Molecular Dynamics

Valerie DaggettBioengineering DepartmentUniversity of Washington

...everything that living things do can be understood in terms of the jigglings and wigglings of atoms.

Richard Feynman

Protein Dynamics

• Proteins are not static• Motion is an incontrovertible consequence of existing @

room temperature (or any T > 0 K)• Kinetic energy per atom is ~ 1 kcal/mole @ 298K (25°C)

several Å/ps• Motion recognized to be important early on. Kendrew

(1950s) solved crystal structure of myoglobin (Perutz, phasing)

Myoglobin

No pathway for

O2 heme!

Protein DynamicsKendrew: “Perhaps the most remarkable features

of the molecule are its complexity and its lack of

symmetry. The arrangement seems to be almost

totally lacking in the kind of regularities which one

instinctively anticipates, and it is more complicated

than has been predicted by any theory of protein structure.”

Situation gets worse when you consider dynamics.

But proteins are dynamic and dynamic behavior critical for function. So, static, average structures are only part of the story.

Function from static structure?

Dynamics necessary for function

Snapshots

Static and/or average structures may not be representative of

conformations critical to function

Theory

• Experiment clearly demonstrates that proteins are mobile, but no single experiment or combination of experiments can provide an all-inclusive view of the dynamic behavior of all atoms in a protein.

• Computer simulations can however– ea. atom as a function of time

Why Molecular Dynamics?

• Most realistic simulation method available

• Can provide structural and dynamic information

unobtainable by experiment, but is experimentally

testable

• Native and nonnative interactions apparent

But• Sampling is limited, the goal is to sample

experimentally relevant regions of conformational

space, not all of conformational space

Molecular Dynamics

• Potential function for MD1,2 [sum of following terms]

U = Bond + Angle + Dihedral + van der Waals + Electrostatic

1. Levitt M. Hirshberg M. Sharon R. Daggett V. Comp. Phys. Comm. (1995) 91: 215-231

2. Levitt M. et al. J. Phys. Chem. B (1997) 101: 5051-5061

Molecular Dynamics

• Potential function for MD

U = Bond + Angle + Dihedral + van der Waals + Electrostatic

Molecular Dynamics

• Potential function for MD

U = Bond + Angle + Dihedral + van der Waals + Electrostatic

b0

Molecular Dynamics

• Potential function for MD

U = Bond + Angle + Dihedral + van der Waals + Electrostatic

θ0

Molecular Dynamics

• Potential function for MD

U = Bond + Angle + Dihedral + van der Waals + Electrostatic

Φ0

Molecular Dynamics

• Potential function for MD

U = Bond + Angle + Dihedral + van der Waals + Electrostatic

Molecular Dynamics

• Non-bonded components of potential function

Unb = van der Waals + Electrostatic

• To a large degree, protein structure is dependent on non-bonded atomic interactions

Molecular Dynamics• Non-bonded components of potential function

Unb = van der Waals + Electrostatic

Molecular Dynamics• Non-bonded components of potential function

Unb = van der Waals + Electrostatic

Molecular Dynamics• Non-bonded components of potential

function

+ -

Molecular Dynamics• Non-bonded components of potential

function

+ +

Molecular Dynamics

• Non-bonded components of potential function

NOTE:Sum over all pairs of N atoms, or

pairs

2

1NN

N is often between 5x105 to 5x106

For 5x105 that is 1.25x1011 pairs

THAT IS A LOT OF POSSIBLE PAIRS!

What can you do with a force field?

Generation of ‘experimental’ structuresRefinement of ‘experimental’ structuresMonte CarloScoring functionsEnergy minimizationAnalysisPerform MD simulationsetc.

Molecular Dynamics• Time dependent integration of classical

equations of motion

Molecular Dynamics

• Time dependent integration

Molecular Dynamics

• Time dependent integration

Molecular Dynamics

• Time dependent integration

Molecular Dynamics

• Time dependent integration

Molecular Dynamics

• Time dependent integration

Molecular Dynamics

• Time dependent integration

Molecular Dynamics• Time dependent integration

Evaluate forces and perform integration for every atom

Each picosecond of simulation time requires 500 iterations of cycle

E.g. w/ 50,000 atoms, each ps (10-12 s) involves 25,000,000 evaluations

8)]()(6)(3[)()(

8)]()(5[)()()(

2

tttatattatvttv

tttatattvtxttx

iiiii

iiiii

Molecular Dynamics

Actual integration the equations of motion

Conserves energySmooth, robust

b

ii

Nk

vmT

3

2

Molecular Dynamics

Determination of temperature

Methods• Molecular dynamics (MD)

– Brooks-Beeman integration algorithm– Microcanonical ensemble (NVE)

• Number of atoms, box volume, & energy are conserved

• Energy conservation is naturally satisfied with classical equations of motion

• Energy conservation is an inherent check on the implementation

• Free from coupling the microscopic system to macroscopic variables as do NVT and NPT

Molecular dynamics

Microcanonical ensemble, all atoms, solvent,

fully flexible molecules, continuous trajectories,

no restraints/biases

predictive MD---expt to check

No Ewald --- artificial periodicity, altered conformational and dynamical properties

No fictitious bonds between H atoms of water

No Shake

Correct masses

Good simple, flexible water correct D and RDF

Methods• Molecular dynamics (MD)

– Temperature in NVE• Mean T over hundreds of steps• Energy drift in ilmm is primarily kinetic

resulting from numerical round-off– Over thousands of steps mean T can be monitored

for energy conservation– Velocity rescales once per 10ns

Implementation• Written in C

– Ubiquitous, standardized, optimized language• 64 bit math• Software design

– Kernel• Compiles user’s molecular mechanics programs• Schedules execution across processor and machines

– Modules, e.g.• Energy minimization• Molecular Dynamics• Monte Carlo• Analysis• REMD• RDCs • + others

Implementation• Dual mode parallelization

– Standardized tools available on modern platforms• POSIX threads

– Distribute computations across multiple CPUs in a single computer

• Message Passing Interface (MPI)– Distribute computations across multiple

computers on a high speed network

– Benefit is scalability

State of the Art MD

What can be done with PCs?

Environment:• Possible to characterize solvent-dependent conformational behavior • Proteins in membranes

Size: •≤ 500 residues (more possible if willing to dedicate resources to it, our record is 2519 residues in solvated membrane)

Timescale:• Multiple 20-100 ns simulations fairly routine for proteins• s possible if willing to dedicate resources to it

Molecular Dynamics• MD provides atomic resolution of native

dynamics

PDB ID: 3chy, E. coli CheY 1.66 Å X-ray crystallography

Molecular Dynamics• MD provides atomic resolution of native

dynamics

PDB ID: 3chy, E. coli CheY 1.66 Å X-ray crystallography

Molecular Dynamics• MD provides atomic resolution of native

dynamics

3chy, hydrogens added

Molecular Dynamics• MD provides atomic resolution of native

dynamics

3chy, waters added (i.e. solvated)

Molecular Dynamics• MD provides atomic resolution of native

dynamics

3chy, waters and hydrogens hidden

Molecular Dynamics• MD provides atomic resolution of native

dynamics

native state simulation of 3chy at 298 Kelvin, waters and hydrogens hidden

Molecular Dynamics• MD provides atomic resolution of native

dynamics

native state simulation of 3chy at 298 Kelvin, waters and hydrogens hidden

Average may not be representive

Storch et al., Biochem, 1995, 1999a,b, 2000

Dynamic cleft discovered through MD

Cytochrome b5

Storch et al., Biochem,1999

Bill Atkins, PatriciaCampbell

S18 R47

Construction of mutants to testwhether cleftforms

Construction of cyt c – cyt b5 complexes

Changes in cyt b5 upon binding cyt c

= Predicted binding surface

= Change in chemical shift

Hom et al., Biochem, 2000

Cleft allows for electron transfer through the protein in channel lined with aromatics

= Nonpolar

= Polar

Validation

• Validation, how do you know if a simulation is correct?

• How do you know it is done?

Starting a Molecular Dynamics Simulation

Crystal orNMR Structure

Solvate with water or other solvent 8 - 14 Åfrom protein

Heat to desiredtemperature andallow motion to evolve over time

T = 298 K r = 0.997 gm/ml

T = 498 Kr = 0.829 gm/ml

All atoms presentFully flexible waterNVE

Native Dynamics at 25 ºC

All C atoms

C R

MS

D (

Å)

Time (ns)

Active site loop and N-terminus removed

3

0.5

2

0

2.5

1.5

1

5 15 4035302520 5045100

<RMSD> = 1.7 Å

<RMSD> = 0.7 Å

Crystal structure

NOEsreproduced

Crystal Structure After 50 ns of MD

Structural Changes to Native State During MD

= Xtal Structure= 50 ns MD

Active Site Loop

N-terminus

Turn

Chymotrypsin Inhibitor 2

• Expt = Shaw et al (1995) Biochem 34:2225

• MD = Li & Daggett (1995) Prot. Eng. 8(11)

N15–H

no mobility

high

Temperature

(K)

Cutoff range(Å)

Timea

(ns)NOEs satisfiedb

(% of 603)

Xtal 91.00

298 8 100 98.18

298 8 50 98.51

298 8 50 98.00

298 8 20 97.18

298 8 20 97.84

298 8 20 98.18

298 10 100 98.84

298 10 50 97.68

298 10 50 97.51

298 10 20 97.84

298 10 20 97.54

298 10 20 97.35

310 8 100 98.51

310 8 50 98.01

310 8 50 98.51

310 10 100 98.68

310 10 50 98.01

310 10 50 98.51

323 8 100 98.34

323 8 50 97.35

323 10 100 98.34

323 10 50 98.01

No restraints22 simulations>1.2 s

NOEs courtesy of Stefan Freund& Trevor Rutherford, ARF

Pushing to high temperature

• Taking excursions farther from the native state, will the force fields and methods hold?

• Thermal unfolding of proteins

Thermal Denaturation of CI2 at 498 K

Ca

RM

SD

(Å

)

Time (ns)

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

N298

MD1

MD2

MD3

MD4

Li & Daggett, 1994, PNAS; JMB, 1996

N

TS?

D

•TS not localized to single bond, distributed and ensemble•From MD cannot calculate DG along reaction coordinate•Structure-based definition of TS:

•Kinetically, protein will not succeed in every attempt to cross TS but will change rapidly afterwards•A process with a large change in energy but small change in entropy large change in free energy

NTS

D

Reaction Coordinate

H

G

-TS

Identifying Transition States in MD Trajectories

TS

D

Projection of Trajectory in RMSD Space

Calculate the RMSD between all structures---15,000 x 15,000 dimensional spaceReduce to 3-dimensions, distance between points RMSD between structures

Clusters indicate similar conformations, conformational states

Li & Daggett, 1994, 1996

N

Main-chain Fold Preserved in Transition State

a

Crystal Structure Average TS1 Structure~4 Å, 43 % native H-Bonds

b1

b2

b3

Packing is Disrupted in Transition State

Crystal StructureWT TS132 % SASA

Structure of TS from Experiment

= 1 site of mutation native-like in TS = 0 site of mutation unfolded in TSFractional values partial structure in TS

= GTS-D / GN-D = 1

N

TS

D

GN-D

GTS-D

GN-D

= GTS-D / GN-D = 0

N

TS

D

GN-D

GTS-D

GN-D

N TS D

Matouschek, Kellis, Serrano, Fersht, 1989, Nature, Fersht, Leatherbarrow, Wells, 1986, Nature; 1987 Biochem, +++

Calculation of S Values for Comparison with Experimental F Values

S = structure index = (S2º ) (S3º )

% native secondary structure ()

% tertiarystructure,contacts

For each residue calculate:

Daggett & Li, 1994, PNAS; Daggett, Li, Itzhaki, Otzen & Fersht, 1996, JMB

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Phi ValueS Value

Residue Number

Ph

i or

S

Valu

e

S V

alu

ePhi Value

Comparison of Calculated S Values and Experimental Values

Otzen et al., 1994, PNAS; Itzhaki et al., 1995, JMB; Li & Daggett, 1994,1996, JMB; Daggett et al., 1996, JMB

R = 0.90

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Overall TS Structure and Unfolding Pathway are Independent of Temperature

373 K

398 K

448 K

473 K

498 K

(21 ns)

(0.3 ns)

(0.57 ns)

(1.44 ns)

(8.26 ns)

TS1

TS2

TS3

TS4

498 K

(0.225 ns)

(0.335 ns)

(0.1 ns)

(0.07 ns)

T

Rotate 90

to right

Conformational Heterogeneity of TS

= Crystal Structure= TS1-4, 498 K= TS5-9, T

<RMSD>XTAL<RMSD>MD 4.5 Å

N TS

D

Free Energy Calculations for Direct Determination of

WT

Mutant

N TS D

N' TS' D'

GNTS GTSD

G'TSDG'NTS

GN GTS GD

GND

G'ND

GN-TS = GN-TS - G'N-TS = GN - GTS

GTS-D = GTS-D - G'TS-D = GTS - GD

GN-D = GN-D - G’N-D = GN - GD

F = GTS-D / GN-D

GN-D GN-TS GTS-D

F Mut GN GTS

GD Calc Expt Calc Expt Calc Expt Calc Expt A16G 2.28 2.17 1.10 1.18 1.09 0.11 -0.20 1.07 1.15 0.91 1.06 V19A 1.68 1.03 1.09 0.59 0.49 0.77 0.86 -0.06 -0.13 -0.10 -0.26 I20V 2.64 1.71 1.42 1.22 1.30 0.93 0.57 0.29 0.52 0.24 0.40 I29V 2.45 1.75 1.04 1.41 1.11 0.70 0.69 0.71 0.19 0.50 0.37 V47A 2.60 0.87 0.74 1.86 4.93 0.73 3.81 1.13 1.02 0.61 0.21 L49A 2.64 0.78 0.63 2.01 3.80 0.86 1.45 1.15 2.11 0.57 0.53 V51A 4.24 2.77 1.92 2.32 1.98 1.47 1.00 0.85 0.49 0.37 0.25 I57A 5.42 2.60 1.80 3.62 4.29 2.82 3.46 0.80 0.36 0.22 0.08

R = 0.85 R = 0.91 (no V47A)

FEP Calculations for Hydrophobic Core Mutants

(kcal/mole)

Pan & Daggett, Biochem, 2001

Extended peptide NOT a good model of D

DA 23 TS

A23

K2

E7

WT TS RF48 TS

K2

E7

D23

R48

R62

F50F48

R62

F50

Ladurner, Itzhaki, Daggett, Fersht, 1998, PNAS

Designing Faster Folding Forms of CI2Based on MD-Generated TS Models

Protein kf

H2O (s-1) GTS-D (kcal/mole)

ku

H2O (s-1) GTS-N (kcal/mole)

Wild type

56.5

1.2

DA23 83.9 -0.23 3.8 -0.69

RF48 2300 -2.18 nd -0.75

Removal of Unfavorable Interactions Identified in TS Models Accelerates Folding

Removal of charge repulsion and improvement of packingin the TS yields fastest-folding form of CI2.

Thermal Denaturation of CI2 at 498 K

C R

MS

D (

Å)

Time (ns)

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

N298

MD1

MD2

MD3

MD4

D40,000structures

The Denatured State of CI2

Distances in N:W5-V14 15 ÅI30-Y42 13 ÅP33-I37 11 Å

V14

Y42

I37

I57

I30

L49

P33

Kazmirski et al., PNAS, 2001

Experimental Results

<3JNH-CH>expt = 7.2 Hz

<3JNH-CH>MD = 7.0 Hz

V19-L21, I30-T36

Nearly random coil

(res. 17-21)

hydrophobic clustering

W5

Summary of CI2 Simulations• N is well behaved and in good agreement with experiment.

•TS is an expanded version of N with disrupted core and loops and frayed secondary structure.

Validity of MD-generated TS models tested through indirect comparison with experimental values, direct comparison of Gs, behavior when T is quenched, and design of faster folding mutants.

•WT D is very disrupted with only minor amounts of hydrophobic clustering and fluctuating helical structure. Nearly random coil.

25 50 75 100

10-9

10-8

10-7

10-6

10-5

10-4

103

104

105

106

107

108

Temperature ( oC)t 1

/2

(s)

k ob

s (s

-1)

kunf

kf

En-HD unfolds at 348 & 373 K on the same timescale by simulation and experiment

47,000 s-1

Time to reachTS in MD simulation

But, it is notenough to get the timescaleright, must get pathway too!

10 °C

Mayor et al., Nature 2003

Development of information-rich property space

Low information content property:Main-chain non-polar SASANo discrimination between native & non-native states

Native

Nonnative

Development of information-rich property spaceHigh information content property:

CONGENEAL structural dissimilarity score1

Excellent discrimination between native & non-native states

1. Yee and Dill, Protein Science, 1993.

Native

Nonnative

• Mean distance in PS (32 -> 10 properties) for a given conformation to the folded or native state ‘cluster’ is acceptable reaction coordinate– Value increases with distance from native cluster– Native cluster is bounded

Development of a reaction coordinate from property space

PSr

• Foldedness ↔ location along folding reaction coordinate