Molecular Dynamics Valerie Daggett Bioengineering Department University 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 room temperature (or any T > 0 K) Kinetic energy per atom is ~ 1 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 O 2  heme!

Protein Dynamics Kendrew: “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 MD 1,2 [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: Levitt M. et al. J. Phys. Chem. B (1997) 101:

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 b0b0

Molecular Dynamics Potential function for MD U = Bond + Angle + Dihedral + van der Waals + Electrostatic θ0θ0

Molecular Dynamics Potential function for MD U = Bond + Angle + Dihedral + van der Waals + Electrostatic Φ0Φ0

Molecular Dynamics Potential function for MD U = Bond + Angle + Dihedral + van der Waals + Electrostatic

Molecular Dynamics Non-bonded components of potential function U nb = 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 U nb = van der Waals + Electrostatic

Molecular Dynamics Non-bonded components of potential function U nb = 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 N is often between 5x10 5 to 5x10 6 For 5x10 5 that is 1.25x10 11 pairs THAT IS A LOT OF POSSIBLE PAIRS!

What can you do with a force field? Generation of ‘experimental’ structures Refinement of ‘experimental’ structures Monte Carlo Scoring functions Energy minimization Analysis Perform MD simulations etc.

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 ( s) involves 25,000,000 evaluations

Molecular Dynamics Actual integration the equations of motion Conserves energy Smooth, robust

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 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

Molecular Dynamics MD provides atomic resolution of folding / unfolding unfolding simulation (reversed) of 3chy at 498 Kelvin, waters & hydrogens hidden

Average may not be representive

Storch et al., Biochem, 1995, 1999a,b, 2000 Dynamic cleft discovered through MD Cytochrome b 5

Storch et al., Biochem, 1999 Bill Atkins, Patricia Campbell S18 R47 Construction of mutants to test whether cleft forms

Construction of cyt c – cyt b 5 complexes

Changes in cyt b 5 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 or NMR Structure Solvate with water or other solvent Å from protein Heat to desired temperature and allow motion to evolve over time T = 298 K r = gm/ml T = 498 K r = gm/ml All atoms present Fully flexible water NVE

Native Dynamics at 25 ºC All C  atoms C  RMSD (Å) Time (ns) Active site loop and N-terminus removed = 1.7 Å = 0.7 Å Crystal structure NOEs reproduced

Crystal StructureAfter 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) N 15 –H no mobility high

Temperatur e (K) Cutoff range (Å) Time a (ns) NOEs satisfied b (% of 603) Xtal No restraints 22 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 RMSD (Å) Time (ns) 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 N TS 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 space Reduce 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 b3b3

Packing is Disrupted in Transition State Crystal Structure WT TS1 32 % SASA

Structure of TS from Experiment  = 1  site of mutation native-like in TS  = 0  site of mutation unfolded in TS Fractional  values  partial structure in TS  =  G TS-D /  G N-D = 1 N TS D  G N-D  G TS-D  G N- D  =  G TS-D /  G N-D = 0 N TS D  G N-D  G TS-D  G N- D NTSD 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 = (S 2º ) (S 3º ) % native secondary structure (  ) % tertiary structure, contacts For each residue calculate: Daggett & Li, 1994, PNAS; Daggett, Li, Itzhaki, Otzen & Fersht, 1996, JMB

Residue Number Phi or S Value  S Value Phi 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

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T

Rotate 90 to right Conformational Heterogeneity of TS = Crystal Structure = TS1-4, 498 K = TS5-9,  T XTAL  MD  4.5 Å

N TS D Free Energy Calculations for Direct Determination of  WT Mutant NTSD N'TS'D'  G N  TS  G TS  D  G' TS  D  G' N  TS GNGN  G TS GDGD GNDGND  G' N  D  G N-TS =  G N-TS -  G' N-TS =  G N -  G TS  G TS-D =  G TS-D -  G' TS-D =  G TS -  G D  G N-D =  G N-D -  G’ N-D =  G N -  G D  F =  G TS-D /  G N-D

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 TSRF48 TS K2 E7 D23 R48 R62 F50 F48 R62 F50 Ladurner, Itzhaki, Daggett, Fersht, 1998, PNAS Designing Faster Folding Forms of CI2 Based on MD-Generated TS Models

Removal of Unfavorable Interactions Identified in TS Models Accelerates Folding Removal of charge repulsion and improvement of packing in the TS yields fastest-folding form of CI2.

Thermal Denaturation of CI2 at 498 K C  RMSD (Å) Time (ns) D 40,000 structures

 The Denatured State of CI2 Distances in N: W5-V14 15 Å I30-Y42 13 Å P33-I37 11 Å V14 Y42 I37 I57 I30 L49 P33 Kazmirski et al., PNAS, 2001 Experimental Results expt = 7.2 Hz MD = 7.0 Hz  V19-L21, I30-T36 Nearly random coil (res ) 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.

k obs (s -1 ) k unf kfkf En-HD unfolds at 348 & 373 K on the same timescale by simulation and experiment 47,000 s -1 Time to reach TS in MD simulation But, it is not enough to get the timescale right, 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 SASA No discrimination between native & non-native states Native Nonnative

Development of information-rich property space High information content property: CONGENEAL structural dissimilarity score 1 Excellent discrimination between native & non-native states 1. Yee and Dill, Protein Science, 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 Foldedness ↔ location along folding reaction coordinate