Polypeptide folding with off-lattice Monte Carlo dynamics: the method

The MC dynamics of an off-lattice all-atom protein backbone model with rigid amide planes are studied. The only degrees of freedom are the dihedral angle pairs of the C-atoms. Conformational changes are generated by Monte Carlo (MC) moves. The MC moves considered are single rotations (simple moves, SM's) giving rise to global conformational changes or, alternatively, cooperative rotations in a window of amide planes (window moves, WM's) generating local conformational changes in the window. Outside the window the protein conformation is kept invariant by constraints. These constraints produce a bias in the distribution of dihedral angles. The WM's are corrected for this bias by suitable Jacobians. The energy function used is derived from the CHARMM force field. In a first application to polyalanine it is demonstrated that WM's sample the conformational space more efficiently than SM's.Abbreviations CPU Central Processing Unit - MC Monte Carlo - MCD Monte Carlo Dynamics - MD Molecular Dynamics - RMS Root-Mean-Square - RMSD Root-Mean-Square-Deviation - SM Simple Move - WM Window Move     

Directional structural features of globular proteins

We describe a novel presentation of the conformation of the backbone atoms for proteins of known structure. Given the C^α atom cartesian co-ordinates from X-ray crystallography, a matrix is calculated, where the ijth element of the matrix is the cosine of the angle between the direction of the chain at residue i and the direction of the chain at residue j. These “direction matrices” have distinctive patterns which correspond to α-helix, extended structure, straight or bent segments, “superhelix”, and many other important structural features. We discuss the direction matrices for a number of proteins, and make some generalizations on the basic principles of protein folding.     

A computationally efficient Monte-Carlo model for biomedical Raman spectroscopy

Monte Carlo (MC) modeling is a valuable tool to gain fundamental understanding of light-tissue interactions, provide guidance and assessment to optical instrument designs, and help analyze experimental data. It has been a major challenge to efficiently extend MC towards modeling of bulk-tissue Raman spectroscopy (RS) due to the wide spectral range, relatively sharp spectral features, and presence of background autofluorescence. Here, we report a computationally efficient MC approach for RS by adapting the massively-parallel Monte Carlo eXtreme (MCX) simulator. Simulation efficiency is achieved through “isoweight,” a novel approach that combines the statistical generation of Raman scattered and Fluorescence emission with a lookup-table-based technique well-suited for parallelization. The MC model uses a graphics processor to produce dense Raman and fluorescence spectra over a range of 800 − 2000 cm⁻¹ with an approximately 100× increase in speed over prior RS Monte Carlo methods. The simulated RS signals are compared against experimentally collected spectra from gelatin phantoms, showing a strong correlation.     

Generalized-ensemble algorithms for molecular simulations of biopolymers.

In complex systems with many degrees of freedom such as peptides and proteins, there exists a huge number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present three new generalized-ensemble algorithms that combine the merits of the above methods. The effectiveness of the methods for molecular simulations in the protein folding problem is tested with short peptide systems.     

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.     

Commitment and nucleation in the protein G transition state

An accurate characterization of the transition state ensemble (TSE) is central to furthering our understanding of the protein folding reaction. We have extensively tested a recently reported method for studying a protein's TSE, utilizing phi-value data from protein engineering experiments and computational studies as restraints in all-atom Monte Carlo (MC) simulations. The validity of interpreting experimental phi-values as the fraction of native contacts made by a residue in the TSE was explored, revealing that this definition is unable to uniquely specify a TSE. The identification of protein G's second hairpin, in both pre and post-transition conformations demonstrates that high experimental phi-values do not guarantee a residue's importance in the TSE. An analysis of simulations based on structures restrained by experimental phi-values is necessary to yield this result, which is not obvious from a simplistic interpretation of individual phi-values. The TSE that we obtain corresponds to a single, specific nucleation event, characterized by six residues common to all three observed, convergent folding pathways. The same specific nucleus was independently identified from computational and experimental data, and "Conservation of Conservation" analysis in the protein G fold. When associated strictly with complete nucleus formation and concomitant chain collapse, folding is a well-defined two state event. Once the nucleus has formed, the folding reaction enters a slow relaxation process associated with side-chain packing and small, local backbone rearrangements. A detailed analysis of phi-values and their relationship to the transition state ensemble allows us to construct a unified theoretical model of protein G folding.     

Metadynamic sampling of the free-energy landscapes of proteins coupled with a Monte Carlo algorithm

Metadynamics is a powerful computational tool to obtain the free-energy landscape of complex systems. The Monte Carlo algorithm has proven useful to calculate thermodynamic quantities associated with simplified models of proteins, and thus to gain an ever-increasing understanding on the general principles underlying the mechanism of protein folding. We show that it is possible to couple metadynamics and Monte Carlo algorithms to obtain the free energy of model proteins in a way which is computationally very economical.     

Monte Carlo replica‐exchange based ensemble docking of protein conformations

A replica‐exchange Monte Carlo (REMC) ensemble docking approach has been developed that allows efficient exploration of protein–protein docking geometries. In addition to Monte Carlo steps in translation and orientation of binding partners, possible conformational changes upon binding are included based on Monte Carlo selection of protein conformations stored as ordered pregenerated conformational ensembles. The conformational ensembles of each binding partner protein were generated by three different approaches starting from the unbound partner protein structure with a range spanning a root mean square deviation of 1–2.5 Å with respect to the unbound structure. Because MC sampling is performed to select appropriate partner conformations on the fly the approach is not limited by the number of conformations in the ensemble compared to ensemble docking of each conformer pair in ensemble cross docking. Although only a fraction of generated conformers was in closer agreement with the bound structure the REMC ensemble docking approach achieved improved docking results compared to REMC docking with only the unbound partner structures or using docking energy minimization methods. The approach has significant potential for further improvement in combination with more realistic structural ensembles and better docking scoring functions. Proteins 2017; 85:924–937. © 2016 Wiley Periodicals, Inc.     

Stochastic roadmap simulation: an efficient representation and algorithm for analyzing molecular motion.

Classic molecular motion simulation techniques, such as Monte Carlo (MC) simulation, generate motion pathways one at a time and spend most of their time in the local minima of the energy landscape defined over a molecular conformation space. Their high computational cost prevents them from being used to compute ensemble properties (properties requiring the analysis of many pathways). This paper introduces stochastic roadmap simulation (SRS) as a new computational approach for exploring the kinetics of molecular motion by simultaneously examining multiple pathways. These pathways are compactly encoded in a graph, which is constructed by sampling a molecular conformation space at random. This computation, which does not trace any particular pathway explicitly, circumvents the local-minima problem. Each edge in the graph represents a potential transition of the molecule and is associated with a probability indicating the likelihood of this transition. By viewing the graph as a Markov chain, ensemble properties can be efficiently computed over the entire molecular energy landscape. Furthermore, SRS converges to the same distribution as MC simulation. SRS is applied to two biological problems: computing the probability of folding, an important order parameter that measures the "kinetic distance" of a protein's conformation from its native state; and estimating the expected time to escape from a ligand-protein binding site. Comparison with MC simulations on protein folding shows that SRS produces arguably more accurate results, while reducing computation time by several orders of magnitude. Computational studies on ligand-protein binding also demonstrate SRS as a promising approach to study ligand-protein interactions.     

Effects of Topology and Sequence in Protein Folding Linked via Conformational Fluctuations

Experiments have compared the folding of proteins with different amino acid sequences but the same basic structure, or fold. Results indicate that folding is robust to sequence variations for proteins with some nonlocal folds, such as all-β, whereas the folding of more local, all-α proteins typically exhibits a stronger sequence dependence. Here, we use a coarse-grained model to systematically study how variations in sequence perturb the folding energy landscapes of three model sequences with 3α, 4β + α, and β-barrel folds, respectively. These three proteins exhibit folding features in line with experiments, including expected rank order in the cooperativity of the folding transition and stability-dependent shifts in the location of the free-energy barrier to folding. Using a generalized-ensemble simulation approach, we determine the thermodynamics of around 2000 sequence variants representing all possible hydrophobic or polar single- and double-point mutations. From an analysis of the subset of stability-neutral mutations, we find that folding is perturbed in a topology-dependent manner, with the β-barrel protein being the most robust. Our analysis shows, in particular, that the magnitude of mutational perturbations of the transition state is controlled in part by the size or “width” of the underlying conformational ensemble. This result suggests that the mutational robustness of the folding of the β-barrel protein is underpinned by its conformationally restricted transition state ensemble, revealing a link between sequence and topological effects in protein folding.     

Lattice simulations of aggregation funnels for protein folding.

A computer model of protein aggregation competing with productive folding is proposed. Our model adapts techniques from lattice Monte Carlo studies of protein folding to the problem of aggregation. However, rather than starting with a single string of residues, we allow independently folding strings to undergo collisions and consider their interactions in different orientations. We first present some background into the nature and significance of protein aggregation and the use of lattice Monte Carlo simulations in understanding other aspects of protein folding. The results of a series of simulation experiments involving simple versions of the model illustrate the importance of considering aggregation in simulations of protein folding and provide some preliminary understanding of the characteristics of the model. Finally, we discuss the value of the model in general and of our particular design decisions and experiments. We conclude that computer simulation techniques developed to study protein folding can provide insights into protein aggregation, and that a better understanding of aggregation may in turn provide new insights into and constraints on the more general protein folding problem.     

Moments of the distribution of the amino acid residues in tertiary structures of globular proteins

Moments of the distributions of the C^α and “side-chain atoms” and associated properties were examined in 22 globular proteins, considered as statistical aggregates of atoms. Although the distributions are generally anisotropic, the densities of the evaluated distributions are highly uniform in the interior of a single protein, as well as among the proteins investigated. The tertiary structure of proteins is characterized by a compact and uniform distribution of amino acids, independent of their molecular weight and hydrophobic character, and by an isotropic distribution of the virtual bond directions in the polypeptide folding. While the general uniformity of the density of distributions in the bulk of proteins can be justified by the architectural requirements of high thermodynamic stability, significant differences in the distribution of the C^α with respect to the “side-chain atoms” suggest a plausible explanation of the general anisotropic morphology of the proteins. The invariance of the density of distributions allows easy recognition of proteinlike domains in more complex proteins and suggests a practical way to predict the following path in single proteins.     

Unbiased,scalable sampling of protein loop conformations from probabilistic priors

Background

Protein loops are flexible structures that are intimately tied to function, but understanding loop motion and generating loop conformation ensembles remain significant computational challenges. Discrete search techniques scale poorly to large loops, optimization and molecular dynamics techniques are prone to local minima, and inverse kinematics techniques can only incorporate structural preferences in adhoc fashion. This paper presents Sub-Loop Inverse Kinematics Monte Carlo (SLIKMC), a new Markov chain Monte Carlo algorithm for generating conformations of closed loops according to experimentally available, heterogeneous structural preferences.

Results

Our simulation experiments demonstrate that the method computes high-scoring conformations of large loops (> 10 residues) orders of magnitude faster than standard Monte Carlo and discrete search techniques. Two new developments contribute to the scalability of the new method. First, structural preferences are specified via a probabilistic graphical model (PGM) that links conformation variables, spatial variables (e.g., atom positions), constraints and prior information in a unified framework. The method uses a sparse PGM that exploits locality of interactions between atoms and residues. Second, a novel method for sampling sub-loops is developed to generate statistically unbiased samples of probability densities restricted by loop-closure constraints.

Conclusion

Numerical experiments confirm that SLIKMC generates conformation ensembles that are statistically consistent with specified structural preferences. Protein conformations with 100+ residues are sampled on standard PC hardware in seconds. Application to proteins involved in ion-binding demonstrate its potential as a tool for loop ensemble generation and missing structure completion.

     

Use of a quantitative structure-property relationship to design larger model proteins that fold rapidly

A quantitative structure-property relationship (QSPR) was used to design model protein sequences that fold repeatedly and relatively rapidly to stable target structures. The specific model was a 125-residue heteropolymer chain subject to Monte Carlo dynamics on a simple cubic lattice. The QSPR was derived from an analysis of a database of 200 sequences by a statistical method that uses a genetic algorithm to select the sequence attributes that are most important for folding and a neural network to determine the corresponding functional dependence of folding ability on the chosen attributes. The QSPR depends on the number of anti-parallel sheet contacts, the energy gap between the native state and quasi-continuous part of the spectrum and the total energy of the contacts between surface residues. Two Monte Carlo procedures were used in series to optimize both the target structures and the sequences. We generated 20 fully optimized sequences and 60 partially optimized control sequences and tested each for its ability to fold in dynamic MC simulations. Although sequences in which either the number of anti-parallel sheet contacts or the energy of the surface residues is non-optimal are capable of folding almost as well as fully optimized ones, sequences in which only the energy gap is optimized fold markedly more slowly. Implications of the results for the design of proteins are discussed.     

The tree structural organization of proteins

We offer an objective definition of the domains of a protein, given its C^α coordinates from high-resolution X-ray crystal studies. This is done by an algorithm which groups segments of the polypeptide chain together when there are a relatively large number of contacts between the two segments. The result is an organizational tree showing a hierarchy of segments grouping together, then clusters merging until all parts of the chain are included. In this view the highest level clusters correspond well to more subjective definitions of folding domains and the lowest level, the segments, roughly match the usual assignments of pieces of secondary structure. The intermediate level clusters suggest possible folding mechanisms, which are discussed.     

Replica-exchange methods and predictions of helix configurations of membrane proteins

A simulation in generalized ensemble is based on a non-Boltzmann weight factor and performs a random walk in potential energy space, which allows the simulation to avoid getting trapped in states of local-minimum energy states. In this article, we review uses of the generalized-ensemble algorithms. Three well-known methods, namely, multicanonical algorithm (MUCA), simulated tempering (ST) and replica-exchange method (REM), are described first. Both Monte Carlo (MC) and molecular dynamics (MD) versions of the algorithms are given. We then present the results of the application of replica-exchange MC method to the predictions of membrane protein structures.     

Recognition of native structure from complete enumeration of low-resolution models with constraints

Complete sets of low-resolution conformations are generated for eight small proteins by rotating the C_α-C_α virtual bonds at selected flexible regions, while the remaining structural elements are assumed to move in rigid blocks. Several filtering criteria are used to reduce the ensemble size and to ensure the sampling of well-constructed conformations. These filters, based on structure and energy constraints deduced from knowledge-based studies, include the excluded volume requirement, the radius of gyration constraint, and the occurrence of sufficiently strong attractive inter-residue potentials to stabilize compact forms. About 8,000 well-constructed decoys or “probable folds” (PFs) are constructed for each protein. A correlation between root-mean-square (rms) deviations from X-ray structure and total energies is observed, revealing a decrease in energy as the rms deviation decreases. The conformation with the lowest energy exhibits an rms deviation smaller than 3.0 Å, in most of the proteins considered. The results are highly sensitive to the choice of flexible regions. A strong tendency to assume native state rotational angles is revealed for some flexible bonds from the analysis of the distributions of dihedral angles in the PFs, suggesting the formation of foldons near these locally stable regions at early folding pathway. Proteins 32:211–222, 1998. © 1998 Wiley-Liss, Inc.     

Improved genetic algorithm for the protein folding problem by use of a Cartesian combination operator.

We have devised a Cartesian combination operator and coding scheme for improving the performance of genetic algorithms applied to the protein folding problem. The genetic coding consists of the C alpha Cartesian coordinates of the protein chain. The recombination of the genes of the parents is accomplished by: (1) a rigid superposition of one parent chain on the other, to make the relation of Cartesian coordinates meaningful, then, (2) the chains of the children are formed through a linear combination of the coordinates of their parents. The children produced with this Cartesian combination operator scheme have similar topology and retain the long-range contacts of their parents. The new scheme is significantly more efficient than the standard genetic algorithm methods for locating low-energy conformations of proteins. The considerable superiority of genetic algorithms over Monte Carlo optimization methods is also demonstrated. We have also devised a new dynamic programming lattice fitting procedure for use with the Cartesian combination operator method. The procedure finds excellent fits of real-space chains to the lattice while satisfying bond-length, bond-angle, and overlap constraints.     

Application of Enhanced Sampling Monte Carlo Methods for High-Resolution Protein-Protein Docking in Rosetta

The high-resolution refinement of docked protein-protein complexes can provide valuable structural and mechanistic insight into protein complex formation complementing experiment. Monte Carlo (MC) based approaches are frequently applied to sample putative interaction geometries of proteins including also possible conformational changes of the binding partners. In order to explore efficiency improvements of the MC sampling, several enhanced sampling techniques, including temperature or Hamiltonian replica exchange and well-tempered ensemble approaches, have been combined with the MC method and were evaluated on 20 protein complexes using unbound partner structures. The well-tempered ensemble method combined with a 2-dimensional temperature and Hamiltonian replica exchange scheme (WTE-H-REMC) was identified as the most efficient search strategy. Comparison with prolonged MC searches indicates that the WTE-H-REMC approach requires approximately 5 times fewer MC steps to identify near native docking geometries compared to conventional MC searches.