1.021 , 3.021, 10.333, 22.00 I ntroduc tion to Modeling and Simulation

Spring 2011

Part I – C ontinuum and partic le me thods

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

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 1-13

Lectures 14-26

Overview: Material covered so far…

 Lecture 1: Broad introduction to IM/S

 Lecture 2 : Introduction to atomistic and continuum modeling (multi-scale m odeling paradigm, difference between continuum and at omistic approac h , c a se st ud y: diff us ion)

 Lecture 3 : Basic statistical mechani cs – p roperty calculation I (property calc ulat ion: microsc o pi c stat es vs. macro scopic properties, ensembles, probability density and partition function)

 Lecture 4 : Property calcula tion II (Mont e Carlo, a d v a nc e d prope rty calcul ati on, introduction to chemical interactions)

 Lectu r e 5: Ho w to m odel ch em i c al interactions I ( e xa mp le: mo vie o f coppe r def ormation/ di sloc ati o ns, etc.)

 Lectu r e 6: Ho w to m odel ch em i c al interactions I I (EAM, a bit of ReaxFF—chemical reactions)

 Lectu r e 7: Ap pli c atio n to modeling brittle materials I

 Lectu r e 8: Ap pli c atio n to modeling brittle materials II

 Lectu r e 9: Ap pli c atio n – A p pli catio ns to m a teri al s fai l ure

 Lecture 10: Applications to biophysics and bionanomec hanics 3

Lecture 10: Applications to biophysics and bionanomechanics

Outline:

1. Protein force fields

2. Single molecule mechanics

3. Fracture of protein domains – B ell model

Goal of today’s lecture:

 Force fields for organic materials, and specifically proteins

 Basic introduction into modeling of biological materials

 Fracture model for protein domains

1. Force fields for organic chemistry - how to model proteins

Significance of proteins

 Proteins are basic building blocks of life

 Define tissues, organs, cells

 Provide a variety of functions and properties , su ch as mechanical stability (strength), elasticity , catalytic activity (enzyme), electrochemical properties, optica l properties, energy conversion

 Molecular simulation is an important tool in the analysis of pr otein structures and protein materials

Goal here: To train you in the fundamental s of modeling techniques for proteins, to enable you to carry out protein simulations

Explain the significance of proteins ( application )

Human body: Composed of diverse array of protein materials

Eye’s cornea (collagen material)

Muscle tissue (motor proteins)

Skin (complex composite of collagen, elastin)

Cells (complex material/system based on proteins)

Imag e removed du e t o co pyrigh t restric t ion s. Human Bod y 3D View ™ i m a ge o f w h ole bo die s .

Nerve cells Blood vessels

Tendon (links bone, muscles)

Cartilage (reduce friction in joints)

Bone (structural stability)

Imag e court e sy of NIH.

Cellular structure: Protein networks

Cell nucleus

Actin network Microtubulus

(e.g. cargo)

Vimentin (extensible, flexible, provide strength)

= cyto skeleton Image co u r te s y of NI H.

Protein structures define the cellular architecture

Intermediate filaments

Imag e removed du e t o copyri gh t restri ct ion s ; see i m age now: http://ww w .nanower k.com/spot ligh t/id2 878_1. jpg . S o urc e : Fig. 2.17 in Bueh ler, Mar k us J. Atomi s ti c Mod e lin g of Materials Failure. Sp ring er, 2008.

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 fol ded 3D structures with specific properties

1D structure transforms into co mplex 3D folded configuration

ACGT

Four letter code “DNA”

Transcription/ translation

.. - P roline - S erine – Proline - Alanine - . .

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

Folding (3D structure)

Chemical structure of peptides/proteins

Typically short

sequence

… of amino acids

side chains

Longer sequence of amino acids, often complex 3D structure

Peptide bond …

© s ourc e u n known. All right s res e rved. This con t en t is excluded fr om our Creat i ve Co mmo ns li cense . Fo r m o re inf o rma t io n , see h ttp:/ /ocw.mi t . e du/fai ruse .

R = side chain, one of the 20 natural amino acids

20 natural amino acids differ in their side chain chemistry 11

Nonpolar Amino Acids

CH 3

CH 3

CH 3

H CH 3

CH 3 CH 3 CH

CH

CH 2

CH 3

CH 2 R

CH

Forms peptide bond

+

H 3 N

C COO -

+

H 3 N

C COO -

+

H 3 N

C COO -

+

H 3 N

C COO -

+

H 3 N

C COO -

H

Glycine (Gly) G

6.0

NE

CH 2

+ -

H

Alanine (Ala) A

6.0

NE

CH 3 S

CH 2 CH 3

+ -

H

V aline (V al) V

6.0

E

CH 2 CH 2

+ -

H

Leucine (Leu) L

6.0

E

H N

CH 2

+ -

H

Isoleucine (lle) l

6.0

E

H 3 N C COO

H

H 3 N C COO

H

H 2 N C COO

H

H 3 N C COO

H

There are 20 natural

Phenylalanine (Phe) F 5.5

E

Methionine (Met) M 5.7

E

Proline (Pro) P 6.3

NE

T ryptophan (T rp) W 5.9

E

amino acids

Polar Amino Acids (Neutral) OH

O NH 2

O NH 2

C

CH 2

CH 3

HCO H

CH 2

SH

CH 2

C

CH 2

CH 2

CH 2

Difference in side

chain, R

+

H 3 N

C COO - H

+

H 3 N

C COO - H

+

H 3 N

C COO - H

+

H 3 N

C COO - H

+

H 3 N

C COO - H

+

H 3 N

C COO - H

Serine (Ser) S 5.7

NE

Threonine (Thr) T 5.6

E

T yrosine (T yr) Y 5.7

NE

Cysteine (Cys) C 5.1

NE

Asparagine (Asn) N 5.4

NE

Glutamine (Gln) Q 5.7

NE

Acidic Amino Acids

O

O -  -

Basic Amino Acids

charges +

NH 2

+

O O -

C

CH 2

+ -

C CH 2

CH 2

+ -

NH 3

 CH 2

+ CH 2

NH CH

C NH

CH 2

CH

NH 2

H 3 N C COO

H

H 3 N C COO

H

CH 2

2

CH 2

2

CH 2

Aspartic acid (Asp) D 2.8

NE

Glutamic acid (Glu) E 3.2

NE

+

H 3 N

C COO - H

+

H 3 N

C COO -

H

+

H 3 N

C COO -

H

Histidine (His) H

7.6

E

L ysine (L ys) K

9.7

E

Ar ginine (Ar g) R

10.8

E

Image by MIT OpenCou r seWare. 12

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)

Concept: split energy contributions

=0 for proteins

U total

 U Metallic

 U vdW

 U H  bond

Covalent bond described as

Ethane C 2 H 6

1. Bond stretching part (energy penalty for bond stretching)

2. Bending part (energy penal ty for bending three atoms)

3. Rotation part (energy penalty for bond rotation, N ≥ 4)

Consider ethane molec u le as “ elastic structure ”

U Covalent

 U stretch

 U bend

 U rotate

Force fields for organics: Basic approach

=0 for proteins

U total

 U Elec

 U Covalent

 U Metallic

 U vdW

 U H  bond

B ond stretching



Angle Bending



B ond R otation

U Covalent

 U stretch

 U bend

 U rot

 stretch

 1 k

2

stretch

( r  r 0 )

U stretch

   stretch

pairs

 bend

 1 k

2

bend ( 

  0 )

U bend

   bend

triplets

 rot

 1 k

2

rot

( 1  cos (  ))

U rot

   rot

quadruplet s

Image by MIT OpenCou r seWare. 15

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 c ati on & Trai ni ng Commi t t ee. Use d wi th pe rmi s si o n .

Images cr eated for the CHARMM tutori a l by Dr. Dmit ry Kuz n e ts o v (Swi ss In sti t ute o f B i oi nformati c s ) for the EMBnet Edu c ati o n & Trai ni ng committee ( h ttp://www.embn et.org )

http://ww w.ch.embne t.org/MD_tutorial/pages/MD. P art2.html

Force fields for organics: Basic approach

=0 for proteins

U total

 U Elec

 U Covalent

 U Metallic

 U vdW

 U H  bond

U Elec

partial charges

q i q

j

U Elec

: Coulomb potential

 ( r ij ) 

q i q j

 1 r ij







Electrostatic inter actions



vdW Inter actions

electrostatic constant

distance

q i q j

Coulomb for c es

F ( r ij )    r 2

1 ij

 1  4  0   1 . 602  10 C

 19

Image by MIT OpenCou r seWare. 17

Force fields for organics: Basic approach

U total

 U Elec

 U Covalent

=0 for proteins

 U Metallic

 U H  bond

vdW Inter actions

U vdW

Image by MIT OpenCou r seWare.

  

 12

   6 

U vdW :

LJ potential

 ( r ij ) 

4   

 

   

  ij   

LJ potential is particularly good m odel for vdW interactions (Argon) 18

H- bond model

=0 for proteins

U total

 U Elec

 U Covalent

 U Metallic

 U vdW

 U H  bond

U H  bond

Evaluated between acceptor (A) /donor(D) pairs

Between electronegative atom and a H- atom that is bonded to another electronegative atom

Slightly modified LJ, different parameters

  R

 12  R

 10 

U H  bond :

 ( r

)  D 

 5  H  bon d 

 6  H  bon d 

 cos 4 (  )

r ij

ij

= distance between D-A

H bond 

 

r ij 

 r ij

  

DHA

19

Summary

=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

( 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 

U vdW :

LJ potential

 ( r ij ) 

4   

 

   

  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

20

The need for atom typing

 Limited transferability of potential expressions: Must use different potential for different chemistry

 Different chemistry is captured in dif ferent “t ags” for atoms: Element type is expanded by additional information on particular chemical state

 Tags spec ify if a C-atom is in sp 3 , sp 2 , 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, …

Atom typing in CHARMM

VMD analysis of protein structure

Common empirical force fields for organics and proteins

Class I (experiment derived , simple form)

Harmonic terms;

 CHARMM

pset #3

Derived from

 CHARMm (Accelrys)

 AMBER

 OPLS/AMBER/Schrödinger

 ECEPP (free energy force field)

Class II (more complex, derive d fro m QM )

 CFF95 (Biosym/Accelrys)

 MM3

 MMFF94 (CHA R MM, Macromodel…)

vibrational

spectroscopy, gas- phase molecular structures

Very system- specific

Include anharmonic terms

Derived from QM, more general

http://ww w.ch.embne t.org/MD_tutorial/pages/MD. P art2.html

CHARMM force field

 Widely used and accepted m odel for protein structures

 Programs such as NAMD have implemented the CH ARMM force field

Problem set #3, nanoHUB stretchmol module, study of a pr otein domain that is part of human vimentin intermediate filaments

Application – protein folding

ACGT

Four letter code “DNA”

Transcription/ translation

.. - P roline - S erine – Proline - Alanine - . .

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

Folding (3D structure)

Goal of protein folding simulations:

Predic t folded 3D structure based on poly peptide sequence

Movie: protein folding with CHARMM

 de novo Folding of a Transmembrane fd Coat Protein

http://ww w.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://ww w.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 th e replic a-exchange simulations and experimental structures from both solid-state NMR in lipid bilayers and solution-phas e NMR on the protein in micelles)

Movies in equilibrium (temperature 300 K)

Dimer

Tetramer (increased effective bending stiffness,

interaction via overlap & head/tail domain)

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 Control s Nan o mec han i c al

Pro p erti es of Vi menti n Interm edi ate Filaments.” PLoS ONE (200 9) . L i c ens e CC BY . 28

2. Single molecule mechanics

Structure and mechanics of protein, DNA, etc. molecules

Cooking spaghetti

Photo c o urtes y of Ha tM o n F lic kr .

Publ i c domai n i m age.

Photo c o urtes y of Ha tM o n F lic kr .

stiff rods cooking soft, flexible rods ( like many protein molecules )

Single molecule tensile test – “ optical tweezer”

molec u le

one end of molec u le fixed at surface

bead trapped in laser light (moves with laser)

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 .

So u r ce: T s kho v r ebo va , L . , J . T r in ick, e t a l . "Ela s t i cit y a n d Un fo l d ing o f Sin g le M o l ecu les of t h e G i a n t M u sc le Protei n Ti ti n." Na ture 387, no. 6 6 3 0 (1997 ): 308 - 1 2 . © 1 997 .

Example 1: Elasticity of tropocollagen molecules

300 nm length

14

12

10

Experimental data

Theoretical model

8

6

4

2

0

-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 molecul e length and persistence length of this sample is 300 and 7.6 nm, respectively .

Entropic elasticity leads to strongly nonlinear elasticity

P h ot o co urt e sy of HatM on F lic k r .

Image by MIT OpenCou r seWare.

Courtesy of Elsevier, Inc., http://www.scien c edirect. com .

Us ed w i t h p e r m iss i on . 32

Example 2: Single protein molecule mechanics

Optical tweezers experiment

Protein structure (I27 multidomain titin in muscle)

Repr inted by p e rmi s si o n from M a c m ill an Publi s he r s L t d: N a t u r e . So urce: Tskho v rebova, L. , J . Tri n ick, e t al . "El a s t i c ity and Unfo ldin g of Single M o l e cule s of th e Gi an t Mu s cl e P r ot ein Titin . " Nat ur e 387, no. 66 30 ( 1 9 9 7 ) :

308 - 12. © 19 97.

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 .

So u r ce: M a r s za lek, P . , H . L u , e t a l . "M ech a n i c a l Un fo ld i n g I n t e rme d ia t e s in T i t i n M o du les . " Na ture 402, no. 6 75 7 (1999 ): 100 -3 . © 1 999 .

http://ww w.nature.com/nature /journal/v387/n6630 /pdf/387308a0.pdf http://ww w.nature.com/nature /journal/v402/n6757 /pdf/402100a0.pdf 33

Example 3: Single DNA molecule mechanics

plateau regime (break ing of bonds)

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.

Plots of stretching force against relative extension of the single DNA molecule (experimental results)

Structural makeup of protein materials

Although very diverse , all protein materials have universal “protocols” o f how they are made

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 fol ded 3D structures with specific properties

1D structure transforms into co mplex 3D folded configuration

Four letter code “DNA”

Transcription/ translation

.. - P roline - S erine – Proline - Alanine - . .

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 H 2 O Bet w een N and O in proteins

Drives formation of helical structures AHs found in: hair, cells, wool, skin, e t c.

A dapted f r om Bal l , D., Hil l , J., et al. T h e Basi c s of Ge n e ra l, Org ani c, and Bi o l og ic al Ch e m is t r y. Fla t worl d Knowl edge, 2011. C o urtesy of Flatworld Kn owl e dge.

Sour c e : Qi n, Z . , L . Krepl a k, and M. Bu ehl e r. “Hi e rarc hi cal str u ctu re co n trol s nan o mec ha ni c al pr operti es of vi menti n in t e r m ed ia t e fi la me n t s . ” PL o S O N E (2009 ) . Li c e n s e C C BY.

Primary, secondary, tertiary structure

A dapted f r om Bal l , D., Hil l , J., and R. S c ott. T h e Ba sics o f G e ne ra l , O r g a nic , a n d Bi o l og ic al Ch e m is t r y . Fl atworl d Kn owl e dg e, 20 1 1 . Courte s y o f Fl atworl d Kn owl e dge.

38

Beta-sheets (abbreviated as BS)

Beta-sheet

Images r e mo ve d d u e to c o py ri ght restri cti o n s .

Found in many mechanic ally relevant proteins

Spider silk Fibronectin

Titin (muscle tissue)

Amy loids (Alz heimer’s disease) 39

Amyloid proteins (Alzheimer’s disease)

Pl ease s e e Fig. 8 from htt p :/ /w eb.mi t .edu /m bu e h l e r/www / p aper s /fi nal _ JCTN _ p r e pri n t.pdf .

3. Fracture of protein domains – Bell model

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

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

X

X

Titin I27 domain: Very resistant to unfolding due to parallel H-bonded strands

Displacement (A)

Keten and Buehler, 2007

Protein unfolding - R eaxFF

F

AHs

PnIB 1AKG

F

ReaxFF modeling

Buehl e r, M. " Hi erar c h i c al Chemo - nan o mecha n i c s of Protei n s : E n tropi c El a s ti ci ty, Protei n Unfol d i n g

a n d Mo lecu lar F r a c t u re . " J o urnal of M e chani c s and Mate rial s and S t ruc t ures 2 , no. 6 ( 2 007). 48

Protein unfolding - CHARMM

Covalent bonds don’t break

CHARMM modeling