Non-equilibrium generalized Langevin equation for multi-dimensional observables

Abstract

The Mori-Zwanzig formalism is a powerful theoretical framework for deriving equations of motion for coarse-grained observables in the form of generalized Langevin equations (GLEs) involving evolution and projection operators. Using a time-dependent many-body Hamiltonian and a multi-dimensional Mori projection operator, we derive a non-equilibrium Mori GLE for a multi-dimensional observable of interest $\vec{A}$ that consists of a Markovian force, a running integral over time of a non-Markovian friction force, and an orthogonal force that is often interpreted as a random force. We study the structure of the derived GLE in three limiting cases: when the components of $\vec{A}$ are uncorrelated, when the Hamiltonian is time-independent and thus the system is at equilibrium, and when both conditions are simultaneously satisfied. We highlight the presence of a contribution to the Markovian force that takes the form of an instantaneous friction force which only vanishes when the components of $\vec{A}$ are uncorrelated. Our non-Markovian framework is an important step towards the systematic modeling of the coupled kinetics of coarse-grained reaction coordinates in biological complex systems, exemplified for the coupled intra- and inter-protein folding during fibril formation of the human islet amyloid polypeptide (IAPP).

Figures (1)

• Figure 1: We show results from a 10 microsecond unconstrained molecular dynamics equilibrium simulation of a periodically replicated fibril of human islet amyloid polypeptide (IAPP) in explicit water. A Side and top views of a human islet amyloid polypeptide (IAPP) fibril composed of 12 IAPP monomers, shown alongside B a schematic representation of the IAPP fibril formation process, in which initially disordered monomers form $\beta$-sheet-rich structures and assemble into amyloid fibrils. Experimental TEM micrographs of IAPP fibrils formed at pH values of 6.8 (C) and 7.4 (D). E Two-dimensional free energy landscape $U(A_{1}, A_{2}) \equiv - \frac{1}{\beta} \log \langle \delta ( A_{1}(0, \vec{r}) - A_{1} ) \delta ( A_{2}(0, \vec{r}) - A_{2} ) \rangle$ as a function of the number of intra-layer hydrogen bonds $A_1$ and the distance between the centers of mass of two adjacent layers $A_2$. F Time correlation function $\hat{C}_{12}^{eq}(t) \equiv \langle ( A_{1}(t, \hat{\vec{w}}) - \langle A_{1}(0, \Tilde{\vec{r}}) \rangle ) ( A_{2}(0, \hat{\vec{r}}) - \langle A_{2}(0, \Tilde{\vec{r}}) \rangle ) \rangle$ between the two reaction coordinates $A_1$ and $A_2$.