Go-ing for the prediction of protein folding mechanisms

The Wako-Saitô-Muñoz-Eaton Model for Predicting Protein Folding and Dynamics

Despite the recent advances in the prediction of protein structures by deep neutral networks, the elucidation of protein-folding mechanisms remains challenging. A promising theory for describing protein folding is a coarse-grained statistical mechanical model called the Wako-Saitô-Muñoz-Eaton (WSME) model. The model can calculate the free-energy landscapes of proteins based on a three-dimensional structure with low computational complexity, thereby providing a comprehensive understanding of the folding pathways and the structure and stability of the intermediates and transition states involved in the folding reaction. In this review, we summarize previous and recent studies on protein folding and dynamics performed using the WSME model and discuss future challenges and prospects. The WSME model successfully predicted the folding mechanisms of small single-domain proteins and the effects of amino-acid substitutions on protein stability and folding in a manner that was consistent with experimental results. Furthermore, extended versions of the WSME model were applied to predict the folding mechanisms of multi-domain proteins and the conformational changes associated with protein function. Thus, the WSME model may contribute significantly to solving the protein-folding problem and is expected to be useful for predicting protein folding, stability, and dynamics in basic research and in industrial and medical applications.

The RED scheme: Rate-constant estimation from pre-steady state weighted ensemble simulations

We present the Rate from Event Durations (RED) scheme, a new scheme that more efficiently calculates rate constants using the weighted ensemble path sampling strategy. This scheme enables rate-constant estimation from shorter trajectories by incorporating the probability distribution of event durations, or barrier-crossing times, from a simulation. We have applied the RED scheme to weighted ensemble simulations of a variety of rare-event processes that range in complexity: residue-level simulations of protein conformational switching, atomistic simulations of Na+/Cl- association in explicit solvent, and atomistic simulations of protein-protein association in explicit solvent. Rate constants were estimated with up to 50% greater efficiency than the original weighted ensemble scheme. Importantly, our scheme accounts for the systematic error that results from statistical bias toward the observation of events with short durations and reweights the event duration distribution accordingly. The RED scheme is relevant to any simulation strategy that involves unbiased trajectories of similar length to the most probable event duration, including weighted ensemble, milestoning, and standard simulations as well as the construction of Markov state models.

Double-Well Ultra-Coarse-Grained Model to Describe Protein Conformational Transitions

The double-well model is usually used to describe the conformational transition between two states of a protein. Since conformational changes usually occur within a relatively large time scale, coarse-grained models are often used to accelerate the dynamic process due to their inexpensive computational cost. In this work, we develop a double-well ultra-coarse-grained (DW-UCG) model to describe the conformational transitions of the adenylate kinase, glutamine-binding protein, and lactoferrin. The coarse-grained simulation results show that the DW-UCG model of adenylate kinase captures the crucial intermediate states in the LID-closing and NMP-closing pathways, reflecting the key secondary structural changes in the conformational transition. A comparison of the different DW-UCG models of adenylate kinase indicates that an appropriate choice of bead resolution could generate the free energy landscape that is comparable to that from the residue-based model. The coarse-grained simulations for the glutamine-binding protein and lactoferrin also demonstrate that the DW-UCG model is valid in reproducing the correct two-state behavior for their functional study, which indicates the potential application of the DW-UCG model in investigating the mechanism of conformational changes of large proteins.

AutoDock CrankPep: combining folding and docking to predict protein-peptide complexes

MOTIVATION

Protein-peptide interactions mediate a wide variety of cellular and biological functions. Methods for predicting these interactions have garnered a lot of interest over the past few years, as witnessed by the rapidly growing number of peptide-based therapeutic molecules currently in clinical trials. The size and flexibility of peptides has shown to be challenging for existing automated docking software programs.

RESULTS

Here we present AutoDock CrankPep or ADCP in short, a novel approach to dock flexible peptides into rigid receptors. ADCP folds a peptide in the potential field created by the protein to predict the protein-peptide complex. We show that it outperforms leading peptide docking methods on two protein-peptide datasets commonly used for benchmarking docking methods: LEADS-PEP and peptiDB, comprised of peptides with up to 15 amino acids in length. Beyond these datasets, ADCP reliably docked a set of protein-peptide complexes containing peptides ranging in lengths from 16 to 20 amino acids. The robust performance of ADCP on these longer peptides enables accurate modeling of peptide-mediated protein-protein interactions and interactions with disordered proteins.

AVAILABILITY AND IMPLEMENTATION

ADCP is distributed under the LGPL 2.0 open source license and is available at http://adcp.scripps.edu. The source code is available at https://github.com/ccsb-scripps/ADCP.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Life in Phases: Intra- and Inter- Molecular Phase Transitions in Protein Solutions

Proteins, these evolutionarily-edited biological polymers, are able to undergo intramolecular and intermolecular phase transitions. Spontaneous intramolecular phase transitions define the folding of globular proteins, whereas binding-induced, intra- and inter- molecular phase transitions play a crucial role in the functionality of many intrinsically-disordered proteins. On the other hand, intermolecular phase transitions are the behind-the-scenes players in a diverse set of macrosystemic phenomena taking place in protein solutions, such as new phase nucleation in bulk, on the interface, and on the impurities, protein crystallization, protein aggregation, the formation of amyloid fibrils, and intermolecular liquid-liquid or liquid-gel phase transitions associated with the biogenesis of membraneless organelles in the cells. This review is dedicated to the systematic analysis of the phase behavior of protein molecules and their ensembles, and provides a description of the major physical principles governing intramolecular and intermolecular phase transitions in protein solutions.

Gō model revisited

This review discusses Gō models broadly used in biomolecular simulations. I start with a brief description of the original lattice model study by Nobuhiro Gō. Then, the theory of protein folding behind Gō model, free energy approaches, and off-lattice Gō models are reviewed. I also mention a stringent test for the assumption in Gō models given from all-atom molecular dynamics simulations. Subsequently, I move to application of Gō models to protein dynamical functions. Various extension of Gō models is also reviewed. Finally, some publicly available tools to use Gō models are listed.

Go-Kit: A Tool To Enable Energy Landscape Exploration of Proteins

Coarse-grained Go̅-like models, based on the principle of minimal frustration, provide valuable insight into fundamental questions in the field of protein folding and dynamics. In conjunction with commonly used molecular dynamics (MD) simulations, energy landscape exploration methods like discrete path sampling (DPS) with Go̅-like models can provide quantitative details of the thermodynamics and kinetics of proteins. Here we present Go-kit, a software that facilitates the setup of MD and DPS simulations of several flavors of Go̅-like models. Go-kit is designed for use with MD (GROMACS) and DPS (PATHSAMPLE) simulation engines that are open source. The Go-kit code is written in python2.7 and is also open source. A case study for the ribosomal protein S6 is discussed to illustrate the utility of the software, which is available at https://github.com/gokit1/gokit .

Choice of Adaptive Sampling Strategy Impacts State Discovery, Transition Probabilities, and the Apparent Mechanism of Conformational Changes

Interest in atomically detailed simulations has grown significantly with recent advances in computational hardware and Markov state modeling (MSM) methods, yet outstanding questions remain that hinder their widespread adoption. Namely, how do alternative sampling strategies explore conformational space and how might this influence predictions generated from the data? Here, we seek to answer these questions for four commonly used sampling methods: (1) a single long simulation, (2) many short simulations run in parallel, (3) adaptive sampling, and (4) our recently developed goal-oriented sampling algorithm, FAST. We first develop a theoretical framework for analytically calculating the probability of discovering select states on simple landscapes, where we uncover the drastic effects of varying the number and length of simulations. We then use kinetic Monte Carlo simulations on a variety of physically inspired landscapes to characterize the probability of discovering particular states and transition pathways for each of the four methods. Consistently, we find that FAST simulations discover each target state with the highest probability, while traversing realistic pathways. Furthermore, we uncover the potential pathology that short parallel simulations sometimes predict an incorrect transition pathway by crossing large energy barriers that long simulations would typically circumnavigate. We refer to this pathology as "pathway tunneling". To protect against this phenomenon when using adaptive-sampling and FAST simulations, we introduce the FAST-string method. This method enhances sampling along the highest-flux transition paths to refine an MSMs transition probabilities and discriminate between competing pathways. Additionally, we compare the performance of a variety of MSM estimators in describing accurate thermodynamics and kinetics. For adaptive sampling, we recommend simply normalizing the transition counts out of each state after adding small pseudocounts to avoid creating sources or sinks. Lastly, we evaluate whether our insights from simple landscapes hold for all-atom molecular dynamics simulations of the folding of the λ-repressor protein. Remarkably, we find that FAST-contacts predicts the same folding pathway as a set of long simulations but with orders of magnitude less simulation time.

Flexibility vs Preorganization: Direct Comparison of Binding Kinetics for a Disordered Peptide and Its Exact Preorganized Analogues

Many intrinsically disordered proteins, which are prevalent in nature, fold only upon binding their structured partner proteins. Such proteins have been hypothesized to have a kinetic advantage over their folded, preorganized analogues in binding their partner proteins. Here we determined the effects of ligand preorganization on the k_on for a biomedically important system: an intrinsically disordered p53 peptide ligand and the MDM2 protein receptor. Based on direct simulations of binding pathways, computed k_on values for fully disordered and preorganized p53 peptide analogues were within error of each other, indicating little if any kinetic advantage to being disordered or preorganized for binding the MDM2 protein. We also examined the effects of increasing the concentration of MDM2 on the extent to which its mechanism of binding to the p53 peptide is induced fit vs conformational selection. Results predict that the mechanism is solely induced fit if the unfolded state of the peptide is more stable than its folded state; otherwise, the mechanism shifts from being dominated by conformational selection at low MDM2 concentration to induced fit at high MDM2 concentration. Taken together, our results are relevant to any protein binding process that involves a disordered peptide of a similar length that forms a single α-helix upon binding a partner protein. Such disorder-to-helix transitions are common among protein interactions of disordered proteins and are therefore of fundamental biological interest.

Topological transformations in proteins: effects of heating and proximity of an interface

Using a structure-based coarse-grained model of proteins, we study the mechanism of unfolding of knotted proteins through heating. We find that the dominant mechanisms of unfolding depend on the temperature applied and are generally distinct from those identified for folding at its optimal temperature. In particular, for shallowly knotted proteins, folding usually involves formation of two loops whereas unfolding through high-temperature heating is dominated by untying of single loops. Untying the knots is found to generally precede unfolding unless the protein is deeply knotted and the heating temperature exceeds a threshold value. We then use a phenomenological model of the air-water interface to show that such an interface can untie shallow knots, but it can also make knots in proteins that are natively unknotted.

Structural fluctuations and thermal stability of proteins in crowded environments: effects of the excluded volume

We perform molecular dynamics simulations for a simple coarse-grained model of a protein placed inside of a softly repulsive sphere of radius R. The protein is surrounded either by a number of same molecules or a number of spherical crowding particles that immitate other biomolecules such as the osmolytes. The two descriptions are shown to lead to distinct results when testing thermal stability as assessed by studying the unfolding times as a function of temperature. We consider three examples of proteins and show that crowding increases the thermal stability provided the inter-protein or protein-crowder interactions are repulsive. On the other hand, an introduction of attraction between the proteins is found to destabilize the proteins. Crowding by repulsive crowder particles is seen to enhance the RMSF in certain exposed regions. The effect grows on decreasing the size of the crowding particles. In the absence of crowding the RMSF anticorrelates with the coordination number related to the residue-residue interaction.

Dual binding mode in cohesin-dockerin complexes as assessed through stretching studies

A recent experimental study by Jobst et al. of stretching of a wild-type (WT) cohesin-dockerin complex has identified two kinds of the force-displacement patterns, with a single or double-peaked final rupture, which are termed "short" and "long" here. This duality has been interpreted as arising from the existence of two kinds of binding. Here, we analyze the separation of two cohesin-dockerin complexes of C. thermocellum theoretically. We use a coarse-grained structure-based model and the values of the pulling speeds are nearly experimental. In their native states, the two systems differ in the mutual binding orientations of the molecules in the complex. We demonstrate that the WT complex (PDB:1OHZ) unravels along two possible pathways that are qualitatively consistent with the presence of the short and long patterns observed experimentally. On the other hand, the mutated complex (PDB:2CCL) leads only to short trajectories. The short and long stretching pathways also appear in the cohesin-dockerin-Xmodule complex (PDB:4IU3, WT) of R. flavefaciens. Thus the duality in the stretching patterns need not be necessarily due to the duality in binding.

Unfolding knots by proteasome-like systems: simulations of the behaviour of folded and neurotoxic proteins

Knots in proteins have been proposed to resist proteasomal degradation, thought in turn to be related to neurodegenerative diseases such as Huntington.

Highly Efficient Computation of the Basal kon using Direct Simulation of Protein-Protein Association with Flexible Molecular Models

An essential baseline for determining the extent to which electrostatic interactions enhance the kinetics of protein–protein association is the “basal” k_on, which is the rate constant for association in the absence of electrostatic interactions. However, since such association events are beyond the milliseconds time scale, it has not been practical to compute the basal k_on by directly simulating the association with flexible models. Here, we computed the basal k_on for barnase and barstar, two of the most rapidly associating proteins, using highly efficient, flexible molecular simulations. These simulations involved (a) pseudoatomic protein models that reproduce the molecular shapes, electrostatic, and diffusion properties of all-atom models, and (b) application of the weighted ensemble path sampling strategy, which enhanced the efficiency of generating association events by >130-fold. We also examined the extent to which the computed basal k_on is affected by inclusion of intermolecular hydrodynamic interactions in the simulations.

Modeling Structural Dynamics of Biomolecular Complexes by Coarse-Grained Molecular Simulations

Due to hierarchic nature of biomolecular systems, their computational modeling calls for multiscale approaches, in which coarse-grained (CG) simulations are used to address long-time dynamics of large systems. Here, we review recent developments and applications of CG modeling methods, focusing on our methods primarily for proteins, DNA, and their complexes. These methods have been implemented in the CG biomolecular simulator, CafeMol. Our CG model has resolution such that ∼10 non-hydrogen atoms are grouped into one CG particle on average. For proteins, each amino acid is represented by one CG particle. For DNA, one nucleotide is simplified by three CG particles, representing sugar, phosphate, and base. The protein modeling is based on the idea that proteins have a globally funnel-like energy landscape, which is encoded in the structure-based potential energy function. We first describe two representative minimal models of proteins, called the elastic network model and the classic Go̅ model. We then present a more elaborate protein model, which extends the minimal model to incorporate sequence and context dependent local flexibility and nonlocal contacts. For DNA, we describe a model developed by de Pablo's group that was tuned to well reproduce sequence-dependent structural and thermodynamic experimental data for single- and double-stranded DNAs. Protein-DNA interactions are modeled either by the structure-based term for specific cases or by electrostatic and excluded volume terms for nonspecific cases. We also discuss the time scale mapping in CG molecular dynamics simulations. While the apparent single time step of our CGMD is about 10 times larger than that in the fully atomistic molecular dynamics for small-scale dynamics, large-scale motions can be further accelerated by two-orders of magnitude with the use of CG model and a low friction constant in Langevin dynamics. Next, we present four examples of applications. First, the classic Go̅ model was used to emulate one ATP cycle of a molecular motor, kinesin. Second, nonspecific protein-DNA binding was studied by a combination of elaborate protein and DNA models. Third, a transcription factor, p53, that contains highly fluctuating regions was simulated on two perpendicularly arranged DNA segments, addressing intersegmental transfer of p53. Fourth, we simulated structural dynamics of dinucleosomes connected by a linker DNA finding distinct types of internucleosome docking and salt-concentration-dependent compaction. Finally, we discuss many of limitations in the current approaches and future directions. Especially, more accurate electrostatic treatment and a phospholipid model that matches our CG resolutions are of immediate importance.

A polarizable coarse-grained protein model for dissipative particle dynamics

We present a new coarse-grained polarizable protein model for the dissipative particle dynamics (DPD) method.

Evolution, energy landscapes and the paradoxes of protein folding

Protein folding has been viewed as a difficult problem of molecular self-organization. The search problem involved in folding however has been simplified through the evolution of folding energy landscapes that are funneled. The funnel hypothesis can be quantified using energy landscape theory based on the minimal frustration principle. Strong quantitative predictions that follow from energy landscape theory have been widely confirmed both through laboratory folding experiments and from detailed simulations. Energy landscape ideas also have allowed successful protein structure prediction algorithms to be developed. The selection constraint of having funneled folding landscapes has left its imprint on the sequences of existing protein structural families. Quantitative analysis of co-evolution patterns allows us to infer the statistical characteristics of the folding landscape. These turn out to be consistent with what has been obtained from laboratory physicochemical folding experiments signaling a beautiful confluence of genomics and chemical physics.

Falling down: landscape and kinetics of one-dimensional protein folding

In this issue of Structure, Tsytlonok and colleagues describe the folding landscape of the giant HEAT-repeat protein PR65/A (a molecular adaptor of protein phosphatase 2A) by using experimental and theoretical methods. Both approaches agree in suggesting the presence of parallel folding pathways with several intermediates.

Perspective: Coarse-grained models for biomolecular systems

By focusing on essential features, while averaging over less important details, coarse-grained (CG) models provide significant computational and conceptual advantages with respect to more detailed models. Consequently, despite dramatic advances in computational methodologies and resources, CG models enjoy surging popularity and are becoming increasingly equal partners to atomically detailed models. This perspective surveys the rapidly developing landscape of CG models for biomolecular systems. In particular, this review seeks to provide a balanced, coherent, and unified presentation of several distinct approaches for developing CG models, including top-down, network-based, native-centric, knowledge-based, and bottom-up modeling strategies. The review summarizes their basic philosophies, theoretical foundations, typical applications, and recent developments. Additionally, the review identifies fundamental inter-relationships among the diverse approaches and discusses outstanding challenges in the field. When carefully applied and assessed, current CG models provide highly efficient means for investigating the biological consequences of basic physicochemical principles. Moreover, rigorous bottom-up approaches hold great promise for further improving the accuracy and scope of CG models for biomolecular systems.

Energy landscape and dynamics of proteins: an exact analysis of a simplified lattice model

We present the results of exact numerical studies of the energy landscape and the dynamics of a 12-monomer chain with contact interactions encoding the ground state on a square lattice. In spite of its simplicity, the model is shown to exhibit behavior at odds with the standard picture of proteins.

Scaling of folding properties in go models of proteins

Insights about scaling of folding properties of proteins are obtained bystudying folding in heteropolymers described by Go-like Hamiltonians. Bothlattice and continuum space models are considered. In the latter case, themonomer-monomer interactions correspond to the Lennard-Jones potential.Several statistical ensembles of the two- and three-dimensional targetnative conformations are considered. Among them are maximally compactconformations which are confined to a lattice and those which are obtainedeither through quenching or annealing of homopolymers to their compactlocal energy minima. Characteristic folding times are found to grow aspower laws with the system size. The corresponding exponents are notuniversal. The size related deterioration of foldability is found to beconsistent with the scaling behavior of the characteristic temperatures:asymptotically, the folding temperature becomes much lower than thetemperature at which glassy kinetics become important. The helicalconformations are found to have the lowest overall scaling exponent andthe best foldability among the classes of conformations studied. Thescaling properties of the Go-like models of the protein conformationsstored in the Protein Data Bank suggest that proteins are not optimizedkinetically.

Complex intramolecular mechanics of G-actin--an elastic network study

Systematic numerical investigations of conformational motions in single actin molecules were performed by employing a simple elastic-network (EN) model of this protein. Similar to previous investigations for myosin, we found that G-actin essentially behaves as a strain sensor, responding by well-defined domain motions to mechanical perturbations. Several sensitive residues within the nucleotide-binding pocket (NBP) could be identified, such that the perturbation of any of them can induce characteristic flattening of actin molecules and closing of the cleft between their two mobile domains. Extending the EN model by introduction of a set of breakable links which become effective only when two domains approach one another, it was observed that G-actin can possess a metastable state corresponding to a closed conformation and that a transition to this state can be induced by appropriate perturbations in the NBP region. The ligands were roughly modeled as a single particle (ADP) or a dimer (ATP), which were placed inside the NBP and connected by elastic links to the neighbors. Our approximate analysis suggests that, when ATP is present, it stabilizes the closed conformation of actin. This may play an important role in the explanation why, in the presence of ATP, the polymerization process is highly accelerated.

Dynamic prestress in a globular protein

A protein at equilibrium is commonly thought of as a fully relaxed structure, with the intra-molecular interactions showing fluctuations around their energy minimum. In contrast, here we find direct evidence for a protein as a molecular tensegrity structure, comprising a balance of tensed and compressed interactions, a concept that has been put forward for macroscopic structures. We quantified the distribution of inter-residue prestress in ubiquitin and immunoglobulin from all-atom molecular dynamics simulations. The network of highly fluctuating yet significant inter-residue forces in proteins is a consequence of the intrinsic frustration of a protein when sampling its rugged energy landscape. In beta sheets, this balance of forces is found to compress the intra-strand hydrogen bonds. We estimate that the observed magnitude of this pre-compression is enough to induce significant changes in the hydrogen bond lifetimes; thus, prestress, which can be as high as a few 100 pN, can be considered a key factor in determining the unfolding kinetics and pathway of proteins under force. Strong pre-tension in certain salt bridges on the other hand is connected to the thermodynamic stability of ubiquitin. Effective force profiles between some side-chains reveal the signature of multiple, distinct conformational states, and such static disorder could be one factor explaining the growing body of experiments revealing non-exponential unfolding kinetics of proteins. The design of prestress distributions in engineering proteins promises to be a new tool for tailoring the mechanical properties of made-to-order nanomaterials.

A tensegrity structure is one composed of members that are permanently under either tension or compression, and the balance of these tensile and compressive forces provides the structure with its mechanical stability. Macroscale tensegrity structures, which include Buckminster Fuller's geodesic domes, achieve exceptional structural integrity with a minimal use of resources. The question we address in this work is whether nature makes use of molecular-scale tensegrity in the design of proteins. Using Molecular Dynamics simulations of the protein ubiquitin, we measure the network of pairwise forces connecting the amino acid residues and show that this network does indeed have the character of a tensegrity structure. Furthermore, we find that the arrangement of tensile and compressive forces is such that hydrogen bonds in the protein's beta sheet, which are crucial for bearing mechanical loads, are compressed. This pre-compression is enough to significantly lengthen the lifetime of a bond under a given force, and thus should be an important factor in determining the protein's mechanical strength. The rational design of molecular prestress networks promises to be a new avenue for the engineering of proteins with made-to-order mechanical properties, for applications in medicine, materials and nanotechnology.

The folding transition state of protein L is extensive with nonnative interactions (and not small and polarized)

Progress in understanding protein folding relies heavily upon an interplay between experiment and theory. In particular, readily interpretable experimental data that can be meaningfully compared to simulations are required. According to standard mutational ϕ analysis, the transition state for Protein L contains only a single hairpin. However, we demonstrate here using ψ analysis with engineered metal ion binding sites that the transition state is extensive, containing the entire four-stranded β sheet. Underreporting of the structural content of the transition state by ϕ analysis also occurs for acyl phosphatase [Pandit, A. D., Jha, A., Freed, K. F. & Sosnick, T. R., (2006). Small proteins fold through transition states with native-like topologies. J. Mol. Biol.361, 755-770], ubiquitin [Sosnick, T. R., Dothager, R. S. & Krantz, B. A., (2004). Differences in the folding transition state of ubiquitin indicated by ϕ and ψ analyses. Proc. Natl Acad. Sci. USA 101, 17377-17382] and BdpA [Baxa, M., Freed, K. F. & Sosnick, T. R., (2008). Quantifying the structural requirements of the folding transition state of protein A and other systems. J. Mol. Biol.381, 1362-1381]. The carboxy-terminal hairpin in the transition state of Protein L is found to be nonnative, a significant result that agrees with our Protein Data Bank-based backbone sampling and all-atom simulations. The nonnative character partially explains the failure of accepted experimental and native-centric computational approaches to adequately describe the transition state. Hence, caution is required even when an apparent agreement exists between experiment and theory, thus highlighting the importance of having alternative methods for characterizing transition states.

αC helix as a switch in the conformational transition of Src/CDK-like kinase domains

One mechanism of regulating the catalytic activity of protein kinases is through conformational transitions. Despite great diversity in the structural changes involved in the transitions, a certain set of changes within the kinase domain (KD) has been observed for many kinases including Src and CDK2. We investigated this conformational transition computationally to identify the topological features that are energetically critical to the transition. Results from both molecular dynamics sampling and transition path optimization highlight the displacement of the αC helix as the major energy barrier, mediating the switch of the KD between the active and down-regulated states. The critical role of the αC helix is noteworthy by providing a rationale for a number of activation and deactivation mechanisms known to occur in cells. We find that kinases with the αC helix displacement exist throughout the kinome, suggesting that this feature may have emerged early in evolution.

Exploring the energy landscapes of protein folding simulations with Bayesian computation

Nested sampling is a Bayesian sampling technique developed to explore probability distributions localized in an exponentially small area of the parameter space. The algorithm provides both posterior samples and an estimate of the evidence (marginal likelihood) of the model. The nested sampling algorithm also provides an efficient way to calculate free energies and the expectation value of thermodynamic observables at any temperature, through a simple post processing of the output. Previous applications of the algorithm have yielded large efficiency gains over other sampling techniques, including parallel tempering. In this article, we describe a parallel implementation of the nested sampling algorithm and its application to the problem of protein folding in a Gō-like force field of empirical potentials that were designed to stabilize secondary structure elements in room-temperature simulations. We demonstrate the method by conducting folding simulations on a number of small proteins that are commonly used for testing protein-folding procedures. A topological analysis of the posterior samples is performed to produce energy landscape charts, which give a high-level description of the potential energy surface for the protein folding simulations. These charts provide qualitative insights into both the folding process and the nature of the model and force field used.

Molecular simulations of mutually exclusive folding in a two-domain protein switch

A major challenge with testing designs of protein conformational switches is the need for experimental probes that can independently monitor their individual protein domains. One way to circumvent this issue is to use a molecular simulation approach in which each domain can be directly observed. Here we report what we believe to be the first molecular simulations of mutually exclusive folding in an engineered two-domain protein switch, providing a direct view of how folding of one protein drives unfolding of the other in a barnase-ubiquitin fusion protein. These simulations successfully capture the experimental effects of interdomain linker length and ligand binding on the extent of unfolding in the less stable domain. In addition, the effect of linker length on the potential for oligomerization, which eliminates switch activity, is in qualitative agreement with analytical ultracentrifugation experiments. We also perform what we believe to be the first study of protein unfolding via progressive localized compression. Finally, we are able to explore the kinetics of mutually exclusive folding by determining the effect of linker length on rates of unfolding and refolding of each protein domain. Our results demonstrate that molecular simulations can provide seemingly novel biological insights on the behavior of individual protein domains, thereby aiding in the rational design of bifunctional switches.

Geometrical Analysis of Cytochrome c Unfolding

We have developed a geometrical model to study the unfolding of iso-1 cytochrome c. The model draws on the crystallographic data reported for this protein. These data were used to calculate the distance between specific residues in the folded state, and in a sequence of extended states defined by n= 3, 5, 7, 9, 11, 13, and 15 residue units. Exact calculations carried out for each of the 103 residues in the polypeptide chain demonstrate that different regions of the chain have different unfolding histories. Regions where there is a persistence of compact structures can be identified, and this geometrical characterization is fully consistent with analyses of time-resolved fluorescence energy-transfer (TrFET) data using dansyl-derivatized cysteine side-chain probes at positions 39, 50, 66, 85, and 99. Our calculations were carried out assuming that different regions of the polypeptide chain unfold synchronously. To test this assumption, we performed lattice Monte Carlo simulations to study systematically the possible importance of asynchronicity. Our calculations show that small departures from synchronous dynamics can arise if displacements of residues in the main body of the chain are much more sluggish than near-terminal residues.

Composition-based effective chain length for prediction of protein folding rates

Folding rate prediction is a useful way to find the key factors affecting folding kinetics of proteins. Structural information is more or less required in the present prediction methods, which limits the application of these methods to various proteins. In this work, an "effective length" is defined solely based on the composition of a protein, namely, the number of specific types of amino acids in a protein. A physical theory based on a minimalist model is employed to describe the relation between the folding rates and the effective length of proteins. Based on the resultant relationship between folding rates and effective length, the optimal sets of amino acids are found through the enumeration over all possible combinations of amino acids. This optimal set achieves a high correlation (with the coefficient of 0.84) between the folding rates and the optimal effective length. The features of these amino acids are consistent with our model and landscape theory. Further comparisons between our effective length and other factors are carried out. The effective length is physically consistent with structure-based prediction methods and has the best predictability for folding rates. These results all suggest that both entropy and energetics contribute importantly to folding kinetics. The ability to accurately and efficiently predict folding rates from composition enables the analysis of the kinetics for various kinds of proteins. The underlying physics in our method may be helpful to stimulate further understanding on the effects of various amino acids in folding dynamics.

Normal modes and phase transition of the protein chain based on the Hamiltonian formalism

We use the torsional angles of the protein chain as generalized coordinates in the canonical formalism, derive canonical equations of motion, and investigate the coordinate dependence of the kinetic energy expressed in terms of the canonical momenta. We use the formalism to compute the normal-frequency distributions of the α helix and the β sheet, under the assumption that they are stabilized purely through hydrogen bonding. In addition, we obtain the free-energy relations of the α helix, the β sheet, and the random coil of a 15-residue polyalanine. Interestingly, our results predict a phase transition from an α helix to a β sheet at a critical temperature.

Multiscale methods for protein folding simulations

Inherently hierarchic nature of proteins makes multiscale computational methods especially useful in the studies of folding and other functional dynamics. With the multiscale strategies, one can achieve improved accuracy and efficiency by coupling the atomistic and the coarse grained simulations. Depending on the problems studied, very different implementation protocols can be used to realize the multiscale idea. Here, we give detailed introductions to the currently used multiscale protocols, together with some recent applications to the protein folding simulations in our group. The advantages and weakness, as well as the application scopes of these multiscale protocols are discussed. The directions for the future developments are also proposed.

Reconstruction and stability of secondary structure elements in the context of protein structure prediction

Efficient and accurate reconstruction of secondary structure elements in the context of protein structure prediction is the major focus of this work. We present a novel approach capable of reconstructing alpha-helices and beta-sheets in atomic detail. The method is based on Metropolis Monte Carlo simulations in a force field of empirical potentials that are designed to stabilize secondary structure elements in room-temperature simulations. Particular attention is paid to lateral side-chain interactions in beta-sheets and between the turns of alpha-helices, as well as backbone hydrogen bonding. The force constants are optimized using contrastive divergence, a novel machine learning technique, from a data set of known structures. Using this approach, we demonstrate the applicability of the framework to the problem of reconstructing the overall protein fold for a number of commonly studied small proteins, based on only predicted secondary structure and contact map. For protein G and chymotrypsin inhibitor 2, we are able to reconstruct the secondary structure elements in atomic detail and the overall protein folds with a root mean-square deviation of <10 A. For cold-shock protein and the SH3 domain, we accurately reproduce the secondary structure elements and the topology of the 5-stranded beta-sheets, but not the barrel structure. The importance of high-quality secondary structure and contact map prediction is discussed.

Thermostability of the N-terminal RNA-binding domain of the SARS-CoV nucleocapsid protein: experiments and numerical simulations

Differential scanning calorimetry, circular dichroism spectroscopy, nuclear magnetic resonance spectroscopy, and numerical simulations were used to study the thermostability of the N-terminal RNA-binding domain (RBD) of the SARS-CoV nucleocapsid protein. The transition temperature of the RBD in a mixing buffer, composed of glycine, sodium acetate, and sodium phosphate with 100 mM sodium chloride, at pH 6.8, determined by differential scanning calorimetry and circular dichroism, is 48.74 degrees C. Experimental results showed that the thermal-induced unfolding-folding transition of the RBD follows a two-state model with a reversibility >90%. Using a simple Gō-like model and Langevin dynamics we have shown that, in agreement with our experiments, the folding of the RBD is two-state. Theoretical estimates of thermodynamic quantities are in reasonable agreement with the experiments. Folding and thermal unfolding pathways of the RBD also were experimentally and numerically studied in detail. It was shown that the strand beta(1) from the N-terminal folds last and unfolds first, while the remaining beta-strands fold/unfold cooperatively.

Periodic forces trigger a complex mechanical response in ubiquitin

Mechanical forces govern physiological processes in all living organisms. Many cellular forces, for example, those generated in cyclic conformational changes of biological machines, have repetitive components. In apparent contrast, little is known about how dynamic protein structures respond to periodic mechanical information. Ubiquitin is a small protein found in all eukaryotes. We developed molecular dynamics simulations to unfold single and multimeric ubiquitins with periodic forces. By using a coarse-grained representation, we were able to model forces with periods about 2 orders of magnitude longer than the protein's relaxation time. We found that even a moderate periodic force weakened the protein and shifted its unfolding pathways in a frequency- and amplitude-dependent manner. A complex dynamic response with secondary structure refolding and an increasing importance of local interactions was revealed. Importantly, repetitive forces with broadly distributed frequencies elicited very similar molecular responses compared to fixed-frequency forces. When testing the influence of pulling geometry on ubiquitin's mechanical stability, it was found that the linkage involved in the mechanical degradation of cellular proteins renders the protein remarkably insensitive to periodic forces. We also devised a complementary kinetic energy landscape model that traces these observations and explains periodic-force, single-molecule measurements. In turn, this analytical model is capable of predicting dynamic protein responses. These results provide new insights into ubiquitin mechanics and a potential mechanical role during protein degradation, as well as first frameworks for dynamic protein stability and the modeling of repetitive mechanical processes.

Confinement effects on the kinetics and thermodynamics of protein dimerization

In the cell, protein complexes form by relying on specific interactions between their monomers. Excluded volume effects due to molecular crowding would lead to correlations between molecules even without specific interactions. What is the interplay of these effects in the crowded cellular environment? We study dimerization of a model homodimer when the mondimers are free and when they are tethered to each other. We consider a structured environment: Two monomers first diffuse into a cavity of size L and then fold and bind within the cavity. The folding and binding are simulated by using molecular dynamics based on a simplified topology based model. The confinement in the cell is described by an effective molecular concentration C approximately L(-3). A two-state coupled folding and binding behavior is found. We show the maximal rate of dimerization occurred at an effective molecular concentration C(op) approximately = 1 mM, which is a relevant cellular concentration. In contrast, for tethered chains the rate keeps at a plateau when C < C(op) but then decreases sharply when C > C(op). For both the free and tethered cases, the simulated variation of the rate of dimerization and thermodynamic stability with effective molecular concentration agrees well with experimental observations. In addition, a theoretical argument for the effects of confinement on dimerization is also made.

Insights from coarse-grained Gō models for protein folding and dynamics

Exploring the landscape of large scale conformational changes such as protein folding at atomistic detail poses a considerable computational challenge. Coarse-grained representations of the peptide chain have therefore been developed and over the last decade have proved extremely valuable. These include topology-based Gō models, which constitute a smooth and funnel-like approximation to the folding landscape. We review the many variations of the Gō model that have been employed to yield insight into folding mechanisms. Their success has been interpreted as a consequence of the dominant role of the native topology in folding. The role of local contact density in determining protein dynamics is also discussed and is used to explain the ability of Gō-like models to capture sequence effects in folding and elucidate conformational transitions.

Folding energy landscape and network dynamics of small globular proteins

The folding energy landscape of proteins has been suggested to be funnel-like with some degree of ruggedness on the slope. How complex the landscape, however, is still rather unclear. Many experiments for globular proteins suggested relative simplicity, whereas molecular simulations of shorter peptides implied more complexity. Here, by using complete conformational sampling of 2 globular proteins, protein G and src SH3 domain and 2 related random peptides, we investigated their energy landscapes, topological properties of folding networks, and folding dynamics. The projected energy surfaces of globular proteins were funneled in the vicinity of the native but also have other quite deep, accessible minima, whereas the randomized peptides have many local basins, including some leading to seriously misfolded forms. Dynamics in the denatured part of the network exhibited basin-hopping itinerancy among many conformations, whereas the protein reached relatively well-defined final stages that led to their native states. We also found that the folding network has the hierarchic nature characterized by the scale-free and the small-world properties.