Plan Last lab will be handed out on 11/22. No more labs/home works after Thanksgiving. 11/29 lab session will be changed to lecture. In-class final (1hour): from 9:30 to 10:30 am 11/30, Tue. Take-home final (48 hours): Due 12 pm 12/03, Fri. by .

Finals In-class final: based on home works/lab reports. (25%) Take-home final: questions based on a research paper and a computer lab session without instructor helps. (25%)

Tricks of the Trade How to make force calculation fast? It can use 99% of the computation time. Avoiding square root Table look-up and spline-fit potentials Multiple time-step methods Proper handling of long-range forces Cutting corner in solvating biomolecules

Multiple Time Steps Reduce the overhead in the force calculation. Is based on the observation that forces originating from distant atoms fluctuate more slowly than forces from atoms nearby. The slowly fluctuating forces may be evaluated less frequently than the fast ones and may be extrapolated at the time steps in between.

Multiple Time Steps

Complexity of Force Calculations Complexity is the scaling with the number of degrees particles. Number of terms in pair potential is N(N-1)/2  O(N 2 ) For short range potential you can use neighbor tables to reduce it to O(N)

Long-range Forces: Ewald Summation

Long-range Forces: Ewald Summation Lattice 1: Seen from a greater distance, a Gaussian charge cloud resembles a delta-like point charge, effectively compensating the original charge it accompanies. The effect of lattice 1 is therefore best computed in real- space, where the summation will converge quite rapidly. Lattice 2: The potential sum can be computed by fast Fourier transformation. By suitably adjusting parameters, optimal convergence of both lattices may be achieved.

Solvation of Biomolecules: Periodic Boundary

Solvation of Biomolecules: Spherical Boundary

Solvation of Biomolecules: Hydration Shell

Solvation of Biomolecules: Implicit Solvent

Solvation of Biomolecules: Implicit Solvents Distance dependent dielectric in Coulomb’s law Generalized Born approximation Poisson-Boltzmann equation

Solvation of Biomolecules: Implicit Solvents Need stochastic dynamics algorithm to incorporate other properties of solvent

Molecular Dynamics

Monte Carlo Simulations Monte Carlo simulation simply imposes relatively large random motions on the system and determines whether or not the altered structure is energetically feasible at the temperature simulated. The system jumps abruptly from conformation to conformation, rather than evolving smoothly through time. It can traverse barriers without feeling them; all that matters is the relative energy of the conformations before and after the jump

Monte Carlo Simulations Goal: Achieving an equilibrium distribution according to Boltzmann’s law R’ R E’E’ E

Monte Carlo: How It Works In any simulation, we can only change the system sequentially. We need to change system configuration R A to R B, to R C, and so on. Now the problem is how to realized the changes. In Monte Carlo methods, random change is attempted every time. How to generate a given equilibrium distribution, such as Boltzmann? This lies in the probability in accepting a random configuration. By definition, equilibrium means that probability of transition_(R A → R B ) = probability of transition_(R B → R A ). This gives the detailed balance condition for the acceptance scheme.

Conf. AConf. B Probability in accepting the move (A→B) Probability distribution of A Probability in accepting the move (B→A) Probability distribution of B Detailed Balance Condition

Metropolis Monte Carlo: Generating Boltzmann Distribution We can choose W(A  B)= 1, when U(A) > U(B), i.e. when it is a downhill energy change. How about W(B  A), when the energy goes up? We should accept the move in the following way: