Why study protein dynamics?

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1 Why study protein dynamics? Protein flexibility is crucial for function. One average structure is not enough. Proteins constantly sample configurational space. Transport - binding and moving molecules (ex: molecular oxygen binding to hemoglobin) Enzyme catalysis - substrate entry and produce release Allosteric regulation - regulation of enzyme activity. Enzyme must be able to flip-flop between on (active) and off (inactive) states Molecular associations - induced fit (ex: transcription complexes)

2 Use of Molecular Dynamics Simulation Kinetics and irreversible processes chemical reaction kinetics (with QM) conformational changes, allosteric mechanisms Protein folding Equilibrium ensemble sampling Flexibility thermodynamics (free energy changes, binding) Modeling tool structure prediction / modeling solvent effects NMR/crystallography (refinement) Electron microscopy (flexible fitting)

3 Atomic Detail Computer Simulation Model System Molecular Mechanics Potential V = + + dihedrals n= 1 i, j bonds k b N 2 ( b b ) + k ( θ θ ) K σij 4ε ij rij 0 θ angles ( n) [ 1+ cos( nφ δ) ] + K ( ω ω ) φ 12 6 σ ij + rij i, j ω impropers qiq Dr ij j 2 0 Energy Surface Exploration by Simulation.. Jeremy Smith

4 Bonded Interactions: Stretching E str represents the energy required to stretch or compress a covalent bond: A bond can be thought of as a spring having its own equilibrium length, r o, and the energy required to stretch or compress it can be approximated by the Hookean potential for an ideal spring: E str = ½ k s,ij ( r ij - r o ) 2

5 Bonded Interactions: Bending E bend is the energy required to bend a bond from its equilibrium angle, θ o : Again this system can be modeled by a spring, and the energy is given by the Hookean potential with respect to angle: E bend = ½ k b,ijk (θ ijk -θ o ) 2

6 Bonded Interactions: Improper Torsion E improper is the energy required to deform a planar group of atoms from its equilibrium angle,ω o, usually equal to zero: k i l j ω Thomas W. Shattuck Again this system can be modeled by a spring, and the energy is given by the Hookean potential with respect to planar angle: E improper = ½ k o,ijkl (ω ijkl -ω o ) 2

7 Bonded Interactions: Torsion E tor is the energy of torsion needed to rotate about bonds: φ A F D C E B Thomas W. Shattuck Dihedral Energy (kcal/mol) Dihedral Angle 300 H H CH 3 H Butane CH 3 H Torsional interactions are modeled by the potential: E tor = ½ k tor,1 (1 - cos φ ) + ½ k tor,2 (1 - cos 2 φ ) + ½ k tor,3 ( 1 - cos 3 φ ) asymmetry (butane) 2-fold groups e.g. COO- standard tetrahedral torsions

8 Non-Bonded Interactions: van der Waals E vdw is the steric exclusion and long-range attraction energy (QM origins): 0.2 V an der Waals Int eract ion for H.....H Ene rg y ( k cal/ mol ) repulsio n att raction -0.2 Thomas W. Shattuck Two frequently used formulas: H... H dist anc e ( Å ) E E

9 Non-Bonded Interactions: Coulomb E qq is the Coulomb potential function for electrostatic interactions of charges: Thomas W. Shattuck Formula: The Q i and Q j are the partial atomic charges for atoms i and j separated by a distance r ij. ε is the relative dielectric constant. For gas phase calculations ε is normally set to 1. Larger values of ε are used to approximate the dielectric effect of intervening solute (ε 60-80) or solvent atoms in solution. k is a units conversion constant; for kcal/mol, k=

10 Newton s Law E steric energy = E str + E bend + E improper + E tor + E vdw + E qq Newton s Law: F = m i i a i Compute a trajectory

11 Deterministic / MD methodology From atom positions, velocities, and accelerations, calculate atom positions and velocities at the next time step. Integrating these infinitesimal steps yields the trajectory of the system for any desired time range. There are efficient methods for integrating these elementary steps with Verlet and leapfrog algorithms being the most commonly used.

12 Timescale Limitations Protein Folding - milliseconds/seconds ( s) Ligand Binding - micro/milliseconds ( s) Enzyme catalysis - micro/milliseconds ( s) Conformational transitions - pico/nanoseconds ( s) Collective vibrations - 1 picosecond (10-12 s) Bond vibrations - 1 femtosecond (10-15 s)

13 Choosing a time step Too short - computation needlessly slow Too long - errors result from approximations Just right - errors acceptable, maximum speed

14 Overlong Timesteps Simulation of the interatomic distance between two Argon atoms at two δts. The difference from the exact path is plotted. Particularly near collisions, The forces change quickly. Errors in these regions are compounded in subsequent steps.

15 Timescale Limitations Molecular dynamics: Integration timestep - 1 fs, set by fastest varying force. Accessible timescale: about 100 nanoseconds.

16 Biased MD - jumping barriers exploring conformations