University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Qun

by Qun Ma In post-genomic computational biology and bioinformatics, long simulations of the dynamics of molecular systems, particularly biological molecules such as proteins and DNA, require advances in time stepping computational methods. The most severe problem of these algorithms is instability. The objective of this dissertation is to present original work in constructing multiscale multiple time stepping (MTS) algorithms for molecular dynamics (MD) that allow large time steps. First, through nonlinear stability analysis and numerical experiments, we reveal that MTS integrators such as Impulse suffer nonlinear overheating when ∆t = T/3 or possibly ∆t = T/4 when constant–energy MD simulations are attempted, where ∆t is the longest step size and T is the shortest period of the modes in the system. Second, we present Targeted MOLLY (TM), a new multiscale integrator for MD simulations. TM combines an efficient implementation of B-spline MOLLY exploiting analytical Hessians of energies and a self–consistent dissipative leapfrog integrator. Results show that TM allows very large time steps for slow forces (and thus multiscale) for the numerically challenging flexible TIP3P water systems (Jorgensen, et al. J. Chem. Phys., vol 79, pp 926–935, 1983) while still computing the dynamical and structural properties accurately. Finally, we show yet another new MOLLY integrator, the Backward Euler (BE) MOLLY in which hydrogen bond forces can easily be included in the averaging and thus stability might be further improved. To my parents, my wife and my kids