We have noted above that current techniques to produce multiscale integrators for MD, such as Verlet-I/r-RESPA, are limited by unstable resonances. Our recent discovery of nonlinear instabilities in Verlet-I/r-RESPA suggests that it has limited applicability to long time simulations, such as those needed for protein dynamics, since the drift in the energy makes the results meaningless. It is worth noting that the longest MD simulation to date, that of Kollman and co-workers for one microsecond [28], used temperature controls, and thus was not a constant energy simulation. Even our current mollified version of the method suffers these nonlinear resonances, though they happen at larger time steps. We have outlined a possible solution by having a parameterized family of methods, where the parameters are adjusted to vanish the terms that induce instability.
Here we present a more radical solution. Our method uses semi-implicit, or mixed
implicit-explicit methods for MD, an MTS formulation with a good separation of
time scales, and constrained or mollified MD to eliminate the fastest
vibrations and preserve the molecular structure with long time steps. We
discuss the motivation for the first two components here, and the third one in
Section 3.2.