
rrk
∑=
−=
tripletsbendbend
20bendbend )(
21
φ
θθφ
U
k
∑=
−=
squadrupletrotrot
rotrot ))cos(1(21
φ
ϑφ
U
k
=0 for proteins
Image by MIT OpenCourseWare.
Bond stretching
Angle Bending
Bond Rotation
θ
ϑ
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Model for covalent bonds
20stretchstretch )(
21 rrk −=φ
20bendbend )(
21 θθφ −= k
))cos(1(21
rotrot ϑφ −= kCourtesy of the EMBnet Education & Training Committee. Used with permission.
Images created for the CHARMM tutorial by Dr. Dmitry Kuznetsov (Swiss Institute of Bioinformatics) for the EMBnet Education & Trainingcommittee (http://www.embnet.org)
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Force fields for organics: Basic approach
bondHvdWMetallicCovalentElec −++++= UUUUUUtotal
ElecU
=0 for proteins
Coulomb potentialij
jiij r
qqr
1
)(ε
φ =
partial charges
distanceelectrostatic constant
21
)(ij
jiij r
qqrF
ε−=
01 4
Coulomb forces
πεε = C10602.1 190
−×=ε
:ElecUiqj
q
Image by MIT OpenCourseWare.
vdW Interactions
Electrostatic interactions
δ−
δ−
δ+δ+
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Force fields for organics: Basic approach
bondHvdWMetallicCovalentElec −++++= UUUUUUtotal
vdWU
:vdWU LJ potential⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛=
612
4)(ijij
ij rrr σσεφ
=0 for proteins
LJ potential is particularly good model for vdW interactions (Argon)
Image by MIT OpenCourseWare.
vdW Interactions
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H-bond model
H2O
:bondH−U
H2O
bondHvdWMetallicCovalentElec −
Evaluated between acceptor (A) /donor(D) pairs
Between electronegative atom and a H- atom that is bonded to another electronegative atom
++++= UUUUUUtotal
=0 for proteins
Slightly modified LJ, different parameters
bondH−U
)(cos65)( DHA4
10
bondH
12
bondHbondH θφ
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛= −−
−ijij
ij rR
rRDr
DHAθH-bond
A
D
H
ijr = distance between D-A
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Summary
bondHvdWMetallicCovalentElec −++++= UUUUUUtotal
=0 for proteins
rotbendstretchCovalent UUUU ++=
20stretchstretch )(
21 rrk −=φ
20bendbend )(
21 θθφ −= k
:ElecU Coulomb potentialij
jiij r
qqr
1
)(ε
φ =
:vdWU LJ potential⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛=
612
4)(ijij
ij rrr σσεφ
:bondH−U )(cos65)( DHA4
10
bondH
12
bondHbondH θφ
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛= −−
−ijij
ij rR
rRDr
))cos(1(21
rotrot ϑφ −= k
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The need for atom typing
Limited transferability of potential expressions: Must use different potential for different chemistry
Different chemistry is captured in different “tags” for atoms: Element type is expanded by additional information on particular chemical state
Tags specify if a C-atom is in sp3, sp2, sp or in aromatic state (that is, to capture resonance effects)
Example atom tags: CA, C_1, C_2, C_3, C…, HN, HO, HC, …
sp3
sp2
sp
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Atom typing in CHARMM
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VMD analysis of protein structure
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Common empirical force fields for organics and proteins
Class I (experiment derived, simple form)CHARMMCHARMm (Accelrys)AMBEROPLS/AMBER/SchrödingerECEPP (free energy force field)GROMOS
Class II (more complex, derived from QM)CFF95 (Biosym/Accelrys)MM3MMFF94 (CHARMM, Macromodel…)UFF, DREIDING
http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html
Harmonic terms;Derived from vibrational spectroscopy, gas-phase molecular structuresVery system-specific
Include anharmonic termsDerived from QM, more general
pset #3
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CHARMM force field
Widely used and accepted model for protein structures
Programs such as NAMD have implemented the CHARMM force field
Problem set #3, nanoHUB stretchmol module, study of a protein domain that is part of human vimentin intermediate filaments
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Application – protein folding
ACGT
Four letter code “DNA”
Combinationof 3 DNA letters equals a amino acid
E.g.: Proline –
CCT, CCC, CCA, CCG
Sequence of amino acids“polypeptide” (1D structure)
Transcription/translation
Folding (3D structure)
.. - Proline - Serine –Proline - Alanine - ..
Goal of protein folding simulations:Predict folded 3D structure based on polypeptide sequence
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Movie: protein folding with CHARMM
de novo Folding of a Transmembrane fd Coat Proteinhttp://www.charmm-gui.org/?doc=gallery&id=23
Quality of predicted structures quite good
Confirmed by comparison of the MSD deviations of a room temperature ensemble of conformations from the replica-exchange simulations and experimental structures from both solid-state NMR in lipid bilayers and solution-phase NMR on the protein in micelles)
Polypeptide chain
Images removed due to copyright restrictions.
Screenshots from protein folding video, which can be found here: http://www.charmm-gui.org/?doc=gallery&id=23.
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Movies in equilibrium (temperature 300 K)Dimer
Tetramer (increased effective bending stiffness, interaction via overlap & head/tail domain)
Source: Qin, Z., L. Kreplak, and M. Buehler. “Hierarchical Structure Controls Nanomechanical Properties of Vimentin Intermediate Filaments.” PLoS ONE (2009). License CC BY.
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2. Single molecule mechanics
Structure and mechanics of protein, DNA, etc. molecules
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Cooking spaghetti
stiff rods cooking soft, flexible rods(like many protein molecules)
Public domain image.Photo courtesy of HatM on Flickr.
Photo courtesy of HatM on Flickr.
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Single molecule tensile test – “optical tweezer”
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bead trapped in laser light(moves withlaser)
one end of molecule fixedat surface
molecule
Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Tskhovrebova, L., J. Trinick, et al. "Elasticity and Unfolding of Single Molecules of the Giant Muscle Protein Titin." Nature 387, no. 6630 (1997): 308- 12. © 1997.
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Example 1: Elasticity of tropocollagen molecules
Entropic elasticityleads to stronglynonlinear elasticity
300 nm length
Photo courtesy of HatM on Flickr.
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
0-2
0
50
The force-extension curve for stretching a single type II collagen molecule.The data were fitted to Marko-Siggia entropic elasticity model. The molecullength and persistence length of this sample is 300 and 7.6 nm, respectively.
2
Forc
e (p
N)
4
6
8
10
12
14
100 150
Extension (nm)
200 250 300 350
Experimental dataTheoretical model
e
Image by MIT OpenCourseWare.
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33http://www.nature.com/nature/journal/v387/n6630/pdf/387308a0.pdfhttp://www.nature.com/nature/journal/v402/n6757/pdf/402100a0.pdf
Example 2: Single protein molecule mechanics
Optical tweezers experimentProtein structure (I27 multidomain titin in muscle)
Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Tskhovrebova, L., J. Trinick, et al. "Elasticity and Unfolding of Single Molecules of the Giant Muscle Protein Titin." Nature 387, no. 6630 (1997): 308- 12. © 1997.
Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Marszalek, P., H. Lu, et al. "Mechanical Unfolding Intermediates in Titin Modules." Nature 402, no. 6757 (1999): 100-3. © 1999.
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34
Plots of stretching force against relative extension of the single DNA molecule (experimental results)
Example 3: Single DNA molecule mechanics
plateau regime (breaking of bonds)
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
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Structural makeup of protein materials
Although very diverse, all protein materials have universal “protocols” of
how they are made
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How protein materials are made–the genetic code
Proteins: Encoded by DNA (three “letters”), utilize 20 basic building blocks (amino acids) to form polypeptides
Polypeptides arrange in complex folded 3D structures with specific properties1D structure transforms into complex 3D folded configuration
ACGT
Four letter code “DNA”
Combinationof 3 DNA letters (=codon)defines one amino acid
E.g.: Proline –
CCT, CCC, CCA, or CCG
Sequence of amino acids“polypeptide” (1D structure)Transcription/
translation
Folding (3D structure)
.. - Proline - Serine –Proline - Alanine - ..
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Concept: hydrogen bonding (H-bonding)e.g. between O and H in H2OBetween N and O in proteinsDrives formation of helical structures
AHs found in: hair, cells, wool, skin, etc.
Alpha-helix (abbreviated as AH)
Adapted from Ball, D., Hill, J., et al. The Basics of General, Organic, and Biological Chemistry. Flatworld Knowledge, 2011. Courtesy of Flatworld Knowledge.
Source: Qin, Z., L. Kreplak, and M. Buehler. “Hierarchical structure controls nanomechanical properties of vimentin intermediate filaments.” PLoS ONE (2009). License CC BY.
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Primary, secondary, tertiary structure
38
Adapted from Ball, D., Hill, J., and R. Scott. The Basics of General, Organic, and Biological Chemistry. Flatworld Knowledge, 2011. Courtesy of Flatworld Knowledge.
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39
Beta-sheet
Found in many mechanically relevant proteins
Spider silk
Fibronectin
Titin (muscle tissue)
Amyloids (Alzheimer’s disease)
Beta-sheets (abbreviated as BS)
Images removed due to copyright restrictions.
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Amyloid proteins (Alzheimer’s disease)
40
Please see Fig. 8 from http://web.mit.edu/mbuehler/www/papers/final_JCTN_preprint.pdf.
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3. Fracture of protein domains –Bell model
41
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How to apply load to a molecule
(in molecular dynamics simulations)
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Virtual atommoves w/ velocity
Steered molecular dynamics used to apply forces to protein structures
Steered molecular dynamics (SMD)
kvx
end point of molecule
v
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44
)( xtvkf −⋅=
Virtual atommoves w/ velocity
Steered molecular dynamics used to apply forces to protein structures
)( xtvkf −⋅=
SMD deformation speed vector
time
Distance between end point of molecule and virtual atom
Steered molecular dynamics (SMD)
kvx
fv
end point of molecule
xtv −⋅SMD spring constant
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45
f
x
SMD mimics AFM single molecule experiments
xk
vAtomic force microscope
k xv
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46
SMD is a useful approach to probe the nanomechanics of proteins (elastic deformation,
“plastic” – permanent deformation, etc.)
Example: titin unfolding (CHARMM force field)
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47
Displacement (A)
Forc
e (p
N)
Unfolding of titin molecule
Titin I27 domain: Very resistant to unfolding due to parallel H-bonded strands
X: breaking
XX
Keten and Buehler, 2007
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48
Protein unfolding - ReaxFF
ReaxFF modeling
PnIB 1AKG
Buehler, M. "Hierarchical Chemo-nanomechanics of Proteins: Entropic Elasticity, Protein Unfolding and Molecular Fracture." Journal of Mechanics and Materials and Structures 2, no. 6 (2007).
F
F
AHs
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49
Protein unfolding - CHARMM
CHARMM modeling
Covalent bonds don’t break
Buehler, M. "Hierarchical Chemo-nanomechanics of Proteins: Entropic Elasticity, Protein Unfolding and Molecular Fracture." Journal of Mechanics and Materials and Structures 2, no. 6 (2007).
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50
Comparison – CHARMM vs. ReaxFF
Buehler, M. "Hierarchical Chemo-nanomechanics of Proteins: Entropic Elasticity, Protein Unfolding and Molecular Fracture." Journal of Mechanics and Materials and Structures 2, no. 6 (2007).
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Application to alpha-helical proteins
51
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Vimentin intermediate filaments
Source: Qin, Z., L. Kreplak, et al. "Hierarchical Structure Controls Nanomechanical Properties of Vimentin Intermediate Filaments." PLoSONE 4, no. 10 (2009). doi:10.1371/journal.pone.0007294. License CC BY.
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Vimentinintermediatefilament
Cells
Filaments
Protein molecule
Chemical bonding
Source: Q
in, Z
., L. Krep
lak, et al. "Hierarch
ical Stru
cture C
ontro
ls N
anom
echan
ical Properties o
f Vim
entin
Interm
ediate Filam
ents."
PLoS O
NE (2
009). Licen
se CC B
Y.
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Intermediate filaments – occurrence neuron cells(brain)
hair, hoof
fibroblast cells(make collagen)
cell nucleus
Image of neuron and cell nucleus © sources unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.
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55
Alpha-helical protein: stretching
M. Buehler, JoMMS, 2007
ReaxFF modeling of AHstretching
A: First H-bonds break (turns open)B: Stretch covalent backboneC: Backbone breaks
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56
What about varying pulling speeds?
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57
Variation of pulling speed
Image by MIT OpenCourseWare. After Ackbarow and Buehler, 2007.
00
4,000
8,000 00 0.2 0.4
500
1,000
1,500
Forc
e (p
N)
12,000
50 100
Strain (%)
150 200
v = 65 m/sv = 45 m/sv = 25 m/sv = 7.5 m/sv = 1 m/smodelmodel 0.1 nm/s
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58
Force at angular point fAP=fracture force
vf ln~AP
See also Ackbarow and Buehler, J. Mat. Sci., 2007
Pulling speed (m/s)
Forc
e at
AP
(pN
)
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59
General results…
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60
Rupture force vs. pulling speed
Buehler et al., Nature Materials, 2009
APf
Reprinted by permission from Macmillan Publishers Ltd: Nature Materials. Source: Buehler, M., and Y. Yung. "Chemomechanical Behaviour of Protein Constituents." Nature Materials 8, no. 3 (2009): 175-88. © 2009.
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61
How to make sense of these results?
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A few fundamental properties of bonds
Bonds have a “bond energy” (energy barrier to break)
Arrhenius relationship gives probability for energy barrier to be overcome, given a temperature
All bonds vibrate at frequency ω
62
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
TkEpB
bexp
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63
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
TkEpB
bexp
Probability for bond rupture (Arrhenius relation)
Bell model
temperatureBoltzmann constant
heightof energy
barrier
distance to energybarrier
“bond”
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64
⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅−−=
TkxfEp
B
Bbexp
Probability for bond rupture (Arrhenius relation)
Bell model
temperatureBoltzmann constant
heightof energy
barrier
distance to energybarrier
force applied(lower energybarrier)
“bond”
APff =
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65
⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅−−=
TkxfEp
B
Bbexp
Probability for bond rupture (Arrhenius relation)
Bell model
Off-rate = probability times vibrational frequency
sec/1101 130 ×=ω
τωωχ 1)(exp00 =⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅
⋅−−⋅=⋅=
TkxfEp
b
bb
bond vibrations
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66
⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅−−=
TkxfEp
B
Bbexp
Probability for bond rupture (Arrhenius relation)
Bell model
Off-rate = probability times vibrational frequency
sec/1101 130 ×=ω
τωωχ 1)(exp00 =⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅
⋅−−⋅=⋅=
TkxfEp
b
bb
“How often bond breaks per unit time”bond vibrations
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67
⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅−−=
TkxfEp
B
Bbexp
Probability for bond rupture (Arrhenius relation)
Bell model
Off-rate = probability times vibrational frequency
sec/1101 130 ×=ω
τωωχ 1)(exp00 =⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅
⋅−−⋅=⋅=
TkxfEp
b
bb
=τ bond lifetime(inverse of off-rate)
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68
Bell model
tΔ↓
pulling speed (at end of molecule)vtx =ΔΔ /
???
tΔ
vtx =ΔΔ /
xΔxΔ→
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69
Bell model
broken turntΔ↓
tΔ
xΔ
vtx =ΔΔ /
pulling speed (at end of molecule)vtx =ΔΔ /
xΔ→
xΔ→
xΔ→
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Structure-energy landscape link
70
bx
bxx =Δ
τ=Δt1
0)(exp
−
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅−−⋅=
TkxfE
b
bbωτ
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71
Bell model
vtxxTk
xfEx bb
bbb =ΔΔ=⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅
⋅−−⋅=⋅ /)(exp0ωχ
Bond breaking at (lateral applied displacement):bx
pulling speed
bxx =Δ
tΔ↓
tΔ
xΔ
vtx =ΔΔ /
τ/1=
broken turn
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72
Bell model
vxTk
xfEb
b
bb =⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅−−⋅
)(exp0ω
Solve this expression for f :
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Bell model
vxTk
xfEb
b
bb =⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅−−⋅
)(exp0ω
Solve this expression for f :
( )( )
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
−⋅⋅⋅
−⋅
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
−⋅⋅
−⋅
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−
⋅⋅
+⋅
=⋅−⋅+
=
⋅−⋅=⋅+−
=⋅+⋅
⋅−−
TkEx
xTkv
xTkf
TkEx
xTkv
xTkf
xTk
Ex
Tkvx
Tkx
xvTkEf
xvTkxfE
vxTk
xfE
b
bb
b
b
b
b
b
bb
b
b
b
b
bb
b
b
b
b
b
b
bbb
bbbb
bb
bb
explnln
)ln(ln
)ln(ln)ln(ln
)ln(ln
ln)ln()(
0
0
00
0
0
ω
ω
ωω
ω
ω ln(..)
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Simplification and grouping of variables
74
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
−⋅⋅⋅⋅
−⋅⋅
=Tk
Exx
Tkvx
TkExvfb
bb
b
b
b
bbb explnln),;( 0ω
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
−⋅⋅==Tk
Exvb
bb exp: 00 ω
Only system parameters,[distance/length]
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Bell model
vxTk
xfEb
b
bb =⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅−−⋅
)(exp0ω
Results in:
bvavx
Tkvx
TkExvfb
b
b
bbb +⋅=⋅
⋅−⋅
⋅= lnlnln),;( 0
0ln vx
Tkb
xTka
b
B
b
B
⋅⋅
−=
⋅=
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behavior of strengthvf ln~
bvaExvf bb +⋅= ln),;(
Pulling speed (m/s)
Eb= 5.6 kcal/mol and xb= 0.17 Ǻ (results obtained from fitting to the simulation data)
Forc
e at
AP
(pN
)
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Pulling speed (m/s)
bvaExvf bb
Forc
e at
AP
(pN
)
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Scaling with Eb : shifts curve
+⋅= ln),;(
↑bE
0ln vx
Tkbx
Tkab
B
b
B ⋅⋅
−=⋅
= ⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
−⋅⋅=Tk
Exvb
bb exp00 ω
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Pulling speed (m/s)
bvaExvf bb +Fo
rce
at A
P (p
N)
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⋅= ln),;(
↓bx
0ln vx
Tkbx
Tkab
B
b
B ⋅⋅
−=⋅
= ⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
−⋅⋅=Tk
Exvb
bb exp00 ω
Scaling with xb: changes slope
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Simulation results
Bertaud, Hester, Jimenez, and Buehler, J. Phys. Cond. Matt., 2010
Courtesy of IOP Publishing, Inc. Used with permission. Source: Fig. 3 from Bertaud, J., Hester, J. et al. "Energy Landscape, Structure andRate Effects on Strength Properties of Alpha-helical Proteins." J Phys.: Condens. Matter 22 (2010): 035102. doi:10.1088/0953-8984/22/3/035102.
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Mechanisms associated with protein fracture
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Change in fracture mechanism
Single AH structure
Simulation span: 250 nsReaches deformation speed O(cm/sec)
FDM: Sequential HB breaking
SDM: Concurrent HB breaking (3..5 HBs)
Courtesy of National Academy of Sciences, U. S. A. Used with permission. Source: Ackbarow, Theodor, et al. "Hierarchies, Multiple Energy Barriers, and Robustness Govern the Fracture Mechanics of Alpha-helical and Beta-sheet Protein Domains." PNAS 104 (October 16, 2007): 16410-5. Copyright 2007 National Academy of Sciences, U.S.A.
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Analysis of energy landscape parameters
Energy single H-bond: ≈3-4 kcal/mol
What does this mean???
Courtesy of National Academy of Sciences, U. S. A. Used with permission. Source: Ackbarow, Theodor, et al. "Hierarchies, Multiple Energy Barriers, and Robustness Govern the Fracture Mechanics of Alpha-helical and Beta-sheet Protein Domains." PNAS 104 (October 16, 2007): 16410-5. Copyright 2007 National Academy of Sciences, U.S.A.
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H-bond rupture dynamics: mechanism
Courtesy of National Academy of Sciences, U. S. A. Used with permission. Source: Ackbarow, Theodor, et al. "Hierarchies, Multiple Energy Barriers, and Robustness Govern the Fracture Mechanics of Alpha-helical and Beta-sheet Protein Domains." PNAS 104 (October 16, 2007): 16410-5. Copyright 2007 National Academy of Sciences, U.S.A.
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I: All HBs are intact
II: Rupture of 3 HBs – simultaneously; within τ ≈ 20 ps
III: Rest of the AH relaxes – slower deformation…
H-bond rupture dynamics: mechanism
Courtesy of National Academy of Sciences, U. S. A. Used with permission. Source: Ackbarow, Theodor, et al. "Hierarchies, Multiple Energy Barriers, and Robustness Govern the Fracture Mechanics of Alpha-helical and Beta-sheet Protein Domains." PNAS 104 (October 16, 2007): 16410-15. Copyright 2007 National Academy of Sciences, U.S.A.
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MIT OpenCourseWarehttp://ocw.mit.edu
3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and SimulatioSpring 2012
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