Analyzing milestoning networks for molecular kinetics: Definitions, algorithms, and examples

Abstract

Network representations are becoming increasingly popular for analyzing kinetic data from techniques like Milestoning, Markov State Models, and Transition Path Theory. Mapping continuous phase space trajectories into a relatively small number of discrete states helps in visualization of the data and in dissecting complex dynamics to concrete mechanisms. However, not only are molecular networks derived from molecular dynamics simulations growing in number, they are also getting increasingly complex, owing partly to the growth in computer power that allows us to generate longer and better converged trajectories. The increased complexity of the networks makes simple interpretation and qualitative insight of the molecular systems more difficult to achieve. In this paper, we focus on various network representations of kinetic data and algorithms to identify important edges and pathways in these networks. The kinetic data can be local and partial (such as the value of rate coefficients between states) or an exact solution to kinetic equations for the entire system (such as the stationary flux between vertices). In particular, we focus on the Milestoning method that provides fluxes as the main output. We proposed Global Maximum Weight Pathways as a useful tool for analyzing molecular mechanism in Milestoning networks. A closely related definition was made in the context of Transition Path Theory. We consider three algorithms to find Global Maximum Weight Pathways: Recursive Dijkstra's, Edge-Elimination, and Edge-List Bisection. The asymptotic efficiency of the algorithms is analyzed and numerical tests on finite networks show that Edge-List Bisection and Recursive Dijkstra's algorithms are most efficient for sparse and dense networks, respectively. Pathways are illustrated for two examples: helix unfolding and membrane permeation. Finally, we illustrate that networks based on local kinetic information can lead to incorrect interpretation of molecular mechanisms.

Permeation of the three aromatic dipeptides through lipid bilayers: Experimental and computational study

“…25 Previous studies of tryptophan using the more advanced method of milestoning report permeation times on the time scale of hours, which are also in agreement with experimental results. 43,44,85,86 Most other studies have also reported much larger and faster permeability coefficients using the inhomogeneous solubilitydiffusion model, which has been discussed in great detail in other works. 18,25,43,44,[87][88][89] Several assumptions underpin the solubility-diffusion model, including memoryless, diffusivetype motion along the reaction coordinate and the presence of only one slow variable describing the motion of the permeant-namely, the translocation along the membrane normal.…”

Calculating Iso-Committor Surfaces as Optimal Reaction Coordinates with Milestoning

“…We are interested in conversions of reactants to products and Milestoning is a technique to sample these transitions efficiently. In the past we showed how Milestoning can be used to determine pathways of maximum flux [13]. Here we study another reaction coordinate (the committor).…”

“…Examples of methods that provides a one-dimensional view of the reaction progress include the Locally Updated Planes (LUP [8]), Nudged Elastic Band (NEB [9]), and the scalar force [10]. Other approaches are the MaxFlux approach [11–13], the String [14], a temperature dependent reaction coordinate [15], and the Dominant Reaction Pathway (DRP [16]). The above approaches provide Minimum Energy Paths, or thermal paths that are locally averaged.…”

“…Both types of reaction coordinates are computed in continuous space, or on a network [13,18,19]. We further comment that the minimum energy path (MEP) and MFEP may encounter the problem of path degeneracy or multiplicity.…”

Technical advances in molecular simulation since the 1980s

“…TPS methods are computationally expensive but they allow, in principle, the computation of both the free energies and rate constants of a process. Other algorithms of similar type include milestoning [86,87], weighted ensemble path sampling [88], and generalizations of the string method [89].…”