Bimolecular reaction simulation using Weighted Ensemble Brownian dynamics and the University of Houston Brownian Dynamics program

Dissecting the general mechanisms of protein cage self-assembly by coarse-grained simulations

The development of artificial protein cages has recently gained massive attention due to their promising application prospect as novel delivery vehicles for therapeutics. These nanoparticles are formed through a process called self-assembly, in which individual subunits spontaneously arrange into highly ordered patterns via non-covalent but specific interactions. Therefore, the first step toward the design of novel engineered protein cages is to understand the general mechanisms of their self-assembling dynamics. Here we have developed a new computational method to tackle this problem. Our method is based on a coarse-grained model and a diffusion-reaction simulation algorithm. Using a tetrahedral cage as test model, we showed that self-assembly of protein cage requires of a seeding process in which specific configurations of kinetic intermediate states are identified. We further found that there is a critical concentration to trigger self-assembly of protein cages. This critical concentration allows that cages can only be successfully assembled under a persistently high concentration. Additionally, phase diagram of self-assembly has been constructed by systematically testing the model across a wide range of binding parameters. Finally, our simulations demonstrated the importance of protein's structural flexibility in regulating the dynamics of cage assembly. In summary, this study throws lights on the general principles underlying self-assembly of large cage-like protein complexes and thus provides insights to design new nanomaterials.

An ergodic theorem for the weighted ensemble method

We study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications of weighted ensemble require the computation of long time averages. We establish the consistency of weighted ensemble in this setting by proving an ergodic theorem for time averages. As part of the proof, we derive explicit variance formulas that could be useful for optimizing the method.

Classification of protein-protein association rates based on biophysical informatics

BACKGROUND

Proteins form various complexes to carry out their versatile functions in cells. The dynamic properties of protein complex formation are mainly characterized by the association rates which measures how fast these complexes can be formed. It was experimentally observed that the association rates span an extremely wide range with over ten orders of magnitudes. Identification of association rates within this spectrum for specific protein complexes is therefore essential for us to understand their functional roles.

RESULTS

To tackle this problem, we integrate physics-based coarse-grained simulations into a neural-network-based classification model to estimate the range of association rates for protein complexes in a large-scale benchmark set. The cross-validation results show that, when an optimal threshold was selected, we can reach the best performance with specificity, precision, sensitivity and overall accuracy all higher than 70%. The quality of our cross-validation data has also been testified by further statistical analysis. Additionally, given an independent testing set, we can successfully predict the group of association rates for eight protein complexes out of ten. Finally, the analysis of failed cases suggests the future implementation of conformational dynamics into simulation can further improve model.

CONCLUSIONS

In summary, this study demonstrated that a new modeling framework that combines biophysical simulations with bioinformatics approaches is able to identify protein-protein interactions with low association rates from those with higher association rates. This method thereby can serve as a useful addition to a collection of existing experimental approaches that measure biomolecular recognition.

Predicting pathological von Willebrand factor unraveling in elongational flow

The globular-to-unraveled conformation transition of von Willebrand factor (vWF), a large polymeric glycoprotein in human blood plasma, is a crucial step in the process of clotting at sites of vascular injury. However, unraveling of vWF multimers in uninjured vasculature can lead to pathology (i.e., thrombus formation or degradation of vWF proteins by enzyme ADAMTS13, making them nonfunctional). To identify blood flow conditions that might induce pathological unraveling of vWF multimers, here we have computed the globular-to-unraveled transition rate of vWF multimers subjected to varying strain rate elongational flow by employing an enhanced sampling technique, the weighted ensemble method. Weighted ensemble sampling was employed instead of standard brute-force simulations because pathological blood flow conditions can induce undesired vWF unraveling on timescales potentially inaccessible to standard simulation methods. Results here indicate that brief but periodic exposure of vWF to the elongational flow of strain rate greater than or equal to 2500 s^-1 represents a source of possible pathology caused by the undesired unraveling of vWF multimers.

Wepy: A Flexible Software Framework for Simulating Rare Events with Weighted Ensemble Resampling

Here, we introduce the open-source software framework wepy (https://github.com/ADicksonLab/wepy) which is a toolkit for running and analyzing weighted ensemble (WE) simulations. The wepy toolkit is in pure Python and as such is highly portable and extensible, making it an excellent platform to develop and use new WE resampling algorithms such as WExplore, REVO, and others while leveraging the entire Python ecosystem. In addition, wepy simplifies WE-specific analyses by defining out-of-core tree-like data structures using the cross-platform HDF5 file format. In this paper, we discuss the motivations and challenges for simulating rare events in biomolecular systems. As has previously been shown, high-dimensional WE resampling algorithms such as WExplore and REVO have been successful at these tasks, especially for rare events that are difficult to describe by one or two collective variables. We explain in detail how wepy facilitates implementation of these algorithms, as well as aids in analyzing the unique structure of WE simulation results. To explain how wepy and WE work in general, we describe the mathematical formalism of WE, an overview of the architecture of wepy, and provide code examples of how to construct, run, and analyze simulation results for a protein-ligand system (T4 Lysozyme in an implicit solvent). This paper is written with a variety of readers in mind, including (1) those curious about how to leverage WE rare-event simulations for their domain, (2) current WE users who want to begin using new high-dimensional resamplers such as WExplore and REVO, and (3) expert users who would like to prototype or implement their own algorithms that can be easily adopted by others.

Ranking of Ligand Binding Kinetics Using a Weighted Ensemble Approach and Comparison with a Multiscale Milestoning Approach

To improve lead optimization efforts in finding the right ligand, pharmaceutical industries need to know the ligand's binding kinetics, such as binding and unbinding rate constants, which often correlate with the ligand's efficacy in vivo. To predict binding kinetics efficiently, enhanced sampling methods, such as milestoning and the weighted ensemble (WE) method, have been used in molecular dynamics (MD) simulations of these systems. However, a comparison of these enhanced sampling methods in ranking ligands has not been done. Hence, a WE approach called the concurrent adaptive sampling (CAS) algorithm that uses MD simulations was used to rank seven ligands for β-cyclodextrin, a system in which a multiscale milestoning approach called simulation enabled estimation of kinetic rates (SEEKR) was also used, which uses both MD and Brownian dynamics simulations. Overall, the CAS algorithm can successfully rank ligands using the unbinding rate constant k_off values and binding free energy ΔG values, as SEEKR did, with reduced computational cost that is about the same as SEEKR. We compare the CAS algorithm simulations with different parameters and discuss the impact of parameters in ranking ligands and obtaining rate constant and binding free energy estimates. We also discuss similarities and differences and advantages and disadvantages of SEEKR and the CAS algorithm for future use.

Understanding the Impacts of Conformational Dynamics on the Regulation of Protein-Protein Association by a Multiscale Simulation Method

Complexes formed among diverse proteins carry out versatile functions in nearly all physiological processes. Association rates which measure how fast proteins form various complexes are of fundamental importance to characterize their functions. The association rates are not only determined by the energetic features at binding interfaces of a protein complex but also influenced by the intrinsic conformational dynamics of each protein in the complex. Unfortunately, how this conformational effect regulates protein association has never been calibrated on a systematic level. To tackle this problem, we developed a multiscale strategy to incorporate the information on protein conformational variations from Langevin dynamic simulations into a kinetic Monte Carlo algorithm of protein-protein association. By systematically testing this approach against a large-scale benchmark set, we found the association of a protein complex with a relatively rigid structure tends to be reduced by its conformational fluctuations. With specific examples, we further show that higher degrees of structural flexibility in various protein complexes can facilitate the searching and formation of intermolecular interactions and thereby accelerate their associations. In general, the integration of conformational dynamics can improve the correlation between experimentally measured association rates and computationally derived association probabilities. Finally, we analyzed the statistical distributions of different secondary structural types on protein-protein binding interfaces and their preference to the change of association rates. Our study, to the best of our knowledge, is the first computational method that systematically estimates the impacts of protein conformational dynamics on protein-protein association. It throws lights on the molecular mechanisms of how protein-protein recognition is kinetically modulated.

Using Coarse-Grained Simulations to Characterize the Mechanisms of Protein-Protein Association

The formation of functionally versatile protein complexes underlies almost every biological process. The estimation of how fast these complexes can be formed has broad implications for unravelling the mechanism of biomolecular recognition. This kinetic property is traditionally quantified by association rates, which can be measured through various experimental techniques. To complement these time-consuming and labor-intensive approaches, we developed a coarse-grained simulation approach to study the physical processes of protein–protein association. We systematically calibrated our simulation method against a large-scale benchmark set. By combining a physics-based force field with a statistically-derived potential in the simulation, we found that the association rates of more than 80% of protein complexes can be correctly predicted within one order of magnitude relative to their experimental measurements. We further showed that a mixture of force fields derived from complementary sources was able to describe the process of protein–protein association with mechanistic details. For instance, we show that association of a protein complex contains multiple steps in which proteins continuously search their local binding orientations and form non-native-like intermediates through repeated dissociation and re-association. Moreover, with an ensemble of loosely bound encounter complexes observed around their native conformation, we suggest that the transition states of protein–protein association could be highly diverse on the structural level. Our study also supports the idea in which the association of a protein complex is driven by a “funnel-like” energy landscape. In summary, these results shed light on our understanding of how protein–protein recognition is kinetically modulated, and our coarse-grained simulation approach can serve as a useful addition to the existing experimental approaches that measure protein–protein association rates.

OPTIMIZING WEIGHTED ENSEMBLE SAMPLING OF STEADY STATES

We propose parameter optimization techniques for weighted ensemble sampling of Markov chains in the steady-state regime. Weighted ensemble consists of replicas of a Markov chain, each carrying a weight, that are periodically resampled according to their weights inside of each of a number of bins that partition state space. We derive, from first principles, strategies for optimizing the choices of weighted ensemble parameters, in particular the choice of bins and the number of replicas to maintain in each bin. In a simple numerical example, we compare our new strategies with more traditional ones and with direct Monte Carlo.

A Multiscale Computational Model for Simulating the Kinetics of Protein Complex Assembly

Proteins fulfill versatile biological functions by interacting with each other and forming high-order complexes. Although the order in which protein subunits assemble is important for the biological function of their final complex, this kinetic information has received comparatively little attention in recent years. Here we describe a multiscale framework that can be used to simulate the kinetics of protein complex assembly. There are two levels of models in the framework. The structural details of a protein complex are reflected by the residue-based model, while a lower-resolution model uses a rigid-body (RB) representation to simulate the process of complex assembly. These two levels of models are integrated together, so that we are able to provide the kinetic information about complex assembly with both structural details and computational efficiency.

Mapping the Ligand Binding Landscape

The interaction between a ligand and a protein involves a multitude of conformational states. To achieve a particular deeply bound pose, the ligand must search across a rough free-energy landscape with many metastable minima. Creating maps of the ligand binding landscape is a great challenge, as binding and release events typically occur on timescales that are beyond the reach of molecular simulation. The WExplore enhanced sampling method is well suited to build these maps because it is designed to broadly explore free-energy landscapes and is capable of simulating ligand release pathways that occur on timescales as long as minutes. WExplore also uses only unbiased trajectory segments, allowing for the construction of Markov state models (MSMs) and conformation space networks that combine the results of multiple simulations. Here, we use WExplore to study two bromodomain-inhibitor systems using multiple docked starting poses (Brd4-MS436 and Baz2B-ICR7) and synthesize our results using a series of MSMs using time-lagged independent component analysis. Ranking the starting poses by exit rate agrees with the crystal structure pose in both cases. We also predict the most stable pose using the equilibrium populations from the MSM but find that the prediction is not robust as a function of MSM parameters. The simulated trajectories are synthesized into network models that visualize the entire binding landscape for each system, and we examine transition paths between deeply bound stable states. We find that, on average, transitions between deeply bound states convert through the unbound state 81% of the time, implying a trial-and-error approach to ligand binding. We conclude with a discussion of the implications of this result for both kinetics-based drug discovery and virtual screening pipelines that incorporate molecular dynamics.

Predicting ligand binding affinity using on- and off-rates for the SAMPL6 SAMPLing challenge

Interest in ligand binding kinetics has been growing rapidly, as it is being discovered in more and more systems that ligand residence time is the crucial factor governing drug efficacy. Many enhanced sampling methods have been developed with the goal of predicting ligand binding rates ([Formula: see text]) and/or ligand unbinding rates ([Formula: see text]) through explicit simulation of ligand binding pathways, and these methods work by very different mechanisms. Although there is not yet a blind challenge for ligand binding kinetics, here we take advantage of experimental measurements and rigorously computed benchmarks to compare estimates of [Formula: see text] calculated as the ratio of two rates: [Formula: see text]. These rates were determined using a new enhanced sampling method based on the weighted ensemble framework that we call "REVO": Reweighting of Ensembles by Variance Optimization. This is a further development of the WExplore enhanced sampling method, in which trajectory cloning and merging steps are guided not by the definition of sampling regions, but by maximizing trajectory variance. Here we obtain estimates of [Formula: see text] and [Formula: see text] that are consistent across multiple simulations, with an average log10-scale standard deviation of 0.28 for on-rates and 0.56 for off-rates, which is well within an order of magnitude and far better than previously observed for previous applications of the WExplore algorithm. Our rank ordering of the three host-guest pairs agrees with the reference calculations, however our predicted [Formula: see text] values were systematically lower than the reference by an average of 4.2 kcal/mol. Using tree network visualizations of the trajectories in the REVO algorithm, and conformation space networks for each system, we analyze the results of our sampling, and hypothesize sources of discrepancy between our [Formula: see text] values and the reference. We also motivate the direct inclusion of [Formula: see text] and [Formula: see text] challenges in future iterations of SAMPL, to further develop the field of ligand binding kinetics prediction and modeling.

Large enhancement of response times of a protein conformational switch by computational design

The design of protein conformational switches—or proteins that change conformations in response to a signal such as ligand binding—has great potential for developing novel biosensors, diagnostic tools, and therapeutic agents. Among the defining properties of such switches, the response time has been the most challenging to optimize. Here we apply a computational design strategy in synergistic combination with biophysical experiments to rationally improve the response time of an engineered protein-based Ca²⁺-sensor in which the switching process occurs via mutually exclusive folding of two alternate frames. Notably, our strategy identifies mutations that increase switching rates by as much as 32-fold, achieving response times on the order of fast physiological Ca²⁺ fluctuations. Our computational design strategy is general and may aid in optimizing the kinetics of other protein conformational switches.

The rational optimization of response times of protein conformational switches is a major challenge for biomolecular switch design. Here the authors present a generally applicable computational design strategy that in combination with biophysical experiments can improve response times using a Ca²⁺-sensor as an example.

Flexibility vs Preorganization: Direct Comparison of Binding Kinetics for a Disordered Peptide and Its Exact Preorganized Analogues

Many intrinsically disordered proteins, which are prevalent in nature, fold only upon binding their structured partner proteins. Such proteins have been hypothesized to have a kinetic advantage over their folded, preorganized analogues in binding their partner proteins. Here we determined the effects of ligand preorganization on the k_on for a biomedically important system: an intrinsically disordered p53 peptide ligand and the MDM2 protein receptor. Based on direct simulations of binding pathways, computed k_on values for fully disordered and preorganized p53 peptide analogues were within error of each other, indicating little if any kinetic advantage to being disordered or preorganized for binding the MDM2 protein. We also examined the effects of increasing the concentration of MDM2 on the extent to which its mechanism of binding to the p53 peptide is induced fit vs conformational selection. Results predict that the mechanism is solely induced fit if the unfolded state of the peptide is more stable than its folded state; otherwise, the mechanism shifts from being dominated by conformational selection at low MDM2 concentration to induced fit at high MDM2 concentration. Taken together, our results are relevant to any protein binding process that involves a disordered peptide of a similar length that forms a single α-helix upon binding a partner protein. Such disorder-to-helix transitions are common among protein interactions of disordered proteins and are therefore of fundamental biological interest.

Weighted Ensemble Simulation: Review of Methodology, Applications, and Software

The weighted ensemble (WE) methodology orchestrates quasi-independent parallel simulations run with intermittent communication that can enhance sampling of rare events such as protein conformational changes, folding, and binding. The WE strategy can achieve superlinear scaling-the unbiased estimation of key observables such as rate constants and equilibrium state populations to greater precision than would be possible with ordinary parallel simulation. WE software can be used to control any dynamics engine, such as standard molecular dynamics and cell-modeling packages. This article reviews the theoretical basis of WE and goes on to describe successful applications to a number of complex biological processes-protein conformational transitions, (un)binding, and assembly processes, as well as cell-scale processes in systems biology. We furthermore discuss the challenges that need to be overcome in the next phase of WE methodological development. Overall, the combined advances in WE methodology and software have enabled the simulation of long-timescale processes that would otherwise not be practical on typical computing resources using standard simulation.

Predicting Protein-protein Association Rates using Coarse-grained Simulation and Machine Learning

Protein–protein interactions dominate all major biological processes in living cells. We have developed a new Monte Carlo-based simulation algorithm to study the kinetic process of protein association. We tested our method on a previously used large benchmark set of 49 protein complexes. The predicted rate was overestimated in the benchmark test compared to the experimental results for a group of protein complexes. We hypothesized that this resulted from molecular flexibility at the interface regions of the interacting proteins. After applying a machine learning algorithm with input variables that accounted for both the conformational flexibility and the energetic factor of binding, we successfully identified most of the protein complexes with overestimated association rates and improved our final prediction by using a cross-validation test. This method was then applied to a new independent test set and resulted in a similar prediction accuracy to that obtained using the training set. It has been thought that diffusion-limited protein association is dominated by long-range interactions. Our results provide strong evidence that the conformational flexibility also plays an important role in regulating protein association. Our studies provide new insights into the mechanism of protein association and offer a computationally efficient tool for predicting its rate.

Efficient Atomistic Simulation of Pathways and Calculation of Rate Constants for a Protein-Peptide Binding Process: Application to the MDM2 Protein and an Intrinsically Disordered p53 Peptide

The characterization of protein binding processes - with all of the key conformational changes - has been a grand challenge in the field of biophysics. Here, we have used the weighted ensemble path sampling strategy to orchestrate molecular dynamics simulations, yielding atomistic views of protein-peptide binding pathways involving the MDM2 oncoprotein and an intrinsically disordered p53 peptide. A total of 182 independent, continuous binding pathways were generated, yielding a kon that is in good agreement with experiment. These pathways were generated in 15 days using 3500 cores of a supercomputer, substantially faster than would be possible with "brute force" simulations. Many of these pathways involve the anchoring of p53 residue F19 into the MDM2 binding cleft when forming the metastable encounter complex, indicating that F19 may be a kinetically important residue. Our study demonstrates that it is now practical to generate pathways and calculate rate constants for protein binding processes using atomistic simulation on typical computing resources.

Analysis of Ligand-Receptor Association and Intermediate Transfer Rates in Multienzyme Nanostructures with All-Atom Brownian Dynamics Simulations

We present the second-generation GeomBD Brownian dynamics software for determining interenzyme intermediate transfer rates and substrate association rates in biomolecular complexes. Substrate and intermediate association rates for a series of enzymes or biomolecules can be compared between the freely diffusing disorganized configuration and various colocalized or complexed arrangements for kinetic investigation of enhanced intermediate transfer. In addition, enzyme engineering techniques, such as synthetic protein conjugation, can be computationally modeled and analyzed to better understand changes in substrate association relative to native enzymes. Tools are provided to determine nonspecific ligand-receptor association residence times, and to visualize common sites of nonspecific association of substrates on receptor surfaces. To demonstrate features of the software, interenzyme intermediate substrate transfer rate constants are calculated and compared for all-atom models of DNA origami scaffold-bound bienzyme systems of glucose oxidase and horseradish peroxidase. Also, a DNA conjugated horseradish peroxidase enzyme was analyzed for its propensity to increase substrate association rates and substrate local residence times relative to the unmodified enzyme. We also demonstrate the rapid determination and visualization of common sites of nonspecific ligand-receptor association by using HIV-1 protease and an inhibitor, XK263. GeomBD2 accelerates simulations by precomputing van der Waals potential energy grids and electrostatic potential grid maps, and has a flexible and extensible support for all-atom and coarse-grained force fields. Simulation software is written in C++ and utilizes modern parallelization techniques for potential grid preparation and Brownian dynamics simulation processes. Analysis scripts, written in the Python scripting language, are provided for quantitative simulation analysis. GeomBD2 is applicable to the fields of biophysics, bioengineering, and enzymology in both predictive and explanatory roles.

Highly Efficient Computation of the Basal kon using Direct Simulation of Protein-Protein Association with Flexible Molecular Models

An essential baseline for determining the extent to which electrostatic interactions enhance the kinetics of protein–protein association is the “basal” k_on, which is the rate constant for association in the absence of electrostatic interactions. However, since such association events are beyond the milliseconds time scale, it has not been practical to compute the basal k_on by directly simulating the association with flexible models. Here, we computed the basal k_on for barnase and barstar, two of the most rapidly associating proteins, using highly efficient, flexible molecular simulations. These simulations involved (a) pseudoatomic protein models that reproduce the molecular shapes, electrostatic, and diffusion properties of all-atom models, and (b) application of the weighted ensemble path sampling strategy, which enhanced the efficiency of generating association events by >130-fold. We also examined the extent to which the computed basal k_on is affected by inclusion of intermolecular hydrodynamic interactions in the simulations.

WESTPA: an interoperable, highly scalable software package for weighted ensemble simulation and analysis

The weighted ensemble (WE) path sampling approach orchestrates an ensemble of parallel calculations with intermittent communication to enhance the sampling of rare events, such as molecular associations or conformational changes in proteins or peptides. Trajectories are replicated and pruned in a way that focuses computational effort on underexplored regions of configuration space while maintaining rigorous kinetics. To enable the simulation of rare events at any scale (e.g., atomistic, cellular), we have developed an open-source, interoperable, and highly scalable software package for the execution and analysis of WE simulations: WESTPA (The Weighted Ensemble Simulation Toolkit with Parallelization and Analysis). WESTPA scales to thousands of CPU cores and includes a suite of analysis tools that have been implemented in a massively parallel fashion. The software has been designed to interface conveniently with any dynamics engine and has already been used with a variety of molecular dynamics (e.g., GROMACS, NAMD, OpenMM, AMBER) and cell-modeling packages (e.g., BioNetGen, MCell). WESTPA has been in production use for over a year, and its utility has been demonstrated for a broad set of problems, ranging from atomically detailed host–guest associations to nonspatial chemical kinetics of cellular signaling networks. The following describes the design and features of WESTPA, including the facilities it provides for running WE simulations and storing and analyzing WE simulation data, as well as examples of input and output.

Simulating rare events using a weighted ensemble-based string method

We introduce an extension to the weighted ensemble (WE) path sampling method to restrict sampling to a one-dimensional path through a high dimensional phase space. Our method, which is based on the finite-temperature string method, permits efficient sampling of both equilibrium and non-equilibrium systems. Sampling obtained from the WE method guides the adaptive refinement of a Voronoi tessellation of order parameter space, whose generating points, upon convergence, coincide with the principle reaction pathway. We demonstrate the application of this method to several simple, two-dimensional models of driven Brownian motion and to the conformational change of the nitrogen regulatory protein C receiver domain using an elastic network model. The simplicity of the two-dimensional models allows us to directly compare the efficiency of the WE method to conventional brute force simulations and other path sampling algorithms, while the example of protein conformational change demonstrates how the method can be used to efficiently study transitions in the space of many collective variables.

Transition path sampling and forward flux sampling. Applications to biological systems

The last decade has seen a rapid growth in the number of simulation methods and applications dealing with the sampling of transition pathways of rare nanoscale events. Such studies are crucial, for example, for understanding the mechanism and kinetics of conformational transitions and enzymatic events associated with the function of biomolecules. In this review, a broad account of transition path sampling approaches is provided, starting from the general concepts, progressing to the specific principles that underlie some of the most important methods, and eventually singling out the so-called forward flux sampling method for a more detailed description. This is done because forward flux sampling, despite its appealing simplicity and potential efficiency, has thus far received limited attention from practitioners. While path sampling methods have a widespread application to many types of rare transitional events, here only recent applications involving biomolecules are reviewed, including isomerization, protein folding, and enzyme catalysis.

Mobile trap algorithm for zinc detection using protein sensors

We present a mobile trap algorithm to sense zinc ions using protein-based sensors such as carbonic anhydrase (CA). Zinc is an essential biometal required for mammalian cellular functions although its intracellular concentration is reported to be very low. Protein-based sensors like CA molecules are employed to sense rare species like zinc ions. In this study, the zinc ions are mobile targets, which are sought by the mobile traps in the form of sensors. Particle motions are modeled using random walk along with the first passage technique for efficient simulations. The association reaction between sensors and ions is incorporated using a probability (p1) upon an ion-sensor collision. The dissociation reaction of an ion-bound CA molecule is modeled using a second, independent probability (p2). The results of the algorithm are verified against the traditional simulation techniques (e.g., Gillespie's algorithm). This study demonstrates that individual sensor molecules can be characterized using the probability pair (p1,p2), which, in turn, is linked to the system level chemical kinetic constants, kon and koff. Further investigations of CA-Zn reaction using the mobile trap algorithm show that when the diffusivity of zinc ions approaches that of sensor molecules, the reaction data obtained using the static trap assumption differ from the reaction data obtained using the mobile trap formulation. This study also reveals similar behavior when the sensor molecule has higher dissociation constant. In both the cases, the reaction data obtained using the static trap formulation reach equilibrium at a higher number of complex molecules (ion-bound sensor molecules) compared to the reaction data from the mobile trap formulation. With practical limitations on the number sensors that can be inserted/expressed in a cell and stochastic nature of the intracellular ionic concentrations, fluorescence from the number of complex sensor molecules at equilibrium will be the measure of the intracellular ion concentration. For reliable detection of zinc ions, it is desirable that the sensors must not bind all the zinc ions tightly, but should rather bind and unbind. Thus for a given fluorescence and with association-dissociation reactions between ions and sensors, the static trap approach will underestimate the number of zinc ions present in the system.

Efficient and verified simulation of a path ensemble for conformational change in a united-residue model of calmodulin

The computational sampling of rare, large-scale, conformational transitions in proteins is a well appreciated challenge-for which a number of potentially efficient path-sampling methodologies have been proposed. Here, we study a large-scale transition in a united-residue model of calmodulin using the "weighted ensemble" (WE) approach of Huber and Kim. Because of the model's relative simplicity, we are able to compare our results with brute-force simulations. The comparison indicates that the WE approach quantitatively reproduces the brute-force results, as assessed by considering (i) the reaction rate, (ii) the distribution of event durations, and (iii) structural distributions describing the heterogeneity of the paths. Importantly, the WE method is readily applied to more chemically accurate models, and by studying a series of lower temperatures, our results suggest that the WE method can increase efficiency by orders of magnitude in more challenging systems.

Robust biased Brownian dynamics for rate constant calculation

A reaction probability is required to calculate the rate constant of a diffusion-dominated reaction. Due to the complicated geometry and potentially high dimension of the reaction probability problem, it is usually solved by a Brownian dynamics simulation, also known as a random walk or path integral method, instead of solving the equivalent partial differential equation by a discretization method. Building on earlier work, this article completes the development of a robust importance sampling algorithm for Brownian dynamics-i.e., biased Brownian dynamics with weight control-to overcome the high energy and entropy barriers in biomolecular association reactions. The biased Brownian dynamics steers sampling by a bias force, and the weight control algorithm controls sampling by a target weight. This algorithm is optimal if the bias force and the target weight are constructed from the solution of the reaction probability problem. In reality, an approximate reaction probability has to be used to construct the bias force and the target weight. Thus, the performance of the algorithm depends on the quality of the approximation. Given here is a method to calculate a good approximation, which is based on the selection of a reaction coordinate and the variational formulation of the reaction probability problem. The numerically approximated reaction probability is shown by computer experiments to give a factor-of-two speedup over the use of a purely heuristic approximation. Also, the fully developed method is compared to unbiased Brownian dynamics. The tests for human superoxide dismutase, Escherichia coli superoxide dismutase, and antisweetener antibody NC6.8, show speedups of 17, 35, and 39, respectively. The test for reactions between two model proteins with orientations shows speedups of 2578 for one set of configurations and 3341 for another set of configurations.

Biomolecular diffusional association

The primary method for the computational study of biomolecular diffusional association is Brownian dynamics. Recent work has seen advances in the efficiency of computing association rates and in the accuracy of simulation models. New areas to which Brownian dynamics has been applied include protein polymerisation and protein adsorption to a surface. There has recently been particularly intense study of protein-protein association, and Brownian dynamics, together with other theoretical and experimental approaches, has led to new insights into the determinants of protein-protein binding kinetics.