Identifying mechanistically distinct pathways in kinetic transition networks

Abstract

We present an implementation of a scalable path deviation algorithm to find the k most kinetically relevant paths in a transition network, where each path is distinguished on the basis of having a distinct rate-limiting edge. The potential of the algorithm to identify distinct pathways that exist in separate regions of the configuration space is demonstrated for two benchmark systems with double-funnel energy landscapes, namely a model 'three-hole' network embedded on a 2D potential energy surface, and the cluster of 38 Lennard-Jones atoms (LJ 38). The path cost profiles for the interbasin transitions of the two systems reflect the contrasting nature of the landscapes. There are multiple well-defined pathway ensembles for the three-hole system, whereas the transition in LJ 38 effectively involves a single ensemble of pathways via disordered structures. A by-product of the algorithm is a set of edges that constitute a cut of the network, which is related to the discrete analogue of a transition dividing surface. The algorithm ought to be useful for determining the existence, or otherwise, of competing mechanisms in large stochastic network models of dynamical processes, and for assessing the kinetic relevance of distinguishable ensembles of pathways. This capability will provide insight into conformational transitions in biomolecules and other complex slow processes.

Energy Landscape for the Membrane Fusion Pathway in Influenza A Hemagglutinin From Discrete Path Sampling

“…The steps we have described above correspond to the fastest sequential path identified from the database. Overall rate constants can be computed from the infinite sum over discrete paths using the graph transformation approach (Trygubenko and Wales, 2006;Wales, 2009;Stevenson and Wales, 2014), and the k distinct paths algorithm (Sharpe and Wales, 2019) FIGURE 4 | Disconnectivity graph for the HA2 system marking the locations of the same selected minima from the pathway shown in Figure 3. can be employed to distinguish pathways with different ratedetermining steps. In this report we simply provide a qualitative account of the essential components we have identified in the fastest pathway.…”

Rare events and first passage time statistics from the energy landscape

“…6 is likely more efficient than using a transfer matrix algorithm, [63][64][65] where the time complexity is strongly dependent on the average connectivity of nodes and the maximum path length for which p(t) converges. Although KTNs are typically sparse, 66 flickering of trajectories within metastable groups of nodes can lead to unfeasibly long path lengths. 67 The detailed balance condition for the individual minimum-to-minimum rate constants, π β k βα = π α k αβ , enables us to define a symmetrised rate matrix 68…”

“…This system has two competing morphologies that define a double funnel landscape (Figure 3), 78,[80][81][82] which has made it a useful benchmark for global optimisation, [83][84][85][86] enhanced thermodynamic sampling to overcome broken ergodicity, 6,81,83,[87][88][89][90][91][92][93][94][95] and rare event dynamics. 4,5,66,81,[96][97][98] This system is amenable to detailed analysis, but it is complex enough to pose challenging problems if we attempt to simplify the configuration space by defining projections such as reaction coordinates. 99 In treating the rates and thermodynamics of this system we normally consider a set of five minima for the funnel containing the global minimum truncated octahedron, point group O h , and 395 minima for the funnel based on an incomplete Mackay icosahedron, which has point group C 5v .…”

“…The pathway making the largest contribution to k SS at different temperatures was determined using Dijkstra's shortest path algorithm 113 for edge weights calculated from the corresponding branching probabilities. 4,66,114 Regrouping minima from these paths, especially the highest energy structures, can affect the overall barrier between groups for the most kinetically relevant paths. Excluding selected minima from these paths therefore enables larger regrouping thresholds to be used without affecting the overall barrier for the fastest pathways.…”

“…The reduction in dimensionality results in improved numerical precision in the eigendecomposition. We note that minima could be excluded from regrouping based on more general analysis of distinct pathways classified by the rate-determining step, 66 and we will test this extension in future work.…”