The molecular dynamics of proteins and peptides is a hierarchical process which involves characteristic time scales ranging from 10-12 seconds to 100 seconds. Although the physical models of the local intramolecular interactions are relatively well developed, and molecular dynamics simulations have proven successful in recovering the dynamics of large-scale biomolecular systems, a mathematical understanding of how local interactions in the molecular root model give rise to a cascade of processes on different time scales is still lacking.
In this project we will investigate how these scaling cascades arise from the physical models of molecular dynamics and develop mathematical tools for their analysis. Our root model is a diffusion in a high-dimensional potential energy landscape that models the local interactions between atoms or groups of atoms. The local interactions in the molecular force .eld (i.e., the gradient of the potential energy) then induce long-range effects and may give rise to the observed long time scales on the order of seconds. Yet the predictability of molecular dynamics with respect to variations in the physical parameters (e.g., force .eld parameters) or boundary conditions (e.g., temperature) is remarkably poor, the reason being the nonlinearity, the large dimensionality of the models and noise present in the systems, which altogether promote large-scale effects induced by small noise or slow collective motions of atoms or groups of atoms.
For molecular systems with reversible dynamics, the relevant so-called implied time scales are related to the dominant eigenvalues of the underlying Markov generator. These eigenvalues can be estimated from molecular dynamics simulations and serve as approximations of experimentally measurable quantities. In molecular dynamics simulations it is possible to selectively tune the strength of a speci.c physical interaction (e.g., strength of long-range forces between different amino acids) or boundary conditions (e.g., temperature or pH), rendering them an ideal tool for analyzing the connection between root model and observed time scales. To investigate how the cascades of time scales arise in molecular dynamics we will extend numerical continuation methods for dynamical systems to stochastic molecular systems in order to study the changes in the implied time scales under variation of force .eld parameters or boundary conditions. We will compare analytical results to results from numerical simulations (classical and ab-initio molecular dynamics) and to results from infrared (IR) spectroscopy.
Despite its popularity in the protein folding community, implied time scales are only one possible way to quantify molecular dynamics time scales. For instance, the exponential convergence rate towards the thermodynamic equilibrium state is closely linked to experimentally measurable quantities. A second focus of the project is therefore to compare quantities which represent these relaxation time scales. To this end we will extend the numerical continuation approach to other observables, such as entropy production rates that, in certain cases, can be related to the shape of the molecular potential or Hankel singular values that characterize the response of the system to the environmental noise and can be related to the typical residence time of a conformation.
The understanding how scaling cascades in protein dynamics originate from the known hierarchy of physical interactions will be crucial for the development of multi-scale models, which consistently capture time scales on any desired level of coarseness. Moreover it will yield insight into biological phenomena such as allosteric regulation mechanisms or pathological misfolding events caused by single-point mutations.