1. Molecular Modelling - Lecture 2 Techniques for Conformational Sampling Uses CHARMM force field Written in C++

2. Protein example

3. Monte Carlo Simulations Technique used to perform first computer simulation of a molecular system “Monte Carlo” = some kind of random sampling

4. Monte Carlo Methods Basis of Monte Carlo methods is the use of random selections in calculations that lead to the solution of numerical and physical problems e.g.  brownian motion  molecular modelling  designing nuclear reactors  predicting the evolution of stars  forecasting the stock market Each calculation is independent of the others and hence amenable to embarrassingly parallel methods

5. Monte Carlo Integration : finding value of π Monte Carlo integration  Compute r by generating random points in a square of side 2 and counting how many of them are in the circle with radius 1 (x 2 +y 2 <1; π=4*ratio). Area= π 2 2 Area of square=4

6. Monte Carlo Simulations Configurations are generated by making random changes to the positions of the atoms Importance sampling  Samples from 3N dimensional space of positions of molecules

7. Monte Carlo Simulations Z= Configurational integral

8. Metropolis Monte Carlo Biases generation of configurations towards those that make the most significant contributions to the integral  Low energy states for most thermodynamic properties  Generates states with probability exp(  x  and counts them equally  (simple Monte Carlo would generate them with equal probability and then weight them by exp(  x  

9. Markov chain of states

10. Monte Carlo Advantages/Disadvantages Advantages:  Does not require a continuous energy function (as in MD)  Number of particles can easily vary (very hard in MD) Disadvantages  Highly correlated motions hard to simulate Poor sampling of large-scale changes

11. Molecular Dynamics Newton’s equations of motions are integrated to propagate the structure through time

12. Molecular Dynamics fast: large systems can be modelled history of molecular motion and interactions  conformational distribution for simulation  t 360 0  P 0 

13. Molecular Dynamics - Integration methods Finite difference methods  Used to generate molecular dynamics trajectories with continuous potentials  Integration broken into stages separated by time  t  Total force on each particle at time t is calculated as a vector sum of all the interactions with other particles  From force can determine accelerations Combined with positions and velocities at time t to calculate positions and velocities at time t +  t Force assumed to be constant during time step

14. Molecular Dynamics - Integration methods Many algorithms:  Verlet  Leap frog method  Predictor-corrector  Velocity-Verlet

15. Molecular Dynamics - Verlet Integration Method Most widely used method

16. Timescale Limitations

17. MD Production Run Protocol Initial coords obtained from experimental data or theoretical model Can be done by randomly selecting from a Maxwell-Boltzmann distribution at the temperature of interest

21. Truncating Long-Range forces and the minimum image convention Non-bonded interactions most time- consuming part of a simulation  N 2 Minimum image convention:  Interaction for a molecule i are only counted between it and it’s closest image Truncation of potential creates problems with consistent potential and force

22. Truncating Long-Range forces and the minimum image convention Use smoothing functions to smoothly switch off the interaction between a “cut- on” and a “cut-off” distance.

24. Simulated Annealing special case of either MD (`quenched' MD) or MC simulation, in which the temperature is gradually reduced during the simulation. Often, the system is first heated and then cooled the system is given the opportunity to surmount energetic barriers in a search for conformations with energies lower than the local-minimum energy found by energy minimization.  can lead to more realistic simulations of dynamics at low temperature  more expensive than energy minimization.