How well can simulation predict protein folding kinetics and thermodynamics?

Reliable protein folding on complex energy landscapes: the free energy reaction path

A theoretical framework is developed to study the dynamics of protein folding. The key insight is that the search for the native protein conformation is influenced by the rate r at which external parameters, such as temperature, chemical denaturant, or pH, are adjusted to induce folding. A theory based on this insight predicts that 1), proteins with complex energy landscapes can fold reliably to their native state; 2), reliable folding can occur as an equilibrium or out-of-equilibrium process; and 3), reliable folding only occurs when the rate r is below a limiting value, which can be calculated from measurements of the free energy. We test these predictions against numerical simulations of model proteins with a single energy scale.

Multiple routes and structural heterogeneity in protein folding

Experimental studies show that many proteins fold along sequential pathways defined by folding intermediates. An intermediate may not always be a single population of molecules but may consist of subpopulations that differ in their average structure. These subpopulations are likely to fold via independent pathways. Parallel folding and unfolding pathways appear to arise because of structural heterogeneity. For some proteins, the folding pathways can effectively switch either because different subpopulations of an intermediate get populated under different folding conditions, or because intermediates on otherwise hidden pathways get stabilized, leading to their utilization becoming discernible, or because mutations stabilize different substructures. Therefore, the same protein may fold via different pathways in different folding conditions. Multiple folding pathways make folding robust, and evolution is likely to have selected for this robustness to ensure that a protein will fold under the varying conditions prevalent in different cellular contexts.

Self-assembly of a simple membrane protein: coarse-grained molecular dynamics simulations of the influenza M2 channel

The transmembrane (TM) domain of the M2 channel protein from influenza A is a homotetrameric bundle of alpha-helices and provides a model system for computational approaches to self-assembly of membrane proteins. Coarse-grained molecular dynamics (CG-MD) simulations have been used to explore partitioning into a membrane of M2 TM helices during bilayer self-assembly from lipids. CG-MD is also used to explore tetramerization of preinserted M2 TM helices. The M2 helix monomer adopts a membrane spanning orientation in a lipid (DPPC) bilayer. Multiple extended CG-MD simulations (5 x 5 micros) were used to study the tetramerization of inserted M2 helices. The resultant tetramers were evaluated in terms of the most populated conformations and the dynamics of their interconversion. This analysis reveals that the M2 tetramer has 2x rotationally symmetrical packing of the helices. The helices form a left-handed bundle, with a helix tilt angle of approximately 16 degrees. The M2 helix bundle generated by CG-MD was converted to an atomistic model. Simulations of this model reveal that the bundle's stability depends on the assumed protonation state of the H37 side chains. These simulations alongside comparison with recent x-ray (3BKD) and NMR (2RLF) structures of the M2 bundle suggest that the model yielded by CG-MD may correspond to a closed state of the channel.

Fast in silico protein folding by introduction of alternating hydrogen bond potentials

We accelerate protein folding in all-atom molecular dynamics simulations by introducing alternating hydrogen bond potentials as a supplement to the force field. The alternating hydrogen bond potentials result in accelerated hydrogen bond reordering, which leads to rapid formation of secondary structure elements. The method does not require knowledge of the native state but generates the potentials based on the development of the tertiary structure in the simulation. In protein folding, the formation of secondary structure elements, especially alpha-helix and beta-sheet, is very important, and we show that our method can fold both efficiently and with great speed.

Estimating free-energy barrier heights for an ultrafast folding protein from calorimetric and kinetic data

Differential scanning calorimetry was used to measure the temperature dependence of the absolute heat capacity of the 35-residue subdomain of the villin headpiece, a protein that folds in 5 mus and is therefore assumed to have a small free-energy barrier separating folded and unfolded states. To obtain an estimate of the barrier height from the calorimetric data, two models, a variable-barrier model and an Ising-like model, were used to fit the heat capacity in excess of the folded state over the temperature range 15-125 degrees C. The variable-barrier model is based on an empirical mathematical form for the density of states, with four adjustable parameters and the enthalpy (H) as a reaction coordinate. The Ising-like model is based on the inter-residue contact map of the X-ray structure with exact enumeration of approximately 10(5) possible conformations, with two adjustable parameters in the partition function, and either the fraction of native contacts (Q) or the number of ordered residues (P) as reaction coordinates. The variable-barrier model provides an excellent fit to the data and yields a barrier height at the folding temperature ranging from 0.4 to 1.1 kcal mol(-1), while the Ising-like model provides a less good fit and yields barrier heights of 2.3 +/- 0.1 kcal mol(-1) and 2.1 +/- 0.1 kcal mol(-1) for the Q and P reaction coordinates, respectively. In both models, the barrier to folding increases with increasing temperature. Assuming a sufficiently large activation energy for diffusion on the free-energy surfaces, both models are consistent with the observation of a temperature-independent folding rate in previously published laser temperature-jump experiments. Analysis of this kinetic data, using an approximate form for the pre-exponential factor of Kramers theory and the 70 ns relaxation time for the fast phase that precedes the unfolding/refolding relaxation to determine the diffusion coefficient, results in a barrier height of 1.6 +/- 0.3 kcal mol-1 for an unspecified reaction coordinate. Although no independent test of the validity of the H, Q, or P reaction coordinates is given, the barrier-height estimates obtained with the three reaction coordinates are in quite good agreement with the value derived from a Kramers analysis of the kinetics that makes no assumptions about the reaction coordinate. However, the higher estimates obtained using Q or P appear more consistent with the finding of barrier-crossing kinetics of a villin mutant that folds in 700 ns, corresponding to a 1.3 kcal mol-1 reduction in the folding barrier relative to wild-type. All of the results suggest that the free-energy barrier to folding is sufficiently low that it should be possible to engineer this protein or find solution conditions that would eliminate the barrier to create the "downhill" folding scenario of Wolynes and Onuchic.

Design of protein-ligand binding based on the molecular-mechanics energy model

While the molecular-mechanics field has standardized on a few potential energy functions, computational protein design efforts are based on potentials that are unique to individual laboratories. Here we show that a standard molecular-mechanics potential energy function without any modifications can be used to engineer protein-ligand binding. A molecular-mechanics potential is used to reconstruct the coordinates of various binding sites with an average root-mean-square error of 0.61 A and to reproduce known ligand-induced side-chain conformational shifts. Within a series of 34 mutants, the calculation can always distinguish between weak (K(d)>1 mM) and tight (K(d)<10 microM) binding sequences. Starting from partial coordinates of the ribose-binding protein lacking the ligand and the 10 primary contact residues, the molecular-mechanics potential is used to redesign a ribose-binding site. Out of a search space of 2 x 10(12) sequences, the calculation selects a point mutant of the native protein as the top solution (experimental K(d)=17 microM) and the native protein as the second best solution (experimental K(d)=210 nM). The quality of the predictions depends on the accuracy of the generalized Born electrostatics model, treatment of protonation equilibria, high-resolution rotamer sampling, a final local energy minimization step, and explicit modeling of the bound, unbound, and unfolded states. The application of unmodified molecular-mechanics potentials to protein design links two fields in a mutually beneficial way. Design provides a new avenue for testing molecular-mechanics energy functions, and future improvements in these energy functions will presumably lead to more accurate design results.

Gating mechanisms of mechanosensitive channels of large conductance, I: a continuum mechanics-based hierarchical framework

A hierarchical simulation framework that integrates information from molecular dynamics (MD) simulations into a continuum model is established to study the mechanical response of mechanosensitive channel of large-conductance (MscL) using the finite element method (FEM). The proposed MD-decorated FEM (MDeFEM) approach is used to explore the detailed gating mechanisms of the MscL in Escherichia coli embedded in a palmitoyloleoylphosphatidylethanolamine lipid bilayer. In Part I of this study, the framework of MDeFEM is established. The transmembrane and cytoplasmic helices are taken to be elastic rods, the loops are modeled as springs, and the lipid bilayer is approximated by a three-layer sheet. The mechanical properties of the continuum components, as well as their interactions, are derived from molecular simulations based on atomic force fields. In addition, analytical closed-form continuum model and elastic network model are established to complement the MDeFEM approach and to capture the most essential features of gating. In Part II of this study, the detailed gating mechanisms of E. coli-MscL under various types of loading are presented and compared with experiments, structural model, and all-atom simulations, as well as the analytical models established in Part I. It is envisioned that such a hierarchical multiscale framework will find great value in the study of a variety of biological processes involving complex mechanical deformations such as muscle contraction and mechanotransduction.

Molecular simulation as an aid to experimentalists

Computer-based molecular simulation techniques are increasingly used to interpret experimental data on biomolecular systems at an atomic level. Direct comparison between experiment and simulation is, however, seldom straightforward. The available experimental data are limited in scope and generally correspond to averages over both time and space. A critical analysis of the various factors that may influence the apparent degree of agreement between the results of simulations and experimentally measured quantities is presented and illustrated using examples from recent literature.

Protein folding studied by single-molecule FRET

A complete understanding of a protein-folding mechanism requires description of the distribution of microscopic pathways that connect the folded and unfolded states. This distribution can, in principle, be described by computer simulations and theoretical models of protein folding, but is hidden in conventional experiments on large ensembles of molecules because only average properties are measured. A long-term goal of single-molecule fluorescence studies is to time-resolve the structural events as individual molecules make transitions between folded and unfolded states. Although such studies are still in their infancy, the work till now shows great promise and has already produced novel and important information on current issues in protein folding that has been impossible or difficult to obtain from ensemble measurements.

Multi-scale computational modelling in biology and physiology

Recent advances in biotechnology and the availability of ever more powerful computers have led to the formulation of increasingly complex models at all levels of biology. One of the main aims of systems biology is to couple these together to produce integrated models across multiple spatial scales and physical processes. In this review, we formulate a definition of multi-scale in terms of levels of biological organisation and describe the types of model that are found at each level. Key issues that arise in trying to formulate and solve multi-scale and multi-physics models are considered and examples of how these issues have been addressed are given for two of the more mature fields in computational biology: the molecular dynamics of ion channels and cardiac modelling. As even more complex models are developed over the coming few years, it will be necessary to develop new methods to model them (in particular in coupling across the interface between stochastic and deterministic processes) and new techniques will be required to compute their solutions efficiently on massively parallel computers. We outline how we envisage these developments occurring.

Molecular dynamics simulations of protein folding

I illustrate the use of the replica exchange molecular dynamics (REMD) algorithm to study the folding of a small (57 amino acids) protein that folds into a three-helix bundle, protein A. The REMD is a trivially parallel method that uses multiple copies of the system of interest to study the canonical ensemble equilibrium properties. Each replica represents a different thermodynamic state, usually at different temperatures. This method enhances the configurational sampling of proteins and allows us to study folding in simulations that are much shorter than the folding timescale for the system at ambient temperature. I show that using REMD and the Amber force field, I can obtain stable configurations of protein A whose backbone root mean square distance (RMSD) is within 0.17 nm of the nuclear magnetic resonance (NMR)-determined structure without biasing the system toward the folded structure. The simulations are done in explicit solvent and starting from nearly extended configurations. This calculation shows that currently available force fields and enhanced sampling methods perform reasonably well in describing the folded structure of small proteins.

Benchmarking of TASSER in the ab initio limit

A significant number of protein sequences in a given proteome have no obvious evolutionarily related protein in the database of solved protein structures, the PDB. Under these conditions, ab initio or template-free modeling methods are the sole means of predicting protein structure. To assess its expected performance on proteomes, the TASSER structure prediction algorithm is benchmarked in the ab initio limit on a representative set of 1129 nonhomologous sequences ranging from 40 to 200 residues that cover the PDB at 30% sequence identity and which adopt alpha, alpha + beta, and beta secondary structures. For sequences in the 40-100 (100-200) residue range, as assessed by their root mean square deviation from native, RMSD, the best of the top five ranked models of TASSER has a global fold that is significantly close to the native structure for 25% (16%) of the sequences, and with a correct identification of the structure of the protein core for 59% (36%). In the absence of a native structure, the structural similarity among the top five ranked models is a moderately reliable predictor of folding accuracy. If we classify the sequences according to their secondary structure content, then 64% (36%) of alpha, 43% (24%) of alpha + beta, and 20% (12%) of beta sequences in the 40-100 (100-200) residue range have a significant TM-score (TM-score > or = 0.4). TASSER performs best on helical proteins because there are less secondary structural elements to arrange in a helical protein than in a beta protein of equal length, since the average length of a helix is longer than that of a strand. In addition, helical proteins have shorter loops and dangling tails. If we exclude these flexible fragments, then TASSER has similar accuracy for sequences containing the same number of secondary structural elements, irrespective of whether they are helices and/or strands. Thus, it is the effective configurational entropy of the protein that dictates the average likelihood of correctly arranging the secondary structure elements.

Implicit modeling of nonpolar solvation for simulating protein folding and conformational transitions

Accurate description of the solvent environment is critical in computer simulations of protein structure and dynamics. An implicit treatment of solvent aims to capture the mean influence of water molecules on the solute via direct estimation of the solvation free energy. It has emerged as a powerful alternative to explicit solvent, and provides a favorable compromise between computational cost and level of detail. We review the current theory and techniques for implicit modeling of nonpolar solvation in the context of simulating protein folding and conformational transitions, and discuss the main directions for further development. It is demonstrated that the current surface area based nonpolar models have severe limitations, including insufficient description of the conformational dependence of solvation, over-estimation of the strength of pair-wise nonpolar interactions, and incorrect prediction of anti-cooperativity for three-body hydrophobic associations. We argue that, to improve beyond current level of accuracy of implicit solvent models, two important aspects of nonpolar solvation need to be incorporated, namely, the length-scale dependence of hydrophobic association and solvent screening of solute-solute dispersion interactions. We recognize that substantial challenges exist in constructing a sufficiently balanced, yet reasonably efficient, implicit solvent protein force field. Nonetheless, most of the fundamental problems are understood, and exciting progress has been made over the last few years. We believe that continual work along the frontiers outlined will greatly improve one's ability to study protein folding and large conformational transitions at atomistic detail.

Essential energy space random walk via energy space metadynamics method to accelerate molecular dynamics simulations

To overcome the possible pseudoergodicity problem, molecular dynamic simulation can be accelerated via the realization of an energy space random walk. To achieve this, a biased free energy function (BFEF) needs to be priori obtained. Although the quality of BFEF is essential for sampling efficiency, its generation is usually tedious and nontrivial. In this work, we present an energy space metadynamics algorithm to efficiently and robustly obtain BFEFs. Moreover, in order to deal with the associated diffusion sampling problem caused by the random walk in the total energy space, the idea in the original umbrella sampling method is generalized to be the random walk in the essential energy space, which only includes the energy terms determining the conformation of a region of interest. This essential energy space generalization allows the realization of efficient localized enhanced sampling and also offers the possibility of further sampling efficiency improvement when high frequency energy terms irrelevant to the target events are free of activation. The energy space metadynamics method and its generalization in the essential energy space for the molecular dynamics acceleration are demonstrated in the simulation of a pentanelike system, the blocked alanine dipeptide model, and the leucine model.

Heterogeneity even at the speed limit of folding: large-scale molecular dynamics study of a fast-folding variant of the villin headpiece

We have performed molecular dynamics simulations on a set of nine unfolded conformations of the fastest-folding protein yet discovered, a variant of the villin headpiece subdomain (HP-35 NleNle). The simulations were generated using a new distributed computing method, yielding hundreds of trajectories each on a time scale comparable to the experimental folding time, despite the large (10,000 atom) size of the simulation system. This strategy eliminates the need to assume a two-state kinetic model or to build a Markov state model. The relaxation to the folded state at 300 K from the unfolded configurations (generated by simulation at 373 K) was monitored by a method intended to reflect the experimental observable (quenching of tryptophan by histidine). We also monitored the relaxation to the native state by directly comparing structural snapshots with the native state. The rate of relaxation to the native state and the number of resolvable kinetic time scales both depend upon starting structure. Moreover, starting structures with folding rates most similar to experiment show some native-like structure in the N-terminal helix (helix 1) and the phenylalanine residues constituting the hydrophobic core, suggesting that these elements may exist in the experimentally relevant unfolded state. Our large-scale simulation data reveal kinetic complexity not resolved in the experimental data. Based on these findings, we propose additional experiments to further probe the kinetics of villin folding.

Free-energy landscape for beta hairpin folding from combined parallel tempering and metadynamics

We develop a new free-energy method, based on the combination of parallel tempering and metadynamics, and apply this method to the calculation of the free-energy landscape of the folding beta hairpin in explicit water. We show that the combined method greatly improves the performance of both parallel tempering and metadynamics. In particular, we are able to sample the high free-energy regions, which are not accessible with conventional parallel tempering. We use our results to calculate the difference in entropy and enthalpy between the folded and the unfolded state and to characterize the most populated configurations in the relevant free-energy basins.

Parallel tempering molecular dynamics folding simulation of a signal peptide in explicit water

Parallel temperature molecular dynamics simulations are used to explore the folding of a signal peptide, a short but functionally independent domain at the N-terminus of proteins. The peptide has been analyzed previously by NMR, and thus a solid reference state is provided with the experimental structure. Particular attention is paid to the role of water considered in full atomic detail. Different partial aspects in the folding process are quantified. The major group of obtained structures matches the NMR structure very closely. An important biological consequence is that in vivo folding of signal peptides seems to be possible within aqueous environments.

Monte Carlo vs molecular dynamics for all-atom polypeptide folding simulations

An efficient Monte Carlo (MC) algorithm including concerted rotations is directly compared to molecular dynamics (MD) in all-atom statistical mechanics folding simulations of small polypeptides. The previously reported algorithm "concerted rotations with flexible bond angles" (CRA) has been shown to successfully locate the native state of small polypeptides. In this study, the folding of three small polypeptides (trpzip2/H1/Trp-cage) is investigated using MC and MD, for a combined sampling time of approximately 10(11) MC configurations and 8 micros, respectively. Both methods successfully locate the experimentally determined native states of the three systems, but they do so at different speed, with 2-2.5 times faster folding of the MC runs. The comparison reveals that thermodynamic and dynamic properties can reliably be obtained by both and that results from folding simulations do not depend on the algorithm used. Similar to previous comparisons of MC and MD, it is found that one MD integration step of 2 fs corresponds to one MC scan, revealing the good sampling of MC. The simplicity and efficiency of the MC method will enable its future use in folding studies involving larger systems and the combination with replica exchange algorithms.

Molecular dynamics simulation of folding of a short helical peptide with many charged residues

A molecular dynamics simulation of the folding of conantokin-T (con-T), a short helical peptide with 5 helical turns of 21 amino acids with 10 charged residues, was carried out to examine folding pathways for this peptide and to predict the folding rate. In the 18 trajectories run at 300 K, 16 trajectories folded, with an averaged folding time of approximately 50 ns. Two trajectories did not fold in up to 200 ns simulation. The folded structure in folded trajectories is in good agreement with experimental structure. An analysis of the trajectories showed that, at the beginning of a few nanoseconds, helix formation started from residues 5-9 with assistance of a hydrophobic clustering involving Tyr5, Met8, and Leu9. The peptide formed a U-shape mainly due to charge-charge interactions between charged residues at the N- and C-terminus segments. In the next approximately 10 ns, several nonnative charge-charge interactions were broken and nonnative Gla10-Lys18 (this denotes a salt bridge between Gal10 and Lys18) and/or Gla10-Lys19 interactions appeared more frequently in this folding step and the peptide became a fishhook J-shape. From this structure, the peptide folded to the folded state in 7 of all 16 folded trajectories in approximately 15 ns. Alternatively, in approximately 30 ns, the con-T went to a conformation in an L-shape with 4 helical turns and a kink at the Arg13 and Gla14 segment in the other 9 trajectories. Con-T in the L-shape then required another approximately 15 ns to fold into the folded state. In addition, in overall folding times, the former 7 trajectories folded faster with the total folding times all shorter than 45 ns, while the latter 9 trajectories folded at a time longer than 45 ns, resulting in an average folding time of approximately 50 ns. Two major folding intermediates found in 2 nonfolded trajectories are stabilized by charge clusters of 5 and 6 charged residues, respectively. With inclusion of friction and solvent-solvent interactions, which were ignored in the present GB/SA solvation model, the folding time obtained above should be multiplied by a factor of 1.25-1.7 according to a previous, similar simulation study. This results in a folding time of 65-105 ns, slightly shorter than the folding time of 127 ns for an alanine-based peptide of the same length. This suggests that the energy barrier of folding for this type of peptides with many charged residues is slightly lower than alanine-based helical peptides by less than 1 kcal/mol.

Unfolding pathways of goat alpha-lactalbumin as revealed in multiple alignment of molecular dynamics trajectories

Molecular dynamics simulations of protein unfolding were performed at an elevated temperature for the authentic and recombinant forms of goat alpha-lactalbumin. Despite very similar three-dimensional structures, the two forms have significantly different unfolding rates due to an extra N-terminal methionine in the recombinant protein. To identify subtle differences between the two forms in the highly stochastic kinetics of unfolding, we classified the unfolding trajectories using the multiple alignment method based on the analogy between the biological sequences and the molecular dynamics trajectories. A dendrogram derived from the multiple trajectory alignment revealed a clear difference in the unfolding pathways of the authentic and recombinant proteins, i.e. the former reached the transition state in an all-or-none manner while the latter unfolded less cooperatively. It was also found in the classification that the two forms of the protein shared a common transition state structure, which was in excellent agreement with the transition state structure observed experimentally in the Phi-value analysis.

Protein folding under confinement: a role for solvent

Although most experimental and theoretical studies of protein folding involve proteins in vitro, the effects of spatial confinement may complicate protein folding in vivo. In this study, we examine the folding dynamics of villin (a small fast folding protein) with explicit solvent confined to an inert nanopore. We have calculated the probability of folding before unfolding (P(fold)) under various confinement regimes. Using P(fold) correlation techniques, we observed two competing effects. Confining protein alone promotes folding by destabilizing the unfolded state. In contrast, confining both protein and solvent gives rise to a solvent-mediated effect that destabilizes the native state. When both protein and solvent are confined we see unfolding to a compact unfolded state different from the unfolded state seen in bulk. Thus, we demonstrate that the confinement of solvent has a significant impact on protein kinetics and thermodynamics. We conclude with a discussion of the implications of these results for folding in confined environments such as the chaperonin cavity in vivo.

Conformational Dynamics and Ensembles in Protein Folding

Recent experimental developments are changing the ways we interpret experimental data in protein folding, leading to a closer connection with theory and an improved understanding of some long-standing questions in the field. We now have a basic roadmap of the types of polypeptide motions and timescales that are relevant to the various folding stages. The folding barriers estimated with a variety of independent methods are consistently small, indicating that several fast-folding proteins are near or within the downhill folding regime. Finally, the structural and statistical analysis of global downhill folding is promising to open a new avenue of research in which folding mechanisms and the networks of noncovalent interactions that stabilize native structures are directly resolved in equilibrium experiments of nonmutated proteins.

Computational design and experimental discovery of an antiestrogenic peptide derived from alpha-fetoprotein

Breast cancer is the most common cancer among women, and tamoxifen is the preferred drug for estrogen receptor-positive breast cancer treatment. Many of these cancers are intrinsically resistant to tamoxifen or acquire resistance during treatment. Consequently, there is an ongoing need for breast cancer drugs that have different molecular targets. Previous work has shown that 8-mer and cyclic 9-mer peptides inhibit breast cancer in mouse and rat models, interacting with an unsolved receptor, while peptides smaller than eight amino acids did not. We show that the use of replica exchange molecular dynamics predicts the structure and dynamics of active peptides, leading to the discovery of smaller peptides with full biological activity. Simulations identified smaller peptide analogues with the same conserved reverse turn demonstrated in the larger peptides. These analogues were synthesized and shown to inhibit estrogen-dependent cell growth in a mouse uterine growth assay, a test showing reliable correlation with human breast cancer inhibition.

Persistent voids: a new structural metric for membrane fusion

MOTIVATION

Membrane fusion constitutes a key stage in cellular processes such as synaptic neurotransmission and infection by enveloped viruses. Current experimental assays for fusion have thus far been unable to resolve early fusion events in fine structural detail. We have previously used molecular dynamics simulations to develop mechanistic models of fusion by small lipid vesicles. Here, we introduce a novel structural measurement of vesicle topology and fusion geometry: persistent voids.

RESULTS

Persistent voids calculations enable systematic measurement of structural changes in vesicle fusion by assessing fusion stalk widths. They also constitute a generally applicable technique for assessing lipid topological change. We use persistent voids to compute dynamic relationships between hemifusion neck widening and formation of a full fusion pore in our simulation data. We predict that a tightly coordinated process of hemifusion neck expansion and pore formation is responsible for the rapid vesicle fusion mechanism, while isolated enlargement of the hemifusion diaphragm leads to the formation of a metastable hemifused intermediate. These findings suggest that rapid fusion between small vesicles proceeds via a small hemifusion diaphragm rather than a fully expanded one.

AVAILABILITY

Software available upon request pending public release.

SUPPLEMENTARY INFORMATION

Supplementary data are available on Bioinformatics online.

What determines protein folding type? An investigation of intrinsic structural properties and its implications for understanding folding mechanisms

Protein folding experiments demonstrate that the folding behaviors of many proteins can be roughly classified into two types: two-state kinetics and multi-state kinetics. Although the two types of protein folding kinetics have been observed for a long time, what determines the folding type of a protein is still largely unclear. The present work performed a comparative study based on a dataset of 43 two-state and 42 multi-state folders at different levels of proteins' intrinsic properties from the simplest sequence length to native structure topology. The results show that protein's amino acids composition and the long-range interaction-based topological complexity rather than secondary structure contents are the major determinants of protein folding type. Furthermore, a sequence-based folding type prediction achieved an accuracy of more than 80%. These findings implicate that there is no clear boundary between secondary and tertiary structure formation during the protein folding process and support the existence of a continuum of folding mechanism between the two ends of hierarchic and nucleation folding scenarios.

Protein-Folding Dynamics: Overview of Molecular Simulation Techniques

Molecular dynamics (MD) is an invaluable tool with which to study protein folding in silico. Although just a few years ago the dynamic behavior of a protein molecule could be simulated only in the neighborhood of the experimental conformation (or protein unfolding could be simulated at high temperature), the advent of distributed computing, new techniques such as replica-exchange MD, new approaches (based on, e.g., the stochastic difference equation), and physics-based reduced models of proteins now make it possible to study protein-folding pathways from completely unfolded structures. In this review, we present algorithms for MD and their extensions and applications to protein-folding studies, using all-atom models with explicit and implicit solvent as well as reduced models of polypeptide chains.

Convergence and sampling efficiency in replica exchange simulations of peptide folding in explicit solvent

Replica exchange methods (REMs) are increasingly used to improve sampling in molecular dynamics (MD) simulations of biomolecular systems. However, despite having been shown to be very effective on model systems, the application of REM in complex systems such as for the simulation of protein and peptide folding in explicit solvent has not been objectively tested in detail. Here we present a comparison of conventional MD and temperature replica exchange MD (T-REMD) simulations of a beta-heptapeptide in explicit solvent. This system has previously been shown to undergo reversible folding on the time scales accessible to MD simulation and thus allows a direct one-to-one comparison of efficiency. The primary properties compared are the free energy of folding and the relative populations of different conformers as a function of temperature. It is found that to achieve a similar degree of precision T-REMD simulations starting from a random set of initial configurations were approximately an order of magnitude more computationally efficient than a single 800 ns conventional MD simulation for this system at the lowest temperature investigated (275 K). However, whereas it was found that T-REMD simulations are more than four times more efficient than multiple independent MD simulations at one temperature (300 K) the actual increase in conformation sampling was only twofold. The overall gain in efficiency using REMD resulted primarily from the ordering of different conformational states over temperature, as opposed to a large increase of conformational sampling. It is also shown that in this system exchanges are accepted primarily based on (random) fluctuations within the solvent and are not strongly correlated with the instantaneous peptide conformation raising questions in regard to the efficiency of T-REMD in larger systems.

Examination of the folding of a short alanine-based helical peptide with salt bridges using molecular dynamics simulation

A molecular dynamics simulation of the folding of a short alanine-based helical peptide of 17 residues with three Glu...Lys (i, i + 4) salt bridge pairs, referred to as the AEK17 peptide, was carried out. The simulation gave an estimated simulation folding time of 2.5 ns, shorter than 12 ns for an alanine-based peptide of 16 residues with three Lys residues only, referred to as the AK16 peptide, simulated previously. After folded, the AEK17 peptide had a helical content of 77%, in excellent agreement with the experimentally determined value of 80%. An examination of the folding pathways of AEK17 indicated that the peptide proceeded via three-turn helix conformations more than the helix-turn-helix conformation in the folding pathways. An analysis of interactions indicated that the formation of hydrogen bonds between Lys residue side chains and backbone carbonyls is a major factor in the abundant conformation of the three-turn helix intermediate. The substitution of three Ala with Glu residues reduces the extent of hydrophobic interaction in alanine-based AK peptides with the result that the breaking of the interactions of Lys epsilon-NH3+(side chain)...C=O(backbone) is a major activation action for the AEK17 to achieve a complete fold, in contrast to the AK16 peptide, in which breaking non-native hydrophobic interaction is the rate-determining step.

Simulated scaling method for localized enhanced sampling and simultaneous “alchemical” free energy simulations: A general method for molecular mechanical, quantum mechanical, and quantum mechanical/molecular mechanical simulations

A potential scaling version of simulated tempering is presented to efficiently sample configuration space in a localized region. The present "simulated scaling" method is developed with a Wang-Landau type of updating scheme in order to quickly flatten the distributions in the scaling parameter lambdam space. This proposal is meaningful for a broad range of biophysical problems, in which localized sampling is required. Besides its superior capability and robustness in localized conformational sampling, this simulated scaling method can also naturally lead to efficient "alchemical" free energy predictions when dual-topology alchemical hybrid potential is applied; thereby simultaneously, both of the chemically and conformationally distinct portions of two end point chemical states can be efficiently sampled. As demonstrated in this work, the present method is also feasible for the quantum mechanical and quantum mechanical/molecular mechanical simulations.

Ultrafast and downhill protein folding

Ultrafast folding proteins have served an important role in benchmarking molecular dynamics simulations and testing protein folding theories. These proteins are simple enough and fold fast enough that realistic simulations are possible, which facilitates the direct comparison of absolute folding rates and folding mechanisms with those observed experimentally. Such comparisons have achieved remarkable success, but have also revealed the shortcomings that remain in experiment, theory and simulation alike. Some ultrafast folding proteins may fold without encountering an activation barrier (downhill folding), allowing the exploration of the molecular timescale of folding and the roughness of the energy landscape. The biological significance of ultrafast folding remains uncertain, but its practical significance is crucial to progress in understanding how proteins fold.

Dynamics of lysozyme structure network: probing the process of unfolding

Recently we showed that the three-dimensional structure of proteins can be investigated from a network perspective, where the amino acid residues represent the nodes in the network and the noncovalent interactions between them are considered for the edge formation. In this study, the dynamical behavior of such networks is examined by considering the example of T4 lysozyme. The equilibrium dynamics and the process of unfolding are followed by simulating the protein at 300 K and at higher temperatures (400 K and 500 K), respectively. The snapshots of the protein structure from the simulations are represented as protein structure networks in which the strength of the noncovalent interactions is considered an important criterion in the construction of edges. The profiles of the network parameters, such as the degree distribution and the size of the largest cluster (giant component), were examined as a function of interaction strength at different temperatures. Similar profiles are seen at all the temperatures. However, the critical strength of interaction (Icritical) and the size of the largest cluster at all interaction strengths shift to lower values at 500 K. Further, the folding/unfolding transition is correlated with contacts evaluated at Icritical and with the composition of the top large clusters obtained at interaction strengths greater than Icritical. Finally, the results are compared with experiments, and predictions are made about the residues, which are important for stability and folding. To summarize, the network analysis presented in this work provides insights into the details of the changes occurring in the protein tertiary structure at the level of amino acid side-chain interactions, in both the equilibrium and the unfolding simulations. The method can also be employed as a valuable tool in the analysis of molecular dynamics simulation data, since it captures the details at a global level, which may elude conventional pairwise interaction analysis.

Fluctuations of primary ubiquitin folding intermediates in a force clamp

Folding experiments of single ubiquitin molecules under force clamp using an atomic force microscope revealed a dynamic long-lived intermediate with nanometer scale end-to-end distance fluctuations along an unexpectedly complex folding pathway. To examine the nature of this intermediate at the atomic level as well as the driving forces that give rise to the observed fluctuations, we performed molecular dynamics refolding simulations of unfolded ubiquitin under constant force. After an initial fast collapse, we find a highly dynamic, broad ensemble of conformations with partial and continuously changing secondary structure and side chain interactions. This ensemble resembles a molten-globule-like state, similar in nature to the previously described non-native state of ubiquitin in solution, but stretched by the external force. The scale of the end-to-end distance fluctuations derived from the simulations compares well with experiment. Transient formation of unspecific and metastable hydrophobic clusters along the chain are found to give rise to the observed end-to-end distance fluctuations. These distinct collapses, interpreted as folding attempts, imply an upper limit for the folding attempt frequency of approximately 10 ns. Our results suggest possible relations between force-induced unfolding and temperature or chemically induced denaturation.

Understanding ensemble protein folding at atomic detail

It has long been known that a protein's amino acid sequence dictates its native structure. However, despite significant recent advances, an ensemble description of how a protein achieves its native conformation from random coil under physiologically relevant conditions remains incomplete. Here we present a detailed all-atom model with a transferable potential that is capable of ab initio folding of entire protein domains using only sequence information. The computational efficiency of this model allows us to perform thousands of microsecond-time scale-folding simulations of the engrailed homeodomain and to observe thousands of complete independent folding events. We apply a graph-theoretic analysis to this massive data set to elucidate which intermediates and intermediary states are common to many trajectories and thus important for the folding process. This method provides an atomically detailed and complete picture of a folding pathway at the ensemble level. The approach that we describe is quite general and could be used to study the folding of proteins on time scales orders of magnitude longer than currently possible.

Finite reservoir replica exchange to enhance canonical sampling in rugged energy surfaces

A "finite reservoir" replica exchange method is presented to further enhance sampling upon the regular replica exchange method (REM) in a rugged energy surface. The present method can facilitate important sampling more efficiently by exchanging structures with configurations randomly selected from a finite-sized reservoir; this finite reservoir is pregenerated and updated by a mechanism of replica exchange with neighboring "temperature" simulations. In practice, this proposal revises exchange schedule in REM simulations in order to make productive exchange for conformational "tunneling" more frequent.

A kinetic theory of tertiary contact formation coupled to the helix-coil transition in polypeptides

The framework model and the hydrophobic collapse model represent two canonical descriptions of the protein folding process. The first places primary reliance on the short-range interactions of secondary structure and the second assigns greater importance to the long-range interactions of tertiary structure. The availability of increasingly detailed information about the folding mechanisms of diverse proteins suggests that both are important and the folding mechanism of most proteins utilizes different combinations of such interactions. A prior report described the XHC model, an extended helix-coil theory, which treats the mutual stabilization of secondary and tertiary structure in simple alpha-helical proteins at equilibrium. In this study, a kinetic scheme describing tertiary contact formation has been developed which relaxes to the XHC equilibrium model. The relaxation is governed by the relative stabilities of the equilibrium states and an additional factor which represents an activation energy for formation of a tertiary contact. The model can be used to simulate time-dependent properties of the ensemble of conformations during the entire folding process, and the resulting predictions are applicable to a range of experimental methods. This XHC kinetic model enables investigation of the relative influence of secondary and tertiary interactions on folding mechanisms.

Direct correlation between proteins' folding rates and their amino acid compositions: An ab initio folding rate prediction

Discovering the mechanism of protein folding, in molecular biology, is a great challenge. A key step to this end is to find factors that correlate with protein folding rates. Over the past few years, many empirical parameters, such as contact order, long-range order, total contact distance, secondary structure contents, have been developed to reflect the correlation between folding rates and protein tertiary or secondary structures. However, the correlation between proteins' folding rates and their amino acid compositions has not been explored. In the present work, we examined systematically the correlation between proteins' folding rates and their amino acid compositions for two-state and multistate folders and found that different amino acids contributed differently to the folding progress. The relation between the amino acids' molecular weight and degeneracy and the folding rates was examined, and the role of hydrophobicity in the protein folding process was also inspected. As a consequence, a new indicator called composition index was derived, which takes no structure factors into account and is merely determined by the amino acid composition of a protein. Such an indicator is found to be highly correlated with the protein's folding rate (r > 0.7). From the results of this work, three points of concluding remarks are evident. (1) Two-state folders and multistate folders have different rate-determining amino acids. (2) The main determining information of a protein's folding rate is largely reflected in its amino acid composition. (3) Composition index may be the best predictor for an ab initio protein folding rate prediction directly from protein sequence from the standpoint of practical application.

Similarity and difference in the unfolding of thermophilic and mesophilic cold shock proteins studied by molecular dynamics simulations

Molecular dynamics simulations were performed to unfold a homologous pair of thermophilic and mesophilic cold shock proteins at high temperatures. The two proteins differ in just 11 of 66 residues and have very similar structures with a closed five-stranded antiparallel beta-barrel. A long flexible loop connects the N-terminal side of the barrel, formed by three strands (beta1-beta3), with the C-terminal side, formed by two strands (beta4-beta5). The two proteins were found to follow the same unfolding pathway, but with the thermophilic protein showing much slower unfolding. Unfolding started with the melting of C-terminal strands, leading to exposure of the hydrophobic core. Subsequent melting of beta3 and the beta-hairpin formed by the first two strands then resulted in unfolding of the whole protein. The slower unfolding of the thermophilic protein could be attributed to ion pair formation of Arg-3 with Glu-46, Glu-21, and the C-terminal. These ion pairs were also found to be important for the difference in folding stability between the pair of proteins. Thus electrostatic interactions appear to play similar roles in the difference in folding stability and kinetics between the pair of proteins.

Electric Fields at the Active Site of an Enzyme: Direct Comparison of Experiment with Theory

The electric fields produced in folded proteins influence nearly every aspect of protein function. We present a vibrational spectroscopy technique that measures changes in electric field at a specific site of a protein as shifts in frequency (Stark shifts) of a calibrated nitrile vibration. A nitrile-containing inhibitor is used to deliver a unique probe vibration to the active site of human aldose reductase, and the response of the nitrile stretch frequency is measured for a series of mutations in the enzyme active site. These shifts yield quantitative information on electric fields that can be directly compared with electrostatics calculations. We show that extensive molecular dynamics simulations and ensemble averaging are required to reproduce the observed changes in field.