applications to biophysics and bionanomechanics to biophysics and bionanomechanics . ... 20 natural...

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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

2

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

6

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)

7

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.

http://www.humanbody3d.com/

http://www.humanbody3d.com/

http://publications.nigms.nih.gov/insidethecell/images/ch1_cellscolor.jpg

8

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.

http://www.nanowerk.com/spotlight/id2878_1.jpg

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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.

http://ocw.mit.edu/fairuse

12

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

13

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:

14

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


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


Bond stretching

Angle Bending

Bond Rotation

θ

ϑ

16http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html

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)

http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html

http://www.embnet.org/

17



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


vdW Interactions

Electrostatic interactions

δ−

δ−

δ+δ+

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vdWU

:vdWU LJ potential⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛=

612

4)(ijij

ij rrr σσεφ

=0 for proteins

LJ potential is particularly good model for vdW interactions (Argon)


vdW Interactions

19

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

20

Summary


=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

21

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

22

Atom typing in CHARMM

23

VMD analysis of protein structure

24

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


Harmonic terms;Derived from vibrational spectroscopy, gas-phase molecular structuresVery system-specific

Include anharmonic termsDerived from QM, more general

pset #3


25

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

26

Application – protein folding

ACGT


Combinationof 3 DNA letters equals a amino acid

E.g.: Proline –

CCT, CCC, CCA, CCG

Sequence of amino acids“polypeptide” (1D structure)

Transcription/translation



Goal of protein folding simulations:Predict folded 3D structure based on polypeptide sequence

27

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.

http://www.charmm-gui.org/?doc=gallery&id=23

http://www.charmm-gui.org/?doc=gallery&id=23

28

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.

29

2. Single molecule mechanics

Structure and mechanics of protein, DNA, etc. molecules

30

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.

http://www.flickr.com/photos/hatm/4076523848/sizes/m/in/photostream/


Single molecule tensile test – “optical tweezer”

31

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.

32

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



http://www.sciencedirect.com/

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.

http://www.nature.com/nature/journal/v387/n6630/pdf/387308a0.pdf

http://www.nature.com/nature/journal/v402/n6757/pdf/402100a0.pdf

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.

http://www.sciencedirect.com/

35

Structural makeup of protein materials

Although very diverse, all protein materials have universal “protocols” of

how they are made

36

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


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



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.

http://www.flatworldknowledge.com/pub/basics-general-organic-and-bio/429852


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.



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.

Amyloid proteins (Alzheimer’s disease)

40

Please see Fig. 8 from http://web.mit.edu/mbuehler/www/papers/final_JCTN_preprint.pdf.

http://web.mit.edu/mbuehler/www/papers/final_JCTN_preprint.pdf

3. Fracture of protein domains –Bell model

41

42

How to apply load to a molecule

(in molecular dynamics simulations)

43

Virtual atommoves w/ velocity

Steered molecular dynamics used to apply forces to protein structures

Steered molecular dynamics (SMD)

kvx

end point of molecule

v

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

45

f

x

SMD mimics AFM single molecule experiments

xk

vAtomic force microscope

k xv

46

SMD is a useful approach to probe the nanomechanics of proteins (elastic deformation,

“plastic” – permanent deformation, etc.)

Example: titin unfolding (CHARMM force field)

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

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

http://msp.berkeley.edu/jomms/2007/2-6/jomms-v2-n6-p04-s.pdf


49

Protein unfolding - CHARMM

CHARMM modeling

Covalent bonds don’t break




50

Comparison – CHARMM vs. ReaxFF




Application to alpha-helical proteins

51

52

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.

53

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.

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.

http://ocw.mit.edu/fairuse

http://www.daviddarling.info/encyclopedia/A/astrocyte.html

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

56

What about varying pulling speeds?

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

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

)

59

General results…

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.

61

How to make sense of these results?

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

63

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

TkEpB

bexp

Probability for bond rupture (Arrhenius relation)

Bell model

temperatureBoltzmann constant

heightof energy

barrier

distance to energybarrier

“bond”

64

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−−=

TkxfEp

B

Bbexp


Bell model

temperatureBoltzmann constant

heightof energy

barrier

distance to energybarrier

force applied(lower energybarrier)

“bond”

APff =

65

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−−=

TkxfEp

B

Bbexp


Bell model

Off-rate = probability times vibrational frequency

sec/1101 130 ×=ω

τωωχ 1)(exp00 =⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

⋅−−⋅=⋅=

TkxfEp

b

bb

bond vibrations

66

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−−=

TkxfEp

B

Bbexp


Bell model


sec/1101 130 ×=ω


⎠

⎞⎜⎜⎝

⎛⋅

⋅−−⋅=⋅=

TkxfEp

b

bb

“How often bond breaks per unit time”bond vibrations

67

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−−=

TkxfEp

B

Bbexp


Bell model


sec/1101 130 ×=ω


⎠

⎞⎜⎜⎝

⎛⋅

⋅−−⋅=⋅=

TkxfEp

b

bb

=τ bond lifetime(inverse of off-rate)

68

Bell model

tΔ↓

pulling speed (at end of molecule)vtx =ΔΔ /

???

tΔ

vtx =ΔΔ /

xΔxΔ→

69

Bell model

broken turntΔ↓

tΔ

xΔ

vtx =ΔΔ /

pulling speed (at end of molecule)vtx =ΔΔ /

xΔ→

xΔ→

xΔ→

Structure-energy landscape link

70

bx

bxx =Δ

τ=Δt1

0)(exp

−

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅−−⋅=

TkxfE

b

bbωτ

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

72

Bell model

vxTk

xfEb

b

bb =⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅−−⋅

)(exp0ω

Solve this expression for f :

73

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(..)

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]

75

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

⋅⋅

−=

⋅=

76

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

)

Pulling speed (m/s)

bvaExvf bb

Forc

e at

AP

(pN

)

77

Scaling with Eb : shifts curve

+⋅= ln),;(

↑bE

0ln vx

Tkbx

Tkab

B

b

B ⋅⋅

−=⋅

= ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

−⋅⋅=Tk

Exvb

bb exp00 ω

Pulling speed (m/s)

bvaExvf bb +Fo

rce

at A

P (p

N)

78

⋅= ln),;(

↓bx

0ln vx

Tkbx

Tkab

B

b

B ⋅⋅

−=⋅

= ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

−⋅⋅=Tk

Exvb

bb exp00 ω

Scaling with xb: changes slope

79

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.

80

Mechanisms associated with protein fracture

81

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.

82

Analysis of energy landscape parameters

Energy single H-bond: ≈3-4 kcal/mol

What does this mean???


83

H-bond rupture dynamics: mechanism


84

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


85

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