Alex Dickson

Enhanced Sampling of Nonequilibrium Steady States

We review recent progress in methods for accelerating the convergence of simulations of nonequilibrium systems, specifically nonequilibrium umbrella sampling (NEUS) and forward flux sampling (FFS). These methods account for statistics of dynamical paths between interfaces to enforce sampling of low probability regions of phase space for computing steady-state averages, including transition rates, for systems driven arbitrarily far from equilibrium. Recent advances in NEUS allow for efficient sampling of complex systems by focusing sampling in the vicinity of a one-dimensional manifold (string) that connects regions of interest in phase space; this procedure can be extended to the case of two strings that describe the forward and backward transition ensembles separately, which is useful, as they do not, in general, coincide. We recast FFS in the framework of NEUS to facilitate comparison of the two methods. We conclude by discussing selected applications of interest.

Flow-Dependent Unfolding and Refolding of an RNA by Nonequilibrium Umbrella Sampling.

Nonequilibrium experiments of single biomolecules such as force-induced unfolding reveal details about a few degrees of freedom of a complex system. Molecular dynamics simulations can provide complementary information, but exploration of the space of possible configurations is often hindered by large barriers in phase space that separate metastable regions. To solve this problem, enhanced sampling methods have been developed that divide a phase space into regions and integrate trajectory segments in each region. These methods boost the probability of passage over barriers and facilitate parallelization since integration of the trajectory segments does not require communication, aside from their initialization and termination. Here, we present a parallel version of an enhanced sampling method suitable for systems driven far from equilibrium: nonequilibrium umbrella sampling (NEUS). We apply this method to a coarse-grained model of a 262-nucleotide RNA molecule that unfolds and refolds in an explicit flow field modeled with stochastic rotation dynamics. Using NEUS, we are able to observe extremely rare unfolding events that have mean first passage times as long as 45 s (1.1 × 10(15) dynamics steps). We examine the unfolding process for a range of flow rates of the medium, and we describe two competing pathways in which different intramolecular contacts are broken.

Binding and Folding of the Small Bacterial Chaperone HdeA

The small pH stress-sensing chaperone HdeA helps pathogenic enteric E. coli survive passage through the severely acidic environment of the mammalian stomach. Under stress conditions, HdeA transitions from an inactive folded dimer to a chaperone-active unfolded monomer to prevent the acid-induced aggregation of periplasmic proteins. Here we use a topology-based Gō-like model to delineate the relationship between dimer interface formation and monomer folding and to better understand the structural details of the chaperone activation mechanism. Free energy surfaces show that dimer interface formation and monomer folding proceed concurrently through an on-pathway dimeric intermediate in which one monomer is partially unfolded. The absence of a preexisting fully folded monomer in the proposed binding mechanism is in agreement with HdeAs rapid chaperone response. Binding between unfolded monomers exhibits an enhancement of molecular recognition reminiscent of the fly-casting mechanism. Overall, our simulations further highlight the efficient nature of HdeAs chaperone response and we anticipate that knowledge of a dimeric intermediate will facilitate the interpretation of experimental studies.

Capturing a Dynamic Chaperone–Substrate Interaction Using NMR-Informed Molecular Modeling

Chaperones maintain a healthy proteome by preventing aggregation and by aiding in protein folding. Precisely how chaperones influence the conformational properties of their substrates, however, remains unclear. To achieve a detailed description of dynamic chaperone-substrate interactions, we fused site-specific NMR information with coarse-grained simulations. Our model system is the binding and folding of a chaperone substrate, immunity protein 7 (Im7), with the chaperone Spy. We first used an automated procedure in which NMR chemical shifts inform the construction of system-specific force fields that describe each partner individually. The models of the two binding partners are then combined to perform simulations on the chaperone-substrate complex. The binding simulations show excellent agreement with experimental data from multiple biophysical measurements. Upon binding, Im7 interacts with a mixture of hydrophobic and hydrophilic residues on Spys surface, causing conformational exchange within Im7 to slow down as Im7 folds. Meanwhile, the motion of Spys flexible loop region increases, allowing for better interaction with different substrate conformations, and helping offset losses in Im7 conformational dynamics that occur upon binding and folding. Spy then preferentially releases Im7 into a well-folded state. Our strategy has enabled a residue-level description of a dynamic chaperone-substrate interaction, improving our understanding of how chaperones facilitate substrate folding. More broadly, we validate our approach using two other binding partners, showing that this approach provides a general platform from which to investigate other flexible biomolecular complexes through the integration of NMR data with efficient computational models.

Ligand Release Pathways Obtained with WExplore: Residence Times and Mechanisms

The binding of ligands with their molecular receptors is of tremendous importance in biology. Although much emphasis has been placed on characterizing binding sites and bound poses that determine the binding thermodynamics, the pathway by which a ligand binds importantly determines the binding kinetics. The computational study of entire unbiased ligand binding and release pathways is still an emerging field, made possible only recently by advances in computational hardware and sampling methodologies. We have developed one such method (WExplore) that is based on a weighted ensemble of trajectories, which we apply to ligand release for the first time, using a set of three previously characterized interactions between low-affinity ligands and the protein FKBP-12 (FK-506 binding protein). WExplore is found to be more efficient that conventional sampling, even for the nanosecond-scale unbinding events observed here. From a nonequilibrium ensemble of unbinding trajectories, we obtain ligand residence times and release pathways without using biasing forces or a Markovian assumption of transitions between regions. We introduce a set of analysis tools for unbinding transition pathways, including using von Mises-Fisher distributions to model clouds of ligand exit points, which provide a quantitative proxy for ligand surface diffusion. Differences between the transition pathway ensembles of the three ligands are identified and discussed.

Coupled folding and binding with 2D Window‐Exchange Umbrella Sampling

Intrinsically disordered regions of proteins can gain structure by binding to a partner. This process, of coupled folding and binding (CFaB), is a fundamental part of many important biological processes. Structure‐based models have proven themselves capable of revealing fundamental aspects of how CFaB occurs, however, typical methods to enhance the sampling of these transitions, such as replica exchange, do not adequately sample the transition state region of this extremely rare process. Here, we use a variant of Umbrella Sampling to enforce sampling of the transition states of CFaB of HdeA monomers at neutral pH, an extremely rare process that occurs over timescales ranging from seconds to hours. Using high resolution sampling in the transition state region, we cluster states along the principal transition path to obtain a detailed description of coupled binding and folding for the HdeA dimer, revealing new insight into the ensemble of states that are accessible to client recognition. We then demonstrate that exchanges between umbrella sampling windows, as done in previous work, significantly improve relaxation in variables orthogonal to the restraints used. Altogether, these results show that Window‐Exchange Umbrella Sampling is a promising approach for systems that exhibit flexible binding, and can reveal transition state ensembles of these systems in high detail.

Efficient in silico exploration of RNA interhelical conformations using Euler angles and WExplore

HIV-1 TAR RNA is a two-helix bulge motif that plays a critical role in HIV viral replication and is an important drug target. However, efforts at designing TAR inhibitors have been challenged by its high degree of structural flexibility, which includes slow large-amplitude reorientations of its helices with respect to one another. Here, we use the recently introduced algorithm WExplore in combination with Euler angles to achieve unprecedented sampling of the TAR conformational ensemble. Our ensemble achieves similar agreement with experimental NMR data when compared with previous TAR computational studies, and is generated at a fraction of the computational cost. It clearly emerges from configuration space network analysis that the intermittent formation of the A22-U40 base pair acts as a reversible switch that enables sampling of interhelical conformations that would otherwise be topologically disallowed. We find that most previously determined ligand-bound structures are found in similar location in the network, and we use a sample-and-select approach to guide the construction of a set of novel conformations which can serve as the basis for future drug development efforts. Collectively, our findings demonstrate the utility of WExplore in combination with suitable order parameters as a method for exploring RNA conformational space.

pH-dependent transient conformational states control optical properties in cyan fluorescent protein.

A recently engineered mutant of cyan fluorescent protein (WasCFP) that exhibits pH-dependent absorption suggests that its tryptophan-based chromophore switches between neutral (protonated) and charged (deprotonated) states depending on external pH. At pH 8.1, the latter gives rise to green fluorescence as opposed to the cyan color of emission that is characteristic for the neutral form at low pH. Given the high energy cost of deprotonating the tryptophan at the indole nitrogen, this behavior is puzzling, even if the stabilizing effect of the V61K mutation in proximity to the protonation/deprotonation site is considered. Because of its potential to open new avenues for the development of optical sensors and photoconvertible fluorescent proteins, a mechanistic understanding of how the charged state in WasCFP can possibly be stabilized is thus important. Attributed to the dynamic nature of proteins, such understanding often requires knowledge of the various conformations adopted, including transiently populated conformational states. Transient conformational states triggered by pH are of emerging interest and have been shown to be important whenever ionizable groups interact with hydrophobic environments. Using a combination of the weighted-ensemble sampling method and explicit-solvent constant pH molecular dynamics (CPHMD(MSλD)) simulations, we have identified a solvated transient state, characterized by a partially open β-barrel where the chromophore pKa of 6.8 is shifted by over 20 units from that of the closed form (6.8 and 31.7, respectively). This state contributes a small population at low pH (12% at pH 6.1) but becomes dominant at mildly basic conditions, contributing as much as 53% at pH 8.1. This pH-dependent population shift between neutral (at pH 6.1) and charged (at pH 8.1) forms is thus responsible for the observed absorption behavior of WasCFP. Our findings demonstrate the conditions necessary to stabilize the charged state of the WasCFP chromophore (namely, local solvation at the deprotonation site and a partial flexibility of the protein β-barrel structure) and provide the first evidence that transient conformational states can control optical properties of fluorescent proteins.

Unbiased Molecular Dynamics of 11 min Timescale Drug Unbinding Reveals Transition State Stabilizing Interactions

Ligand (un)binding kinetics is being recognized as a determinant of drug specificity and efficacy in an increasing number of systems. However, the calculation of kinetics and the simulation of drug unbinding is more difficult than computing thermodynamic quantities, such as binding free energies. Here we present the first full simulations of an unbinding process at pharmacologically relevant time scales (11 min), without the use of biasing forces, detailed prior knowledge, or specialized processors using the weighted ensemble based algorithm, WExplore. These simulations show the inhibitor TPPU unbinding from its enzyme target soluble epoxide hydrolase, which is a clinically relevant target that has attracted interest in kinetics optimization in order to increase efficacy. We make use of conformation space networks that allow us to conceptualize unbinding not just as a linear process, but as a network of interconnected states that connect the bound and unbound states. This allows us to visualize patterns in hydrogen-bonding, solvation, and nonequilibrium free energies, without projection onto progress coordinates. The topology and layout of the network reveal multiple unbinding pathways, and other rare events, such as the reversal of ligand orientation within the binding site. Furthermore, we make a prediction of the transition state ensemble, using transition path theory, and identify protein-ligand interactions which are stabilizing to the transition state. Additionally, we uncover trends in ligand and binding site solvation that corroborate experimental evidence from more classical structure kinetics relationships and generate new questions as to the role of drug modifications in kinetics optimization. Finally, from only 6 μs of simulation time we observed 75 unbinding events from which we calculate a residence time of 42 s, and a standard error range of 23 to 280 s. This nearly encompasses the experimental residence time 11 min (660 s). In addition to the insights to sEH inhibitor unbinding, this study shows that simulations of complex processes on timescales as long as minutes are becoming feasible for more researchers to perform.

Multiple Ligand Unbinding Pathways and Ligand-Induced Destabilization Revealed by WExplore

We report simulations of full ligand exit pathways for the trypsin-benzamidine system, generated using the sampling technique WExplore. WExplore is able to observe millisecond-scale unbinding events using many nanosecond-scale trajectories that are run without introducing biasing forces. The algorithm generates rare events by dividing the coordinate space into regions, on-the-fly, and balancing computational effort between regions through cloning and merging steps, as in the weighted ensemble method. The averaged exit flux yields a ligand exit rate of 180 μs, which is within an order of magnitude of the experimental value. We obtain broad sampling of ligand exit pathways, and visualize our findings using conformation space networks. The analysis shows three distinct exit channels, two of which are formed through large, rare motions of the loop regions in trypsin. This broad set of ligand-bound poses is then used to investigate general properties of ligand binding: we observe both a direct stabilizing effect of ligand-protein interactions and an indirect destabilizing effect on intraprotein interactions that is induced by the ligand. Significantly, the crystallographic binding poses are distinguished not only because their ligands induce large stabilizing effects, but also because they induce relatively low indirect destabilizations.