Introduction to Bioinformatics: Lecture XV Empirical Force Fields and Molecular Dynamics

1. Introduction to Bioinformatics: Lecture XVEmpirical Force Fields and Molecular Dynamics Jarek Meller Division of Biomedical Informatics, Children’s Hospital Research Foundation & Department of Biomedical Engineering, UC JM - http://folding.chmcc.org

2. Outline of the lecture • Motivation: atomistic models of molecular systems • Empirical force fields as effective interaction models for atomistic simulations • Molecular Dynamics algorithm • Kinetics, thermodynamics, conformational search and docking using MD • Limitations of MD: force fields inaccuracy, long range interactions, integration stability and time limitations, ergodicity and sampling problem • Beyond MD: other protocols for atomistic simulations JM - http://folding.chmcc.org

3. Molecular systems and interatomic interactions JM - http://folding.chmcc.org

4. Molecular systems and interatomic interactions b-strand a-helix JM - http://folding.chmcc.org

5. Molecular Dynamics as a way to study molecular motion • What is wrong with the previous pictures? • Real molecules “breathe”: molecular motion is inherent to all chemical processes, “structure” and function of molecular systems • For example, ligand binding (oxygen to hemoglobin, hormone to receptor etc.) require inter- and intra-molecular motions • Another example is protein folding – check out some MD trajectories JM - http://folding.chmcc.org

6. Web watch: folding simulations using MD and distributed computing: Folding@Home Folding@Home Vijay S Pande and colleagues, Stanford Univ. For example, folding simulations of the villin headpiece … http://www.stanford.edu/group/pandegroup/folding/papers.html Some more MD movies from Ron Elber’s group: http://www.cs.cornell.edu/ron/movies.htm JM - http://folding.chmcc.org

7. Two approximations and two families of MD methods • The quantum or first-principles MD simulations (Car and Parinello), take explicitly into account the quantum nature of the chemical bond. The electron density functional for the valence electrons that determine bonding in the system is computed using quantum equations, whereas the dynamics of ions (nuclei with their inner electrons) is followed classically. • In the classical mechanics approach to MD simulations molecules are treated as classical objects, resembling very much the “ball and stick” model. Atoms correspond to soft balls and elastic sticks correspond to bonds. The laws of classical mechanics define the dynamics of the system. JM - http://folding.chmcc.org

8. From quantum models to classical approximations Ab initio methods: computational methods of physics and chemistry that are based on fundamental physical models and, contrary to empirical methods, do not use experimentally derived parameters except for fundamental physical constants such as speed of light c or Planck constant h. Born-Oppenheimer approximation, potential energy surface and empirical force fields, parametrizing atomistic force fields by combination of ab initio, experiment and fitting … The NIH guide to molecular mechanics: http://cmm.info.nih.gov/modeling/guide_documents/molecular_mechanics_document.html JM - http://folding.chmcc.org

9. Force fields for atomistic simulations Definition Empirical potential is a certain functional form of the potential energy of a system of interacting atoms with the parameters derived from ab initio calculations and experimental data. How to get parameters that would have something to do with the physical reality: experiment and ab initio calculations, also just fitting! JM - http://folding.chmcc.org

10. Dispersion interactions and Lennard-Jones potential Dispersion (van der Waals) interactions result from polarization of electron clouds and their range is significantly shorter than that of Coulomb interactions. sij rij -eij Problem Find that the minimum of van der Waals (Lennard-Jones) potential JM - http://folding.chmcc.org

11. Time evolution of the system: Newton’s equations of motion JM - http://folding.chmcc.org

12. Solving EOM: Coulomb interactions and N-body problem Solving EOM for a harmonic oscillator – simple … Potential: U(x)=1/2 k x2 ; Solution: x(t) = A cos(wt+g) Problem Show that w2=k/m Solving EOM for a system with more than two atoms and Coulomb or Lennard-Jones potentials – no analytical solution, numerical integration JM - http://folding.chmcc.org

13. Numerical integration of EOM: the Verlet algorithm DefinitionMolecular Dynamics is a technique for atomistic simulations of complex systems in which the time evolution of the system is followed using numerical integration of the equations of motion. One commonly used method of numerical integration of motion was first proposed by Verlet: Problem Using Taylor’s expansions derive the Verlet formula given above. JM - http://folding.chmcc.org

14. Fast motions and the integration time step For example, O-H bonds vibrate with a period of about 17 fs To preserve stability of the integration, Dt needs to very short - of the order of femtoseconds (even if fastest vibrations are filtered out) Except for very fast processes, nano- and micro-seconds time scales are required: time limitation and long time dynamics JM - http://folding.chmcc.org

15. Long range forces as computational bottleneck Long range interactions: electrostatic and dispersion interactions lead (in straightforward implementations) to summation over all pairs of atoms in the system to compute the forces Environment, e.g. solvent, membranes, complexes Implicit solvent models: from effective pair energies to PB models Explicit solvent models: multiple expansion, periodic boundary conditions (lattice symmetry), PME JM - http://folding.chmcc.org

16. Examples of problems and MD trajectories Thermodynamics: what states are possible, what states are “visited”, statistics and averages for observables, chemical processes as driven by free energy differences between states, MD as a sampling method (different ensembles and the corresponding MD protocols) Kinetics: how fast (and along what trajectory) the system interconverts between states, rates of processes, mechanistic insights, MD provides “real” trajectories and intermediate states, often inaccessible experimentally Specific applications: sampling for energy minimization and structure prediction, homology modeling, sampling for free energy of ligand binding, folding rates and folding intermediates etc. JM - http://folding.chmcc.org

17. Ligand diffusion in myoglobin JM - http://folding.chmcc.org

18. Ligand diffusion in myoglobin JM - http://folding.chmcc.org

19. Molecular Dynamics as a way to study molecular motion • Quantum (first principles) MD is computationally expensive • Empirical force fields as a more effective alternative • No chemical change though, problem with parametrization and numerous approximations (read inherent limitations of empirical force fields) • Commonly used force fields and MD packages: Charmm, AMBER, MOIL, GROMOS, Tinker • Other limitations of MD: long range interactions, integration stability and time limitations, ergodicity and sampling problem JM - http://folding.chmcc.org