Simulation methods for protein structure fluctuations

Three numerical techniques for generating thermally accessible configurations of globular proteins are considered; these techniques are the molecular dynamics method, the Metropolis Monte Carlo method, and a modified Monte Carlo method which takes account of the forces acting on the protein atoms. The molecular dynamics method is shown to be more efficient than either of the Monte Carlo methods. Because it may be necessary to use Monte Carlo methods in certain important types of sampling problems, the behavior of these methods is examined in some detail. It is found that an acceptance ratio close to 1/6 yields optimum efficiency for the Metropolis method, in contrast to what is often assumed. This result, together with the overall inefficiency of the Monte Carlo methods, appears to arise from the anisotropic forces acting on the protein atoms due to their covalent bonding. Possible ways of improving the Monte Carlo methods are suggested.     

Confinement effects on the thermodynamics of protein folding: Monte Carlo simulations

The effects of chaperonin-like cage-induced confinement on protein stability have been studied for molecules of varying sizes and topologies. Minimalist models based on Gō-like interactions are employed for the proteins, and density-of-states-based Monte Carlo simulations are performed to accurately characterize the thermodynamic transitions. This method permits efficient sampling of conformational space and yields precise estimates of free energy and entropic changes associated with protein folding. We find that confinement-driven stabilization is not only dependent on protein size and cage radius, but also on the specific topology. The choice of the confining potential is also shown to have an effect on the observed stabilization and the scaling behavior of the stabilization with respect to the cage size.     

Kinetics analysis methods for approximate folding landscapes

MOTIVATION: Protein motions play an essential role in many biochemical processes. Lab studies often quantify these motions in terms of their kinetics such as the speed at which a protein folds or the population of certain interesting states like the native state. Kinetic metrics give quantifiable measurements of the folding process that can be compared across a group of proteins such as a wild-type protein and its mutants. RESULTS: We present two new techniques, map-based master equation solution and map-based Monte Carlo simulation, to study protein kinetics through folding rates and population kinetics from approximate folding landscapes, models called maps. From these two new techniques, interesting metrics that describe the folding process, such as reaction coordinates, can also be studied. In this article we focus on two metrics, formation of helices and structure formation around tryptophan residues. These two metrics are often studied in the lab through circular dichroism (CD) spectra analysis and tryptophan fluorescence experiments, respectively. The approximated landscape models we use here are the maps of protein conformations and their associated transitions that we have presented and validated previously. In contrast to other methods such as the traditional master equation and Monte Carlo simulation, our techniques are both fast and can easily be computed for full-length detailed protein models. We validate our map-based kinetics techniques by comparing folding rates to known experimental results. We also look in depth at the population kinetics, helix formation and structure near tryptophan residues for a variety of proteins. AVAILABILITY: We invite the community to help us enrich our publicly available database of motions and kinetics analysis by submitting to our server: http://parasol.tamu.edu/foldingserver/.     

Water as a conformational editor in protein folding

As molecules approach one another in aqueous solution, desolvation free energy barriers to association are encountered. Experiments suggest these (de)solvation effects contribute to the free energy barriers separating the folded and unfolded states of protein molecules. To explore their influence on the energy landscapes of protein folding reactions, we have incorporated desolvation barriers into a semi-realistic, off-lattice protein model that uses a simplified physico-chemical force-field determined solely by the sequence of amino acids. Monte Carlo sampling techniques were used to study the effects on the thermodynamics and kinetics of folding of a number of systems, diverse in structure and sequence. In each case, desolvation barriers increase the stability of the native conformation and the cooperativity of the major folding/unfolding transition. The folding times of these systems are reduced significantly upon inclusion of desolvation barriers, demonstrating that the particulate nature of the solvent engenders a more defined route to the native fold.     

Application of the multiensemble sampling to the equilibrium folding of proteins

MOTIVATION: Conventional Monte Carlo and molecular dynamics simulations of proteins in the canonical ensemble are of little use, because they tend to get trapped in states of energy local minima at low temperatures. One way to surmount this difficulty is to use a non-Boltzmann sampling method in which conformations are sampled upon a general weighting function instead of the conventional Boltzmann weighting function. The multiensemble sampling (MES) method is a non-Boltzmann sampling method that was originally developed to estimate free energy differences between systems with different potential energies and/or at different thermodynamic states. The method has not yet been applied to studies of complex molecular systems such as proteins. RESULTS: MES Monte Carlo simulations of small proteins have been carried out using a united-residue force field. The proteins at several temperatures from the unfolded to the folded states were simulated in a single MC run at a time and their equilibrium thermodynamic properties were calculated correctly. The distributions of sampled conformations clearly indicate that, when going through states of energy local minima, the MES simulation did not get trapped in them but escaped from them so quickly that all the relevant parts of conformation space could be sampled properly. A two-step folding process consisting of a collapse transition followed by a folding transition is observed. This study demonstrates that the use of MES alleviates the multiple-minima problem greatly. AVAILABILITY: Available on request from the authors.     

Determination of the conformation of folding initiation sites in proteins by computer simulation

Experimental evidence and theoretical models both suggest that protein folding begins by specific short regions of the polypeptide chain intermittently assuming conformations close to their final ones. The independent folding properties and small size of these folding initiation sites make them suitable subjects for computational methods aimed at deriving structure from sequence. We have used a torsion space Monte Carlo procedure together with an all-atom free energy function to investigate the folding of a set of such sites. The free energy function is derived by a potential of mean force analysis of experimental protein structures. The most important contributions to the total free energy are the local main chain electrostatics, main chain hydrogen bonds, and the burial of nonpolar area. Six proposed independent folding units and four control peptides 11–14 residues long have been investigated. Thirty Monte Carlo simulations were performed on each peptide, starting from different random conformations. Five of the six folding units adopted conformations close to the experimental ones in some of the runs. None of the controls did so, as expected. The generated conformations which are close to the experimental ones have among the lowest free energies encountered, although some less native like low free energy conformations were also found. The effectiveness of the method on these peptides, which have a wide variety of experimental conformations, is encouraging in two ways: First, it provides independent evidence that these regions of the sequences are able to adopt native like conformations early in folding, and therefore are most probably key components of the folding pathways. Second, it demonstrates that available simulation methods and free energy functions are able to produce reasonably accurate structures. Extensions of the methods to the folding of larger portions of proteins are suggested. © 1995 Wiley-Liss, Inc.     

Using robotics to fold proteins and dock ligands

The problems of protein folding and ligand docking have been explored largely using molecular dynamics or Monte Carlo methods. These methods are very compute intensive because they often explore a much wider range of energies, conformations and time than necessary. In addition, Monte Carlo methods often get trapped in local minima. We initially showed that robotic motion planning permitted one to determine the energy of binding and dissociation of ligands from protein binding sites (Singh et al., 1999). The robotic motion planning method maps complicated three-dimensional conformational states into a much simpler, but higher dimensional space in which conformational rearrangements can be represented as linear paths. The dimensionality of the conformation space is of the same order as the number of degrees of conformational freedom in three-dimensional space. We were able to determine the relative energy of association and dissociation of a ligand to a protein by calculating the energetics of interaction for a few thousand conformational states in the vicinity of the protein and choosing the best path from the roadmap. More recently, we have applied roadmap planning to the problem of protein folding (Apaydin et al., 2002a). We represented multiple conformations of a protein as nodes in a compact graph with the edges representing the probability of moving between neighboring states. Instead of using Monte Carlo simulation to simulate thousands of possible paths through various conformational states, we were able to use Markov methods to calculate the steady state occupancy of each conformation, needing to calculate the energy of each conformation only once. We referred to this Markov method of representing multiple conformations and transitions as stochastic roadmap simulation or SRS. We demonstrated that the distribution of conformational states calculated with exhaustive Monte Carlo simulations asymptotically approached the Markov steady state if the same Boltzman energy distribution was used in both methods. SRS permits one to calculate contributions from all possible paths simultaneously with far fewer energy calculations than Monte Carlo or molecular dynamics methods. The SRS method also permits one to represent multiple unfolded starting states and multiple, near-native, folded states and all possible paths between them simultaneously. The SRS method is also independent of the function used to calculate the energy of the various conformational states. In a paper to be presented at this conference (Apaydin et al., 2002b) we have also applied SRS to ligand docking in which we calculate the dynamics of ligand-protein association and dissociation in the region of various binding sites on a number of proteins. SRS permits us to determine the relative times of association to and dissociation from various catalytic and non-catalytic binding sites on protein surfaces. Instead of just following the best path in a roadmap, we can calculate the contribution of all the possible binding or dissociation paths and their relative probabilities and energies simultaneously.     

A knowledge-based move set for protein folding

The free energy landscape of protein folding is rugged, occasionally characterized by compact, intermediate states of low free energy. In computational folding, this landscape leads to trapped, compact states with incorrect secondary structure. We devised a residue-specific, protein backbone move set for efficient sampling of protein-like conformations in computational folding simulations. The move set is based on the selection of a small set of backbone dihedral angles, derived from clustering dihedral angles sampled from experimental structures. We show in both simulated annealing and replica exchange Monte Carlo (REMC) simulations that the knowledge-based move set, when compared with a conventional move set, shows statistically significant improved ability at overcoming kinetic barriers, reaching deeper energy minima, and achieving correspondingly lower RMSDs to native structures. The new move set is also more efficient, being able to reach low energy states considerably faster. Use of this move set in determining the energy minimum state and for calculating thermodynamic quantities is discussed.     

Simulation of peptide folding with explicit water--a mean solvation method

A new approach to efficiently calculate solvent effect in computer simulation of macromolecular systems has been developed. Explicit solvent molecules are included in the simulation to provide a mean solvation force for the solute conformational search. Simulations of an alanine dipeptide in aqueous solution showed that the new approach is significantly more efficient than conventional molecular dynamics method in conformational search, mainly because the mean solvation force reduced the solvent damping effect. This approach allows the solute and solvent to be simulated separately with different methods. For the macromolecule, the rigid fragment constraint dynamics method we developed previously allows large time-steps. For the solvent, a combination of a modified force-bias Monte Carlo method and a preferential sampling can efficiently sample the conformational space. A folding simulation of a 16-residue peptide in water showed high efficiency of the new approach.     

Folding simulations of small proteins

Understanding how a protein folds is a long-standing challenge in modern science. We have used an optimized atomistic model (united-residue force field) to simulate folding of small proteins of various structures: HP-36 (alpha protein), protein A (beta), 1fsd (alpha+beta), and betanova (beta). Extensive Monte Carlo folding simulations (ten independent runs with 10(9) Monte Carlo steps at a temperature) starting from non-native conformations are carried out for each protein. In all cases, proteins fold into their native-like conformations at appropriate temperatures, and glassy transitions occur at low temperatures. To investigate early folding trajectories, 200 independent runs with 10(6) Monte Carlo steps are also performed at a fixed temperature for a protein. There are a variety of possible pathways during non-equilibrium early processes (fast process, approximately 10(4) Monte Carlo steps). Finally, these pathways converge to the point unique for each protein. The convergence point of the early folding pathways can be determined only by direct folding simulations. The free energy surface, an equilibrium thermodynamic property, dictates the rest of the folding (slow process, approximately 10(8) Monte Carlo steps).     

A replica exchange Monte Carlo algorithm for protein folding in the HP model

Background  

The ab initio protein folding problem consists of predicting protein tertiary structure from a given amino acid sequence by minimizing an energy function; it is one of the most important and challenging problems in biochemistry, molecular biology and biophysics. The ab initio protein folding problem is computationally challenging and has been shown to be -hard even when conformations are restricted to a lattice. In this work, we implement and evaluate the replica exchange Monte Carlo (REMC) method, which has already been applied very successfully to more complex protein models and other optimization problems with complex energy landscapes, in combination with the highly effective pull move neighbourhood in two widely studied Hydrophobic Polar (HP) lattice models.     

MC-PHS: A Monte Carlo Implementation of the Primary Hydration Shell for Protein Folding and Design

A primary hydration shell (PHS) approach is developed for Monte Carlo simulations of conformationally rich macromolecular systems in an environment that efficiently captures principal solvation effects. It has been previously demonstrated that molecular dynamics using PHS is an efficient method to study peptide structure and dynamics in aqueous solution. Here, we extend the PHS approach to Monte Carlo simulations, whereby a stable shell of water molecules is maintained with a flexible, nonspherical, half-harmonic potential, tuned to maintain a constant restraining energy, with the difference between the restraint and shell energies used to dynamically adjust the shell radius. Examination of the shell and system size dependence of the restraining potential reveals its robustness. Moreover, its suitability for biomolecular simulations is evaluated using small spheres of water, hydration properties of small biological molecules, and configurational sampling of β-hairpin pentapeptide YPGDV. This method, termed MC-PHS, appears to provide efficient representation of dominant solvation effects and should prove useful in the study of protein folding and design.     

An efficient conformational sampling method for homology modeling

The structural refinement of protein models is a challenging problem in protein structure prediction (Moult et al., Proteins 2003;53(Suppl 6):334-339). Most attempts to refine comparative models lead to degradation rather than improvement in model quality, so most current comparative modeling procedures omit the refinement step. However, it has been shown that even in the absence of alignment errors and using optimal templates, methods based on a single template have intrinsic limitations, and that refinement is needed to improve model accuracy. It is thought that failure of current methods originates on one hand from the inaccuracy of the effective free energy functions adopted, which do not represent properly the energetic balance in the native state, and on the other hand from the difficulty to sample the high dimensional and rugged free energy landscape of protein folding, in the search for the global minimum. Here, we address this second issue. We define the evolutionary and vibrational armonics subspace (EVA), a reduced sampling subspace that consists of a combination of evolutionarily favored directions, defined by the principal components of the structural variation within a homologous family, plus topologically favored directions, derived from the low frequency normal modes of the vibrational dynamics, up to 50 dimensions. This subspace is accurate enough so that the cores of most proteins can be represented within 1 A accuracy, and reduced enough so that Replica Exchange Monte Carlo (Hukushima and Nemoto, J Phys Soc Jpn 1996;65:1604-1608; Hukushima et al., Int J Mod Phys C: Phys Comput 1996;7:337-344; Mitsutake et al., J Chem Phys 2003;118:6664-6675; Mitsutake et al., J Chem Phys 2003;118:6676-6688) (REMC) can be applied. REMC is one of the best sampling methods currently available, but its applicability is restricted to spaces of small dimensionality. We show that the combination of the EVA subspace and REMC can essentially solve the optimization problem for backbone atoms in the reduced sampling subspace, even for rather rugged free energy landscapes. Applications and limitations of this methodology are finally discussed.     

Enhanced sampling simulations to construct free-energy landscape of protein–partner substrate interaction

Molecular dynamics (MD) simulations using all-atom and explicit solvent models provide valuable information on the detailed behavior of protein–partner substrate binding at the atomic level. As the power of computational resources increase, MD simulations are being used more widely and easily. However, it is still difficult to investigate the thermodynamic properties of protein–partner substrate binding and protein folding with conventional MD simulations. Enhanced sampling methods have been developed to sample conformations that reflect equilibrium conditions in a more efficient manner than conventional MD simulations, thereby allowing the construction of accurate free-energy landscapes. In this review, we discuss these enhanced sampling methods using a series of case-by-case examples. In particular, we review enhanced sampling methods conforming to trivial trajectory parallelization, virtual-system coupled multicanonical MD, and adaptive lambda square dynamics. These methods have been recently developed based on the existing method of multicanonical MD simulation. Their applications are reviewed with an emphasis on describing their practical implementation. In our concluding remarks we explore extensions of the enhanced sampling methods that may allow for even more efficient sampling.     

Influence of organization of native protein structure on its folding: Modeling of the folding of α-helical proteins

An important question that is addressed here is whether the modeling of protein folding can catch the difference between the folding of proteins with similar structures but with different folding mechanisms. In this work, the modeling of folding of four α-helical proteins from the homeodomain family, which are similar in size, was done using the Monte Carlo and dynamic programming methods. A frequently observed order of folding of α-helices for each protein was determined using the Monte Carlo method. A correlation between the experimental folding rate and the number of Monte Carlo steps was also demonstrated. Amino acid residues that are important for the folding were determined using the dynamic programming method. The defined regions correlate with the order of folding of secondary-structure elements in the proteins both in experiments and in modeling.     

Exploration of compact protein conformations using the guided replication monte carlo method

We have studied the use of a new Monte Carlo (MC) chain generation algorithm, introduced by T. Garel and H. Orland[(1990) Journal of Physics A, Vol. 23, pp. L621–L626], for examining the thermodynamics of protein folding transitions and for generating candidate C^αbackbone structures as starting points for a de now protein structure paradigm. This algorithm, termed the guided replication Monte Carlo method, allows a rational approach to the introduction of known “native” folded characteristics as constraints in the chain generation process. We have shown this algorithm to be computationally very efficient in generating large ensembles of candidate C^αchains on the face centered cubic lattice, and illustrate its use by calculating a number of thermodynamic quantities related to protein folding characteristics. In particular, we have used this static MC algorithm to compute such temperature-dependent quantities as the ensemble mean energy, ensemble mean free energy, the heat capacity, and the mean-square radius of gyration. We also demonstrate the use of several simple “guide fields” for introducing protein-specific constraints into the ensemble generation process. Several extensions to our current model are suggested, and applications of the method to other folding related problems are discussed. © 1995 John Wiley & Sons, Inc.     

Folding small proteins via annealing stochastic approximation Monte Carlo

Recently, the stochastic approximation Monte Carlo algorithm has been proposed by Liang et al. (2007) as a general-purpose stochastic optimization and simulation algorithm. An annealing version of this algorithm was developed for real small protein folding problems. The numerical results indicate that it outperforms simulated annealing and conventional Monte Carlo algorithms as a stochastic optimization algorithm. We also propose one method for the use of secondary structures in protein folding. The predicted protein structures are rather close to the true structures.     

The folding funnel landscape for the peptide Met-enkephalin

We study the free energy landscape of the small peptide Met-enkephalin. Our data were obtained from a generalized-ensemble Monte Carlo simulation taking the interactions among all atoms into account. We show that the free energy landscape resembles that of a funnel, indicating that this peptide is a good folder. Our work demonstrates that the energy landscape picture and folding concept, developed in the context of simplified protein models, can also be used to describe the folding in more realistic models.