String Method for Calculation of Minimum Free-Energy Paths in Cartesian Space in Freely Tumbling Systems

Abstract

The string method is a molecular-simulation technique that aims to calculate the minimum free-energy path of a chemical reaction or conformational transition, in the space of a pre-defined set of reaction coordinates that is typically highly dimensional. Any descriptor may be used as a reaction coordinate, but arguably the Cartesian coordinates of the atoms involved are the most unprejudiced and intuitive choice. Cartesian coordinates, however, present a non-trivial problem, in that they are not invariant to rigid-body molecular rotations and translations, which ideally ought to be unrestricted in the simulations. To overcome this difficulty, we reformulate the framework of the string method to integrate an on-the-fly structural-alignment algorithm. This approach, referred to as SOMA (String method with Optimal Molecular Alignment), enables the use of Cartesian reaction coordinates in freely tumbling molecular systems. In addition, this scheme permits the dissection of the free-energy change along the most probable path into individual atomic contributions, thus revealing the dominant mechanism of the simulated process. This detailed analysis also provides a physically-meaningful criterion to coarse-grain the representation of the path. To demonstrate the accuracy of the method we analyze the isomerization of the alanine dipeptide in vacuum and the chair-to-inverted-chair transition of β-D mannose in explicit water. Notwithstanding the simplicity of these systems, the SOMA approach reveals novel insights into the atomic mechanism of these isomerizations. In both cases, we find that the dynamics and the energetics of these processes are controlled by interactions involving only a handful of atoms in each molecule. Consistent with this result, we show that a coarse-grained SOMA calculation defined in terms of these subsets of atoms yields nearidentical minimum free-energy paths and committor distributions to those obtained via a highly-dimensional string.

Atomic‐resolution dissection of the energetics and mechanism of isomerization of hydrated ATP‐Mg2+ through the SOMA string method

“…In the SM, a transition path between two states is parameterized as a series of configurations in the space of a set of CVs. SOMA is a variation of this method in which the space of CVs is that of the Cartesian coordinates of a subset of atoms in the system after these atoms have been optimally aligned, via a rigid‐body roto‐translation, onto an arbitrary reference configuration. That is, where n is the number of atoms used to define the string and the superscript r indicates that this set of coordinates are roto‐translated onto the reference frame.…”

“…To address these two drawbacks, we have recently developed an ES methodology, we refer to as string method with optimal molecular alignment (SOMA) . Like other flavors of the String Method (SM) and other ES methods focused on transition paths, SOMA aims to identify and characterize the minimum free‐energy path (MFEP) between two well‐defined states, which may be defined in a highly dimensional space.…”

“…First, an ES simulation method known as bias‐exchange metadynamics is utilized to explore the conformational space of ATP‐Mg 2+ , so as to define all metastable states and their connectivity, and to obtain an approximate ranking in free energy. SOMA is then utilized to characterize the energetics and mechanism of each of the major transitions between states. Finally, the MFEP identified by SOMA are reanalyzed using a related ES method, namely Metadynamics in path collective variables (PCVs), which allows us to examine entropy variations along the MFEP.…”

“…To address these two drawbacks, we have recently developed an ES methodology, we refer to as string method with optimal molecular alignment (SOMA). [13] Like other flavors of the String Method (SM) [14][15][16] and other ES methods focused on transition paths, [17][18][19][20] SOMA aims to identify and characterize the minimum free-energy path (MFEP) between two welldefined states, which may be defined in a highly dimensional space. Unlike other versions of the SM, however, SOMA does not use CVs in the traditional sense; instead, it is rigorously formulated in the space of the Cartesian coordinates of all the atoms considered to be involved in the transition or reaction of interest.…”

Charting molecular free-energy landscapes with an atlas of collective variables

“…We apply the proposed methodology to a different benchmark molecule, β-D-Glucopyranose, 4,5 known to have the topology of the sphere. 52 For this six-membered ring, the Cremer-Pople puckering coordinates 53 reduce to spherical coordinates (Q, θ, φ), and adequately describe its molecular flexibility with small fluctuations about the "radius" Q.…”