The free energy landscape and dynamics of met-enkephalin

Reaction prediction via atomistic simulation: from quantum mechanics to machine learning

It is an ultimate goal in chemistry to predict reaction without recourse to experiment. Reaction prediction is not just the reaction rate determination of known reactions but, more broadly, the reaction exploration to identify new reaction routes. This review briefly overviews the theory on chemical reaction and the current methods for computing/estimating reaction rate and exploring reaction space. We particularly focus on the atomistic simulation methods for reaction exploration, which are benefited significantly by recently emerged machine learning potentials. We elaborate the stochastic surface walking global pathway sampling based on the global neural network (SSW-NN) potential, developed in our group since 2013, which can explore complex reactions systems unbiasedly and automatedly. Two examples, molecular reaction and heterogeneous catalytic reactions, are presented to illustrate the current status for reaction prediction using SSW-NN.

Analytical Method Using a Scaled Hypersphere Search for High-Dimensional Metadynamics Simulations

Metadynamics (MTD) is one of the most effective methods for calculating the free energy surface and finding rare events. Nevertheless, numerous studies using MTD have been carried out using 3D or lower dimensional collective variables (CVs), as higher dimensional CVs require costly computational resources and the obtained results are too complex to understand the important events. The latter issue can be conveniently solved by utilizing the free energy reaction network (FERN), which is a graph structure consisting of edges of minimum free energy paths (MFEPs), nodes of equation (EQ) points, and transition state (TS) points. In the present article, a new method for exploring FERNs on high-dimensional CVs using MTD and the scaled hypersphere search (SHS) method is described. A test calculation based on the MTD-SHS simulation of met-enkephalin in explicit water with 7 CVs was conducted. As a result, 889 EQ points and 1805 TS points were found. The MTD-SHS approach can find MFEPs exhaustively; therefore, the FERNs can be estimated without any a priori knowledge of the EQ and TS points.

Effective Riemannian Diffusion Model for Conformational Dynamics of Biomolecular Systems

We present a Riemannian formalism for effective diffusion of biomolecules in collective variable spaces that provides a robust framework for conformational free energy calculation methods. Unlike their Euclidean counterparts, the Riemannian potential of mean force (PMF) and minimum free energy path (MFEP) are invariant under coordinate transformations. The presented formalism can be readily employed to modify the collective variable based enhanced sampling techniques, such as umbrella sampling (US) commonly used in biomolecular simulations, to take into account the role of intrinsic geometry of collective variable space. Although our model is mathematically equivalent to a Euclidean diffusion with a position-dependent diffusion tensor, the Riemannian formulation provides a more convenient framework for free energy calculation methods and path-finding algorithms aimed at characterizing the effective conformational dynamics of biomolecules. A simple three-dimensional toy model and a pentapeptide (met-enkephalin) simulated in an explicit solvent environment are used to illustrate the workings of the formalism and its implementation.

Conformational Interconversions of Amino Acid Derivatives

Exhaustive conformational interconversions including transition structure analyses of N-acetyl-l-glycine-N-methylamide as well as its alanine, serine, and cysteine analogues have been investigated at the MP2/6-31G** level, yielding a total of 142 transition states. Improved estimates of relative energies were obtained by separately extrapolating the Hartree-Fock and MP2 energies to the basis set limit and adding the difference between CCSD(T) and MP2 results with the cc-pVDZ basis set to the extrapolated MP2 results. The performance of eight empirical force fields (AMBER94, AMBER14SB, MM2, MM3, MMFFs, CHARMM22_CMAP, OPLS_2005, and AMOEBAPRO13) in reproducing ab initio energies of transition states was tested. Our results indicate that commonly used class I force fields employing a fixed partial charge model for the electrostatic interaction provide mean errors in the ∼10 kJ/mol range for energies of conformational transition states for amino acid conformers. Modern reparametrized versions, such as CHARMM22_CMAP, and polarizable force fields, such as AMOEBAPRO13, have slightly lower mean errors, but maximal errors are still in the 35 kJ/mol range. There are differences between the force fields in their ability for reproducing conformational transitions classified according to backbone/side-chain or regions in the Ramachandran angles, but the data set is likely too small to draw any general conclusions. Errors in conformational interconversion barriers by ∼10 kJ/mol suggest that the commonly used force field may bias certain types of transitions by several orders of magnitude in rate and thus lead to incorrect dynamics in simulations. It is therefore suggested that information for conformational transition states should be included in parametrizations of new force fields.

Transition Networks for the Comprehensive Characterization of Complex Conformational Change in Proteins

Functionally relevant transitions between native conformations of a protein can be complex, involving, for example, the reorganization of parts of the backbone fold, and may occur via a multitude of pathways. Such transitions can be characterized by a transition network (TN), in which the experimentally determined end state structures are connected by a dense network of subtransitions via low-energy intermediates. We show here how the computation of a TN can be achieved for a complex protein transition. First, an efficient hierarchical procedure is used to uniformly sample the conformational subspace relevant to the transition. Then, the best path which connects the end states is determined as well as the rate-limiting ridge on the energy surface which separates them. Graph-theoretical algorithms permit this to be achived by computing the barriers of only a small number out of the many subtransitions in the TN. These barriers are computed using the Conjugate Peak Refinement method. The approach is illustrated on the conformational switch of Ras p21. The best and the 12 next-best transition pathways, having rate-limiting barriers within a range of 10 kcal/mol, were identified. Two main energy ridges, which respectively involve rearrangements of the switch I and switch II loops, show that switch I must rearrange by threading Tyr32 underneath the protein backbone before the rate-limiting switch II rearrangement can occur, while the details of the switch II rearrangement differ significantly among the low-energy pathways.

Membrane Protein Structure, Function, and Dynamics: a Perspective from Experiments and Theory

Membrane proteins mediate processes that are fundamental for the flourishing of biological cells. Membrane-embedded transporters move ions and larger solutes across membranes; receptors mediate communication between the cell and its environment and membrane-embedded enzymes catalyze chemical reactions. Understanding these mechanisms of action requires knowledge of how the proteins couple to their fluid, hydrated lipid membrane environment. We present here current studies in computational and experimental membrane protein biophysics, and show how they address outstanding challenges in understanding the complex environmental effects on the structure, function, and dynamics of membrane proteins.

Visualization of protein folding funnels in lattice models

Protein folding occurs in a very high dimensional phase space with an exponentially large number of states, and according to the energy landscape theory it exhibits a topology resembling a funnel. In this statistical approach, the folding mechanism is unveiled by describing the local minima in an effective one-dimensional representation. Other approaches based on potential energy landscapes address the hierarchical structure of local energy minima through disconnectivity graphs. In this paper, we introduce a metric to describe the distance between any two conformations, which also allows us to go beyond the one-dimensional representation and visualize the folding funnel in 2D and 3D. In this way it is possible to assess the folding process in detail, e.g., by identifying the connectivity between conformations and establishing the paths to reach the native state, in addition to regions where trapping may occur. Unlike the disconnectivity maps method, which is based on the kinetic connections between states, our methodology is based on structural similarities inferred from the new metric. The method was developed in a 27-mer protein lattice model, folded into a 3×3×3 cube. Five sequences were studied and distinct funnels were generated in an analysis restricted to conformations from the transition-state to the native configuration. Consistent with the expected results from the energy landscape theory, folding routes can be visualized to probe different regions of the phase space, as well as determine the difficulty in folding of the distinct sequences. Changes in the landscape due to mutations were visualized, with the comparison between wild and mutated local minima in a single map, which serves to identify different trapping regions. The extension of this approach to more realistic models and its use in combination with other approaches are discussed.

The OPEP protein model: from single molecules, amyloid formation, crowding and hydrodynamics to DNA/RNA systems

The OPEP coarse-grained protein model has been applied to a wide range of applications since its first release 15 years ago. The model, which combines energetic and structural accuracy and chemical specificity, allows the study of single protein properties, DNA-RNA complexes, amyloid fibril formation and protein suspensions in a crowded environment. Here we first review the current state of the model and the most exciting applications using advanced conformational sampling methods. We then present the current limitations and a perspective on the ongoing developments.

Visualizing energy landscapes with metric disconnectivity graphs

The visualization of multidimensional energy landscapes is important, providing insight into the kinetics and thermodynamics of a system, as well the range of structures a system can adopt. It is, however, highly nontrivial, with the number of dimensions required for a faithful reproduction of the landscape far higher than can be represented in two or three dimensions. Metric disconnectivity graphs provide a possible solution, incorporating the landscape connectivity information present in disconnectivity graphs with structural information in the form of a metric. In this study, we present a new software package, PyConnect, which is capable of producing both disconnectivity graphs and metric disconnectivity graphs in two or three dimensions. We present as a test case the analysis of the 69-bead BLN coarse-grained model protein and show that, by choosing appropriate order parameters, metric disconnectivity graphs can resolve correlations between structural features on the energy landscape with the landscapes energetic and kinetic properties.

Reconstructing the free-energy landscape of Met-enkephalin using dihedral principal component analysis and well-tempered metadynamics

Well-Tempered Metadynamics (WTmetaD) is an efficient method to enhance the reconstruction of the free-energy surface of proteins. WTmetaD guarantees a faster convergence in the long time limit in comparison with the standard metadynamics. It still suffers, however, from the same limitation, i.e., the non-trivial choice of pertinent collective variables (CVs). To circumvent this problem, we couple WTmetaD with a set of CVs generated from a dihedral Principal Component Analysis (dPCA) on the Ramachandran dihedral angles describing the backbone structure of the protein. The dPCA provides a generic method to extract relevant CVs built from internal coordinates, and does not depend on the alignment to an arbitrarily chosen reference structure as usual in Cartesian PCA. We illustrate the robustness of this method in the case of a reference model protein, the small and very diffusive Met-enkephalin pentapeptide. We propose a justification a posteriori of the considered number of CVs necessary to bias the metadynamics simulation in terms of the one-dimensional free-energy profiles associated with Ramachandran dihedral angles along the amino-acid sequence.

SIMONA 1.0: an efficient and versatile framework for stochastic simulations of molecular and nanoscale systems

Molecular simulation methods have increasingly contributed to our understanding of molecular and nanoscale systems. However, the family of Monte Carlo techniques has taken a backseat to molecular dynamics based methods, which is also reflected in the number of available simulation packages. Here, we report the development of a generic, versatile simulation package for stochastic simulations and demonstrate its application to protein conformational change, protein-protein association, small-molecule protein docking, and simulation of the growth of nanoscale clusters of organic molecules. Simulation of molecular and nanoscale systems (SIMONA) is easy to use for standard simulations via a graphical user interface and highly parallel both via MPI and the use of graphical processors. It is also extendable to many additional simulations types. Being freely available to academic users, we hope it will enable a large community of researchers in the life- and materials-sciences to use and extend SIMONA in the future. SIMONA is available for download under http://int.kit.edu/nanosim/simona.

The Activation-Relaxation Technique: ART Nouveau and Kinetic ART

The evolution of many systems is dominated by rare activated events that occur on timescale ranging from nanoseconds to the hour or more. For such systems, simulations must leave aside the full thermal description to focus specifically on mechanisms that generate a configurational change. We present here the activation relaxation technique (ART), an open-ended saddle point search algorithm, and a series of recent improvements to ART nouveau and kinetic ART, an ART-based on-the-fly off-lattice self-learning kinetic Monte Carlo method.

Protein inherent structures by different minimization strategies

Network-based methods provide an accurate description of the free-energy landscape of peptides and proteins sampled by molecular dynamics simulations. To that end, it is necessary to group the individual snapshots in a meaningful way. The inherent structures (ISs) provide an appropriate discretization of the trajectory into microstates, avoiding problems that can arise in clustering algorithms that have been used previously. In this work, different minimization protocols to obtain the IS of a peptide are investigated on the basis of cut-based free-energy profiles. It is found that a computationally more efficient quasi-Newtonian algorithm provides quantitative agreement to the classical conjugate gradient method in terms of the population of the peptide substates and the energy barriers separating them. That is, despite the fact that the two algorithms can occasionally quench a given peptide snapshot in different potential energy minima, the overall properties of the system are not affected. As reported by others, atom permutations affect the calculation of the IS, requiring an improved implementation of current potential energy functions.

Modelling proteins: conformational sampling and reconstruction of folding kinetics

In the last decades biomolecular simulation has made tremendous inroads to help elucidate biomolecular processes in-silico. Despite enormous advances in molecular dynamics techniques and the available computational power, many problems involve long time scales and large-scale molecular rearrangements that are still difficult to sample adequately. In this review we therefore summarise recent efforts to fundamentally improve this situation by decoupling the sampling of the energy landscape from the description of the kinetics of the process. Recent years have seen the emergence of many advanced sampling techniques, which permit efficient characterisation of the relevant family of molecular conformations by dispensing with the details of the short-term kinetics of the process. Because these methods generate thermodynamic information at best, they must be complemented by techniques to reconstruct the kinetics of the process using the ensemble of relevant conformations. Here we review recent advances for both types of methods and discuss their perspectives to permit efficient and accurate modelling of large-scale conformational changes in biomolecules. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.

Exploring Multidimensional Free Energy Landscapes Using Time-Dependent Biases on Collective Variables

A new implementation of the adaptive biasing force (ABF) method is described. This implementation supports a wide range of collective variables and can be applied to the computation of multidimensional energy profiles. It is provided to the community as part of a code that implements several analogous methods, including metadynamics. ABF and metadynamics have not previously been tested side by side on identical systems. Here, numerical tests are carried out on processes including conformational changes in model peptides and translocation of a halide ion across a lipid membrane through a peptide nanotube. On the basis of these examples, we discuss similarities and differences between the ABF and metadynamics schemes. Both approaches provide enhanced sampling and free energy profiles in quantitative agreement with each other in different applications. The method of choice depends on the dimension of the reaction coordinate space, the height of the barriers, and the relaxation times of degrees of freedom in the orthogonal space, which are not explicitly described by the chosen collective variables.

The SAAP force field: development of the single amino acid potentials for 20 proteinogenic amino acids and Monte Carlo molecular simulation for short peptides

Molecular simulation by using force field parameters has been widely applied in the fields of peptide and protein research for various purposes. We recently proposed a new all-atom protein force field, called the SAAP force field, which utilizes single amino acid potentials (SAAPs) as the fundamental elements. In this article, whole sets of the SAAP force field parameters in vacuo, in ether, and in water have been developed by ab initio calculation for all 20 proteinogenic amino acids and applied to Monte Carlo molecular simulation for two short peptides. The side-chain separation approximation method was employed to obtain the SAAP parameters for the amino acids with a long side chain. Monte Carlo simulation for Met-enkephalin (CHO-Tyr-Gly-Gly-Phe-Met-NH2) by using the SAAP force field revealed that the conformation in vacuo is mainly controlled by strong electrostatic interactions between the amino acid residues, while the SAAPs and the interamino acid Lennard-Jones potentials are predominant in water. In ether, the conformation would be determined by the combination of the three components. On the other hand, the SAAP simulation for chignolin (H-Gly-Tyr-Asp-Pro-Glu-Thr-Gly-Thr-Trp-Gly-OH) reasonably reproduced a native-like beta-hairpin structure in water although the C-terminal and side-chain conformations were different from the native ones. It was suggested that the SAAP force field is a useful tool for analyzing conformations of polypeptides in terms of intrinsic conformational propensities of the single amino acid units.

Exploring the free energy landscape: from dynamics to networks and back

Knowledge of the Free Energy Landscape topology is the essential key to understanding many biochemical processes. The determination of the conformers of a protein and their basins of attraction takes a central role for studying molecular isomerization reactions. In this work, we present a novel framework to unveil the features of a Free Energy Landscape answering questions such as how many meta-stable conformers there are, what the hierarchical relationship among them is, or what the structure and kinetics of the transition paths are. Exploring the landscape by molecular dynamics simulations, the microscopic data of the trajectory are encoded into a Conformational Markov Network. The structure of this graph reveals the regions of the conformational space corresponding to the basins of attraction. In addition, handling the Conformational Markov Network, relevant kinetic magnitudes as dwell times and rate constants, or hierarchical relationships among basins, completes the global picture of the landscape. We show the power of the analysis studying a toy model of a funnel-like potential and computing efficiently the conformers of a short peptide, dialanine, paving the way to a systematic study of the Free Energy Landscape in large peptides.

A complete description of complex polymers, such as proteins, includes information about their structure and their dynamics. In particular it is of utmost importance to answer the following questions: What are the structural conformations possible? Is there any relevant hierarchy among these conformers? What are the transition paths between them? These and other questions can be addressed by analyzing in an efficient way the Free Energy Landscape of the system. With this knowledge, several problems about biomolecular reactions (such as enzymatic activity, protein folding, protein deposition diseases, etc.) can be tackled. In this article we show how to efficiently describe the Free Energy Landscape for small and large peptides. By mapping the trajectories of molecular dynamics simulations into a graph (the Conformational Markov Network) and unveiling its structural organization, we obtain a coarse grained description of the protein dynamics across the Free Energy Landscape in terms of the relevant kinetic magnitudes of the system. Therefore, we show the way to bridge the gap between the microscopic dynamics and the macroscopic kinetics by means of a mesoscopic description of the associated Conformational Markov Network. Along this path the compromise between the physical nature of the process and the magnitudes that characterize the network is carefully kept to assure the reliability of the results shown.

Transition networks for modeling the kinetics of conformational change in macromolecules

The kinetics and thermodynamics of complex transitions in biomolecules can be modeled in terms of a network of transitions between the relevant conformational substates. Such a transition network, which overcomes the fundamental limitations of reaction-coordinate-based methods, can be constructed either based on the features of the energy landscape, or from molecular dynamics simulations. Energy-landscape-based networks are generated with the aid of automated path-optimization methods, and, using graph-theoretical adaptive methods, can now be constructed for large molecules such as proteins. Dynamics-based networks, also called Markov State Models, can be interpreted and adaptively improved using statistical concepts, such as the mean first passage time, reactive flux and sampling error analysis. This makes transition networks powerful tools for understanding large-scale conformational changes.

Folding pathways and rates for the three-stranded beta-sheet peptide Beta3s using discrete path sampling

The discrete path sampling method was used to investigate the folding of a three-stranded antiparallel beta-sheet peptide, Beta3s, described by an empirical potential and implicit solvent model. After application of a coarse-graining scheme that groups together sets of minima in local equilibrium, the calculated folding time was in reasonable agreement with other simulations and consistent with the experimental upper bound. The folding mechanism exhibited by the most significant discrete paths involves early formation of the C-terminal hairpin followed by docking of the N-terminal strand.

Conformational memories with variable bond angles

Conformational Memories (CM) is a simulated annealing/Monte Carlo method that explores peptide and protein dihedral conformational space completely and efficiently, independent of the original conformation. Here we extend the CM method to include the variation of a randomly chosen bond angle, in addition to the standard variation of two or three randomly chosen dihedral angles, in each Monte Carlo trial of the CM exploratory and biased phases. We test the hypothesis that the inclusion of variable bond angles in CM leads to an improved sampling of conformational space. We compare the results with variable bond angles to CM with no bond angle variation for the following systems: (1) the pentapeptide Met-enkephalin, which is a standard test case for conformational search methods; (2) the proline ring pucker in a 17mer model peptide, (Ala)(8)Pro(Ala)(8); and (3) the conformations of the Ser 7.39 chi(1) in transmembrane helix 7 (TMH7) of the cannabinoid CB1 receptor, a 25-residue system. In each case, analysis of the CM results shows that the inclusion of variable bond angles results in sampling of regions of conformational space that are inaccessible to CM calculations with only variable dihedral angles, and/or a shift in conformational populations from those calculated when variable bond angles are not included. The incorporation of variable bond angles leads to an improved sampling of conformational space without loss of efficiency. Our examples show that this improved sampling leads to better exploration of biologically relevant conformations that have been experimentally validated.

Exploring a free energy landscape by means of multidimensional infrared and terahertz spectroscopies

A model for the dipolar crystal system is employed to explore a role of free energy landscape (FEL), in which dipolar molecules are posted on two-dimensional lattice sites with two-state libratinal dynamics. All dipole-dipole interactions are included to have frustrated interactions among the dipoles. For the regular and distorted lattice cases, the FEL is calculated from the interaction energies and the total polarizations for all possible dipolar states at various temperatures. At high temperatures, the shape of the calculated FEL is smooth and parabolic, while it becomes bumpy at low temperatures exhibiting multiple local minima. To study dynamical aspects of the system, the single flip dynamics and the single-double mixed flips dynamics of dipoles are examined from a master equation approach. As the observables of linear absorption and two-dimensional (2D) infrared, far infrared, and terahertz spectroscopies, the first- and third-order response functions of polarization are calculated for different physical conditions characterized by the FEL. While the linear absorption signals decay in time in a similar manner regardless of the FEL profiles, the 2D signals exhibit prominent differences for those profiles. This indicates that we may differentiate the FEL profiles by changing two-time valuables in 2D spectroscopy. As illustrated in the single-double flips case, the FEL study by means of 2D spectroscopy, however, relies on the dynamics which is set independently from the FEL. The Smoluchowski equation is applied to examine the description of the collective dynamics on the microscopically calculated FEL. We found that the one-dimensional and 2D signals calculated from the Smoluchowski equation agree with those from master equation only at temperatures where the FEL becomes parabolic shape.

Potential energy and free energy landscapes

Familiar concepts for small molecules may require reinterpretation for larger systems. For example, rearrangements between geometrical isomers are usually considered in terms of transitions between the corresponding local minima on the underlying potential energy surface, V. However, transitions between bulk phases such as solid and liquid, or between the denatured and native states of a protein, are normally addressed in terms of free energy minima. To reestablish a connection with the potential energy surface we must think in terms of representative samples of local minima of V, from which a free energy surface is projected by averaging over most of the coordinates. The present contribution outlines how this connection can be developed into a tool for quantitative calculations. In particular, stepping between the local minima of V provides powerful methods for locating the global potential energy minimum, and for calculating global thermodynamic properties. When the transition states that link local minima are also sampled we can exploit statistical rate theory to obtain insight into global dynamics and rare events. Visualizing the potential energy landscape helps to explain how the network of local minima and transition states determines properties such as heat capacity features, which signify transitions between free energy minima. The organization of the landscape also reveals how certain systems can reliably locate particular structures on the experimental time scale from among an exponentially large number of local minima. Such directed searches not only enable proteins to overcome Levinthal's paradox but may also underlie the formation of "magic numbers" in molecular beams, the self-assembly of macromolecular structures, and crystallization.

Graph transformation method for calculating waiting times in Markov chains

We describe an exact approach for calculating transition probabilities and waiting times in finite-state discrete-time Markov processes. All the states and the rules for transitions between them must be known in advance. We can then calculate averages over a given ensemble of paths for both additive and multiplicative properties in a nonstochastic and noniterative fashion. In particular, we can calculate the mean first-passage time between arbitrary groups of stationary points for discrete path sampling databases, and hence extract phenomenological rate constants. We present a number of examples to demonstrate the efficiency and robustness of this approach.

Absolute Free-Energy Calculations of Liquids Using a Harmonic Reference State

Absolute free-energy methods provide a potential solution to the overlap problem in free-energy calculations. In this paper, we report an extension of the previously published confinement method (J. Phys. Chem. B 2006, 110, 17212-20) to fluid simulations. Absolute free energies of liquid argon and liquid water are obtained accurately and compared with results from thermodynamic integration. The method works by transforming the liquid state into a harmonic, solid reference state. This is achieved using a special restraint potential that allows molecules to change their restraint position during the simulation, which circumvents the need for the molecules to sample the full extent of their translational freedom. The absolute free energy of the completely restrained reference state is obtained from a normal mode calculation. Because of the generic reference state used, the method is applicable to nonhomogeneous, diffusive systems and could provide an alternative method in situations in which solute annihilation fails due to the size of the solute. Potential applications include calculation of solvation energies of large molecules and free energies of peptide conformational changes in explicit solvent.

Hierarchical analysis of conformational dynamics in biomolecules: transition networks of metastable states

Molecular dynamics simulation generates large quantities of data that must be interpreted using physically meaningful analysis. A common approach is to describe the system dynamics in terms of transitions between coarse partitions of conformational space. In contrast to previous work that partitions the space according to geometric proximity, the authors examine here clustering based on kinetics, merging configurational microstates together so as to identify long-lived, i.e., dynamically metastable, states. As test systems microsecond molecular dynamics simulations of the polyalanines Ala(8) and Ala(12) are analyzed. Both systems clearly exhibit metastability, with some kinetically distinct metastable states being geometrically very similar. Using the backbone torsion rotamer pattern to define the microstates, a definition is obtained of metastable states whose lifetimes considerably exceed the memory associated with interstate dynamics, thus allowing the kinetics to be described by a Markov model. This model is shown to be valid by comparison of its predictions with the kinetics obtained directly from the molecular dynamics simulations. In contrast, clustering based on the hydrogen-bonding pattern fails to identify long-lived metastable states or a reliable Markov model. Finally, an approach is proposed to generate a hierarchical model of networks, each having a different number of metastable states. The model hierarchy yields a qualitative understanding of the multiple time and length scales in the dynamics of biomolecules.

Modeling protein thermodynamics and fluctuations at the mesoscale

We use an extended Go model, in unfrustrated and frustrated variants, to study the energy landscape and the fluctuations of a model protein. The model exhibits two transitions, folding and dynamical transitions, when changing the temperature. The inherent structures corresponding to the minima of the landscape are analyzed and we show how their energy density can be obtained from simulations around the folding temperature. The scaling of this energy density is found to reflect the folding transition. Moreover, this approach allows us to build a reduced thermodynamics in the inherent structure landscape. Equilibrium studies, from full molecular dynamics (MD) simulations and from the reduced thermodynamics, detect the features of a dynamical transition at low temperature and we analyze the location and time scale of the fluctuations of the protein, showing the need of some frustration in the model to get realistic results. The frustrated model also shows the presence of a kinetic trap which strongly affects the dynamics of folding.

Conformational study of Met-enkephalin based on the ECEPP force fields

We report a computational study of the small peptide Met-enkephalin based on the ECEPP/2 and ECEPP/3 force fields using the basin paving method. We have located a new global minimum when using the ECEPP/3 force field with peptide angles omega fixed at 180 degrees. With this new result, we can conclude that the lowest energy configurations of Met-enkephalin predicted based on all four versions of ECEPP have a classic gamma-turn centered at residue Gly3 and a beta-turn at residues Gly3-Phe4. However, minor differences between the structures also exist.

An Efficient, Path-Independent Method for Free-Energy Calculations

Classical free-energy methods depend on the definition of physical or nonphysical integration paths to calculate free-energy differences between states. This procedure can be problematic and computationally expensive when the states of interest do not overlap and are far apart in phase space. Here we introduce a novel method to calculate free-energy differences that is path-independent by transforming each end state into a reference state in which the vibrational entropy is the sole component of the total entropy, thus allowing direct computation of the relative free energy. We apply the method to calculate side-chain entropies of a beta-hairpin-forming peptide in a variety of backbone conformations, demonstrating its importance in determining structural propensities. We find that low-free-energy conformations achieve their stability through optimal trade off between enthalpic gains due to favorable interatomic interactions and entropic losses incurred by the same.

A doubly nudged elastic band method for finding transition states

A modification of the nudged elastic band (NEB) method is presented that enables stable optimizations to be run using both the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) quasi-Newton and slow-response quenched velocity Verlet minimizers. The performance of this new "doubly nudged" DNEB method is analyzed in conjunction with both minimizers and compared with previous NEB formulations. We find that the fastest DNEB approach (DNEB/L-BFGS) can be quicker by up to 2 orders of magnitude. Applications to permutational rearrangements of the seven-atom Lennard-Jones cluster (LJ7) and highly cooperative rearrangements of LJ38 and LJ75 are presented. We also outline an updated algorithm for constructing complicated multi-step pathways using successive DNEB runs.