1.021 , 3.021, 10.333, 22.00 I ntroduc tion to Modeling and Simulation Part I – C ontinuum and partic le me thods
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
Content overview
I. Particle and continuum me thods
1. Atoms, molecul e s, chemistry
2. Continuum modeling approac hes and solution approaches
3. Statistical mechanics
4. Molecular dynamics, Monte Carlo
5. Visualization and data analysis
6. Mechanical proper ties – applic ation: how things fail (and how to prevent it)
7. Multi-scale modeling par adigm
8. Biological systems (simulation in biophysics) – h ow proteins work and how to model them
II. Quantum mechanical methods
1. It’s A Q uantum World: T he Theory of Quantum Mechanics
2. Quantum Mechanics: Practice Makes Perfect
3. The Many-Body Problem: Fr om Many-Body to Single- Particle
4. Quantum modeling of materials
5. From Atoms to Solids
6. Basic pr operties of mater i als
7. Advanced proper ties of materials
8. What else can we do?
Lectures 2-13
Lectures 14-26
Overview: Material covered so far…
Lecture 1: B road introduction to IM/S
Lecture 2 : Introduction t o atomistic and conti nuum modeling (mult i-scale modeling paradigm, difference between continuum and atomis tic approach, case study: d iffusion)
Lecture 3 : Basic statistical mechani cs – p roperty calculation I (property calculati o n: microscopic states vs. macroscopic properties , ensembles, probability density and partition functi on)
Lecture 4 : Prope rty calculation II (Monte Carl o, advanced property calculati o n, introduction to chemical interacti ons)
Lecture 5: How to model chemic al intera ctions I (exampl e: movi e of copper deformation/disl ocations, etc.)
Lecture 6: How to model ch emic al intera ctions II (EAM, a bit of ReaxFF—chemical reacti ons)
Lecture 7: Appli cation to mo del i ng brittle materials I
Lecture 8: Appli cation to mo del i ng brittle materials II
Lecture 9: Appli cation – A pplications to materials failure
Lecture 10: Appl ications to bi ophysics an d bi onanomechanics
Lecture 11: Appl ications to bi oph ysics an d bi onanomechanics (cont’ d)
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 – m aterials 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
Chemistry, structure and properties are linked
Chemical structure
Cartoon
Presence of various chemical bonds:
• C ovalent bonds (C-C, C-O, C-H, C-N..)
• Electrostatic interactions (charged amino acid side chains )
• H -bonds (e.g. between H and O)
• v dW interactions (uncharged parts of molecules)
Model for covalent bonds
1 k
( ) 2
stretch
1 k
2
stretch
( r r ) 2
0
rot
1 k
2
rot
( 1 cos ( ))
bend
2 bend 0
Courte s y of the E M Bn et Ed u cati o n & Trai ning Commit t ee. Used with permissi on.
Images cr eated for the CHARMM tutori a l by Dr. Dmit ry Kuz n e ts o v (Swi ss Insti tute of Bi oi nformati cs ) for t h e E M Bn et Edu c ati on & Trainin g commit t ee ( http://www.embn et.org )
Summary: CHARMM potential (pset #3)
=0 for proteins
U total
U Elec
U Covalent
U Metallic
U vdW
U H bond
U : Coulomb potential
( r )
q i q j
Elec
ij r
stretch
1 k
2
stretch
1 ij
2
( r r 0 )
U U U U
1 k ( ) 2
Covalent stretch
bend
rot
bend 2 bend 0
rot
1 k
2
rot
( 1 cos ( ))
12 6
r ij
U vdW :
LJ potential
( r ij )
4
r
ij
R
12 R
10
U H bond :
( r
) D
5 H bon d
6 H bon d
cos 4 ( )
ij H
bond
r ij
r ij
DHA
8
2. Fracture of protein domains – Bell model
9
Experimental techniques
Courte sy of El se vie r, Inc., h t t p ://www. scien c e dir ec t . c o m . Used wi th permission. 10
How to apply load to a molecule
(in molecular dynamics simulations)
Steered molecular dynamics (SMD)
Steered molecular dynamics used to apply forces to protein structures
v
Virtual atom
moves w/ velocity v k
x
end point of molec u le
Steered molecular dynamics (SMD)
Steered molecular dynamics used to apply forces to protein structures
v
Virtual atom f
moves w/ velocity v k
x
x
f k ( v t x )
v t
end
SMD spring constant
point of
molec u le
f k ( v t x )
SMD
deformation speed vector
time
Distance between end point of molecule and virtual atom
k
x
v
k
x
SMD mimics AFM single molecule experiments
Atomic force microscope
v
f
x
SMD is a useful approach to probe the nanomechanics of proteins (elastic deformation, “plastic” – permanent deformation, etc.)
Example: titin unfolding (CHARMM force field)
Unfolding of titin molecule
X : breaking
Force (pN)
X
X
Titin I27 domain: Very resistant to unfolding due to parallel H-bonded strands
Displacement (A)
16
Keten and Buehler, 2007
Protein unfolding - R eaxFF
F
AHs
M. Buehler, JoMMS, 2007
PnIB 1AKG
F
ReaxFF modeling
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
Vimentin intermediate filaments
Image cour tes y o f Bluebie Pixie on F lic k r . L i cense: C C -BY.
Sour c e : Qi n, Z . , L . Krepl a k, and M. Bu ehl e r. "H i e rar c hi cal Struct ur e Co n t r o l s N a no mech an ic a l P r op er t i es o f Vime n t in I n t e r m e d ia t e F i l a me n t s . " PLoS ON E 4 , no. 1 0 ( 2009) . d o i:10 .1 371 /journal.pone . 0007 294. L i c ense CC BY.
Image cour tes y o f G r een m ons t er on F lic k r .
I m a g e o f n e ur on a n d c e ll nuc leus © s ou rces unkn ow n . All r ig h t s res e rv ed. Thi s c o ntent i s ex cl u d ed from our Creative Common s lic ense . F o r mo re in fo r m a t io n , s ee htt p :/ / o cw.mi t .edu /fai ru se .
Alpha-helical protein: stretching
ReaxFF modeling of AH stretching
M. Buehler, JoMMS, 2007
A: Firs t 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
23
Case study: From nanoscale filaments to micrometer meshworks
Movie: MD simulation of AH coiled coil
Imag e removed du e t o copyrigh t r e s t ri ctio ns . Pl e a se see ht tp://d x.d o i.org/10.1 103/PhysRevLet t.104.1 98304 .
See a l so : Z . Q i n , AC S Nano, 2011, and Z . Q i n Bio N anoSc i ence , 2 010.
What about varying pulling speeds?
Changing the time-scale of observation of fracture
12,000
1,500
1,000
500
8,000
0 0
0.2
0.4
4,000
0 0
50
100
150
200
Strain (%)
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
Force (pN)
Variation of pulling speed
Ima g e by MIT OCW . Aft e r Ackbarow and Bueh ler, 2007.
Force at AP (pN)
Force at angular point f AP =fracture force
f AP ~ ln v
Pulling speed (m /s)
General results…
Rupture force vs. pulling speed
f AP
R ep r i n t e d by p er m i s s i on fr o m M a c m illa n P u b l i s hers L t d : N a t u re M a t e r i a l s .
Sour c e : Buehl e r, M. ,and Yun g , Y . " C hemom e c hani c al Behavi ou r of Protei n C o n s ti tuent s ." Nature Mater ial s 8, no. 3 (2 0 0 9 ) : 1 7 5 - 8 8 . © 2 0 09.
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
E b
p exp
k B T
All bonds vibrate at frequency
Bell model
Probability for bond rupture (Arrhenius relation)
E b
p exp
k B T
Boltzmann constant
temperature
distance
height
Bell model
Probability for bond rupture (Arrhenius relation)
f f AP
p exp
E b f x B
k B T
force applied ( lower energy barrier )
Boltzmann constant
temperature
distance
height
Bell model
Probability for bond rupture (Arrhenius relation)
p exp
E b f x B
k B T
0
Off-rate = probability times vibrational frequency
( E b f x b
exp ) 1
0 k T
b
0 p
1 1 0 13 1 / sec
Bell model
Probability for bond rupture (Arrhenius relation)
p exp
E b f x B
k B T
Off-rate = probability times vibrational frequency
1
( E b f x b ) 13
0 p
0
exp
k b T
0 1 10
1 / sec
“How often bond breaks per unit time”
Bell model
Probability for bond rupture (Arrhenius relation)
p exp
E b f x B
k B T
Off-rate = probability times vibrational frequency
( E b f x b ) 1 13
0 p
0
exp
k b T
0 1 10
1 / sec
bond lifetime (inverse of off-rate)
Bell model
t ???
x
x
x / t v
t
x / t v
pulling speed (at end of molecule)
Bell model
t
x x
broken turn
x / t v
x
x t
x / t v
pulling speed (at end of molecule)
Structure-energy landscape link
x b
x x b 1
t
( E b
f x b )
0
exp
k b T
Bell model
x / t v
x
broken turn
t
x x b t
Bond breaking at
x b (lateral applied displacement):
( E b f x b )
x b
0
exp
k b T
x b
x / t v
1 /
pulling speed
Bell model
( E b f x b )
0 exp
k b T
x b v
Solve this expression for f :
Bell model
( E b f x b )
0 exp
k b T
x b v
Solve this expression for f :
( E b
f x b )
ln(
x )
ln v
ln(..)
0 b
k b T
E b f x b k b T ln v ln( 0
x b )
E b k b
T ln v
ln( 0
x b )
k b T
k b T E b
f
x b
ln v
x b
x b k b T
ln( 0
x b )
k b T
k b T
E b
f ln v
x b
ln( 0
x b
x b )
k b T
k b T
k b T
E b
f ln v
x b
ln 0
x b
x b
exp
k b T 43
Simplification and grouping of variables
Only system parameters, [distance/length]
k b T
k b T
E b
f ( v ; x b , E b )
ln v
x b
ln 0
x b
x b
exp
k b T
: v 0
0
x b
exp
E b
k b T
Bell model
( E b f x b )
0 exp
k b T
x b v
Results in:
f ( v ; x , E
) k b T
ln v
k b T
ln v
a ln v b
x
x
b b 0
b b
a k B T
x b
x
b k B T
ln v
0
b
f ~ ln v behavior of strength
f ( v ; x b , E b ) a ln v b
Force at AP (pN)
Pulling speed (m /s)
E b = 5.6 kcal/mol and x b = 0.17 Ǻ (results obtained from fitting to the simulation data)
f ( v ; x b , E b ) a ln v b
E b
Force at AP (pN)
Scaling with E b : shifts curve
k B T
k B T
Pulling speed (m /s)
E b
a b
x x
ln v 0
v 0 0
x b
exp
k
T
b b 47 b
f ( v ; x b , E b ) a ln v b
x b
Force at AP (pN)
Scaling with x b : changes slope
k B T
k B T
Pulling speed (m /s)
E b
a b
x b x b
ln v 0
v 0 0
x b
exp
4 8
k b T
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.
Mechanisms associated with protein fracture
Change in fracture mechanism
Single AH structure
FDM : Sequential HB break ing
SDM : Concurrent HB break ing
(3..5 HBs)
Simulation span: 250 ns
Reaches deformation speed O(cm/sec)
Courtes y of Nati onal Academ y of Sci e n c e s , U. S. A. Use d w i th permi s si o n . Sour c e : Ackbarow, Theo dor, et al. " H i e rar c hi es, Mul t i p l e Ener gy Barri er s, and Ro bu stn e s s Go v e r n t h e Fractu r e Mec h ani c s of Al pha- hel i cal and Beta- s h eet Protei n D o mai n s. " PN A S 104 ( O c t obe r 1 6 , 20 0 7 ) : 1 6 4 10 - 5 . Copy ri g h t
200 7 National Acad e m y of Scie nces, U. S . A. 51
Analysis of energy landscape parameters
Energy single H-bond: ≈ 3-4 kcal/mol
What does this m e an???
52
Courtes y of Nati onal Academ y of Sci e n c e s , U. S. A. Use d w i th permi s si o n . Sour c e : Ackbarow, Theo dor, et al. " H i e rar c hi es, Mul t i p l e Ener gy Barri er s, and Ro bu stn e s s Go v e r n t h e Fractu r e Mec h ani c s of Al pha- hel i cal and Beta- s h eet Protei n D o mai n s. " PN A S 104 (O ctober 16, 2007): 16410-5. Copy right 200 7 National Acad e m y of Scie nces, U. S . A.
H- bond rupture dynamics: mechanism
Courtes y of Nati onal Academ y of Sci e n c e s , U. S. A. Use d w i th permi s si o n . Sour c e : Ackbarow, Theo dor, et al. " H i e rar c hi es, Mul t i p l e Ener gy Barri er s, and Ro bu stn e s s Go v e r n t h e Fractu r e Mec h ani c s of Al pha- hel i cal and Beta- s h eet Protei n D o mai n s. " PN A S 104 (O ctober 16, 2007): 16410-5. Copy right 200 7 National Acad e m y of Scie nces, U. S . A.
H- bond rupture dynamics: mechanism
I: All HBs are intact
Courtes y of Nati onal Academ y of Sci e n c e s , U. S. A. Use d w i th permi s si o n . Sour c e : Ackbarow, Theo dor, et al. " H i e rar c hi es, Mul t i p l e Ener gy Barri er s, and Ro bu stn e s s Go v e r n t h e Fractu r e Mec h ani c s of Al pha- hel i cal and Beta- s h eet Protei n D o mai n s. " PN A S 104 (O ctober 16, 2007): 16410-5. Copy right 200 7 National Acad e m y of Scie nces, U. S . A.
II: Rupture of 3 HBs – s imultaneous ly ; within ≈ 20 ps
III: Rest of the AH relaxes – s lower deformation…
3. Examples – materials and applications
E.g. disease diagnosis, mechanisms, etc.
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
.. - P roline - S erine – Proline - Alanine - . .
Sequence of amino acids “polypeptide”
(1D structure)
CHANGED
Folding (3D structure) STRUCTURAL
DEFECT
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 – H GPS ( Nature , 2003; Nature , 2006, PNAS , 2006)
Cell nucleus loses stability under mechanical (e.g. cyclic) loading, failure occurs at heart (fatigue)
Genetic defect:
Imag e of pat i ent removed du e t o c o pyrigh t r e stri ctio ns .
substitution of a single DNA base: Amino acid guanine is switched to adenine
Structural change in protein molecules can lead to fatal diseases
Single point mutations in IF structur e causes severe diseases such as rapid aging disease progeria – H GPS ( 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 rem ov e d due to co pyri ght restri cti o n s.
Fractures
Courtes y of Nati onal Academ y of Sci e n c e s , U. S. A. Use d w i th permi s si o n . Sour c e : Dahl, et al. "Di stin ct Struct ura l and Mec h an i cal Properti es of the Nu cl ear L a min a in Hu t c h i nson –G ilf o r d P r og er ia Syn d ro me ." PNAS 10 3 ( 2 0 0 6) : 1 0 27 1 - 6.
C o py ri g h t 20 0 6 Nati onal Aca d em y of S c i e n c e s , U.S.A. 58
Mechanisms of progeria
Images co u rtesy of Nat i on al Academy o f Sc ien c es, U. S. A. Used with permissi on.
Sour c e : Dahl, et al. "Di stin ct Struct ura l and Mec h an i cal Properti es of the Nu cl ear Lami na in Hu t c h i nson–G ilfo r d P r og er ia Syn d r o me." PNA S 1 03 ( 2 0 06): 1 0271 -6 . Cop y ri ght 2 0 0 6 N a tional Acad e m y of Scie nces, U. S . A.
Deformation of red blood cells
Courte sy of El se vie r, Inc., h t t p ://www. scien c e dir ec t . c o m . Used wi th permission.
Stages of malaria and effect on cell stiffness
Disease stages
H-R BC (healthy)
Courte sy of El se vie r, Inc., h t t p ://www. scien c e dir ec t . c o m . Used wi th permission.
Pf-U-RBC (exposed but not infected) Pf-R-pRBC (ring stage)
Pf-T-pRBC
( trophozoite s tage)
Pf-S-pRBC
(schizont stage)
Consequence: Due to r ig idi ty, RBCs can not move easi ly th rough
capillar i es in the lung 61
Cell deformation
Courte sy of El se vie r, Inc., h t t p ://www. scien c e dir ec t . c o m . Used wi th permission.
Deformation of red blood cells
Courte sy of El se vie r, Inc., h t t p ://www. scien c e dir ec t . c o m . Used wi th permission.
Mechanical signature of cancer cells (AFM)
Healthy cells
=stiff
Cancer cells
=soft
R e print e d by permis sio n from Macmillan Publish e rs L td: Nature Nanot e chn o logy.
Source: Cross, S., Y . Ji n, et al . "Nanomechan i cal Anal ysis of Cells from Cancer P a t i ent s." Nature Nanot e chn o l o gy 2, no. 12 (2007): 780-3. © 2007.