Weighted-ensemble Brownian dynamics simulations for protein association reactions

From thermodynamics to kinetics: enhanced sampling of rare events

Despite great advances in molecular dynamics simulations, there remain large gaps between the simulations and experimental observations in terms of the time and length scales that can be approached. Developing fast and accurate algorithms and methods is of ultimate importance to bridge these gaps. In this Account, we briefly summarize recent efforts in such directions. In particular, we focus on integrated tempering sampling. The efficiency of this sampling method has been demonstrated by applications to a range of chemical and biological problems: protein folding, molecular cluster structure searches, and chemical reactions. The combination of integrated tempering sampling and a trajectory sampling method allows the calculation of rate constants and reaction pathways without predefined collective coordinates.

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.

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.

Internal coordinate molecular dynamics: a foundation for multiscale dynamics

Internal coordinates such as bond lengths, bond angles, and torsion angles (BAT) are natural coordinates for describing a bonded molecular system. However, the molecular dynamics (MD) simulation methods that are widely used for proteins, DNA, and polymers are based on Cartesian coordinates owing to the mathematical simplicity of the equations of motion. However, constraints are often needed with Cartesian MD simulations to enhance the conformational sampling. This makes the equations of motion in the Cartesian coordinates differential-algebraic, which adversely impacts the complexity and the robustness of the simulations. On the other hand, constraints can be easily placed in BAT coordinates by removing the degrees of freedom that need to be constrained. Thus, the internal coordinate MD (ICMD) offers an attractive alternative to Cartesian coordinate MD for developing multiscale MD method. The torsional MD method is a special adaptation of the ICMD method, where all the bond lengths and bond angles are kept rigid. The advantages of ICMD simulation methods are the longer time step size afforded by freezing high frequency degrees of freedom and performing a conformational search in the more important low frequency torsional degrees of freedom. However, the advancements in the ICMD simulations have been slow and stifled by long-standing mathematical bottlenecks. In this review, we summarize the recent mathematical advancements we have made based on spatial operator algebra, in developing a robust long time scale ICMD simulation toolkit useful for various applications. We also present the applications of ICMD simulations to study conformational changes in proteins and protein structure refinement. We review the advantages of the ICMD simulations over the Cartesian simulations when used with enhanced sampling methods and project the future use of ICMD simulations in protein dynamics.

Computing ensembles of transitions with molecular dynamics simulations

A molecular understanding of conformational change is important for connecting structure and function. Without the ability to sample on the meaningful large-scale conformational changes, the ability to infer biological function and to understand the effect of mutations and changes in environment is not possible. Our Dynamic Importance Sampling method (DIMS), part of the CHARMM simulation package, is a method that enables sampling over ensembles of transition intermediates. This chapter outlines the context for the method and the usage within the program.

Fluctuation Flooding Method (FFM) for accelerating conformational transitions of proteins

A powerful conformational sampling method for accelerating structural transitions of proteins, "Fluctuation Flooding Method (FFM)," is proposed. In FFM, cycles of the following steps enhance the transitions: (i) extractions of largely fluctuating snapshots along anisotropic modes obtained from trajectories of multiple independent molecular dynamics (MD) simulations and (ii) conformational re-sampling of the snapshots via re-generations of initial velocities when re-starting MD simulations. In an application to bacteriophage T4 lysozyme, FFM successfully accelerated the open-closed transition with the 6 ns simulation starting solely from the open state, although the 1-μs canonical MD simulation failed to sample such a rare event.

Protein folding pathways extracted by OFLOOD: Outlier FLOODing method

The Outlier FLOODing method (OFLOOD) is proposed as an efficient conformational sampling method to extract biologically rare events such as protein folding. In OFLOOD, sparse distributions (outliers in the conformational space) were regarded as relevant states for the transitions. Then, the transitions were enhanced through conformational resampling from the outliers. This evidence indicates that the conformational resampling of the sparse distributions might increase chances for promoting the transitions from the outliers to other meta-stable states, which resembles a conformational flooding from the outliers to the neighboring clusters. OFLOOD consists of (i) detections of outliers from conformational distributions and (ii) conformational resampling from the outliers by molecular dynamics (MD) simulations. Cycles of (i) and (ii) are simply repeated. As demonstrations, OFLOOD was applied to folding of Chignolin and HP35. In both cases, OFLOOD automatically extracted folding pathways from unfolded structures with ns-order computational costs, although µs-order canonical MD failed to extract them.

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.

AWE-WQ: fast-forwarding molecular dynamics using the accelerated weighted ensemble

A limitation of traditional molecular dynamics (MD) is that reaction rates are difficult to compute. This is due to the rarity of observing transitions between metastable states since high energy barriers trap the system in these states. Recently the weighted ensemble (WE) family of methods have emerged which can flexibly and efficiently sample conformational space without being trapped and allow calculation of unbiased rates. However, while WE can sample correctly and efficiently, a scalable implementation applicable to interesting biomolecular systems is not available. We provide here a GPLv2 implementation called AWE-WQ of a WE algorithm using the master/worker distributed computing WorkQueue (WQ) framework. AWE-WQ is scalable to thousands of nodes and supports dynamic allocation of computer resources, heterogeneous resource usage (such as central processing units (CPU) and graphical processing units (GPUs) concurrently), seamless heterogeneous cluster usage (i.e., campus grids and cloud providers), and support for arbitrary MD codes such as GROMACS, while ensuring that all statistics are unbiased. We applied AWE-WQ to a 34 residue protein which simulated 1.5 ms over 8 months with peak aggregate performance of 1000 ns/h. Comparison was done with a 200 μs simulation collected on a GPU over a similar timespan. The folding and unfolded rates were of comparable accuracy.

Using Markov state models to study self-assembly

Markov state models (MSMs) have been demonstrated to be a powerful method for computationally studying intramolecular processes such as protein folding and macromolecular conformational changes. In this article, we present a new approach to construct MSMs that is applicable to modeling a broad class of multi-molecular assembly reactions. Distinct structures formed during assembly are distinguished by their undirected graphs, which are defined by strong subunit interactions. Spatial inhomogeneities of free subunits are accounted for using a recently developed Gaussian-based signature. Simplifications to this state identification are also investigated. The feasibility of this approach is demonstrated on two different coarse-grained models for virus self-assembly. We find good agreement between the dynamics predicted by the MSMs and long, unbiased simulations, and that the MSMs can reduce overall simulation time by orders of magnitude.

Exploring the conformational transitions of biomolecular systems using a simple two-state anisotropic network model

Biomolecular conformational transitions are essential to biological functions. Most experimental methods report on the long-lived functional states of biomolecules, but information about the transition pathways between these stable states is generally scarce. Such transitions involve short-lived conformational states that are difficult to detect experimentally. For this reason, computational methods are needed to produce plausible hypothetical transition pathways that can then be probed experimentally. Here we propose a simple and computationally efficient method, called ANMPathway, for constructing a physically reasonable pathway between two endpoints of a conformational transition. We adopt a coarse-grained representation of the protein and construct a two-state potential by combining two elastic network models (ENMs) representative of the experimental structures resolved for the endpoints. The two-state potential has a cusp hypersurface in the configuration space where the energies from both the ENMs are equal. We first search for the minimum energy structure on the cusp hypersurface and then treat it as the transition state. The continuous pathway is subsequently constructed by following the steepest descent energy minimization trajectories starting from the transition state on each side of the cusp hypersurface. Application to several systems of broad biological interest such as adenylate kinase, ATP-driven calcium pump SERCA, leucine transporter and glutamate transporter shows that ANMPathway yields results in good agreement with those from other similar methods and with data obtained from all-atom molecular dynamics simulations, in support of the utility of this simple and efficient approach. Notably the method provides experimentally testable predictions, including the formation of non-native contacts during the transition which we were able to detect in two of the systems we studied. An open-access web server has been created to deliver ANMPathway results.

Many biomolecules are like tiny molecular machines that need to change their shapes and visit many states to perform their biological functions. For a complete molecular understanding of a biological process, one needs to have information on the relevant stable states of the system in question, as well as the pathways by which the system travels from one state to another. We report here an efficient computational method that uses the knowledge of experimental structures of a pair of stable states in order to construct an energetically favoravle pathway between them. We adopt a simple representation of the molecular system by replacing the atoms with beads connected by springs and constructing an energy function with two minima around the end-states. We searched for the structure with highest energy that the system is most likely to visit during the transition and created two paths starting from this structure and proceeding toward the end-states. The combined result of these two paths is the minimum energy pathway between the two stable states. We apply this method to study important structural changes in one enzyme and three large proteins that transport small molecules and ions across the cell membrane.

Simultaneous Computation of Dynamical and Equilibrium Information Using a Weighted Ensemble of Trajectories

Equilibrium formally can be represented as an ensemble of uncoupled systems undergoing unbiased dynamics in which detailed balance is maintained. Many nonequilibrium processes can be described by suitable subsets of the equilibrium ensemble. Here, we employ the “weighted ensemble” (WE) simulation protocol [Huber and Kim, Biophys. J.1996, 70, 97–110] to generate equilibrium trajectory ensembles and extract nonequilibrium subsets for computing kinetic quantities. States do not need to be chosen in advance. The procedure formally allows estimation of kinetic rates between arbitrary states chosen after the simulation, along with their equilibrium populations. We also describe a related history-dependent matrix procedure for estimating equilibrium and nonequilibrium observables when phase space has been divided into arbitrary non-Markovian regions, whether in WE or ordinary simulation. In this proof-of-principle study, these methods are successfully applied and validated on two molecular systems: explicitly solvated methane association and the implicitly solvated Ala4 peptide. We comment on challenges remaining in WE calculations.

WExplore: hierarchical exploration of high-dimensional spaces using the weighted ensemble algorithm

As most relevant motions in biomolecular systems are inaccessible to conventional molecular dynamics simulations, algorithms that enhance sampling of rare events are indispensable. Increasing interest in intrinsically disordered systems and the desire to target ensembles of protein conformations (rather than single structures) in drug development motivate the need for enhanced sampling algorithms that are not limited to "two-basin" problems, and can efficiently determine structural ensembles. For systems that are not well-studied, this must often be done with little or no information about the dynamics of interest. Here we present a novel strategy to determine structural ensembles that uses dynamically defined sampling regions that are organized in a hierarchical framework. It is based on the weighted ensemble algorithm, where an ensemble of copies of the system ("replicas") is directed to new regions of configuration space through merging and cloning operations. The sampling hierarchy allows for a large number of regions to be defined, while using only a small number of replicas that can be balanced over multiple length scales. We demonstrate this algorithm on two model systems that are analytically solvable and examine the 10-residue peptide chignolin in explicit solvent. The latter system is analyzed using a configuration space network, and novel hydrogen bonds are found that facilitate folding.

A practical perspective on the implementation of hyperdynamics for accelerated simulation

Consideration is given to several practical issues arising during the implementation of hyperdynamics, a methodology that extends the time scale of the conventional molecular dynamics simulation potentially by orders of magnitude. First, the methodology is reformulated in terms of the transition rate based on the buffer region approach (buffer rate), which can describe transitions in more general contexts than the transition state theory (TST). It will be shown that hyperdynamics can exactly preserve the buffer rate as well as the TST rate, which broadens the scope of the method. Next, the originally proposed scheme to compute the boost factor on-the-fly is reviewed and some alternative methods, one of which uses the umbrella sampling method, are presented. Finally, the methodology is validated in the context of a 1-dimensional example potential and a 3-dimensional simulation of the motion of an atomic force microscope tip moving along a surface.

Mapping conformational dynamics of proteins using torsional dynamics simulations

All-atom molecular dynamics simulations are widely used to study the flexibility of protein conformations. However, enhanced sampling techniques are required for simulating protein dynamics that occur on the millisecond timescale. In this work, we show that torsional molecular dynamics simulations enhance protein conformational sampling by performing conformational search in the low-frequency torsional degrees of freedom. In this article, we use our recently developed torsional-dynamics method called Generalized Newton-Euler Inverse Mass Operator (GNEIMO) to study the conformational dynamics of four proteins. We investigate the use of the GNEIMO method in simulations of the conformationally flexible proteins fasciculin and calmodulin, as well as the less flexible crambin and bovine pancreatic trypsin inhibitor. For the latter two proteins, the GNEIMO simulations with an implicit-solvent model reproduced the average protein structural fluctuations and sample conformations similar to those from Cartesian simulations with explicit solvent. The application of GNEIMO with replica exchange to the study of fasciculin conformational dynamics produced sampling of two of this protein's experimentally established conformational substates. Conformational transition of calmodulin from the Ca(2+)-bound to the Ca(2+)-free conformation occurred readily with GNEIMO simulations. Moreover, the GNEIMO method generated an ensemble of conformations that satisfy about half of both short- and long-range interresidue distances obtained from NMR structures of holo to apo transitions in calmodulin. Although unconstrained all-atom Cartesian simulations have failed to sample transitions between the substates of fasciculin and calmodulin, GNEIMO simulations show the transitions in both systems. The relatively short simulation times required to capture these long-timescale conformational dynamics indicate that GNEIMO is a promising molecular-dynamics technique for studying domain motion in proteins.

Efficient stochastic simulation of chemical kinetics networks using a weighted ensemble of trajectories

We apply the "weighted ensemble" (WE) simulation strategy, previously employed in the context of molecular dynamics simulations, to a series of systems-biology models that range in complexity from a one-dimensional system to a system with 354 species and 3680 reactions. WE is relatively easy to implement, does not require extensive hand-tuning of parameters, does not depend on the details of the simulation algorithm, and can facilitate the simulation of extremely rare events. For the coupled stochastic reaction systems we study, WE is able to produce accurate and efficient approximations of the joint probability distribution for all chemical species for all time t. WE is also able to efficiently extract mean first passage times for the systems, via the construction of a steady-state condition with feedback. In all cases studied here, WE results agree with independent "brute-force" calculations, but significantly enhance the precision with which rare or slow processes can be characterized. Speedups over "brute-force" in sampling rare events via the Gillespie direct Stochastic Simulation Algorithm range from ~10(12) to ~10(18) for characterizing rare states in a distribution, and ~10(2) to ~10(4) for finding mean first passage times.

Learning Kinetic Distance Metrics for Markov State Models of Protein Conformational Dynamics

Statistical modeling of long timescale dynamics with Markov state models (MSMs) has been shown to be an effective strategy for building quantitative and qualitative insight into protein folding processes. Existing methodologies, however, rely on geometric clustering using distance metrics such as root mean square deviation (RMSD), assuming that geometric similarity provides an adequate basis for the kinetic partitioning of phase space. Here, inspired by advances in the machine learning community, we introduce a new approach for learning a distance metric explicitly constructed to model kinetic similarity. This approach enables the construction of models, especially in the regime of high anisotropy in the diffusion constant, with fewer states than was previously possible. Application of this technique to the analysis of two ultralong molecular dynamics simulations of the FiP35 WW domain identifies discrete near-native relaxation dynamics in the millisecond regime that were not resolved in previous analyses.

Weighted-ensemble Brownian dynamics simulation: sampling of rare events in nonequilibrium systems

We provide an algorithm based on weighted-ensemble (WE) methods, to accurately sample systems at steady state. Applying our method to different one- and two-dimensional models, we succeed in calculating steady-state probabilities of order 10(-300) and reproduce the Arrhenius law for rates of order 10(-280). Special attention is payed to the simulation of nonpotential systems where no detailed balance assumption exists. For this large class of stochastic systems, the stationary probability distribution density is often unknown and cannot be used as preknowledge during the simulation. We compare the algorithm's efficiency with standard Brownian dynamics simulations and the original WE method.

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.

Folding Proteins at 500 ns/hour with Work Queue

Molecular modeling is a field that traditionally has large computational costs. Until recently, most simulation techniques relied on long trajectories, which inherently have poor scalability. A new class of methods is proposed that requires only a large number of short calculations, and for which minimal communication between computer nodes is required. We considered one of the more accurate variants called Accelerated Weighted Ensemble Dynamics (AWE) and for which distributed computing can be made efficient. We implemented AWE using the Work Queue framework for task management and applied it to an all atom protein model (Fip35 WW domain). We can run with excellent scalability by simultaneously utilizing heterogeneous resources from multiple computing platforms such as clouds (Amazon EC2, Microsoft Azure), dedicated clusters, grids, on multiple architectures (CPU/GPU, 32/64bit), and in a dynamic environment in which processes are regularly added or removed from the pool. This has allowed us to achieve an aggregate sampling rate of over 500 ns/hour. As a comparison, a single process typically achieves 0.1 ns/hour.

An adaptive weighted ensemble procedure for efficient computation of free energies and first passage rates

We introduce an adaptive weighted-ensemble procedure (aWEP) for efficient and accurate evaluation of first-passage rates between states for two-state systems. The basic idea that distinguishes aWEP from conventional weighted-ensemble (WE) methodology is the division of the configuration space into smaller regions and equilibration of the trajectories within each region upon adaptive partitioning of the regions themselves into small grids. The equilibrated conditional∕transition probabilities between each pair of regions lead to the determination of populations of the regions and the first-passage times between regions, which in turn are combined to evaluate the first passage times for the forward and backward transitions between the two states. The application of the procedure to a non-trivial coarse-grained model of a 70-residue calcium binding domain of calmodulin is shown to efficiently yield information on the equilibrium probabilities of the two states as well as their first passage times. Notably, the new procedure is significantly more efficient than the canonical implementation of the WE procedure, and this improvement becomes even more significant at low temperatures.

Tunable, mixed-resolution modeling using library-based Monte Carlo and graphics processing units

Building on our recently introduced library-based Monte Carlo (LBMC) approach, we describe a flexible protocol for mixed coarse-grained (CG)/all-atom (AA) simulation of proteins and ligands. In the present implementation of LBMC, protein side chain configurations are pre-calculated and stored in libraries, while bonded interactions along the backbone are treated explicitly. Because the AA side chain coordinates are maintained at minimal run-time cost, arbitrary sites and interaction terms can be turned on to create mixed-resolution models. For example, an AA region of interest such as a binding site can be coupled to a CG model for the rest of the protein. We have additionally developed a hybrid implementation of the generalized Born/surface area (GBSA) implicit solvent model suitable for mixed-resolution models, which in turn was ported to a graphics processing unit (GPU) for faster calculation. The new software was applied to study two systems: (i) the behavior of spin labels on the B1 domain of protein G (GB1) and (ii) docking of randomly initialized estradiol configurations to the ligand binding domain of the estrogen receptor (ERα). The performance of the GPU version of the code was also benchmarked in a number of additional systems.

Coarse-grained simulations of transitions in the E2-to-E1 conformations for Ca ATPase (SERCA) show entropy-enthalpy compensation

SERCA is a membrane transport protein that has been extensively studied. There are a large number of highly resolved X-ray structures and several hundred mutations that have been characterized functionally. Despite this, the molecular details of the catalytic cycle, a cycle that includes large conformational changes, is not fully understood. In this computational study, we provide molecular dynamics descriptions of conformational changes during the E2→E1 transitions. The motivating point for these calculations was a series of insertion mutants in the A-M3 linker region that led to significant shifts in measured rates between the E2 and E1 states, as shown by experimental characterization. Using coarse-grained dynamic importance sampling within the context of a population shift framework, we sample on the intermediates along the transition pathway to address the mechanism for the conformational changes and the effects of the insertion mutations on the kinetics of the transition. The calculations define an approximation for the relative changes in entropy and enthalpy along the transition. These are found to be important for understanding the experimentally observed differences in rates. In particular, the interactions between cytoplasmic domains, water interactions, and the shifts in protein degrees of freedom with the insertion mutations show mutual compensation for the E2→E1 transitions in wild-type and mutant systems.

The role of domain: domain interactions versus domain: water interactions in the coarse-grained simulations of the E1P to E2P transitions in Ca-ATPase (SERCA)

SERCA is an important model system for understanding the molecular details of conformational change in membrane transport systems. This reflects the large number of solved X-ray structures and the equally large database of mutations that have been assayed. In this computational study, we provide a molecular dynamics description of the conformational changes during the E1P → E2P transitions. This set of states further changes with insertion mutants in the A-M3 linker region. These mutants were experimentally shown to lead to significant shifts in rates between the E1P → E2P states. Using the population shift framework and dynamic importance sampling method along with coarse-grained representations of the protein, lipid, and water, we suggest why these changes are found. The calculations sample on intermediates and suggest that changes in interactions, individual helix interactions, and water behavior are key elements in the molecular compositions that underlie shifts in kinetics. In particular, as the insertion length grows, it attracts more water and disrupts domain interactions, creating changes as well at the sites of key helix interactions between the A-Domain and the P-Domain. This provides a conceptual picture that aids understanding of the experimental results.

Simulations of the alternating access mechanism of the sodium symporter Mhp1

Sodium coupled cotransporters of the five-helix inverted repeat (5HIR) superfamily use an alternating access mechanism to transport a myriad of small molecules across the cell membrane. One of the primary steps in this mechanism is the conformational transition from a state poised to bind extracellular substrates to a state that is competent to deliver substrate to the cytoplasm. Here, we construct a coarse-grained model of the 5HIR benzylhydantoin transporter Mhp1 that incorporates experimental structures of the outward- and inward-open states to investigate the mechanism of this conformational change. Using the weighted ensemble path-sampling method, we rigorously sample the outward- to inward-facing transition path ensemble. The transition path ensemble reveals a heterogeneous set of pathways connecting the two states and identifies two modes of transport: one consistent with a strict alternating access mechanism and another where decoupling of the inner and outer gates causes the transient formation of a continuous permeation pathway through the transporter. We also show that the conformational switch between the outward- and inward-open states results from rigid body motions of the hash motif relative to the substrate bundle, supporting the rocking bundle hypothesis. Finally, our methodology provides the groundwork for more chemically detailed investigations of the alternating mechanism.

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.

Computing ensembles of transitions from stable states: Dynamic importance sampling

There is an increasing dataset of solved biomolecular structures in more than one conformation and increasing evidence that large-scale conformational change is critical for biomolecular function. In this article, we present our implementation of a dynamic importance sampling (DIMS) algorithm that is directed toward improving our understanding of important intermediate states between experimentally defined starting and ending points. This complements traditional molecular dynamics methods where most of the sampling time is spent in the stable free energy wells defined by these initial and final points. As such, the algorithm creates a candidate set of transitions that provide insights for the much slower and probably most important, functionally relevant degrees of freedom. The method is implemented in the program CHARMM and is tested on six systems of growing size and complexity. These systems, the folding of Protein A and of Protein G, the conformational changes in the calcium sensor S100A6, the glucose-galactose-binding protein, maltodextrin, and lactoferrin, are also compared against other approaches that have been suggested in the literature. The results suggest good sampling on a diverse set of intermediates for all six systems with an ability to control the bias and thus to sample distributions of trajectories for the analysis of intermediate states.

Browndye: A Software Package for Brownian Dynamics

A new software package, Browndye, is presented for simulating the diffusional encounter of two large biological molecules. It can be used to estimate second-order rate constants and encounter probabilities, and to explore reaction trajectories. Browndye builds upon previous knowledge and algorithms from software packages such as UHBD, SDA, and Macrodox, while implementing algorithms that scale to larger systems.

Heterogeneous path ensembles for conformational transitions in semi-atomistic models of adenylate kinase

We performed "weighted ensemble" path-sampling simulations of adenylate kinase, using several semi-atomistic protein models. The models have an all-atom backbone with various levels of residue interactions. The primary result is that full statistically rigorous path sampling required only a few weeks of single-processor computing time with these models, indicating the addition of further chemical detail should be readily feasible. Our semi-atomistic path ensembles are consistent with previous biophysical findings: the presence of two distinct pathways, identification of intermediates, and symmetry of forward and reverse pathways.

PagP crystallized from SDS/cosolvent reveals the route for phospholipid access to the hydrocarbon ruler

Enzymatic reactions involving bilayer lipids occur in an environment with strict physical and topological constraints. The integral membrane enzyme PagP transfers a palmitoyl group from a phospholipid to lipid A in order to assist Escherichia coli in evading host immune defenses during infection. PagP measures the palmitoyl group with an internal hydrocarbon ruler that is formed in the interior of the eight-stranded antiparallel β barrel. The access and egress of the palmitoyl group is thought to take a lateral route from the bilayer phase to the barrel interior. Molecular dynamics, mutagenesis, and a 1.4 A crystal structure of PagP in an SDS / 2-methyl-2,4-pentanediol (MPD) cosolvent system reveal that phospholipid access occurs at the crenel present between strands F and G of PagP. In this way, the phospholipid head group can remain exposed to the cell exterior while the lipid acyl chain remains in a predominantly hydrophobic environment as it translocates to the protein interior.

Steady-state simulations using weighted ensemble path sampling

We extend the weighted ensemble (WE) path sampling method to perform rigorous statistical sampling for systems at steady state. A straightforward steady-state implementation of WE is directly practical for simple landscapes, but not when significant metastable intermediates states are present. We therefore develop an enhanced WE scheme, building on existing ideas, which accelerates attainment of steady state in complex systems. We apply both WE approaches to several model systems, confirming their correctness and efficiency by comparison with brute-force results. The enhanced version is significantly faster than the brute force and straightforward WE for systems with WE bins that accurately reflect the reaction coordinate(s). The new WE methods can also be applied to equilibrium sampling, since equilibrium is a steady state.

Connective Tissue Growth Factor (CTGF/CCN2) enhances lactogenic differentiation of mammary epithelial cells via integrin-mediated cell adhesion

Background

Connective Tissue Growth Factor (CTGF/CCN2), a known matrix-associated protein, is required for the lactogenic differentiation of mouse mammary epithelial cells. An HC11 mammary epithelial cell line expressing CTGF/CCN2 was constructed to dissect the cellular responses to CTGF/CCN2 that contribute to this differentiation program.

Results

Tetracycline-regulated expression of CTGF/CCN2 in HC11 cells enhanced multiple markers of lactogenic differentiation including β-casein transcription and mammosphere formation. In a separate measure of mammary differentiation the addition of CTGF/CCN2 to cultures of MCF10A cells increased the development of acini in vitro. In HC11 cells the elevated levels of CTGF/CCN2 diminished the requirement for extracellular matrix proteins in the activation of β-casein transcription, indicating that CTGF/CCN2 contributed to lactogenic differentiation through the regulation of matrix dependent cell adhesion. CTGF/CCN2 expression in HC11 cells increased expression of extracellular matrix proteins and integrins, enhanced the formation of focal adhesion complexes, and increased survival signaling. In addition, HC11 cells adhered to immobilized CTGF/CCN2 and this was inhibited by function-blocking antibodies to the integrins α6 and β1, and to a lesser degree by antibody to β3 integrin.

Conclusions

CTGF/CCN2 expression in HC11 cells led to an increase in multiple markers of lactogenic differentiation. The mechanisms by which CTGF/CCN2 contributed to lactogenic differentiation include direct binding of CTGF/CCN2 to integrin complexes and CTGF/CCN2-induced matrix protein expression resulting in elevated integrin functionality.