Capturing Subdiffusive Solute Dynamics and Predicting Selectivity in Nanoscale Pores with Time Series Modeling

Abstract

Fitting mathematical models with a direct connection to experimental observables to the outputs of molecular simulations can be a powerful tool for extracting important physical information from them. In this study, we present two new approaches that use stochastic time series modeling to predict long-time-scale behavior and macroscopic properties from molecular simulation, which can be generalized to other molecular systems where complex diffusion occurs. In our previous work, we studied long molecular dynamics (MD) simulation trajectories of a cross-linked H II phase lyotropic liquid crystal (LLC) membrane, where we observed subdiffusive solute transport behavior characterized by intermittent hops separated by periods of entrapment. In this work, we use our models to parameterize the behavior of the same systems, so we can generate characteristic trajectory realizations that can be used to predict solute mean-squared displacements (MSDs), solute flux, and solute selectivity in macroscopic length pores. First, using anomalous diffusion theory, we show how solute dynamics can be modeled as a fractional diffusion process subordinate to a continuous time random walk. From the MD simulations, we parameterize the distribution of dwell times, hop lengths between dwells, and correlation between hops. We explore two variations of the anomalous diffusion modeling approach. The first variation applies a single set of parameters to the solute displacements and the second applies two sets of parameters based on the solute's radial distance from the closest pore center. Next, we present an approach that generalizes Markov state models, treating the configurational states of the system as a Markov process where each state has distinct transport properties. For each state and transition between states, we parameterize the distribution and temporal correlation structure of positional fluctuations as a means of characterization and to allow us to predict solute MSDs. We show that both stochastic models reasonably reproduce the MSDs calculated from MD simulations. However, qualitative differences between MD and Markov state-dependent model-generated trajectories may in some cases limit their usefulness. With these parameterized stochastic models, we demonstrate how one can estimate the flux of a solute across a macroscopic length pore and, based on these quantities, the membrane's selectivity toward each solute. This work therefore helps to connect microscopic, chemically dependent solute motions that do not follow simple diffusive behavior with long-time-scale behavior, in an approach generalizable to many types of molecular systems with complex dynamics.

Statistical Inference of Transport Mechanisms and Long Time Scale Behavior from Time Series of Solute Trajectories in Nanostructured Membranes

“…In our previous work, we showed that in systems without convective flux, such as our MD simulations, selectivity toward component i over j is simply: where J is solute flux and ΔC is the trans-membrane concentration difference . The concentrations of solutes are the same across our simulations, therefore eq reduces to the ratio of solute fluxes.…”

“…11 Based on the chemical intuition gained from our previous qualitative MD studies of solute transport in an H II phase LLC membrane, we have contributed our own stochastic modeling approaches with goals similar to both TEKMC and DBP. 7,12 In previous studies, we identified three classes of trapping behavior which result in subdiffusion out to the microsecond time scale. In our first approach to stochastically model this behavior, we treated the system's dynamics as a sequence of anticorrelated hops between power law distributed periods of entrapment, a framework called subordinated fractional Brownian or Levy motion.…”

“…Based on the chemical intuition gained from our previous qualitative MD studies of solute transport in an H II phase LLC membrane, we have contributed our own stochastic modeling approaches with goals similar to both TEKMC and DBP. , In previous studies, we identified three classes of trapping behavior which result in subdiffusion out to the microsecond time scale. In our first approach to stochastically model this behavior, we treated the system’s dynamics as a sequence of anticorrelated hops between power law distributed periods of entrapment, a framework called subordinated fractional Brownian or Lévy motion. , In a second approach, we treated solute motion as a Markov state model with state-dependent dynamics, where we parametrized the state transition probabilities between each of eight discrete states, defined by the observed trapping mechanisms, as well as the solute dynamics within each of these states.…”

“…where J is solute flux and ΔC is the trans-membrane concentration difference. 12 The concentrations of solutes are the same across our simulations, therefore eq 4 reduces to the ratio of solute fluxes. Solute flux, as written, can be expressed in terms of the product of solute diffusivity, D i , within the membrane and a sorption coefficient, K i , that quantifies the ability of the solute to move between the bulk solution and the membrane:…”

“…We can apply a similar analysis to the other solutes in this study and learn new information by comparing their behavior. Where referenced in the following analysis, we define the predominant states exhibited by each solute as any state visited by at least half (12) of the solute trajectories. By our definition and based on our choice to study 10 clusters for each solute, methanol, ethylene glycol, urea, and acetic acid are characterized by 5, 6, 5, and 4 predominant states respectively (see Figure S16 of the Supporting Information).…”