Population analyses of kinetic partitioning in protein folding

A simple lattice model of protein folding is studied in order to analyze the kinetic partitioning phenomena in the energy landscape perspective. By restricting the area of conformational space, it becomes possible to follow many Monte Carlo trajectories until they reach equilibrium. Alteration of population of trajectories is monitored and the relations between the energy landscape and kinetics are examined. Kinetic partitioning phenomena are categorized into different types in terms of characteristic time constants and partitioning ratio. In a specific partitioning process, refolding proceeds along the parallel pathways; the time constants have a temperature dependence similar to that observed in hen lysozyme. High-energy conformations are classified into groups according to the probability that the trajectories starting from those conformations will reach each energy valley. The partitioning ratio is determined by the way in which the conformational space is organized into these groups.     

Elucidating the ensemble of functionally-relevant transitions in protein systems with a robotics-inspired method

Background

Many proteins tune their biological function by transitioning between different functional states, effectively acting as dynamic molecular machines. Detailed structural characterization of transition trajectories is central to understanding the relationship between protein dynamics and function. Computational approaches that build on the Molecular Dynamics framework are in principle able to model transition trajectories at great detail but also at considerable computational cost. Methods that delay consideration of dynamics and focus instead on elucidating energetically-credible conformational paths connecting two functionally-relevant structures provide a complementary approach. Effective sampling-based path planning methods originating in robotics have been recently proposed to produce conformational paths. These methods largely model short peptides or address large proteins by simplifying conformational space.

Methods

We propose a robotics-inspired method that connects two given structures of a protein by sampling conformational paths. The method focuses on small- to medium-size proteins, efficiently modeling structural deformations through the use of the molecular fragment replacement technique. In particular, the method grows a tree in conformational space rooted at the start structure, steering the tree to a goal region defined around the goal structure. We investigate various bias schemes over a progress coordinate for balance between coverage of conformational space and progress towards the goal. A geometric projection layer promotes path diversity. A reactive temperature scheme allows sampling of rare paths that cross energy barriers.

Results and conclusions

Experiments are conducted on small- to medium-size proteins of length up to 214 amino acids and with multiple known functionally-relevant states, some of which are more than 13Å apart of each-other. Analysis reveals that the method effectively obtains conformational paths connecting structural states that are significantly different. A detailed analysis on the depth and breadth of the tree suggests that a soft global bias over the progress coordinate enhances sampling and results in higher path diversity. The explicit geometric projection layer that biases the exploration away from over-sampled regions further increases coverage, often improving proximity to the goal by forcing the exploration to find new paths. The reactive temperature scheme is shown effective in increasing path diversity, particularly in difficult structural transitions with known high-energy barriers.

     

Trajectory analysis of NMR structure calculations

Summary NMR as well as X-ray crystallography are used to determine the three-dimensional structures of macromolecules at atomic resolution. Structure calculation generates coordinates that are compatible with NMR data from randomly generated initial structures. We analyzed the trajectory taken by structures during NMR structure calculation in conformational space, assuming that the distance between two structures in conformational space is the root-mean-square deviation between the two structures. The coordinates of a structure in conformational space were obtained by applying the metric multidimensional scaling method. As an example, we used a 22-residue peptide, -Conotoxin GIIIA, and a simulated annealing protocol of XPLOR. We found that the three-dimensional solution of the multidimensional scaling analysis is sufficient to describe the overall configuration of the trajectories in conformational space. By comparing the trajectories of the entire calculation with those of the converged calculation, random sampling of conformational space is readily discernible. Trajectory analysis can also be used for optimization of protocols of NMR structure calculation, by examining individual trajectories.Abbreviations MD molecular dynamics - MDS multidimensional scaling - rmsd root-mean-square deviation - armsd angular rmsd - R multiple correlation coefficient - YASAP yet another simulated annealing protocol - PCA principal component analysis     

Can morphing methods predict intermediate structures?

Movement is crucial to the biological function of many proteins, yet crystallographic structures of proteins can give us only a static snapshot. The protein dynamics that are important to biological function often happen on a timescale that is unattainable through detailed simulation methods such as molecular dynamics as they often involve crossing high-energy barriers. To address this coarse-grained motion, several methods have been implemented as web servers in which a set of coordinates is usually linearly interpolated from an initial crystallographic structure to a final crystallographic structure. We present a new morphing method that does not extrapolate linearly and can therefore go around high-energy barriers and which can produce different trajectories between the same two starting points. In this work, we evaluate our method and other established coarse-grained methods according to an objective measure: how close a coarse-grained dynamics method comes to a crystallographically determined intermediate structure when calculating a trajectory between the initial and final crystal protein structure. We test this with a set of five proteins with at least three crystallographically determined on-pathway high-resolution intermediate structures from the Protein Data Bank. For simple hinging motions involving a small conformational change, segmentation of the protein into two rigid sections outperforms other more computationally involved methods. However, large-scale conformational change is best addressed using a nonlinear approach and we suggest that there is merit in further developing such methods.     

Exploring the Conformational Transitions of Biomolecular Systems Using a Simple Two-State Anisotropic Network Model

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

Filtering molecular dynamics trajectories to reveal low-frequency collective motions: phospholipase A2

A novel method for analysing molecular dynamics trajectories has been developed, which filters out high frequencies using digital signal processing techniques and facilitates focusing on the low-frequency collective motions of proteins. These motions involve low energy slow motions, which lead to important biological phenomena such as domain closure and allosteric effects in enzymes. The filtering method treats each of the atomic trajectories obtained from the molecular dynamics simulation as a "signal". The trajectories of each of the atoms in the system (or any subset of interest) are Fourier transformed to the frequency domain, a filtering function is applied and then an inverse transformation back to the time domain yields the filtered trajectory. The filtering method has been used to study the dynamics of the enzyme phospholipase A2. In the filtered trajectory, all the high frequency bond and valence angle vibrations were eliminated, leaving only low-frequency motion, mainly fluctuations in torsions and conformational transitions. Analysis of this trajectory revealed interesting motions of the protein, including concerted movements of helices, and changes in shape of the active site cavity. Unlike normal mode analysis, which has been used to study the motion of proteins, this method does not require converged minimizations or diagonalization of a matrix of second derivatives. In addition, anharmonicity, multiple minima and conformational transitions are treated explicitly. Thus, the filtering method avoids most of the approximations implicit in other investigations of the dynamic behaviour of large systems.     

Locally accessible conformations of proteins: multiple molecular dynamics simulations of crambin.

Multiple molecular dynamics (MD) simulations of crambin with different initial atomic velocities are used to sample conformations in the vicinity of the native structure. Individual trajectories of length up to 5 ns sample only a fraction of the conformational distribution generated by ten independent 120 ps trajectories at 300 K. The backbone atom conformational space distribution is analyzed using principal components analysis (PCA). Four different major conformational regions are found. In general, a trajectory samples only one region and few transitions between the regions are observed. Consequently, the averages of structural and dynamic properties over the ten trajectories differ significantly from those obtained from individual trajectories. The nature of the conformational sampling has important consequences for the utilization of MD simulations for a wide range of problems, such as comparisons with X-ray or NMR data. The overall average structure is significantly closer to the X-ray structure than any of the individual trajectory average structures. The high frequency (less than 10 ps) atomic fluctuations from the ten trajectories tend to be similar, but the lower frequency (100 ps) motions are different. To improve conformational sampling in molecular dynamics simulations of proteins, as in nucleic acids, multiple trajectories with different initial conditions should be used rather than a single long trajectory.     

Continuous metadynamics in essential coordinates as a tool for free energy modelling of conformational changes

Modelling of conformational changes in biopolymers is one of the greatest challenges of molecular biophysics. Metadynamics is a recently introduced free energy modelling technique that enhances sampling of configurational (e.g. conformational) space within a molecular dynamics simulation. This enhancement is achieved by the addition of a history-dependent bias potential, which drives the system from previously visited regions. Discontinuous metadynamics in the space of essential dynamics eigenvectors (collective motions) has been proposed and tested in conformational change modelling. Here, we present an implementation of two continuous formulations of metadynamics in the essential subspace. The method was performed in a modified version of the molecular dynamics package GROMACS. These implementations were tested on conformational changes in cyclohexane, alanine dipeptide (terminally blocked alanine, Ace-Ala-Nme) and SH3 domain. The results illustrate that metadynamics in the space of essential coordinates can accurately model free energy surfaces associated with conformational changes. Figure The conformational free energy surface of cyclohexane in the space of the two most intensive collective motions.

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Reconciling Mediating and Slaving Roles of Water in Protein Conformational Dynamics

Proteins accomplish their physiological functions with remarkably organized dynamic transitions among a hierarchical network of conformational substates. Despite the essential contribution of water molecules in shaping functionally important protein dynamics, their exact role is still controversial. Water molecules were reported either as mediators that facilitate or as masters that slave protein dynamics. Since dynamic behaviour of a given protein is ultimately determined by the underlying energy landscape, we systematically analysed protein self energies and protein-water interaction energies obtained from extensive molecular dynamics simulation trajectories of barstar. We found that protein-water interaction energy plays the dominant role when compared with protein self energy, and these two energy terms on average have negative correlation that increases with increasingly longer time scales ranging from 10 femtoseconds to 100 nanoseconds. Water molecules effectively roughen potential energy surface of proteins in the majority part of observed conformational space and smooth in the remaining part. These findings support a scenario wherein water on average slave protein conformational dynamics but facilitate a fraction of transitions among different conformational substates, and reconcile the controversy on the facilitating and slaving roles of water molecules in protein conformational dynamics.     

Molecular dynamics of the anticodon domain of yeast tRNA(Phe): codon-anticodon interaction

We have studied the effect of codon-anticodon interaction on the structure and dynamics of transfer RNAs using molecular dynamics simulations over a nanosecond time scale. From our molecular dynamical investigations of the solvated anticodon domain of yeast tRNA(Phe) in the presence and absence of the codon trinucleotides UUC and UUU, we find that, although at a gross level the structures are quite similar for the free and the bound domains, there are small but distinct differences in certain parts of the molecule, notably near the Y37 base. Comparison of the dynamics in terms of interatomic or inter-residual distance fluctuation for the free and the bound domains showed regions of enhanced rigidity in the loop region in the presence of codons. Because fluorescence experiments suggested the existence of multiple conformers of the anticodon domain, which interconvert on a much larger time scale than our simulations, we probed the conformational space using five independent trajectories of 500 ps duration. A generalized ergodic measure analysis of the trajectories revealed that at least for this time scale, all the trajectories populated separate parts of the conformational space, indicating a need for even longer simulations or enhanced sampling of the conformational space to give an unequivocal answer to this question.     

Mass-weighted molecular dynamics simulation and conformational analysis of polypeptide.

Atomic motions in protein molecules have been studied by molecular dynamics (MD) simulations; dynamics simulation methods have also been employed in conformational studies of polypeptide molecules. It was found that when atomic masses are weighted, the molecular dynamics method can significantly increase the sampling of dihedral conformation space in such studies, compared to a conventional MD simulation of the same total simulation time length. Herein the theoretical study of molecular conformation sampling by the molecular dynamics-based simulation method in which atomic masses are weighted is reported in detail; moreover, a numerical scheme for analyzing the extensive conformational sampling in the simulation of a tetrapeptide amide molecule is presented. From numerical analyses of the mass-weighted molecular dynamics trajectories of backbone dihedral angles, low-resolution structures covering the entire backbone dihedral conformation space of the molecule were determined, and the distribution of rotationally stable conformations in this space were analyzed quantitatively. The theoretical analyses based on the computer simulation and numerical analytical methods suggest that distinctive regimes in the conformational space of the peptide molecule can be identified.     

Can molecular dynamics simulations help in discriminating correct from erroneous protein 3D models?

Background  

Recent approaches for predicting the three-dimensional (3D) structure of proteins such asde novoor fold recognition methods mostly rely on simplified energy potential functions and a reduced representation of the polypeptide chain. These simplifications facilitate the exploration of the protein conformational space but do not permit to capture entirely the subtle relationship that exists between the amino acid sequence and its native structure. It has been proposed that physics-based energy functions together with techniques for sampling the conformational space, e.g., Monte Carlo or molecular dynamics (MD) simulations, are better suited to the task of modelling proteins at higher resolutions than those of models obtained with the former type of methods. In this study we monitor different protein structural properties along MD trajectories to discriminate correct from erroneous models. These models are based on the sequence-structure alignments provided by our fold recognition method, FROST. We define correct models as being built from alignments of sequences with structures similar to their native structures and erroneous models from alignments of sequences with structures unrelated to their native structures.     

Enhanced sampling of molecular dynamics simulation of peptides and proteins by double coupling to thermal bath

Low sampling efficiency in conformational space is the well-known problem for conventional molecular dynamics. It greatly increases the difficulty for molecules to find the transition path to native state, and costs amount of CPU time. To accelerate the sampling, in this paper, we re-couple the critical degrees of freedom in the molecule to environment temperature, like dihedrals in generalized coordinates or nonhydrogen atoms in Cartesian coordinate. After applying to ALA dipeptide model, we find that this modified molecular dynamics greatly enhances the sampling behavior in the conformational space and provides more information about the state-to-state transition, while conventional molecular dynamics fails to do so. Moreover, from the results of 16 independent 100?ns simulations by the new method, it shows that trpzip2 has one-half chances to reach the naive state in all the trajectories, which is greatly higher than conventional molecular dynamics. Such an improvement would provide a potential way for searching the conformational space or predicting the most stable states of peptides and proteins.     

Conformation spaces of proteins

We report a simple method for measuring the accessible conformational space explored by an ensemble of protein structures. The method is useful for diverse ensembles derived from molecular dynamics trajectories, molecular modeling, and molecular structure determinations. It can be used to examine a wide range of time scales. The central tactic we use, which has been previously employed, is to replace the true mechanical degrees of freedom of a molecular system with the conformationally effective degrees of freedom as measured by the root-mean squared cartesian distances among all pairs of conformations. Each protein conformation is treated as a point in a high dimensional euclidean space. In this article, we model this space in a novel way by representing it as an N-dimensional hypercube, describable with only two parameters: the number of dimensions and the edge length. To validate this approach, we provide a number of elementary test cases and then use the N-cube method for measuring the size and shape of conformational space covered by molecular dynamics trajectories spanning 10 orders of magnitude in time. These calculations were performed on a small protein, the villin headpiece subdomain, exploring both the native state and the misfolded/folding regime. Distinct features include single, vibrationally averaged, substate minima on the 0.1-1-ps time scale, thermally averaged conformational states that persist for 1-100 ps and transitions between these local minima on nanosecond time scales. Large-scale refolding modes appear to become uncorrelated on the microsecond time scale. Associated length scales for these events are 0.2 A for the vibrational minima; 0.5 A for the conformational minima; and 1-2 A for the nanosecond events. We find that the conformational space that is dynamically accessible during folding of villin has enough volume for approximately 10(9) minima of the variety that persist for picoseconds. Molecular dynamics trajectories of the native protein and experimentally derived solution ensembles suggest the native state to be composed of approximately 10(2) of these thermally accessible minima. Thus, based on random exploration of accessible folding space alone, protein folding for a small protein is predicted to be a milliseconds time scale event. This time can be compared with the experimental folding time for villin of 10-100 micros. One possible explanation for the 10-100-fold discrepancy is that the slope of the "folding funnel" increases the rate 1-2 orders of magnitude above random exploration of substates.     

Molecular conformational space analysis using computer graphics: Going beyond FRODO

The molecular graphics program FRODO has been modified to support analytical animation of molecular dynamics trajectories. The enhanced program, mdFRODO, supports all features available in FRODO and is interfaced to GROMOS. A variety of analytical animation modes is included. Extensive coloring and atom selection features are implemented to aid the user in distinguishing features of interest in a set of conformations. Molecular conformational space can be analyzed efficiently and comprehended. Animations may be viewed in stereo, and the animated object can be overlaid with any of the standard FRODO objects. The mdFRODO program is of wide use in molecular dynamics, X-ray crystallography and two-dimensional NMR work. Examples illustrating various aspects of collective motion in protein molecules are given and discussed.     

A coupling of homology modeling with multiple molecular dynamics simulation for identifying representative conformation of GPCR structures: a case study on human bombesin receptor subtype-3

In this study, a computational pipeline was therefore devised to overcome homology modeling (HM) bottlenecks. The coupling of HM with molecular dynamics (MD) simulation is useful in that it tackles the sampling deficiency of dynamics simulations by providing good-quality initial guesses for the native structure. Indeed, HM also relaxes the severe requirement of force fields to explore the huge conformational space of protein structures. In this study, the interaction between the human bombesin receptor subtype-3 and MK-5046 was investigated integrating HM, molecular docking, and MD simulations. To improve conformational sampling in typical MD simulations of GPCRs, as in other biomolecules, multiple trajectories with different initial conditions can be employed rather than a single long trajectory. Multiple MD simulations of human bombesin receptor subtype-3 with different initial atomic velocities are applied to sample conformations in the vicinity of the structure generated by HM. The backbone atom conformational space distribution of replicates is analyzed employing principal components analysis. As a result, the averages of structural and dynamic properties over the twenty-one trajectories differ significantly from those obtained from individual trajectories.     

An approach to the multiple-minima problem in protein folding by relaxing dimensionality. Tests on enkephalin

An algorithm for locating the region in conformational space containing the global energy minimum of a polypeptide is described. Distances are used as the primary variables in the minimization of an objective function that incorporates both energetic and distance-geometric terms. The latter are obtained from geometry and energy functions, rather than nuclear magnetic resonance experiments, although the algorithm can incorporate distances from nuclear magnetic resonance data if desired. The polypeptide is generated originally in a space of high dimensionality. This has two important consequences. First, all interatomic distances are initially at their energetically most favorable values; i.e. the polypeptide is initially at a global minimum-energy conformation, albeit a high-dimensional one. Second, the relaxation of dimensionality constraints in the early stages of the minimization removes many potential energy barriers that exist in three dimensions, thereby allowing a means of escaping from three-dimensional local minima. These features are used in an algorithm that produces short trajectories of three-dimensional minimum-energy conformations. A conformation in the trajectory is generated by allowing the previous conformation in the trajectory to evolve in a high-dimensional space before returning to three dimensions. The resulting three-dimensional structure is taken to be the next conformation in the trajectory, and the process is iterated. This sequence of conformations results in a limited but efficient sampling of conformational space. Results for test calculations on Met-enkephalin, a pentapeptide with the amino acid sequence H-Tyr-Gly-Gly-Phe-Met-OH, are presented. A tight cluster of conformations (in three-dimensional space) is found with ECEPP energies (Empirical Conformational Energy Program for Peptides) lower than any previously reported. This cluster of conformations defines a region in conformational space in which the global-minimum-energy conformation of enkephalin appears to lie.     

Reorganization energy of the initial electron-transfer step in photosynthetic bacterial reaction centers.

The reorganization energy (lambda) for electron transfer from the primary electron donor (P*) to the adjacent bacteriochlorophyll (B) in photosynthetic bacterial reaction centers is explored by molecular-dynamics simulations. Relatively long (40 ps) molecular-dynamics trajectories are used, rather than free energy perturbation techniques. When the surroundings of the reaction center are modeled as a membrane, lambda for P* B --> P+ B- is found to be approximately 1.6 kcal/mol. The results are not sensitive to the treatment of the protein's ionizable groups, but surrounding the reaction center with water gives higher values of lambda (approximately 6.5 kcal/mol). In light of the evidence that P+ B- lies slightly below P* in energy, the small lambda obtained with the membrane model is consistent with the speed and temperature independence of photochemical charge separation. The calculated reorganization energy is smaller than would be expected if the molecular dynamics trajectories had sampled the full conformational space of the system. Because the system does not relax completely on the time scale of electron transfer, the lambda obtained here probably is more pertinent than the larger value that would be obtained for a fully equilibrated system.     

Irrelevance of the Power Stroke for the Directionality,Stopping Force,and Optimal Efficiency of Chemically Driven Molecular Machines

A simple model for a chemically driven molecular walker shows that the elastic energy stored by the molecule and released during the conformational change known as the power-stroke (i.e., the free-energy difference between the pre- and post-power-stroke states) is irrelevant for determining the directionality, stopping force, and efficiency of the motor. Further, the apportionment of the dependence on the externally applied force between the forward and reverse rate constants of the power-stroke (or indeed among all rate constants) is irrelevant for determining the directionality, stopping force, and efficiency of the motor. Arguments based on the principle of microscopic reversibility demonstrate that this result is general for all chemically driven molecular machines, and even more broadly that the relative energies of the states of the motor have no role in determining the directionality, stopping force, or optimal efficiency of the machine. Instead, the directionality, stopping force, and optimal efficiency are determined solely by the relative heights of the energy barriers between the states. Molecular recognition—the ability of a molecular machine to discriminate between substrate and product depending on the state of the machine—is far more important for determining the intrinsic directionality and thermodynamics of chemo-mechanical coupling than are the details of the internal mechanical conformational motions of the machine. In contrast to the conclusions for chemical driving, a power-stroke is very important for the directionality and efficiency of light-driven molecular machines and for molecular machines driven by external modulation of thermodynamic parameters.