1.021, 3.021, 10.333, 22.00 Introduction to Modeling and Simulation
Spring 2011
Part I – Continuum and particle methods
Applications to biophysics and
bionanomechanics
Lecture 10
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 1-13
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…
ƒ
Lecture 1: Broad introduction to IM/S
ƒ
Lecture 2: Introduction to atomistic and continuum modeling (multi-scale modeling
paradigm, difference between continuum and atomistic approach, case study: diffusion)
ƒ
Lecture 3: Basic statistical mechanics – property calculation I (property
calculation: microscopic states vs. macroscopic properties, ensembles, probability
density and partition function)
ƒ
Lecture 4: Property calculation II (Monte Carlo, advanced property calculation,
introduction to chemical interactions)
ƒ
Lecture 5: How to model chemical interactions I (example: movie of copper
deformation/dislocations, etc.)
ƒ
Lecture 6: How to model chemical interactions II (EAM, a bit of ReaxFF—chemical
reactions)
ƒ
Lecture 7: Application to modeling brittle materials I
ƒ
Lecture 8: Application to modeling brittle materials II
ƒ
Lecture 9: Application – Applications to materials failure
ƒ
Lecture 10: Applications to biophysics and bionanomechanics
3
Lecture 10: Applications to biophysics and
bionanomechanics
Outline:
1. Protein force fields
2. Single molecule mechanics
3. Fracture of protein domains – Bell model
Goal of today’s lecture:
ƒ
Force fields for organic materials, and specifically
proteins
ƒ
Basic introduction into modeling of biological materials
ƒ
Fracture model for protein domains
4
1. Force fields for organic chemistry how to model proteins
5
Significance of proteins
ƒ Proteins are basic building blocks of life
ƒ Define tissues, organs, cells
ƒ Provide a variety of functions and properties, such as mechanical
stability (strength), elasticity, catalytic activity (enzyme),
electrochemical properties, optical properties, energy conversion
ƒ 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)
6
Human body: Composed of diverse array of
protein materials
Eye’s cornea
(collagen
material)
Muscle tissue
(motor
proteins)
Skin (complex
composite of
collagen,
elastin)
Nerve cells
Blood vessels
Image removed due to copyright restrictions.
Cells (complex
material/system
based on
proteins)
Human Body 3D View™ image of whole bodies.
Tendon
(links bone,
muscles)
Cartilage (reduce
friction in joints)
Bone (structural
stability)
Image courtesy of NIH.
http://www.humanbody3d.com/ and http://publications.nigms.nih.gov/insidethecell/images/ch1_cellscolor.jpg
7
Cellular structure: Protein networks
Cell nucleus
Actin network
Microtubulus
(e.g. cargo)
Vimentin
(extensible,
flexible, provide
strength)
= cytoskeleton
Image courtesy of NIH.
8
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 Modeling
of Materials Failure. Springer, 2008.
9
How protein materials are made – the genetic code
ƒ Proteins: Encoded by DNA (three “letters”), utilize 20 basic building blocks
(amino acids) to form polypeptides
ƒ Polypeptides arrange in complex folded 3D structures with specific
properties
1D structure transforms into complex 3D folded configuration
ACGT
Four letter
code “DNA”
Combination
of 3 DNA
letters equals
a amino acid
E.g.: Proline –
.. - Proline - Serine –
Proline - Alanine - ..
CCT, CCC,
CCA, CCG
Transcription/
translation
Sequence of amino acids
“polypeptide” (1D
structure)
Folding
(3D structure)
10
Chemical structure of peptides/proteins
Typically short
sequence
of amino acids
…
Longer sequence of
amino acids, often
complex 3D
structure
side chains
Peptide bond
…
© source unknown. All rights reserved. This content is
excluded from our Creative Commons license. For more
information, see http://ocw.mit.edu/fairuse.
R = side chain, one of the 20 natural amino acids
20 natural amino acids differ in their side chain chemistry
11
Nonpolar Amino Acids
CH3
H
Forms peptide bond
+
H3N
C
CH3
COO-
+
H3N
+
COO-
C
H
CH
H3N
H
+
COO-
H3N
H
Glycine (Gly) G
6.0
Alanine (Ala) A
6.0
Valine (Val) V
6.0
NE
NE
E
H3N
There are 20 natural
amino acids
C
H3N
H
Isoleucine (lle) l
6.0
E
E
N
Difference in side
chain, R
CH2 CH2
+
COO-
C
H2N
C
+
COO-
H3N
CH2
C
COO-
H
H
H
H
Phenylalanine (Phe) F
5.5
Methionine (Met) M
5.7
Proline (Pro) P
6.3
Tryptophan (Trp) W
5.9
E
E
NE
E
Polar Amino Acids (Neutral)
H3N
CH2
C
SH
CH2
HCOH
COO-
+
H3N
C
+
COO-
H3N
H3N
C
CH2
C
CH2
+
COO-
C
C
NH2
O
+
NH2
O
OH
CH3
OH
COO-
CH2
CH3
+
C
H
CH2
COO-
H3N
H
S
CH2
+
COO-
C
CH
Leucine (Leu) L
6.0
CH3
+
CH3
CH2
C
R
CH2
CH
CH3
CH3
CH3
CH3
CH2
COO-
+
H3N
CH2
COO-
C
+
H3N
C
COO-
H
H
H
H
H
H
Serine (Ser) S
5.7
Threonine (Thr) T
5.6
Tyrosine (Tyr) Y
5.7
Cysteine (Cys) C
5.1
Asparagine (Asn) N
5.4
Glutamine (Gln) Q
5.7
NE
E
NE
NE
NE
NE
Acidic Amino Acids
Basic Amino Acids
O-
O
O-
O
C
CH2
H3N
C
+
H3N
C
+
δ+
CH2
COO-
NH2
charges
CH2
C
+
δ-
+
HN
NH
COO-
H
H
Aspartic acid (Asp) D
2.8
Glutamic acid (Glu) E
3.2
NE
NE
H3N
C
C
CH2
NH
CH2
CH2
CH2
CH2
CH2
CH2
+
NH3
COO-
+
H3N
C
+
NH2
CH2
COO-
+
H3N
C
COO-
H
H
Histidine (His) H
7.6
Lysine (Lys) K
9.7
Arginine (Arg) R
10.8
E
E
E
H
Image by MIT OpenCourseWare.
12
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)
13
Concept: split energy contributions
=0 for proteins
U total = U Elec + U Covalent + U Metallic + U vdW + U H − bond
Ethane
C2H6
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”
U Covalent = U stretch + U bend + U rotate
14
Force fields for organics: Basic approach
=0 for proteins
U total = U Elec + U Covalent + U Metallic + U vdW + U H − bond
U Covalent = U stretch + U bend + U rot
1
2
φstretch = kstretch ( r − r0 ) 2
U stretch = ∑ φstretch
pairs
1
φbend = k bend (θ − θ0 ) 2
2
U bend = ∑ φbend
triplets
1
φrot = krot (1 − cos(ϑ ))
2
U rot = ∑ φrot
quadruplets
Bond stretching
θ
Angle Bending
ϑ
Bond Rotation
Image by MIT OpenCourseWare.
15
Model for covalent bonds
φstretch
1
= kstretch ( r − r0 ) 2
2
φbend
1
= k bend (θ − θ 0 ) 2
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
16
Force fields for organics: Basic approach
=0 for proteins
U total = U Elec + U Covalent + U Metallic + U vdW + U H − bond
partial charges
U Elec
qi
qj
U Elec :Coulomb potential φ ( rij ) =
qi q j
δ−
δ+
δ+
Electrostatic
interactions
electrostatic constant
ε1rij
distance
δ−
Coulomb forces
vdW Interactions
F ( rij ) = −
qi q j
ε1rij2
ε1 = 4πε 0 ε 0 = 1.602 × 10−19 C
Image by MIT OpenCourseWare.
17
Force fields for organics: Basic approach
=0 for proteins
U total = U Elec + U Covalent + U Metallic + U vdW + U H − bond
U vdW
vdW Interactions
Image by MIT OpenCourseWare.
U vdW :
⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤
LJ potential φ ( rij ) = 4ε ⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥
⎢⎝ rij ⎠
⎝ rij ⎠ ⎥⎦
⎣
LJ potential is particularly good model for vdW interactions (Argon)
18
H-bond model
=0 for proteins
U total = U Elec + U Covalent + U Metallic + U vdW + U H − bond
H 2O
D
U H −bond
θ DHA
H
Evaluated between acceptor (A) /donor(D) pairs
H-bond
H 2O
Between electronegative atom and a H- atom
that is bonded to another electronegative atom
A
Slightly modified LJ, different parameters
U H − bond :
12
10
⎡ ⎛R
⎤
⎞
⎛
⎞
R
φ ( rij ) = DH − bond ⎢5⎜⎜ H − bond ⎟⎟ − 6⎜⎜ H − bond ⎟⎟ ⎥ cos4 (θ DHA )
rij ⎠ ⎥
⎢ ⎝ rij ⎠
⎝
⎣
⎦
rij = distance between D-A
19
Summary
=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
20
The need for atom typing
ƒ Limited transferability of potential expressions: Must use different
potential for different chemistry
ƒ Different chemistry is captured in different “tags” for atoms: Element
type is expanded by additional information on particular chemical
state
ƒ Tags specify if a C-atom is in sp3, sp2, sp or in aromatic state (that is,
to capture resonance effects)
ƒ Example atom tags: CA, C_1, C_2, C_3, C…, HN, HO, HC, …
sp3
sp2
sp
21
Atom typing in CHARMM
22
VMD analysis of protein structure
23
Common empirical force fields for organics and
proteins
Class I (experiment derived, simple form)
pset #3
ƒ CHARMM
ƒ CHARMm (Accelrys)
ƒ AMBER
ƒ OPLS/AMBER/Schrödinger
ƒ ECEPP (free energy force field)
ƒ GROMOS
Harmonic terms;
Derived from
vibrational
spectroscopy, gasphase molecular
structures
Very systemspecific
Class II (more complex, derived from QM)
ƒ CFF95 (Biosym/Accelrys)
ƒ MM3
ƒ MMFF94 (CHARMM, Macromodel…)
ƒ UFF, DREIDING
Include anharmonic
terms
Derived from QM,
more general
http://www.ch.embnet.org/MD_tutorial/pages/MD.Part2.html
24
CHARMM force field
ƒ Widely used and accepted model for protein structures
ƒ 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
25
Application – protein folding
Combination
of 3 DNA
letters equals
a amino acid
ACGT
Four letter
code “DNA”
E.g.: Proline –
.. - Proline - Serine –
Proline - Alanine - ..
CCT, CCC,
CCA, CCG
Transcription/
translation
Sequence of amino acids
“polypeptide” (1D
structure)
Folding
(3D structure)
Goal of protein folding simulations:
Predict folded 3D structure based on polypeptide sequence
26
Movie: protein folding with CHARMM
ƒ de novo Folding of a Transmembrane fd Coat Protein
http://www.charmm-gui.org/?doc=gallery&id=23
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.
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)
27
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.
28
2. Single molecule mechanics
Structure and mechanics of
protein, DNA, etc. molecules
29
Cooking spaghetti
Photo courtesy of HatM on Flickr.
Public domain image.
Photo courtesy of HatM on Flickr.
stiff rods
cooking
soft, flexible rods
(like many protein
molecules)
30
Single molecule tensile test – “optical tweezer”
molecule
one end of
molecule fixed
at surface
bead trapped
in laser light
(moves with
laser)
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.
31
Example 1: Elasticity of tropocollagen molecules
300 nm length
Entropic elasticity
leads to strongly
nonlinear elasticity
14
12
Experimental data
Theoretical model
Force (pN)
10
8
6
4
2
0
Photo courtesy of HatM on Flickr.
-2
0
50
100
150
200
250
300
350
Extension (nm)
The force-extension curve for stretching a single type II collagen molecule.
The data were fitted to Marko-Siggia entropic elasticity model. The molecule
length and persistence length of this sample is 300 and 7.6 nm, respectively.
Image by MIT OpenCourseWare.
Courtesy of Elsevier, Inc., http://www.sciencedirect.com.
Used with permission.
32
Example 2: Single protein molecule mechanics
Optical tweezers experiment
Protein 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
33
Example 3: Single DNA molecule mechanics
plateau regime (breaking of bonds)
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Plots of stretching force against relative extension of
the single DNA molecule (experimental results)
34
Structural makeup of protein materials
Although very diverse, all protein
materials have universal “protocols” of
how they are made
35
How protein materials are made–the genetic code
ƒ Proteins: Encoded by DNA (three “letters”), utilize 20 basic building blocks
(amino acids) to form polypeptides
ƒ Polypeptides arrange in complex folded 3D structures with specific
properties
1D structure transforms into complex 3D folded configuration
ACGT
Four letter
code “DNA”
Combination
of 3 DNA
letters
(=codon)
defines one
amino acid
.. - Proline - Serine –
Proline - Alanine - ..
E.g.: Proline –
CCT, CCC,
CCA, or CCG
Transcription/
translation
Sequence of amino acids
“polypeptide” (1D
structure)
Folding
(3D structure)
36
Alpha-helix (abbreviated as AH)
Concept: hydrogen bonding (H-bonding)
e.g. between O and H in H2O
Between N and O in proteins
Drives formation of helical structures
AHs found in: hair, cells, wool, skin, etc.
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.
Primary, secondary, tertiary structure
Adapted from Ball, D., Hill, J., and R. Scott. The Basics of General, Organic,
and Biological Chemistry. Flatworld Knowledge, 2011. Courtesy of Flatworld
Knowledge.
38
Beta-sheets (abbreviated as BS)
Beta-sheet
Images removed due to copyright restrictions.
Found in many mechanically relevant
proteins
Spider silk
Fibronectin
Titin (muscle tissue)
Amyloids (Alzheimer’s disease)
39
Amyloid proteins (Alzheimer’s disease)
Please see Fig. 8 from http://web.mit.edu/mbuehler/www/papers/final_JCTN_preprint.pdf.
40
3. Fracture of protein domains –
Bell model
41
How to apply load to a molecule
(in molecular dynamics
simulations)
42
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
43
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
44
SMD mimics AFM single molecule experiments
G
v
Atomic force microscope
k
G
v
x
k
x
f
x
45
SMD is a useful approach to probe the
nanomechanics of proteins (elastic deformation,
“plastic” – permanent deformation, etc.)
Example: titin unfolding (CHARMM force field)
46
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
47
Protein unfolding - ReaxFF
F
AHs
PnIB 1AKG
F
ReaxFF modeling
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).
48
Protein unfolding - CHARMM
Covalent bonds don’t break
CHARMM modeling
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).
49
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).
50
Application to alpha-helical proteins
51
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.
52
Cells
Vimentin
intermediate
filament
Protein molecule
Source: Qin, Z., L. Kreplak, et al. "Hierarchical Structure Controls
Nanomechanical Properties of Vimentin Intermediate Filaments."
PLoS ONE (2009). License CC BY.
Filaments
Chemical bonding
53
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.
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
55
What about varying pulling speeds?
56
Variation of pulling speed
1,500
Force (pN)
12,000
1,000
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 OpenCourseWare. After Ackbarow and Buehler, 2007.
57
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
58
General results…
59
Rupture force vs. pulling speed
f AP
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.
Buehler et al., Nature Materials, 2009
60
How to make sense of these results?
61
A few fundamental properties of bonds
ƒ Bonds have a “bond energy” (energy barrier to break)
ƒ Arrhenius relationship gives probability for energy barrier
to be overcome, given a temperature
⎛ Eb ⎞
⎟⎟
p = exp ⎜⎜ −
⎝ k BT ⎠
ƒ All bonds vibrate at frequency ω
62
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”
63
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”
64
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
65
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
66
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)
67
Bell model
→ Δx
Δx
↓
Δt
???
Δ x / Δt = v
Δt
Δ x / Δt = v
pulling speed (at end of molecule)
68
Bell model
→ Δx
Δx
↓
Δt
broken turn
→ Δx
Δ x / Δt = v
→ Δx
Δ x / Δt = v
Δt
pulling speed (at end of molecule)
69
Structure-energy landscape link
xb
Δx = xb
Δt = τ
⎡
⎛ ( Eb − f ⋅ xb ) ⎞⎤
⎟⎟⎥
τ = ⎢ω0 ⋅ exp⎜⎜ −
kb ⋅ T
⎝
⎠⎦
⎣
−1
70
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
71
Bell model
⎛ ( Eb − f ⋅ xb ) ⎞
⎟⎟ ⋅ xb = v
ω0 ⋅ exp⎜⎜ −
kb ⋅ T
⎝
⎠
Solve this expression for f :
72
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
⎝
⎞⎞
⎟⎟ ⎟
⎟
⎠⎠
73
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 ⎠
74
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
75
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)
76
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 77⎠
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 78⎠
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
79
Mechanisms associated with protein
fracture
80
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.
81
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.
82
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.
83
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): 16410-15.
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…
84
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