Simulation of an Ensemble of Conformational Transitions in a United-Residue Model of Calmodulin

Inherent flexibility determines the transition mechanisms of the EF-hands of calmodulin

We explore how inherent flexibility of a protein molecule influences the mechanism controlling allosteric transitions by using a variational model inspired from work in protein folding. The striking differences in the predicted transition mechanism for the opening of the two domains of calmodulin (CaM) emphasize that inherent flexibility is key to understanding the complex conformational changes that occur in proteins. In particular, the C-terminal domain of CaM (cCaM), which is inherently less flexible than its N-terminal domain (nCaM), reveals "cracking" or local partial unfolding during the open/closed transition. This result is in harmony with the picture that cracking relieves local stresses caused by conformational deformations of a sufficiently rigid protein. We also compare the conformational transition in a recently studied even-odd paired fragment of CaM. Our results rationalize the different relative binding affinities of the EF-hands in the engineered fragment compared with the intact odd-even paired EF-hands (nCaM and cCaM) in terms of changes in flexibility along the transition route. Aside from elucidating general theoretical ideas about the cracking mechanism, these studies also emphasize how the remarkable intrinsic plasticity of CaM underlies conformational dynamics essential for its diverse functions.

Switching-path distribution in multidimensional systems

We explore the distribution of paths followed in fluctuation-induced switching between coexisting stable states. We introduce a quantitative characteristic of the path distribution in phase space that does not require a priori knowledge of system dynamics. The theory of the distribution is developed and its direct measurement is performed in a micromechanical oscillator driven into parametric resonance. The experimental and theoretical results on the shape and position of the distribution are in excellent agreement, with no adjustable parameters. In addition, the experiment provides the first demonstration of the lack of time-reversal symmetry in switching of systems far from thermal equilibrium. The results open the possibility of efficient control of the switching probability based on the measured narrow path distribution.

Single molecule conformational dynamics of adenylate kinase: energy landscape, structural correlations, and transition state ensembles

We developed a coarse grained two-well model to study the single molecule protein conformational dynamics in microscopic detail at the residue level, overcoming the often encountered computational bottleneck. In particular, we explored the underlying conformational energy landscape of adenylate kinase, a crucial protein for signal transduction in the cell, and identified two major kinetic pathways for the conformational switch between open and closed states through either the intermediate state or the transient state. Based on the parameters fitted to the room-temperature experimental data, we predicted open and closed kinetic rates at the whole temperature ranges from 10 to 50 degrees C, which agree well with the experimental turnover numbers. After uncovering the underlying mechanism for conformational dynamics and exploring the structural correlations, we found the crucial dynamical interplay between the nucleoside monophosphate binding domain (NMP) and the ATP-binding domain (LID) in controlling the conformational switch. The key residues and contacts responsible for the conformational transitions are identified by following the time evolution of the two-dimensional spatial contact maps and characterizing the transition state as well as intermediate structure ensembles through phi value analysis. Our model provides a general framework to study the conformational dynamics of biomolecules and can be applied to many other systems.

On the structural convergence of biomolecular simulations by determination of the effective sample size

Although atomistic simulations of proteins and other biological systems are approaching microsecond timescales, the quality of simulation trajectories has remained difficult to assess. Such assessment is critical not only for establishing the relevance of any individual simulation but also in the extremely active field of developing computational methods. Here we map the trajectory assessment problem onto a simple statistical calculation of the "effective sample size", that is, the number of statistically independent configurations. The mapping is achieved by asking the question, "How much time must elapse between snapshots included in a sample for that sample to exhibit the statistical properties expected for independent and identically distributed configurations?" Our method is more general than autocorrelation methods in that it directly probes the configuration-space distribution without requiring a priori definition of configurational substates and without any fitting parameters. We show that the method is equally and directly applicable to toy models, peptides, and a 72-residue protein model. Variants of our approach can readily be applied to a wide range of physical and chemical systems.

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.

Protein conformational transitions explored by mixed elastic network models

We develop a mixed elastic network model (MENM) to study large-scale conformational transitions of proteins between two (or more) known structures. Elastic network potentials for the beginning and end states of a transition are combined, in effect, by adding their respective partition functions. The resulting effective MENM energy function smoothly interpolates between the original surfaces, and retains the beginning and end structures as local minima. Saddle points, transition paths, potentials of mean force, and partition functions can be found efficiently by largely analytic methods. To characterize the protein motions during a conformational transition, we follow "transition paths" on the MENM surface that connect the beginning and end structures and are invariant to parameterizations of the model and the mathematical form of the mixing scheme. As illustrations of the general formalism, we study large-scale conformation changes of the motor proteins KIF1A kinesin and myosin II. We generate possible transition paths for these two proteins that reveal details of their conformational motions. The MENM formalism is computationally efficient and generally applicable even for large protein systems that undergo highly collective structural changes.

Coarse-grained free energy functions for studying protein conformational changes: a double-well network model

In this work, a double-well network model (DWNM) is presented for generating a coarse-grained free energy function that can be used to study the transition between reference conformational states of a protein molecule. Compared to earlier work that uses a single, multidimensional double-well potential to connect two conformational states, the DWNM uses a set of interconnected double-well potentials for this purpose. The DWNM free energy function has multiple intermediate states and saddle points, and is hence a "rough" free energy landscape. In this implementation of the DWNM, the free energy function is reduced to an elastic-network model representation near the two reference states. The effects of free energy function roughness on the reaction pathways of protein conformational change is demonstrated by applying the DWNM to the conformational changes of two protein systems: the coil-to-helix transition of the DB-loop in G-actin and the open-to-closed transition of adenylate kinase. In both systems, the rough free energy function of the DWNM leads to the identification of distinct minimum free energy paths connecting two conformational states. These results indicate that while the elastic-network model captures the low-frequency vibrational motions of a protein, the roughness in the free energy function introduced by the DWNM can be used to characterize the transition mechanism between protein conformations.

Structural change and nucleotide dissociation of Myosin motor domain: dual go model simulation

We investigated the structural relaxation of myosin motor domain from the pre-power stroke state to the near-rigor state using molecular dynamics simulation of a coarse-grained protein model. To describe the spontaneous structural change, we propose a dual Gō-model-a variant of the Gō-like model that has two reference structures. The nucleotide dissociation process is also studied by introducing a coarse-grained nucleotide in the simulation. We found that the myosin structural relaxation toward the near-rigor conformation cannot be completed before the nucleotide dissociation. Moreover, the relaxation and the dissociation occurred cooperatively when the nucleotide was tightly bound to the myosin head. The result suggested that the primary role of the nucleotide is to suppress the structural relaxation.

Simulating Protein Motions with Rigidity Analysis

Protein motions, ranging from molecular flexibility to large-scale conformational change, play an essential role in many biochemical processes. Despite the explosion in our knowledge of structural and functional data, our understanding of protein movement is still very limited. In previous work, we developed and validated a motion planning based method for mapping protein folding pathways from unstructured conformations to the native state. In this paper, we propose a novel method based on rigidity theory to sample conformation space more effectively, and we describe extensions of our framework to automate the process and to map transitions between specified conformations. Our results show that these additions both improve the accuracy of our maps and enable us to study a broader range of motions for larger proteins. For example, we show that rigidity-based sampling results in maps that capture subtle folding differences between protein G and its mutants, NuG1 and NuG2, and we illustrate how our technique can be used to study large-scale conformational changes in calmodulin, a 148 residue signaling protein known to undergo conformational changes when binding to Ca(2+). Finally, we announce our web-based protein folding server which includes a publicly available archive of protein motions: (http://parasol.tamu.edu/foldingserver/).

Multiscale modeling of biomolecular systems: in serial and in parallel

Considerable progress has been recently achieved in the multiscale modeling of complex biological processes. Multiscale models have now investigated the structure and dynamics of lipid membranes, proteins, peptides and DNA over length and time scales ranging from the atomic to the macroscopic. Serial multiscale methods that parameterize low-resolution coarse-grained models with data from high-resolution models have studied long time or length scale phenomena that cannot be investigated with atomically detailed models. Parallel multiscale methods that directly couple high- and low-resolution models have efficiently explored slow structural transitions and the importance of long-wavelength fluctuations for biological molecules. The success of such models relies upon new theories and methods for constructing accurate multiscale bridges that transfer information between models with different resolutions.

An Approximate Method in Using Molecular Mechanics Simulations To Study Slow Protein Conformational Changes

The broad range of characteristic motions in proteins has limited the applicability of molecular dynamics simulations in studying large-scale conformational transitions. We present an approximate method, making use of standard MD simulations and using a much larger integration time step, to obtain the structural changes for slow systematic motions of large complex systems. We show the applicability of this method by simulating the open to closed Calmodulin calcium binding domain conformational changes. Starting with the Ca2+-bound X-ray structure, and after the removal of the Ca2+ ions, our calculation yielded intermediate conformations during the rearrangement of helices in each Ca2+ binding pocket, leading to a structure with a lowest rmsd of 1.56 A compared to the NMR apo-calmodulin structure.

Transition-event durations in one-dimensional activated processes

Despite their importance in activated processes, transition-event durations--which are much shorter than first passage times--have not received a complete theoretical treatment. The authors therefore study the distribution rhob(t) of durations of transition events over a barrier in a one-dimensional system undergoing overdamped Langevin dynamics. The authors show that rhob(t) is determined by a Fokker-Planck equation with absorbing boundary conditions and obtain a number of results, including (i) the analytic form of the asymptotic short-time transient behavior, which is universal and independent of the potential function; (ii) the first nonuniversal correction to the short-time behavior leading to an estimate of a key physical time scale; (iii) following previous work, a recursive formulation for calculating, exactly, all moments of rhob based solely on the potential function-along with approximations for the distribution based on a small number of moments; and (iv) a high-barrier approximation to the long-time (t-->infinity) behavior of rhob(t). The authors also find that the mean event duration does not depend simply on the barrier-top frequency (curvature) but is sensitive to details of the potential. All of the analytic results are confirmed by transition-path-sampling simulations implemented in a novel way. Finally, the authors discuss which aspects of the duration distribution are expected to be general for more complex systems.

Introduction to the membrane protein reviews: the interplay of structure, dynamics, and environment in membrane protein function

In our review, we introduce an organizational scheme for membrane protein function. It is the relationship between structure, dynamics, and environment that endows the membrane and its constituents with remarkable sensitivity and robustness. Our understanding begins with landmark advances like those presented in the following chapters. Membrane proteins are notoriously difficult to study, and so the work presented here on the ADP/ATP carrier [Nury et al. ( 2 )], rhodopsin [Palczewski ( 24 )], and the cytochrome b6f complex [Cramer et al. ( 35 )] represents incredible progress in this now blossoming field.

Conformational transitions of adenylate kinase: switching by cracking

Conformational heterogeneity in proteins is known to often be the key to their function. We present a coarse grained model to explore the interplay between protein structure, folding and function which is applicable to allosteric or non-allosteric proteins. We employ the model to study the detailed mechanism of the reversible conformational transition of Adenylate Kinase (AKE) between the open to the closed conformation, a reaction that is crucial to the protein's catalytic function. We directly observe high strain energy which appears to be correlated with localized unfolding during the functional transition. This work also demonstrates that competing native interactions from the open and closed form can account for the large conformational transitions in AKE. We further characterize the conformational transitions with a new measure Phi(Func), and demonstrate that local unfolding may be due, in part, to competing intra-protein interactions.

Multiple-basin energy landscapes for large-amplitude conformational motions of proteins: Structure-based molecular dynamics simulations

Biomolecules often undergo large-amplitude motions when they bind or release other molecules. Unlike macroscopic machines, these biomolecular machines can partially disassemble (unfold) and then reassemble (fold) during such transitions. Here we put forward a minimal structure-based model, the "multiple-basin model," that can directly be used for molecular dynamics simulation of even very large biomolecular systems so long as the endpoints of the conformational change are known. We investigate the model by simulating large-scale motions of four proteins: glutamine-binding protein, S100A6, dihydrofolate reductase, and HIV-1 protease. The mechanisms of conformational transition depend on the protein basin topologies and change with temperature near the folding transition. The conformational transition rate varies linearly with driving force over a fairly large range. This linearity appears to be a consequence of partial unfolding during the conformational transition.

Pathways of activated escape in periodically modulated systems

We investigate dynamics of activated escape in periodically modulated systems. The trajectories followed in escape form diffusion-broadened tubes, which are periodically repeated in time. We show that these tubes can be directly observed and find their shape. Quantitatively, the tubes are characterized by the distribution of trajectories that, after escape, pass through a given point in phase space for a given modulation phase. This distribution may display several peaks separated by the modulation period. Analytical results agree with the results of simulations of a Brownian particle in a model modulated potential.

Ensemble-based convergence analysis of biomolecular trajectories

Assessing the convergence of a biomolecular simulation is an essential part of any careful computational investigation, because many fundamental aspects of molecular behavior depend on the relative populations of different conformers. Here we present a physically intuitive method to self-consistently assess the convergence of trajectories generated by molecular dynamics and related methods. Our approach reports directly and systematically on the structural diversity of a simulation trajectory. Straightforward clustering and classification steps are the key ingredients, allowing the approach to be trivially applied to systems of any size. Our initial study on met-enkephalin strongly suggests that even fairly long trajectories (approximately 50 ns) may not be converged for this small--but highly flexible--system.

Resolution exchange simulation

We extend replica-exchange simulation in two ways and apply our approaches to biomolecules. The first generalization permits exchange simulation between models of differing resolution--i.e., between detailed and coarse-grained models. Such "resolution exchange" can be applied to molecular systems or spin systems. The second extension is to "pseudoexchange" simulations, which require little CPU usage for most levels of the exchange ladder and also substantially reduce the need for overlap between levels. Pseudoexchanges can be used in either replica or resolution exchange simulations. We perform efficient, converged simulations of a 50-atom peptide to illustrate the new approaches.

Calculation of the entropy of random coil polymers with the hypothetical scanning Monte Carlo method

Hypothetical scanning Monte Carlo (HSMC) is a method for calculating the absolute entropy S and free energy F from a given MC trajectory developed recently and applied to liquid argon, TIP3P water, and peptides. In this paper HSMC is extended to random coil polymers by applying it to self-avoiding walks on a square lattice--a simple but difficult model due to strong excluded volume interactions. With HSMC the probability of a given chain is obtained as a product of transition probabilities calculated for each bond by MC simulations and a counting formula. This probability is exact in the sense that it is based on all the interactions of the system and the only approximation is due to finite sampling. The method provides rigorous upper and lower bounds for F, which can be obtained from a very small sample and even from a single chain conformation. HSMC is independent of existing techniques and thus constitutes an independent research tool. The HSMC results are compared to those obtained by other methods, and its application to complex lattice chain models is discussed; we emphasize its ability to treat any type of boundary conditions for which a reference state (with known free energy) might be difficult to define for a thermodynamic integration process. Finally, we stress that the capability of HSMC to extract the absolute entropy from a given sample is important for studying relaxation processes, such as protein folding.

Thermodynamic properties of lattice hard-sphere models

Thermodynamic properties of several lattice hard-sphere models were obtained from grand canonical histogram- reweighting Monte Carlo simulations. Sphere centers occupy positions on a simple cubic lattice of unit spacing and exclude neighboring sites up to a distance sigma. The nearestneighbor exclusion model, sigma = radical2, was previously found to have a second-order transition. Models with integer values of sigma = 1 or 2 do not have any transitions. Models with sigma = radical3 and sigma = 3 have weak first-order fluid-solid transitions while those with sigma = 2 radical2, 2 radical3, and 3 radical2 have strong fluid-solid transitions. Pressure, chemical potential, and density are reported for all models and compared to the results for the continuum, theoretical predictions, and prior simulations when available.