Applications to biophysics and bionanomechanics (cont’d) · See also: Z. Qin, ACS Nano, 2011, and Z. Qin BioNanoScience, 2010. 26 What about varying pulling speeds? Changing the

1

1.021, 3.021, 10.333, 22.00 Introduction to Modeling and Simulation

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 (cont’d)Lecture 11

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 2-13

Lectures 14-26

3

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

Lecture 11: Applications to biophysics and bionanomechanics (cont’d)

4

Lecture 11: Applications to biophysics and bionanomechanics (cont’d)

Outline:1. Force fields for proteins: (brief) review2. Fracture of protein domains – Bell model 3. Examples – materials and applications

Goal of today’s lecture: Fracture model for protein domains: “Bell model”Method to apply loading in molecular dynamics simulation (nanomechanics of single molecules)Applications to disease and other aspects

1. Force fields for proteins: (brief) review

5

6

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:

7http://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 & Training committee (http://www.embnet.org)

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

http://www.embnet.org/

8

Summary: CHARMM potential (pset #3)

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

2. Fracture of protein domains –Bell model

9

10

Experimental techniques

Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.

http://www.sciencedirect.com/

11

How to apply load to a molecule

(in molecular dynamics simulations)

12

Virtual atommoves w/ velocity

Steered molecular dynamics used to apply forces to protein structures

Steered molecular dynamics (SMD)

kvx

end point of molecule

v

13

)( 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

14

f

x

SMD mimics AFM single molecule experiments

xk

vAtomic force microscope

k xv

15

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

“plastic” – permanent deformation, etc.)

Example: titin unfolding (CHARMM force field)

16

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

17

Protein unfolding - ReaxFF

PnIB 1AKG

ReaxFF modelingM. Buehler, JoMMS, 2007

F

F

AHs

18

Protein unfolding - CHARMM

CHARMM modeling

Covalent bonds don’t break

M. Buehler, JoMMS, 2007

19

Comparison – CHARMM vs. ReaxFF


Application to alpha-helical proteins

20

Vimentin intermediate filaments

Image courtesy of Greenmonster on Flickr.

Image courtesy of Bluebie Pixie on Flickr.

License: CC-BY.

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.

Sourc

e: Q

in,

Z.,

L.

Kre

pla

k, a

nd M

. Bueh

ler.

"H

iera

rchic

al S

truct

ure

Contr

ols

Nan

om

echanic

al P

roper

ties

of Vim

entin I

nte

rmed

iate

Fila

men

ts."

PL

oSO

NE 4

, no.

10 (

2009).

doi:

10.1

371/j

ourn

al.pone.

0007294.

Lice

nse

CC B

Y.

http://ocw.mit.edu/fairuse

22

Alpha-helical protein: stretching


ReaxFF modeling of AHstretching

A: First H-bonds break (turns open)B: Stretch covalent backboneC: Backbone breaks

Coarse-graining approach

23

Describe interaction between “beads” and not “atoms”

Same concept as force fields for atoms

See also: http://dx.doi.org/10.1371/journal.pone.0006015

http://dx.doi.org/10.1371/journal.pone.0006015

Case study: From nanoscale filaments to micrometer meshworks

24

Movie: MD simulation of AH coiled coil

25

Image removed due to copyright restrictions. Please see http://dx.doi.org/10.1103/PhysRevLett.104.198304.

See also: Z. Qin, ACS Nano, 2011, and Z. Qin BioNanoScience, 2010.

http://dx.doi.org/10.1103/PhysRevLett.104.198304

26

What about varying pulling speeds?

Changing the time-scale of observation of fracture

27

Variation of pulling speed

Image by MIT OCW. 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

28

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

)

29

General results…

30

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 Yung, Y. "Chemomechanical Behaviour of Protein Constituents." Nature Materials 8, no. 3 (2009): 175-88. © 2009.

31

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

32

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

TkEpB

bexp

33

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

TkEpB

bexp

Probability for bond rupture (Arrhenius relation)

Bell model

temperatureBoltzmann constant

heightof energy

barrier

distance to energybarrier

“bond”

34

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−−=

TkxfEp

B

Bbexp


Bell model

temperatureBoltzmann constant

heightof energy

barrier

distance to energybarrier

force applied(lower energybarrier)

“bond”

APff =

35

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−−=

TkxfEp

B

Bbexp


Bell model

Off-rate = probability times vibrational frequency

sec/1101 130 ×=ω

τωωχ 1)(exp00 =⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

⋅−−⋅=⋅=

TkxfEp

b

bb

bond vibrations

36

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−−=

TkxfEp

B

Bbexp


Bell model


sec/1101 130 ×=ω


⎠

⎞⎜⎜⎝

⎛⋅

⋅−−⋅=⋅=

TkxfEp

b

bb

“How often bond breaks per unit time”bond vibrations

37

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−−=

TkxfEp

B

Bbexp


Bell model


sec/1101 130 ×=ω


⎠

⎞⎜⎜⎝

⎛⋅

⋅−−⋅=⋅=

TkxfEp

b

bb

=τ bond lifetime(inverse of off-rate)

38

Bell model

tΔ↓

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

???

tΔ

vtx =ΔΔ /

xΔxΔ→

39

Bell model

broken turntΔ↓

tΔ

xΔ

vtx =ΔΔ /

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

xΔ→

xΔ→

xΔ→

Structure-energy landscape link

40

bx

bxx =Δ

τ=Δt1

0)(exp

−

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅−−⋅=

TkxfE

b

bbωτ

41

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

42

Bell model

vxTk

xfEb

b

bb =⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

⋅−−⋅

)(exp0ω

Solve this expression for f :

43

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

44

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

−⋅⋅⋅⋅

−⋅⋅

=Tk

Exx

Tkvx

TkExvfb

bb

b

b

b

bbb explnln),;( 0ω

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

−⋅⋅==Tk

Exvb

bb exp: 00 ω

Only system parameters,[distance/length]

45

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

⋅⋅

−=

⋅=

46

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

)

47

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)

48

⋅= ln),;(

↓bx

0ln vx

Tkbx

Tkab

B

b

B ⋅⋅

−=⋅

= ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

−⋅⋅=Tk

Exvb

bb exp00 ω

Scaling with xb: changes slope

49

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.

50

Mechanisms associated with protein fracture

51

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.

52

Analysis of energy landscape parameters

Energy single H-bond: ≈3-4 kcal/mol

What does this mean???


53

H-bond rupture dynamics: mechanism


54

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


3. Examples – materials and applications

E.g. disease diagnosis, mechanisms, etc.

55

56

Genetic diseases – defects in protein materials

Defect at DNA level causes structure modification

Question: how does such a structure modification influence material behavior / material properties?

Four letter code “DNA”

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

CHANGED

Folding (3D structure)STRUCTURAL

DEFECT

.. - Proline - Serine –Proline - Alanine - ..ACGT

DEFECT IN SEQUENCE

57

Structural change in protein molecules can lead to fatal diseases

Single point mutations in IF structure causes severe diseases such as rapid aging disease progeria – HGPS (Nature, 2003; Nature, 2006, PNAS, 2006)Cell nucleus loses stability under mechanical (e.g. cyclic) loading, failure occurs at heart (fatigue)

Genetic defect:

substitution of a single DNA base: Amino acid guanine is switched to adenine

Image of patient removed due to copyright restrictions.

58

Structural change in protein molecules can lead to fatal diseases

Single point mutations in IF structure causes severe diseases such as rapid aging disease progeria – HGPS (Nature, 2003; Nature, 2006, PNAS, 2006)Cell nucleus loses stability under cyclic loadingFailure occurs at heart (fatigue)

Experiment suggests that mechanical properties of nucleus change

Fractures

Image of patient removed due to copyright restrictions.

Courtesy of National Academy of Sciences, U. S. A. Used with permission. Source: Dahl, et al. "Distinct Structural and Mechanical Properties of the Nuclear Lamina in Hutchinson–Gilford Progeria Syndrome." PNAS 103 (2006): 10271-6. Copyright 2006 National Academy of Sciences, U.S.A.

59

Mechanisms of progeria

Images courtesy of National Academy of Sciences, U. S. A. Used with permission. Source: Dahl, et al. "Distinct Structural and Mechanical Properties of the Nuclear Lamina in Hutchinson–Gilford Progeria Syndrome." PNAS 103 (2006): 10271-6. Copyright 2006 National Academy of Sciences, U.S.A.

60

Deformation of red blood cells



61

Stages of malaria and effect on cell stiffness

Disease stagesH-RBC (healthy)Pf-U-RBC (exposed but not infected)Pf-R-pRBC (ring stage)Pf-T-pRBC(trophozoite stage)Pf-S-pRBC (schizont stage)Consequence: Due to rigidity, RBCs can not move easily through capillaries in the lung



62

Cell deformation



63

Deformation of red blood cells



64

Mechanical signature of cancer cells (AFM)

Cancer cells=soft

Healthy cells=stiff

Reprinted by permission from Macmillan Publishers Ltd: Nature Nanotechnology. Source: Cross, S., Y. Jin, et al. "Nanomechanical Analysis of Cells from Cancer Patients." Nature Nanotechnology 2, no. 12 (2007): 780-3. © 2007.

65

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Documents

Applications to biophysics and bionanomechanics (cont’d) · See also: Z. Qin, ACS Nano, 2011, and Z. Qin BioNanoScience, 2010. 26 What about varying pulling speeds? Changing the