Accurate estimates of dynamical statistics using memory.

Many chemical reactions and molecular processes occur on time scales that are significantly longer than those accessible by direct simulations. One successful approach to estimating dynamical statistics for such processes is to use many short time series of observations of the system to construct a Markov state model, which approximates the dynamics of the system as memoryless transitions between a set of discrete states. The dynamical Galerkin approximation (DGA) is a closely related framework for estimating dynamical statistics, such as committors and mean first passage times, by approximating solutions to their equations with a projection onto a basis. Because the projected dynamics are generally not memoryless, the Markov approximation can result in significant systematic errors. Inspired by quasi-Markov state models, which employ the generalized master equation to encode memory resulting from the projection, we reformulate DGA to account for memory and analyze its performance on two systems: a two-dimensional triple well and the AIB9 peptide. We demonstrate that our method is robust to the choice of basis and can decrease the time series length required to obtain accurate kinetics by an order of magnitude.

Inexact iterative numerical linear algebra for neural network-based spectral estimation and rare-event prediction.

Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics, such as the likelihood and average time of events (predictions). Here, we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a dataset of short trajectories sampled at finite intervals. We demonstrate the methods on a low-dimensional model that facilitates visualization and a high-dimensional model of a biomolecular system. Implications for the prediction problem in reinforcement learning are discussed.

Predicting rare events using neural networks and short-trajectory data.

Estimating the likelihood, timing, and nature of events is a major goal of modeling stochastic dynamical systems. When the event is rare in comparison with the timescales of simulation and/or measurement needed to resolve the elemental dynamics, accurate prediction from direct observations becomes challenging. In such cases a more effective approach is to cast statistics of interest as solutions to Feynman-Kac equations (partial differential equations). Here, we develop an approach to solve Feynman-Kac equations by training neural networks on short-trajectory data. Our approach is based on a Markov approximation but otherwise avoids assumptions about the underlying model and dynamics. This makes it applicable to treating complex computational models and observational data. We illustrate the advantages of our method using a low-dimensional model that facilitates visualization, and this analysis motivates an adaptive sampling strategy that allows on-the-fly identification of and addition of data to regions important for predicting the statistics of interest. Finally, we demonstrate that we can compute accurate statistics for a 75-dimensional model of sudden stratospheric warming. This system provides a stringent test bed for our method.

Kinetics of Phenol Escape from the Insulin R6 Hexamer.

Therapeutic preparations of insulin often contain phenolic molecules, which can impact both pharmacokinetics and shelf life. Thus, understanding the interactions of insulin and phenolic molecules can aid in designing improved therapeutics. In this study, we use molecular dynamics to investigate phenol release from the insulin hexamer. Leveraging recent advances in methods for analyzing molecular dynamics data, we expand on existing simulation studies to identify and quantitatively characterize six phenol binding/unbinding pathways for wild-type and A10 Ile → Val and B13 Glu → Gln mutant insulins. A number of these pathways involve large-scale opening of the primary escape channel, suggesting that the hexamer is much more dynamic than previously appreciated. We show that phenol unbinding is a multipathway process, with no single pathway representing more than 50% of the reactive current and all pathways representing at least 10%. We use the mutant simulations to show how the contributions of specific pathways can be rationally manipulated. Predicting the net effects of mutations is more challenging because the kinetics depend on all of the pathways, demanding quantitatively accurate simulations and experiments.

Long-Time-Scale Predictions from Short-Trajectory Data: A Benchmark Analysis of the Trp-Cage Miniprotein.

Elucidating physical mechanisms with statistical confidence from molecular dynamics simulations can be challenging owing to the many degrees of freedom that contribute to collective motions. To address this issue, we recently introduced a dynamical Galerkin approximation (DGA) [Thiede, E. H. J. Chem. Phys., 150, 2019, 244111], in which chemical kinetic statistics that satisfy equations of dynamical operators are represented by a basis expansion. Here, we reformulate this approach, clarifying (and reducing) the dependence on the choice of lag time. We present a new projection of the reactive current onto collective variables and provide improved estimators for rates and committors. We also present simple procedures for constructing suitable smoothly varying basis functions from arbitrary molecular features. To evaluate estimators and basis sets numerically, we generate and carefully validate a data set of short trajectories for the unfolding and folding of the trp-cage miniprotein, a well-studied system. Our analysis demonstrates a comprehensive strategy for characterizing reaction pathways quantitatively.

Modulating absorption and charge transfer in bodipy-carbazole donor-acceptor dyads through molecular design.

Three bodipy-based (BDP = 4,4-difluoro-4-bora-3a,4a-diaza-s-indacene) donor-acceptor dyads were designed and synthesized, and their ground-state and photophysical properties were systematically characterized. The electronic coupling between the BDP chromophore and an electron-donating carbazole (Carb) moiety was tuned by attachment via the meso and the beta positions on the BDP core, and through the use of various chemical linkers (phenyl and alkynyl) to afford mesoBDP-Carb, mesoBDP-phen-Carb, and betaBDP-alk-Carb. meso-Substituted dyads were found to retain ground-state absorption features of the unsubstituted BDP. However, variation of the linkage between the donor and acceptor moieties modulated the photophysical behavior of excited-state deactivation by controlling the rate of photoinduced internal charge transfer (ICT). The beta-substituted dyad dramatically tuned (red shifted) the absorption spectrum, while retaining desired features of the BDP, specifically stability and high extinction coefficients, however the ICT kinetics were accelerated compared to the meso-substituted dyads. Density functional theory (DFT) and time-dependent DFT (TDDFT) were carried out on the six potential dyads formed between BDP and Carb (attachment using the beta and meso positions for all three connections: direct, phenyl and alkynyl) to support the experimental observations. DFT and TDDFT showed molecular orbital density spread across the HOMO level only when attachment occurred through the beta position of BDP. In the meso-substituted BDP-Carb dyads, the molecular orbitals resembled those of the unsubstituted BDP. This work reveals several possible synthetic paradigms to tune photophysical properties with directed synthetic modifications and provides a mechanistic understanding of the ground- and excited- state behavior in these small-molecule donor-acceptor dyads.

Excited State Characterization of Carborane-Containing Poly(dihexyl fluorene)s.

Carborane-containing poly(dihexylfluorene)s experience drastic solvatochromism in both the solution and solid states, a characteristic that is advantageous for use in environmental and biological sensing applications. Understanding the intrinsic decay mechanisms that give rise to such sensitive emission properties is important for designing responsive sensors. The solution-state photophysical properties of homopolymer, poly(9,9-dihexyl(bisfluorenyl)carborane) (PFCY), and alternating copolymer, poly(9,9-dihexyl-2,7-fluorene- alt-9,9-dihexyl(bisfluorenyl)carborane) (PFCS), were deciphered using steady-state, electrochemical, spectroelectrochemical, and time-resolved spectroscopic methods. From these techniques, it was discovered that following excitation the conjugated fluorene local excited state (LES) donates an electron to the carborane molecule, forming an intramolecular charge transfer (ICT) state between a radical cation on the fluorene moiety and a radical anion on the carborane moiety. From the global analysis of transient absorption data, it was discovered that the rate of electron transfer from the fluorene to the carborane is heavily influenced by solvent polarity and is significantly faster in more polar solvents. Once formed, the ICT state can decay through radiative or nonradiative mechanisms and is more likely to undergo radiative decay in nonpolar solvents, due to an intramolecular restriction of the polar ICT state. This study elucidates the effects that polarity has on the excited-state formation and subsequent decay mechanisms of fluorene-carborane systems, conclusively explaining the solvatochromism and steady-state emission properties exhibited by this system.