1.021, 3.021, 10.333, 22.00 Introduction to Modeling and Simulation
Part I – Continuum and particle methods
Applications to biophysics and
bionanomechanics (cont’d)
Lecture 11
Markus J. Buehler
Laboratory for Atomistic and Molecular Mechanics
Department of Civil and Environmental Engineering
Massachusetts Institute of Technology
1
Content overview
I. Particle and continuum methods
1.
2.
3.
4.
5.
6.
7.
8.
Atoms, molecules, chemistry
Continuum modeling approaches and solution approaches
Statistical mechanics
Molecular dynamics, Monte Carlo
Visualization and data analysis
Mechanical properties – application: how things fail (and
how to prevent it)
Multi-scale modeling paradigm
Biological systems (simulation in biophysics) – how
proteins work and how to model them
II. Quantum mechanical methods
1.
2.
3.
4.
5.
6.
7.
8.
Lectures 2-13
Lectures 14-26
It’s A Quantum World: The Theory of Quantum Mechanics
Quantum Mechanics: Practice Makes Perfect
The Many-Body Problem: From Many-Body to SingleParticle
Quantum modeling of materials
From Atoms to Solids
Basic properties of materials
Advanced properties of materials
What else can we do?
2
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)
3
Lecture 11: Applications to biophysics and
bionanomechanics (cont’d)
Outline:
1. Force fields for proteins: (brief) review
2. 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
4
1. Force fields for proteins: (brief) review
5
Chemistry, structure and properties are linked
Chemical structure
Cartoon
Presence of various chemical bonds:
• 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)
6
Model for covalent bonds
φstretch
1
= kstretch ( r − r0 ) 2
2
1
2
φbend = k bend (θ − θ 0 ) 2
1
φrot = k rot (1 − cos(ϑ ))
2
Courtesy 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
7
Summary: CHARMM potential (pset #3)
=0 for proteins
U total = U Elec + U Covalent + U Metallic + U vdW + U H − bond
U Elec :
Coulomb potential
φ ( rij ) =
qi q j
ε1rij
1
2
1
2
φ
=
k
(
θ
−
θ
)
UCovalent= Ustretch+Ubend +Urot
bend
bend
0
2
1
φrot = k rot (1 − cos(ϑ ))
2
⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤
U vdW : LJ potential φ ( rij ) = 4ε ⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥
rij ⎠ ⎥
⎢⎝ rij ⎠
⎝
⎣
⎦
12
10
⎡ ⎛R
⎤
⎞
⎛
⎞
R
U H − bond : φ ( rij ) = DH −bond ⎢5⎜⎜ H −bond ⎟⎟ − 6⎜⎜ H −bond ⎟⎟ ⎥ cos4 (θ DHA )
rij ⎠ ⎥
⎢ ⎝ rij ⎠
⎝
⎣
⎦
φstretch = kstretch ( r − r0 ) 2
8
2. Fracture of protein domains –
Bell model
9
Experimental techniques
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
10
How to apply load to a molecule
(in molecular dynamics
simulations)
11
Steered molecular dynamics (SMD)
Steered molecular
dynamics used to apply
forces to protein
structures
G
v
Virtual atom
moves w/ velocity
G
v
k
G
x
end point of
molecule
12
Steered molecular dynamics (SMD)
G
v
Steered molecular
dynamics used to apply
forces to protein
structures
Virtual atom
moves w/ velocity
f = k (v ⋅ t − x )
G
v
G
G
v ⋅t − x
SMD spring constant
G
G
G
f = k (v ⋅ t − x )
SMD
deformation
speed vector
time
f
k
G
x
end
point of
molecule
Distance between end
point of molecule and
virtual atom
13
SMD mimics AFM single molecule experiments
G
v
Atomic force microscope
k
G
v
x
k
x
f
x
14
SMD is a useful approach to probe the
nanomechanics of proteins (elastic deformation,
“plastic” – permanent deformation, etc.)
Example: titin unfolding (CHARMM force field)
15
Unfolding of titin molecule
Force (pN)
X: breaking
X
X
Titin I27 domain: Very
resistant to unfolding due to
parallel H-bonded strands
Displacement (A)
Keten and Buehler, 2007
16
Protein unfolding - ReaxFF
F
AHs
PnIB 1AKG
F
ReaxFF modeling
M. Buehler, JoMMS, 2007
17
Protein unfolding - CHARMM
Covalent bonds don’t break
CHARMM modeling
M. Buehler, JoMMS, 2007
18
Comparison – CHARMM vs. ReaxFF
M. Buehler, JoMMS, 2007
19
Application to alpha-helical proteins
20
Source: Qin, Z., L. Kreplak, and M. Buehler. "Hierarchical Structure
Controls Nanomechanical Properties of Vimentin Intermediate Filaments."
PLoSONE 4, no. 10 (2009). doi:10.1371/journal.pone.0007294. License
CC BY.
Vimentin intermediate filaments
Image courtesy of
Bluebie Pixie on Flickr.
License: CC-BY.
Image courtesy of Greenmonster on Flickr.
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.
Alpha-helical protein: stretching
ReaxFF modeling of AH
stretching
M. Buehler, JoMMS, 2007
A: First H-bonds break (turns open)
B: Stretch covalent backbone
C: Backbone breaks
22
Coarse-graining approach
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
23
Case study: From nanoscale filaments to
micrometer meshworks
24
Movie: MD simulation of AH coiled coil
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.
25
What about varying pulling speeds?
Changing the time-scale of
observation of fracture
26
Variation of pulling speed
1,500
1,000
Force (pN)
12,000
500
0
8,000
0
0.2
v = 65 m/s
v = 45 m/s
v = 25 m/s
v = 7.5 m/s
v = 1 m/s
model
model 0.1 nm/s
0.4
4,000
0
0
50
100
150
200
Strain (%)
Image by MIT OCW. After Ackbarow and Buehler, 2007.
27
Force at angular point fAP=fracture force
Force at AP (pN)
f AP ~ ln v
Pulling speed (m/s)
See also Ackbarow and Buehler, J. Mat. Sci., 2007
28
General results…
29
Rupture force vs. pulling speed
f AP
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.
Buehler et al., Nature Materials, 2009
30
How to make sense of these results?
31
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
⎛ Eb ⎞
⎟⎟
p = exp ⎜⎜ −
⎝ k BT ⎠
All bonds vibrate at frequency ω
32
Bell model
Probability for bond rupture (Arrhenius relation)
⎛ Eb ⎞
⎟⎟
p = exp ⎜⎜ −
⎝ k BT ⎠
Boltzmann constant
temperature
distance
to energy
barrier
height
of energy
barrier
“bond”
33
Bell model
Probability for bond rupture (Arrhenius relation)
⎛ Eb − f ⋅ x B ⎞
⎟⎟
p = exp ⎜⎜ −
k BT
⎠
⎝
Boltzmann constant
f = f AP
force applied
(lower energy
barrier)
temperature
distance
to energy
barrier
height
of energy
barrier
“bond”
34
Bell model
Probability for bond rupture (Arrhenius relation)
⎛ Eb − f ⋅ x B ⎞
⎟⎟
p = exp ⎜⎜ −
k BT
⎠
⎝
Off-rate = probability times
vibrational frequency
⎛ ( Eb − f ⋅ xb ) ⎞ 1
⎟⎟ =
χ = ω0 ⋅ p = ω0 ⋅ exp⎜⎜ −
kb ⋅ T
⎠ τ
⎝
ω0 = 1 × 1013 1 / sec
bond vibrations
35
Bell model
Probability for bond rupture (Arrhenius relation)
⎛ Eb − f ⋅ x B ⎞
⎟⎟
p = exp ⎜⎜ −
k BT
⎠
⎝
Off-rate = probability times
vibrational frequency
⎛ ( Eb − f ⋅ xb ) ⎞ 1
⎟⎟ =
χ = ω0 ⋅ p = ω0 ⋅ exp⎜⎜ −
kb ⋅ T
⎠ τ
⎝
ω0 = 1 × 1013 1 / sec
“How often bond breaks per unit time”
bond vibrations
36
Bell model
Probability for bond rupture (Arrhenius relation)
⎛ Eb − f ⋅ x B ⎞
⎟⎟
p = exp ⎜⎜ −
k BT
⎠
⎝
Off-rate = probability times
vibrational frequency
⎛ ( Eb − f ⋅ xb ) ⎞ 1
⎟⎟ =
χ = ω0 ⋅ p = ω0 ⋅ exp⎜⎜ −
kb ⋅ T
⎠ τ
⎝
ω0 = 1 × 1013 1 / sec
τ = bond lifetime
(inverse of off-rate)
37
Bell model
→ Δx
Δx
↓
Δt
???
Δ x / Δt = v
Δt
Δ x / Δt = v
pulling speed (at end of molecule)
38
Bell model
→ Δx
Δx
↓
Δt
broken turn
→ Δx
Δ x / Δt = v
→ Δx
Δ x / Δt = v
Δt
pulling speed (at end of molecule)
39
Structure-energy landscape link
xb
Δx = xb
Δt = τ
⎡
⎛ ( Eb − f ⋅ xb ) ⎞⎤
⎟⎟⎥
τ = ⎢ω0 ⋅ exp⎜⎜ −
kb ⋅ T
⎝
⎠⎦
⎣
−1
40
Bell model
Δx
↓
Δt
broken turn
Δ x / Δt = v
Δx = xb
Δt
Bond breaking at xb (lateral applied displacement):
⎛ ( Eb − f ⋅ xb ) ⎞
⎟⎟ ⋅ xb = Δx / Δt = v
χ ⋅ xb = ω0 ⋅ exp⎜⎜ −
kb ⋅ T
⎝
⎠
= 1 /τ
pulling speed
41
Bell model
⎛ ( Eb − f ⋅ xb ) ⎞
⎟⎟ ⋅ xb = v
ω0 ⋅ exp⎜⎜ −
kb ⋅ T
⎝
⎠
Solve this expression for f :
42
Bell model
⎛ ( Eb − f ⋅ xb ) ⎞
⎟⎟ ⋅ xb = v
ω0 ⋅ exp⎜⎜ −
kb ⋅ T
⎝
⎠
Solve this expression for f :
( E b − f ⋅ xb )
−
+ ln(ω0 ⋅ xb ) = ln v
kb ⋅ T
ln(..)
− Eb + f ⋅ xb = kb ⋅ T (ln v − ln(ω0 ⋅ xb ) )
Eb + kb ⋅ T (ln v − ln(ω0 ⋅ xb ) ) kb ⋅ T
kb ⋅ T
f =
=
ln v +
xb
xb
xb
kb ⋅ T
kb ⋅ T
f =
ln v −
xb
xb
⎛
Eb
⎜⎜ ln(ω0 ⋅ xb ) −
kb ⋅ T
⎝
⎛ Eb
⎞
⎜⎜
− ln(ω0 ⋅ xb ) ⎟⎟
⎝ kb ⋅ T
⎠
⎞
⎟⎟
⎠
⎛
kb ⋅ T
kb ⋅ T ⎛
Eb
⎜
f =
ln v −
ln ⎜ ω0 ⋅ xb ⋅ exp ⎜⎜ −
xb
xb
⎝ kb ⋅ T
⎝
⎞⎞
⎟⎟ ⎟
⎟
⎠⎠
43
Simplification and grouping of variables
Only system parameters,
[distance/length]
⎛
⎛ Eb ⎞ ⎞
kb ⋅ T
kb ⋅ T
⎟⎟ ⎟
⋅ ln v −
⋅ ln⎜⎜ ω0 ⋅ xb ⋅ exp⎜⎜ −
f (v; xb , Eb ) =
⎟
xb
xb
k
⋅
T
⎝ b ⎠⎠
⎝
⎛ Eb ⎞
⎟⎟
=: v0 = ω0 ⋅ xb ⋅ exp⎜⎜ −
⎝ kb ⋅ T ⎠
44
Bell model
⎛ ( Eb − f ⋅ xb ) ⎞
⎟⎟ ⋅ xb = v
ω0 ⋅ exp⎜⎜ −
kb ⋅ T
⎝
⎠
Results in:
kb ⋅ T
kb ⋅ T
f ( v; xb , Eb ) =
⋅ ln v −
⋅ ln v0 = a ⋅ ln v + b
xb
xb
kB ⋅ T
a=
xb
kB ⋅ T
b=−
⋅ ln v0
xb
45
f ~ ln v behavior of strength
Force at AP (pN)
f ( v; xb , Eb ) = a ⋅ ln v + b
Pulling speed (m/s)
Eb= 5.6 kcal/mol and xb= 0.17 Ǻ (results obtained from fitting
to the simulation data)
46
Scaling with Eb : shifts curve
Force at AP (pN)
f ( v; xb , Eb ) = a ⋅ ln v + b
Eb ↑
Pulling speed (m/s)
kB ⋅ T
a=
xb
kB ⋅ T
b=−
⋅ ln v0
xb
⎛ Eb ⎞
⎟⎟
v0 = ω0 ⋅ xb ⋅ exp⎜⎜ −
kb ⋅ T ⎠
47 ⎝
Scaling with xb: changes slope
Force at AP (pN)
f (v; xb , Eb ) = a ⋅ ln v + b
xb ↓
Pulling speed (m/s)
kB ⋅ T
a=
xb
kB ⋅ T
b=−
⋅ ln v0
xb
⎛ Eb ⎞
⎟⎟
v0 = ω0 ⋅ xb ⋅ exp⎜⎜ −
kb ⋅ T ⎠
48⎝
Simulation results
Courtesy of IOP Publishing, Inc. Used with permission. Source: Fig. 3 from Bertaud, J., Hester, J. et al. "Energy Landscape, Structure and
Rate Effects on Strength Properties of Alpha-helical Proteins." J Phys.: Condens. Matter 22 (2010): 035102. doi:10.1088/0953-8984/22/3/035102.
Bertaud, Hester, Jimenez, and Buehler, J. Phys. Cond. Matt., 2010
49
Mechanisms associated with protein
fracture
50
Change in fracture mechanism
Single AH structure
FDM: Sequential
HB breaking
SDM: Concurrent
HB breaking
(3..5 HBs)
Simulation span: 250 ns
Reaches deformation speed O(cm/sec)
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 Betasheet Protein Domains." PNAS 104 (October 16, 2007): 16410-5. Copyright
2007 National Academy of Sciences, U.S.A.
51
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 Betasheet Protein Domains." PNAS 104 (October 16, 2007): 16410-5. Copyright
2007 National Academy of Sciences, U.S.A.
52
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 Betasheet Protein Domains." PNAS 104 (October 16, 2007): 16410-5. Copyright
2007 National Academy of Sciences, U.S.A.
53
H-bond rupture dynamics: mechanism
I: All HBs are intact
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 Betasheet Protein Domains." PNAS 104 (October 16, 2007): 16410-5. Copyright
2007 National Academy of Sciences, U.S.A.
II: Rupture of 3 HBs – simultaneously; within τ ≈ 20 ps
III: Rest of the AH relaxes – slower deformation…
54
3. Examples – materials and applications
E.g. disease diagnosis,
mechanisms, etc.
55
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?
ACGT
Four letter
code “DNA”
DEFECT IN
SEQUENCE
.. - Proline - Serine –
Proline - Alanine - ..
Sequence of amino acids
“polypeptide”
(1D structure)
CHANGED
Folding
(3D structure)
STRUCTURAL
DEFECT
56
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:
Image of patient removed due to copyright
restrictions.
substitution of a single
DNA base: Amino acid
guanine is switched to
adenine
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 cyclic loading
Failure occurs at heart (fatigue)
Experiment suggests that mechanical properties of
nucleus change
Image of patient removed due to copyright restrictions.
Fractures
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.
58
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.
59
Deformation of red blood cells
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
60
Stages of malaria and effect on cell stiffness
Disease stages
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
H-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
61
capillaries in the lung
Cell deformation
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
62
Deformation of red blood cells
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
63
Mechanical signature of cancer cells (AFM)
Healthy cells
=stiff
Cancer cells
=soft
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.
64
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