Adaptive dimensionality reduction of stochastic differential equations for protein dynamics

The dynamics of proteins can be described as the superposition of motions at a continuum of time scales. In the special case of a protein immersed in an implicit solvent, a stochastic differential equation (SDE) can model the dynamics of the solute protein. Traditional model reduction techniques fail because a priori characterization of the slow variables in these SDEs is nearly impossible. We present an approach that instead, does a local dimensionality reduction of the SDE in a neighborhood of phase space, which is adaptively performed when the reduced model is no longer valid. The local slow variables, which we call approximate normal modes (ANM), are found using the diagonalization of a coarse-grained Hessian (CGH) from the potential energy function. We call this procedure coarsegrained normal mode analysis, or CNMA. Diagonalization of the CGH can be achieved in O(N log N) time and O(N) memory rather than O(N) time and O(N) memory of ordinary diagonalization. CNMA is able to capture the low frequency motions of the protein. An SDE on the ANM is found by using a saddle-point approximation of the mean fast-frequency force experienced by the slow variables, and an implicit solvent model that considers the protein as a Brownian particle. This mean force can be computed at a cost no greater than a fine-grained force evaluation. Discretization of the resulting SDE achieves very long time steps compared to the discretization of the fine-grained SDE. A metric is used to monitor the validity of the ANM as slow variables and prompt re-diagonalization of the CGH or adaptation of the time step used. I will present numerical results on small peptide and protein system that show that this coarse-graining scheme allows up to three orders of magnitude speedup due to increase in the SDE discretization time step, and that the scheme is able to preserve kinetics when compared to the fine-grained SDE.