Hierarchical Order Parameters for Macromolecular Assembly Simulations. 1. Construction and Dynamical Properties of Order Parameters

Abstract

Macromolecular assemblies often display a hierarchical organization of macromolecules or their sub-assemblies. To model this, we have formulated a space warping method that enables capturing overall macromolecular structure and dynamics via a set of coarse-grained order parameters (OPs). This article is the first of two describing the construction and computational implementation of an additional class of OPs that has built into them the hierarchical architecture of macromolecular assemblies. To accomplish this, first, the system is divided into subsystems, each of which is described via a representative set of OPs. Then, a global set of variables is constructed from these subsystem-centered OPs to capture overall system organization. Dynamical properties of the resulting OPs are compared to those of our previous nonhierarchical ones, and implied conceptual and computational advantages are discussed for a 100ns, 2 million atom solvated Human Papillomavirus-like particle simulation. In the second article, the hierarchical OPs are shown to enable a multiscale analysis that starts with the N-atom Liouville equation and yields rigorous Langevin equations of stochastic OP dynamics. The latter is demonstrated via a force-field based simulation algorithm that probes key structural transition pathways, simultaneously accounting for all-atom details and overall structure.

A Review of Two Multiscale Methods for the Simulation of Macromolecular Assemblies: Multiscale Perturbation and Multiscale Factorization

“…Here, µ k serves as an effective mass associated with Φ k and is proportional to the square of the basis vector's length. The masses primarily decrease with increasing complexity of U S k [32,44]. Thus, the OPs with higher k probe smaller regions in space.…”

“…As in [19,44], OPs labeled by indices k = {000, 100, 010, 001} are denoted as lower-order, while k > {000, 100, 010, 001} are higher-order. Notice that basis functions do not depend on each atomic position r i , but rather on the intermediate scale variables, R, thereby ensuring a hierarchical foundation.…”

“…In particular, the OP and CM velocity autocorrelation functions provide a criterion for the applicability of the present multiscale approach. If the reduced description is complete, i.e., the set of OPs and CMs considered do not couple strongly with other slow variables, then the correlation functions decay on a time scale much shorter than the characteristic time(s) of OP evolution [44]. However, if some slow modes are not included in the set of OPs, then these correlation functions can decay on timescales comparable to those of OP dynamics [44,45].…”

“…If the reduced description is complete, i.e., the set of OPs and CMs considered do not couple strongly with other slow variables, then the correlation functions decay on a time scale much shorter than the characteristic time(s) of OP evolution [44]. However, if some slow modes are not included in the set of OPs, then these correlation functions can decay on timescales comparable to those of OP dynamics [44,45]. This is because the missing slow modes, now expressed through the all-atom dynamics, couple with the adopted set of OPs, and the present approach fails under such conditions.…”

“…This ensemble is employed to construct the diffusion factors, D kk , and thermal-averaged forces, f k . Further details regarding the ensemble generation procedure and construction of these factors are provided in [44]. Using the forces and diffusions, the OPs and subsystem CMs are evolved via Langevin equations to capture overall assembly deformation.…”