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

Using pretrained graph neural networks with token mixers as geometric featurizers for conformational dynamics

Identifying informative low-dimensional features that characterize dynamics in molecular simulations remains a challenge, often requiring extensive manual tuning and system-specific knowledge. Here, we introduce geom2vec, in which pretrained graph neural networks (GNNs) are used as universal geometric featurizers. By pretraining equivariant GNNs on a large dataset of molecular conformations with a self-supervised denoising objective, we obtain transferable structural representations that are useful for learning conformational dynamics without further fine-tuning. We show how the learned GNN representations can capture interpretable relationships between structural units (tokens) by combining them with expressive token mixers. Importantly, decoupling training the GNNs from training for downstream tasks enables analysis of larger molecular graphs (that can represent small proteins at all-atom resolution) with limited computational resources. In these ways, geom2vec eliminates the need for manual feature selection and increases the robustness of simulation analyses.

Using pretrained graph neural networks with token mixers as geometric featurizers for conformational dynamics

Identifying informative low-dimensional features that characterize dynamics in molecular simulations remains a challenge, often requiring extensive manual tuning and system-specific knowledge. Here, we introduce geom2vec, in which pretrained graph neural networks (GNNs) are used as universal geometric featurizers. By pretraining equivariant GNNs on a large dataset of molecular conformations with a self-supervised denoising objective, we obtain transferable structural representations that are useful for learning conformational dynamics without further fine-tuning. We show how the learned GNN representations can capture interpretable relationships between structural units (tokens) by combining them with expressive token mixers. Importantly, decoupling training the GNNs from training for downstream tasks enables analysis of larger molecular graphs (such as small proteins at all-atom resolution) with limited computational resources. In these ways, geom2vec eliminates the need for manual feature selection and increases the robustness of simulation analyses.

Analysis of the Dynamics of a Complex, Multipathway Reaction: Insulin Dimer Dissociation

The protein hormone insulin forms a homodimer that must dissociate to bind to its receptor. Understanding the kinetics and mechanism of dissociation is essential for the rational design of therapeutic analogs. In addition to its physiological importance, this dissociation process serves as a paradigm for coupled (un)folding and (un)binding. Based on previous free energy simulations, insulin dissociation is thought to involve multiple pathways with comparable free energy barriers. Here, we analyze the mechanism of insulin dimer dissociation using a recently developed computational framework for estimating kinetic statistics from short-trajectory data. These statistics indicate that the likelihood of dissociation (the committor) closely tracks the decrease in the number of (native and nonnative) intermonomer contacts and the increase in the number of water contacts at the dimer interface; the transition state with equal likelihood of association and dissociation corresponds to an encounter complex with relatively few native contacts and many nonnative contacts. We identify four pathways out of the dimer state and quantify their contributions to the rate as well as their exchange by computing reactive fluxes. We show that both the pathways and their extents of exchange can be understood in terms of rotations around three axes of the dimer structure. Our results provide insights into the kinetics of insulin analogs and, more generally, how to characterize complex, multipathway processes.

Temperature-dependent fold-switching mechanism of the circadian clock protein KaiB

The oscillator of the cyanobacterial circadian clock relies on the ability of the KaiB protein to switch reversibly between a stable ground-state fold (gsKaiB) and an unstable fold-switched fold (fsKaiB). Rare fold-switching events by KaiB provide a critical delay in the negative feedback loop of this posttranslational oscillator. In this study, we experimentally and computationally investigate the temperature dependence of fold switching and its mechanism. We demonstrate that the stability of gsKaiB increases with temperature compared to fsKaiB and that the Q10 value for the gsKaiB → fsKaiB transition is nearly three times smaller than that for the reverse transition in a construct optimized for NMR studies. Simulations and native-state hydrogen-deuterium exchange NMR experiments suggest that fold switching can involve both partially and completely unfolded intermediates. The simulations predict that the transition state for fold switching coincides with isomerization of conserved prolines in the most rapidly exchanging region, and we confirm experimentally that proline isomerization is a rate-limiting step for fold switching. We explore the implications of our results for temperature compensation, a hallmark of circadian clocks, through a kinetic model.

Committor Guided Estimates of Molecular Transition Rates

The probability that a configuration of a physical system reacts, or transitions from one metastable state to another, is quantified by the committor function. This function contains richly detailed mechanistic information about transition pathways, but a full parametrization of the committor requires the construction of a high-dimensional function, a generically challenging task. Recent efforts to leverage neural networks as a means to solve high-dimensional partial differential equations, often called "physics-informed" machine learning, have brought the committor into computational reach. Here, we build on the semigroup approach to learning the committor and assess its utility for predicting dynamical quantities such as transition rates. We show that a careful reframing of the objective function and improved adaptive sampling strategies provide highly accurate representations of the committor. Furthermore, by directly applying the Hill relation, we show that these committors provide accurate transition rates for molecular systems.

Analysis of the dynamics of a complex, multipathway reaction: Insulin dimer dissociation

The protein hormone insulin forms a homodimer that must dissociate to bind to its receptor. Understanding the kinetics and mechanism of dissociation is essential for rational design of therapeutic analogs. In addition to its physiological importance, this dissociation process serves as a paradigm for coupled (un)folding and (un)binding. Based on previous free energy simulations, insulin dissociation is thought to involve multiple pathways with comparable free energy barriers. Here, we analyze the mechanism of insulin dimer dissociation using a recently developed computational framework for estimating kinetic statistics from short-trajectory data. These statistics indicate that the likelihood of dissociation (the committor) closely tracks the decrease in the number of (native and nonnative) intermonomer contacts and the increase in the number of water contacts at the dimer interface; the transition state with equal likelihood of association and dissociation corresponds to an encounter complex with relatively few native contacts and many nonnative contacts. We identify four pathways out of the dimer state and quantify their contributions to the rate, as well as their exchange, by computing reactive fluxes. We show that both the pathways and their extents of exchange can be understood in terms of rotations around three axes of the dimer structure. Our results provide insights into the kinetics of insulin analogues and, more generally, how to characterize complex, multipathway processes.

BAD-NEUS: Rapidly converging trajectory stratification

An issue for molecular dynamics simulations is that events of interest often involve timescales that are much longer than the simulation time step, which is set by the fastest timescales of the model. Because of this timescale separation, direct simulation of many events is prohibitively computationally costly. This issue can be overcome by aggregating information from many relatively short simulations that sample segments of trajectories involving events of interest. This is the strategy of Markov state models (MSMs) and related approaches, but such methods suffer from approximation error because the variables defining the states generally do not capture the dynamics fully. By contrast, once converged, the weighted ensemble (WE) method aggregates information from trajectory segments so as to yield unbiased estimates of both thermodynamic and kinetic statistics. Unfortunately, errors decay no faster than unbiased simulation in WE as originally formulated and commonly deployed. Here, we introduce a theoretical framework for describing WE that shows that the introduction of an approximate stationary distribution on top of the stratification, as in nonequilibrium umbrella sampling (NEUS), accelerates convergence. Building on ideas from MSMs and related methods, we generalize the NEUS approach in such a way that the approximation error can be reduced systematically. We show that the improved algorithm can decrease the simulation time required to achieve the desired precision by orders of magnitude.

Temperature-Dependent Fold-Switching Mechanism of the Circadian Clock Protein KaiB

The oscillator of the cyanobacterial circadian clock relies on the ability of the KaiB protein to switch reversibly between a stable ground-state fold (gsKaiB) and an unstable fold-switched fold (fsKaiB). Rare fold-switching events by KaiB provide a critical delay in the negative feedback loop of this post-translational oscillator. In this study, we experimentally and computationally investigate the temperature dependence of fold switching and its mechanism. We demonstrate that the stability of gsKaiB increases with temperature compared to fsKaiB and that the Q10 value for the gsKaiB → fsKaiB transition is nearly three times smaller than that for the reverse transition. Simulations and native-state hydrogen-deuterium exchange NMR experiments suggest that fold switching can involve both subglobally and near-globally unfolded intermediates. The simulations predict that the transition state for fold switching coincides with isomerization of conserved prolines in the most rapidly exchanging region, and we confirm experimentally that proline isomerization is a rate-limiting step for fold switching. We explore the implications of our results for temperature compensation, a hallmark of circadian clocks, through a kinetic model.

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.

Dynamics of activation in the voltage-sensing domain of Ciona intestinalis phosphatase Ci-VSP

The Ciona intestinalis voltage-sensing phosphatase (Ci-VSP) is a membrane protein containing a voltage-sensing domain (VSD) that is homologous to VSDs from voltage-gated ion channels responsible for cellular excitability. Previously published crystal structures of Ci-VSD in putative resting and active conformations suggested a helical-screw voltage sensing mechanism in which the S4 helix translocates and rotates to enable exchange of salt-bridge partners, but the microscopic details of the transition between the resting and active conformations remained unknown. Here, by combining extensive molecular dynamics simulations with a recently developed computational framework based on dynamical operators, we elucidate the microscopic mechanism of the resting-active transition at physiological membrane potential. Sparse regression reveals a small set of coordinates that distinguish intermediates that are hidden from electrophysiological measurements. The intermediates arise from a noncanonical helical-screw mechanism in which translocation, rotation, and side-chain movement of the S4 helix are only loosely coupled. These results provide insights into existing experimental and computational findings on voltage sensing and suggest ways of further probing its mechanism.

Effect of Main and Side Chains on the Folding Mechanism of the Trp-Cage Miniprotein

Proteins that do not fold into their functional native state have been linked to diseases. In this study, the influence of the main and side chains of individual amino acids on the folding of the tryptophan cage (Trp-cage), a designed 20-residue miniprotein, was analyzed. For this purpose, we calculated the solvation free energy (SFE) contributions of individual atoms by using the 3D-reference interaction site model with the atomic decomposition method. The mechanism by which the Trp-cage is stabilized during the folding process was examined by calculating the total energy, which is the sum of the conformational energy and SFE. The folding process of the Trp-cage resulted in a stable native state, with a total energy that was 62.4 kcal/mol lower than that of the unfolded state. The solvation entropy, which is considered to be responsible for the hydrophobic effect, contributed 31.3 kcal/mol to structural stabilization. In other words, the contribution of the solvation entropy accounted for approximately half of the total contribution to Trp-cage folding. The hydrophobic core centered on Trp6 contributed 15.6 kcal/mol to the total energy, whereas the solvation entropy contribution was 6.3 kcal/mol. The salt bridge formed by the hydrophilic side chains of Asp9 and Arg16 contributed 10.9 and 5.0 kcal/mol, respectively. This indicates that not only the hydrophobic core but also the salt bridge of the hydrophilic side chains gain solvation entropy and contribute to stabilizing the native structure of the Trp-cage.

An interoperable implementation of collective-variable based enhanced sampling methods in extended phase space within the OpenMM package

Collective variable (CV)-based enhanced sampling techniques are widely used today for accelerating barrier-crossing events in molecular simulations. A class of these methods, which includes temperature accelerated molecular dynamics (TAMD)/driven-adiabatic free energy dynamics (d-AFED), unified free energy dynamics (UFED), and temperature accelerated sliced sampling (TASS), uses an extended variable formalism to achieve quick exploration of conformational space. These techniques are powerful, as they enhance the sampling of a large number of CVs simultaneously compared to other techniques. Extended variables are kept at a much higher temperature than the physical temperature by ensuring adiabatic separation between the extended and physical subsystems and employing rigorous thermostatting. In this work, we present a computational platform to perform extended phase space enhanced sampling simulations using the open-source molecular dynamics engine OpenMM. The implementation allows users to have interoperability of sampling techniques, as well as employ state-of-the-art thermostats and multiple time-stepping. This work also presents protocols for determining the critical parameters and procedures for reconstructing high-dimensional free energy surfaces. As a demonstration, we present simulation results on the high dimensional conformational landscapes of the alanine tripeptide in vacuo, tetra-N-methylglycine (tetra-sarcosine) peptoid in implicit solvent, and the Trp-cage mini protein in explicit water.

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.

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.

Committor-Consistent Variational String Method

The treatment of slow and rare transitions in the simulation of complex systems poses a great computational challenge. A powerful approach to tackle this challenge is the string method, which represents the transition path as a one-dimensional curve in a multidimensional space of collective variables. Commonly used strategies for pathway optimization include aligning the tangent of the string to the local mean force or to the mean drift determined from swarms of short trajectories. Here, a novel strategy is proposed, allowing the string to be optimized based on a variational principle involving the unidirectional reactive flux expressed in terms of the time-correlation function of the committor. The method is illustrated with model systems and then probed with the alanine dipeptide and a coarse-grained model of the barstar-barnase protein complex. Successive iterations variationally refine the string toward an optimal transition pathway following the gradient of the committor between two metastable states.

Augmented transition path theory for sequences of events

Transition path theory provides a statistical description of the dynamics of a reaction in terms of local spatial quantities. In its original formulation, it is limited to reactions that consist of trajectories flowing from a reactant set A to a product set B. We extend the basic concepts and principles of transition path theory to reactions in which trajectories exhibit a specified sequence of events and illustrate the utility of this generalization on examples.

Computing transition path theory quantities with trajectory stratification

Transition path theory computes statistics from ensembles of reactive trajectories. A common strategy for sampling reactive trajectories is to control the branching and pruning of trajectories so as to enhance the sampling of low probability segments. However, it can be challenging to apply transition path theory to data from such methods because determining whether configurations and trajectory segments are part of reactive trajectories requires looking backward and forward in time. Here, we show how this issue can be overcome efficiently by introducing simple data structures. We illustrate the approach in the context of nonequilibrium umbrella sampling, but the strategy is general and can be used to obtain transition path theory statistics from other methods that sample segments of unbiased trajectories.

Boosting the conformational sampling by combining replica exchange with solute tempering and well-sliced metadynamics

Methods that combine collective variable (CV) based enhanced sampling and global tempering approaches are used in speeding-up the conformational sampling and free energy calculation of large and soft systems with a plethora of energy minima. In this paper, a new method of this kind is proposed in which the well-sliced metadynamics approach (WSMTD) is united with replica exchange with solute tempering (REST2) method. WSMTD employs a divide-and-conquer strategy wherein high-dimensional slices of a free energy surface are independently sampled and combined. The method enables one to accomplish a controlled exploration of the CV-space with a restraining bias as in umbrella sampling, and enhance-sampling of one or more orthogonal CVs using a metadynamics like bias. The new hybrid method proposed here enables boosting the sampling of more slow degrees of freedom in WSMTD simulations, without the need to specify associated CVs, through a replica exchange scheme within the framework of REST2. The high-dimensional slices of the probability distributions of CVs computed from the united WSMTD and REST2 simulations are subsequently combined using the weighted histogram analysis method to obtain the free energy surface. We show that the new method proposed here is accurate, improves the conformational sampling, and achieves quick convergence in free energy estimates. We demonstrate this by computing the conformational free energy landscapes of solvated alanine tripeptide and Trp-cage mini protein in explicit water.

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.