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B05 - Origin of scaling cascades in protein dynamics

Head(s): Prof. Dr. Bettina Keller (FU Berlin), PD Dr. Marcus Weber (ZIB)
Project member(s): Dr. Daniel Baum, Dr. Luca Donati, Irtaza Hassan, Dr. Shreetama Karmakar, Stefanie Kieninger
Participating institution(s): FU Berlin, ZIB

Project Summary

Proteins are characterized by a hierarchy of dynamical processes which shape their macroscopic properties. The aim of B05 is to develop a mathematical understanding of how a small and local variation in the molecular interaction potential gives rise to a cascade of processes and ultimately the slow processes of the system.
Chemists already have a good working knowledge on how a change in a molecular structure translates into a change in the stationary populations of the molecular conformations. The relation between molecular structure and stationary density is usually rationalized in terms of the thermodynamic state functions enthalpy, entropy and free energy. Additionally, Arrhenius’ equation or Kramers’ equation provide a framework to link molecular structures to kinetics, but only for two-state-processes separated by a barrier along a clear reaction coordinate. For multi-state dynamics on a rugged (free) energy surface with a hierarchy of energy barriers a
similar framework is lacking.
Numerical techniques to link the molecular structure to kinetics for molecules with complex multi-state dynamics have become accessible in the past two decades. They rely on discretizing the dynamical operator of the molecular dynamics and evaluating its matrix elements through simulations of the associated stochastic process. For example, in Markov state models molecular-dynamics simulations are conducted at the molecular interaction potential and the matrix elements are estimated as time-lagged correlation function. However, the waiting time to cross a barrier increases exponentially with barrier height, and the sampled estimates of the matrix elements are consquently subject to large variances. Waiting times can be decreased by biasing the simulation, but this distorts the estimate of the matrix elements. Path reweighting is an in principle exact method to recover the correct dynamics from biased simulations. In the past two funding periods, we have shown that path reweighting is tractable for molecular simulations biased by a metadynamics potential and thermostatted by a Langevin thermostat. This changes the role of molecular simulations. In so far as MD simulations are used as a numerical technique to estimate matrix elements, there is no need to sample at the actual molecular potential. One can, and in fact often should, estimate the matrix elements by biasing the simulation and reweighting the correlation functions using path reweighting.
In the third funding period, we focus on a more direct approach to discretize a dynamical operator.  In project B05, we developed the Square-root approximation (SqRA) of an infinitesimal generator. In this method, the molecular state space is discretized by a grid, and the transition probability density between adjacent grid cells are approximated using Gauss’s divergence theorem. One thus avoids sampling the matrix elements. The matrix elements represent transition rates and the discretized infinitesimal generator is a rate matrix. During the second funding period, we applied the SqRA to alanine dipeptide, and we derived expressions for the rates for different grid geometries. In the third funding period, we will extend the SqRA to highdimensional systems and thus larger molecules by following two approaches: (i) discretizing the full state space using a sparse grid, (ii) projecting the dynamics onto a low-dimensional space of collective variables and discretizing the effective dynamics in this space. We will analyze how the effective dynamics depends on the choice of collective variables. This will yield a numerical method to obtain a rate matrix which is (largely) independent of long-term simulations.
Moreover, the SqRA offers two leverage points to answer the motivating question of this project. First, the algebraic structure of the resulting rate matrix can be decomposed into terms that represents the connectivity of molecular states and terms that depend on the stationary density. We will relate these terms to the thermodynamics state functions entropy, enthalpy and/or free energy, thereby establishing an intuitive link between molecular structure and rate matrix. Second, given a connectivity of the molecular states, the rate matrix only depends on the stationary density on this state graph. Thus, also the dominant eigenfunctions of the dynamical operator are entirely determined by the stationary density. We will use the Min-Max theorem by Courant and Fischer to establish an intuitive connection between features of the stationary density (e.g., critical points) and the slow molecular processes.

Publications B05