Search

The snake cube puzzle made of a linear array of 27 cubes and its modified and extended versions are used as theoretical models to study the mechanism of folding of proteins into their sequence-specific native three-dimensional structures. Each of the three versions is characterized by the respective set of characteristics attributed to each of its constituent cubes and an array is characterized by its specific sequence of the cube characteristics. The aim of the puzzles is to fold the cube array into a compact 3×3×3 cubic structure. In all three versions, out of all possible sequences, only a limited fraction of sequences are found foldable into the compact cube. Even among foldable sequences, the structures folded into the compact 3×3×3 cube are found often not uniquely determined from the sequence. By comparing the results obtained for the three versions of models, we conclude that the power of the hydrophobic interactions to make the folded structure unique to the sequence is much weaker than the geometrical varieties of constituent cubes as modelled in the original snake cube puzzle. However, when this weak cube attribute is compounded to that of the original snake cube puzzle, the power is enhanced very effectively. This is a strong manifestation of the consistency principle: The sequence-specific native structure of protein is realized as a result of consistency of various types of interactions working in protein.

Computational coarse-grained models play a fundamental role as a research tool in protein folding, and they are important in bridging theory and experiments. Folding mechanisms are generally discussed using the energy landscape framework, which is well mapped within a class of simplified structure-based models. In this chapter, simplified computer models are discussed with special focus on structure-based ones.

Cryo-electron microscopy (EM) requires molecular modeling to refine structural details from data. Ensemble models arrive at low free-energy molecular structures, but are computationally expensive and limited to resolving only small proteins that cannot be resolved by cryo-EM. Here, we introduce CryoFold - a pipeline of molecular dynamics simulations that determines ensembles of protein structures directly from sequence by integrating density data of varying sparsity at 3-5 Å resolution with coarse-grained topological knowledge of the protein folds. We present six examples showing its broad applicability for folding proteins between 72 to 2000 residues, including large membrane and multi-domain systems, and results from two EMDB competitions. Driven by data from a single state, CryoFold discovers ensembles of common low-energy models together with rare low-probability structures that capture the equilibrium distribution of proteins constrained by the density maps. Many of these conformations, unseen by traditional methods, are experimentally validated and functionally relevant. We arrive at a set of best practices for data-guided protein folding that are controlled using a Python GUI.

The computer-aided investigation of protein folding has greatly benefited from coarse-grained models, that is, simplified representations at a resolution level lower than atomistic, providing access to qualitative and quantitative details of the folding process that would be hardly attainable, via all-atom descriptions, for medium to long molecules. Nonetheless, the effectiveness of low-resolution models is itself hampered by the presence, in a small but significant number of proteins, of nontrivial topological self-entanglements. Features such as native state knots or slipknots introduce conformational bottlenecks, affecting the probability to fold into the correct conformation; this limitation is particularly severe in the context of coarse-grained models. In this work, we tackle the relationship between folding probability, protein folding pathway, and protein topology in a set of proteins with a nontrivial degree of topological complexity. To avoid or mitigate the risk of incurring in kinetic traps, we make use of the elastic folder model, a coarse-grained model based on angular potentials optimized toward successful folding via a genetic procedure. This light-weight representation allows us to estimate in silico folding probabilities, which we find to anti-correlate with a measure of topological complexity as well as to correlate remarkably well with experimental measurements of the folding rate. These results strengthen the hypothesis that the topological complexity of the native state decreases the folding probability and that the force-field optimization mimics the evolutionary process these proteins have undergone to avoid kinetic traps.

Conformational sampling of biomolecules using molecular dynamics simulations often produces large amount of high dimensional data that makes it difficult to interpret using conventional analysis techniques. Dimensionality reduction methods are thus required to extract useful and relevant information. Here we devise a machine learning method, Gaussian mixture variational autoencoder (GMVAE) that can simultaneously perform dimensionality reduction and clustering of biomolecular conformations in an unsupervised way. We show that GMVAE can learn a reduced representation of the free energy landscape of protein folding with highly separated clusters that correspond to the metastable states during folding. Since GMVAE uses a mixture of Gaussians as the prior, it can directly acknowledge the multi-basin nature of protein folding free-energy landscape. To make the model end-to-end differentialble, we use a Gumbel-softmax distribution. We test the model on three long-timescale protein folding trajectories and show that GMVAE embedding resembles the folding funnel with folded states down the funnel and unfolded states outer in the funnel path. Additionally, we show that the latent space of GMVAE can be used for kinetic analysis and Markov state models built on this embedding produce folding and unfolding timescales that are in close agreement with other rigorous dynamical embeddings such as time independent component analysis (TICA).

Significance Protein-folding kinetics is often described as Markovian (i.e., memoryless) diffusion in a one-dimensional free energy landscape, governed by an instantaneous friction coefficient that is fitted to reproduce experimental or simulated folding times. For the α-helix forming polypeptide alanine9 and a specific reaction coordinate that consists of the summed native hydrogen-bond lengths, we demonstrate that the friction extracted from molecular dynamics simulations exhibits significant memory with a decay time that is in the nanosecond range and thus, of the same order as the folding and unfolding times. Our non-Markovian modeling not only reproduces the molecular dynamics simulations accurately but also demonstrates that memory friction effects lead to anomalous and drastically accelerated protein kinetics., We extract the folding free energy landscape and the time-dependent friction function, the two ingredients of the generalized Langevin equation (GLE), from explicit-water molecular dynamics (MD) simulations of the α-helix forming polypeptide alanine9 for a one-dimensional reaction coordinate based on the sum of the native H-bond distances. Folding and unfolding times from numerical integration of the GLE agree accurately with MD results, which demonstrate the robustness of our GLE-based non-Markovian model. In contrast, Markovian models do not accurately describe the peptide kinetics and in particular, cannot reproduce the folding and unfolding kinetics simultaneously, even if a spatially dependent friction profile is used. Analysis of the GLE demonstrates that memory effects in the friction significantly speed up peptide folding and unfolding kinetics, as predicted by the Grote–Hynes theory, and are the cause of anomalous diffusion in configuration space. Our methods are applicable to any reaction coordinate and in principle, also to experimental trajectories from single-molecule experiments. Our results demonstrate that a consistent description of protein-folding dynamics must account for memory friction effects.

International audience; The predictive capabilities of deep neural networks (DNNs) continue to evolve to increasingly impressive levels. However, it is still unclear how training procedures for DNNs succeed in finding parameters that produce good results for such high-dimensional and nonconvex loss functions. In particular, we wish to understand why simple optimization schemes, such as stochastic gradient descent, do not end up trapped in local minima with high loss values that would not yield useful predictions. We explain the optimizability of DNNs by characterizing the local minima and transition states of the loss-function landscape (LFL) along with their connectivity. We show that the LFL of a DNN in the shallow network or data-abundant limit is funneled, and thus easy to optimize. Crucially, in the opposite low-data/deep limit, although the number of minima increases, the landscape is characterized by many minima with similar loss values separated by low barriers. This organization is different from the hierarchical landscapes of structural glass formers and explains why minimization procedures commonly employed by the machine-learning community can navigate the LFL successfully and reach low-lying solutions.

Geometry and topology are the main factors that determine the functional properties of proteins. In this work, we show how to use the Gauss linking integral (GLN) in the form of a matrix diagram - for a pair of a loop and a tail - to study both the geometry and topology of proteins with closed loops e.g. lassos. We show that the GLN method is a significantly faster technique to detect entanglement in lasso proteins in comparison with other methods. Based on the GLN technique, we conduct comprehensive analysis of all proteins deposited in the PDB and compare it to the statistical properties of the polymers. We found that there are significantly more lassos with negative crossings than those with positive ones in proteins, the average value of maxGLN (maximal GLN between loop and pieces of tail) depends logarithmically on the length of a tail similarly as in the polymers. Next, we show the how high and low GLN values correlate with the internal exibility of proteins, and how the GLN in the form of a matrix diagram can be used to study folding and unfolding routes. Finally, we discuss how the GLN method can be applied to study entanglement between two structures none of which are closed loops. Since this approach is much faster than other linking invariants, the next step will be evaluation of lassos in much longer molecules such as RNA or loops in a single chromosome., 25 pages, 8 figures, submitted to the journal and waiting for reviews, having Supplementary Material

Inspired by the NP-hardness of string folding problems modeling the natural process of protein folding, we discuss the idea of solving instances of NP-hard problems (e.g., string folding problems) of moderate size by letting artificially assembled proteins to fold. The accuracy with which one can combinatorially model the protein folding process, e.g., by string folding, as well as the precision with which one could experimentally estimate the energy of folded artificial proteins are crucial issues.

Over the past two decades, one of the standard models of protein folding has been the "two-state" model, in which a protein only resides in the folded or fully unfolded states with a single pathway between them. Recent advances in spatial and temporal resolution of biophysical measurements have revealed "beyond-two-state" complexity in protein folding, even for small, single-domain proteins. In this work, we used high-resolution optical tweezers to investigate the folding/unfolding kinetics of the B1 domain of immunoglobulin-binding protein G (GB1), a well-studied model system. Experiments were performed for GB1 both in and out of equilibrium using force spectroscopy. When the force was gradually ramped, simple single-peak folding force distributions were observed, while multiple rupture peaks were seen in the unfolding force distributions, consistent with multiple force-dependent parallel unfolding pathways. Force-dependent folding and unfolding rate constants were directly determined by both force-jump and fixed-trap measurements. Monte Carlo modeling using these rate constants was in good agreement with the force ramp data. The unfolding rate constants exhibited two different behaviors at low vs high force. At high force, the unfolding rate constant increased with increasing force, as previously reported by high force, high pulling speed force ramp measurements. However, at low force, the situation reversed and the unfolding rate constant decreased with increasing force. Taken together, these data indicate that this small protein has multiple distinct pathways to the native state on the free energy landscape.

Made available in DSpace on 2020-12-10T19:41:27Z (GMT). No. of bitstreams: 0 Previous issue date: 2019-11-01 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) NSF CAREER award Center for Theoretical Biological Physics - NSF Protein folding occurs in a high dimensional phase space, and the representation of the associated energy landscape is nontrivial. A widely applied approach to studying folding landscapes is to describe the dynamics along a small number of reaction coordinates. However, other strategies involve more elaborate analysis of the complex phase space. There have been many attempts to obtain a more detailed representation of all available conformations for a given system. In this work, we address this problem using a metric based on internal distances between amino acids to describe the differences between any two conformations. Using an effective projection method, we are able to go beyond the typical one-dimensional representation and provide intuitive two dimensional visualizations of the landscape. We refer to this method as the energy landscape visualization method (ELViM). We have applied this methodology using a C-alpha structure-based model to study the folding of two well-known proteins: SH3 domain and protein-A. Our visualization method yields a detailed description of the folding process, making possible the identification of transition state regions, and establishing the paths that lead to the native state. For SH3, we have analyzed structural differences in the distribution of folding routes. The competition between the native and mirror structures in protein A is also discussed. Finally, the method is applied to study conformational changes in the protein elongation factor thermally unstable. Distinct features of ELViM are that it does not require or assume a reaction coordinate, and it does not require analysis of kinetic aspects of the system. Univ Estadual Paulista, Inst Biociencias Letras & Ciencias Exatas, Dept Fis, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil Northeastern Univ, Dept Phys, Boston, MA 02115 USA Rice Univ, Ctr Theoret Biol Phys, Houston, TX 77005 USA Univ Estadual Paulista, Inst Biociencias Letras & Ciencias Exatas, Dept Fis, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil FAPESP: 2018/18668-1 FAPESP: 2016/19766-1 FAPESP: 2014/50739-5 NSF CAREER award: MCB-1350312 Center for Theoretical Biological Physics - NSF: PHY-1427654

Coarse-grained models are an attractive tool for studying the long time scale dynamics of large macromolecules at a level that cannot be studied directly by experiment and is still out of reach for atomistic simulation. However, coarse models involve approximations that may affect their predictive power. We propose a modeling framework that allows us to design simplified models to accurately reproduce experimental observables. We demonstrate the approach on the folding mechanism of a WW domain. We show that when the correct coarsening resolution is used not only do the optimized models match the Reference model simulated experimental data accurately but additional observables not directly targeted during the optimization procedure are also reproduced. Additionally, the analysis of the results shows that localized frustration plays an important role in the folding mechanism of this protein and suggests that nontrivial aspects of the protein dynamics are evolutionary conserved.

Protein folding is a central challenge in computational biology, with important applications in molecular biology, drug discovery and catalyst design. As a hard combinatorial optimisation problem, it has been studied as a potential target problem for quantum annealing. Although several experimental implementations have been discussed in the literature, the computational scaling of these approaches has not been elucidated. In this article, we present a numerical study of quantum annealing applied to a large number of small peptide folding problems, aiming to infer useful insights for near-term applications. We present two conclusions: that even naive quantum annealing, when applied to protein lattice folding, has the potential to outperform classical approaches, and that careful engineering of the Hamiltonians and schedules involved can deliver notable relative improvements for this problem. Overall, our results suggest that quantum algorithms may well offer improvements for problems in the protein folding and structure prediction realm., Comment: 45 pages, 18 figures

It has been a long-standing and intriguing issue to develop robust methods to identify metastable states and interstate transitions from simulations or experimental data to understand the functional conformational changes of proteins. It is usually hard to define the complicated boundaries of the states in the conformational space using most of the existing methods, and they often lead to parameter-sensitive results. Here, we present a new approach, visualized Trajectory Map (vTM), to identify the metastable states and the rare interstate transitions, by considering both the conformational similarity and the temporal successiveness of conformations. The vTM is able to give a nonambiguous description of slow dynamics. The case study of a β-hairpin peptide shows that the vTM can reveal the states and transitions from all-atom MD trajectory data even when a single observable (i.e, one-dimensional reaction coordinate) is used. We also use the vTM to refine the folding/unfolding mechanism of HP35 in explicit water by analyzing a 125 μs all-atom MD trajectory and obtain folding/unfolding rates of about 1/μs, which are in good agreement with the experimental values.

Single-molecule force spectroscopy has proven extremely beneficial in elucidating folding pathways for membrane proteins. Here, we simulate these measurements, conducting hundreds of unfolding trajectories using our fast Upside algorithm for slow enough speeds to reproduce key experimental features that may be missed using all-atom methods. The speed also enables us to determine the logarithmic dependence of pulling velocities on the rupture levels to better compare to experimental values. For simulations of atomic force microscope measurements in which force is applied vertically to the C-terminus of bacteriorhodopsin, we reproduce the major experimental features including even the back-and-forth unfolding of single helical turns. When pulling laterally on GlpG to mimic the experiment, we observe quite different behavior depending on the stiffness of the spring. With a soft spring, as used in the experimental studies with magnetic tweezers, the force remains nearly constant after the initial unfolding event, and a few pathways and a high degree of cooperativity are observed in both the experiment and simulation. With a stiff spring, however, the force drops to near zero after each major unfolding event, and numerous intermediates are observed along a wide variety of pathways. Hence, the mode of force application significantly alters the perception of the folding landscape, including the number of intermediates and the degree of folding cooperativity, important issues that should be considered when designing experiments and interpreting unfolding data.

The folding transition of a Wako-Saito-Mu˜noz-Eaton (WSME) α-helix model is studied by using several methods. The transition temperature of a finite-size system is defined by using the extrapolation of partition function zeros, a microcanonical analysis, and the average fraction of unbroken native contacts, and the result is applied to a WSME α-helix model. The analysis results in a phase diagram for the helix, whose boundaries depend on the method used for defining the transition temperature. However, the qualitative behaviors are quite similar for all of these methods, and any definition can lead to a reasonable phase diagram for a finite-size system.

A central goal of protein-folding theory is to predict the stochastic dynamics of transition paths — the rare trajectories that transit between the folded and unfolded ensembles — using only thermodynamic information, such as a low-dimensional equilibrium free-energy landscape. However, commonly used one-dimensional landscapes typically fall short of this aim, because an empirical coordinate-dependent diffusion coefficient has to be fit to transition-path trajectory data in order to reproduce the transition-path dynamics. We show that an alternative, first-principles free-energy landscape predicts transition-path statistics that agree well with simulations and single-molecule experiments without requiring dynamical data as an input. This ‘topological configuration’ model assumes that distinct, native-like substructures assemble on a timescale that is slower than native-contact formation but faster than the folding of the entire protein. Using only equilibrium simulation data to determine the free energies of these coarse-grained intermediate states, we predict a broad distribution of transition-path transit times that agrees well with the transition-path durations observed in simulations. We further show that both the distribution of finite-time displacements on a one-dimensional order parameter and the ensemble of transition-path trajectories generated by the model are consistent with the simulated transition paths. These results indicate that a landscape based on transient folding intermediates, which are often hidden by one-dimensional projections, can form the basis of a predictive model of protein-folding transition-path dynamics.

Background Accelerated molecular dynamics (aMD) has been proven to be a powerful biasing method for enhanced sampling of biomolecular conformations on general-purpose computational platforms. Biologically important long timescale events that are beyond the reach of standard molecular dynamics can be accessed without losing the detailed atomistic description of the system in aMD. Over other biasing methods, aMD offers the advantages of tuning the level of acceleration to access the desired timescale without any advance knowledge of the reaction coordinate. Scope of review Recent advances in the implementation of aMD and its applications to small peptides and biological macromolecules are reviewed here along with a brief account of all the aMD variants introduced in the last decade. Major conclusions In comparison to the original implementation of aMD, the recent variant in which all the rotatable dihedral angles are accelerated (RaMD) exhibits faster convergence rates and significant improvement in statistical accuracy of retrieved thermodynamic properties. RaMD in conjunction with accelerating diffusive degrees of freedom, i.e. dual boosting, has been rigorously tested for the most difficult conformational sampling problem, protein folding. It has been shown that RaMD with dual boosting is capable of efficiently sampling multiple folding and unfolding events in small fast folding proteins. General significance RaMD with the dual boost approach opens exciting possibilities for sampling multiple timescales in biomolecules. While equilibrium properties can be recovered satisfactorily from aMD-based methods, directly obtaining dynamics and kinetic rates for larger systems presents a future challenge. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

Summary Models of symmetric protein evolution typically invoke gene duplication and fusion events, in which repetition of a structural motif generates foldable, stable symmetric protein architecture. Success of such evolutionary processes suggests that the duplicated structural motif must be capable of nucleating protein folding. If correct, symmetric expansion of a folding nucleus sequence derived from an extant symmetric fold may be an elegant and computationally tractable solution to de novo protein design. We report the efficient de novo design of a β-trefoil protein by symmetric expansion of a β-trefoil folding nucleus, previously identified by ɸ-value analysis. The resulting protein, having exact sequence symmetry, exhibits superior folding properties compared to its naturally evolved progenitor—with the potential for redundant folding nuclei. In principle, folding nucleus symmetric expansion can be applied to any given symmetric protein fold (that is, nearly one-third of the known proteome) provided information of the folding nucleus is available.

Nonequilibrium drivings applied in molecular dynamics (MD) simulations can efficiently extend the visiting range of protein conformations, but might compel systems to go far away from equilibrium and thus mainly explore irrelevant conformations. Here we propose a general method, called adaptive nonequilibrium simulation (ANES), to automatically adjust the external driving on the fly, based on the feedback of the short-time average response of system. Thus, the ANES approximately keeps the local equilibrium but efficiently accelerates the global motion. We illustrate the capability of the ANES in highly efficiently exploring metastable conformations in the deca-alanine peptide and find that the $0.2\ensuremath{-}\ensuremath{\mu}\mathrm{s}$ ANES approximately captures the important states and folding and unfolding pathways in the HP35 solution by comparing with the result of the recent $398\ensuremath{-}\ensuremath{\mu}\mathrm{s}$ equilibrium MD simulation on Anton [S. Piana et al., Proc. Natl. Acad. Sci. USA 109, 17845 (2012)].

We have modeled the protein folding process with cellular automata using the Face-Centered Cubic lattice model. An artificial neural network implements a cellular automaton-like scheme that defines the moves of each of the amino acids of the protein chain and through several time iterations until a folded protein is obtained. Differential Evolution was used to evolve these neural cellular automata, which take the information for defining the folding process from the energy space considered in the lattice model. Different proteins were used for testing the process, comparing the results of the folded structures against other methods of direct prediction of the final folded conformation.

The Bias Functional (BF) approach is a variational method which enables one to efficiently generate ensembles of reactive trajectories for complex biomolecular transitions, using ordinary computer clusters. For example, this scheme was applied to simulate in atomistic detail the folding of proteins consisting of several hundreds of amino acids and with experimental folding time of several minutes. A drawback of the BF approach is that it produces trajectories which do not satisfy microscopic reversibility. Consequently, this method cannot be used to directly compute equilibrium observables, such as free energy landscapes or equilibrium constants. In this work, we develop a statistical analysis which permits us to compute the potential of mean-force (PMF) along an arbitrary collective coordinate, by exploiting the information contained in the reactive trajectories calculated with the BF approach. We assess the accuracy and computational efficiency of this scheme by comparing its results with the PMF obtained for a small protein by means of plain molecular dynamics.

In this article, we present an enhanced sampling method based on a hybrid Hamiltonian which combines experimental distance restraints with a bias dependent from multiple path-dependent variables. This simulation method determines the bias-coordinates on the fly and does not require a priori knowledge about reaction coordinates. The hybrid Hamiltonian accelerates the sampling of proteins, and, combined with experimental distance information, the technique considers the restraints adaptively and in dependency of the system&rsquo, s intrinsic dynamics. We validate the methodology on the dipole relaxation of two water models and the conformational landscape of dialanine. Using experimental NMR-restraint data, we explore the folding landscape of the TrpCage mini-protein and in a second example apply distance restraints from chemical crosslinking/mass spectrometry experiments for the sampling of the conformation space of the Killer Cell Lectin-like Receptor Subfamily B Member 1A (NKR-P1A). The new methodology has the potential to adaptively introduce experimental restraints without affecting the conformational space of the system along an ergodic trajectory. Since only a limited number of input- and no-order parameters are required for the setup of the simulation, the method is broadly applicable and has the potential to be combined with coarse-graining methods.

In this work the temporal and dynamic folding of proteins was modeled with neural cellular automata, on the contrary to the ample research performed on the prediction of the final protein structure. Using the Rosetta environment and its coarse-grained representation, starting from an unfolded or partially folded chain, a connectionist model acts like a cellular automaton to define the moves of the dihedral angles of the protein chain. The process is repeated for all the angles of the amino acids and through several time steps until the protein is folded. The neural cellular automaton uses as input information a partial view of the energy landscape, obtained through the consequences in the energy changes when an angle is moved. The neural model learns to decide the best move in each angle in order to minimize the energy of the final folded conformation. The neural cellular automata are automatically obtained by means of differential evolution. Initial results with short proteins are presented and discussed.

Proteins fold using a two-state or multi-state kinetic mechanisms, but up to now there is not a first-principle model to explain this different behavior. We exploit the network properties of protein structures by introducing novel observables to address the problem of classifying the different types of folding kinetics. These observables display a plain physical meaning, in terms of vibrational modes, possible configurations compatible with the native protein structure and folding cooperativity. The relevance of these observables is supported by a classification performance up to 90%, even with simple classifiers such as discriminant analysis.

The space accessed by a folding macromolecule is vast, and how to best project computer simulations of protein folding trajectories into an interpretable sequence of discrete states is an open research problem. There are numerous alternative ways of associating individual configurations into collective states, and in deciding on the number of such clustered states there is a trade-off between human interpretability (smaller number of states) and accuracy of representation (larger number of states). Here we introduce a trajectory likelihood measure for assessing alternative discrete state models of protein folding. We find that widely used rmsd-based clustering methods require large numbers of initial states and a second agglomeration step based on kinetic connectivity to produce models with high predictive power; this is the approach taken in elegant recent work with Markov State Models of protein folding. In contrast, we find that grouping of states based on secondary structure pairings or contact maps, when refined with K-means clustering, yields higher likelihood models with many fewer states. Using the most predictive contact map representation to study the folding transitions of the WW domain in very long molecular dynamics simulations, we identify new states and transitions. The methods should be generally useful for investigating the structural transitions in protein folding simulations for larger proteins.

The free energy landscape of a protein is a function of many interdependent degrees of freedom. For this reason, conceptual constructs (e.g., funnels) have been useful to visualize these landscapes. One relatively new construct is the idea of a hub-like native state that is the final destination of many non-interconverting folding pathways. This is in contrast to the idea of a single predominant folding pathway connecting the native state to a rapidly interconverting ensemble of unfolded states. The key quantity to distinguish between these two ideas is the connectivity of the unfolded ensemble. We present a metric to determine this connectivity for a given network, which can be calculated either from continuous folding trajectories, or a Markov model. The metric determines how often a region of space is used as an intermediate on transition paths that connect two other regions of space, and we use it here to determine how often two parts of the unfolded ensemble are connected directly, versus how often these transitions are mediated by the native state.

A simple model is presented that describes general features of protein folding, in good agreement with experimental results and detailed all-atom simulations. Starting from microscopic physics, and with no free parameters, this model predicts that protein folding occurs remarkably quickly because native-like states are kinetic hubs. A hub-like network arises naturally out of microscopic physical concerns, specifically the kinetic longevity of native contacts during a search of globular conformations. The model predicts folding times scaling as τ(f) ~ e(ξN) in the number of residues, but because the model shows ξ is small, the folding times are much faster than Levinthal's approximation. Importantly, the folding time scale is found to be small due to the topology and structure of the network. We show explicitly how our model agrees with generic experimental features of the folding process, including the scaling of τ(f) with N, two-state thermodynamics, a sharp peak in C(V), and native-state fluctuations.

The phenomenon of protein folding is a fundamental issue in the field of the computational molecular biology. The protein folding inside the cells is performed in a highly inhomogeneous, tortuous, and correlated environment. Therefore, it is important to include in the theoretical studies the medium where the protein folding is developed. In this work we present the combination of three models to mimic the protein folding inside of an inhomogeneous medium. The models used here are Hydrophobic-Polar (HP) in 2D square arrangement, Evolutionary Algorithms (EA), and the Dual Site Bond Model (DSBM). The DSBM model is used to simulate the environment where the HP beads are folded; in this case the medium is correlated and is fractal-like. The analysis of five benchmark HP sequences shows that the inhomogeneous space provided with a given correlation length and fractal dimension plays an important role for correct folding of these sequences, which does not occur in a homogeneous space.

Among the unsolved problems in computational biology, protein folding is one of the most interesting challenges. To study this folding, tools like neural networks and genetic algorithms have received a lot of attention, mainly due to the NP completeness of the folding process. The background idea that has given rise to the use of these algorithms is obviously that the folding process is predictable. However, this important assumption is disputable as chaotic properties of such a process have been recently highlighted. In this paper, which is an extension of a former work accepted to the 2011 International Joint Conference on Neural Networks (IJCNN11), the topological behavior of a well-known dynamical system used for protein folding prediction is evaluated. It is mathematically established that the folding dynamics in the 2D hydrophobic–hydrophilic (HP) square lattice model, simply called the 2D model in this document, is indeed a chaotic dynamical system as defined by Devaney. Furthermore, the chaotic behavior of this model is qualitatively and quantitatively deepened, by studying other mathematical properties of disorder, namely: the indecomposability, instability, strong transitivity, and constants of expansivity and sensitivity. Some consequences for both biological paradigms and structure prediction using this model are then discussed. In particular, it is shown that some neural networks seems to be unable to predict the evolution of this model with accuracy, due to its complex behavior.

Protein folding is formulated as a moving horizon optimal control problem. We present a novel method that combines coarse-grained optimal control and molecular dynamics to generate dynamical folding trajectories with full atomic details. The method is applied to a benchmark protein, Villin headpiece.

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.

We investigate various topological and energy parameters from the protein native structure and find combinations of some parameters that are well correlated with the rate of folding/unfolding. For folding, the topological quantity that combines the clustering coefficient and the long-range order (or total contact distance/contact order) has a high correlation with the folding rate, expressed as ln k F , obtained from standard experimental conditions. For unfolding, a combination of the impact of edge removal, obtained from the protein structure, and the stability of the native protein structure, as expressed by the free energy change ΔG, gives a good correlation with unfolding rate, ln k U .

We investigated the folding kinetics of a series of two-state proteins by using the model of general random walk in native contact number space, and derive the observed linear relationship between the logarithms of the folding rate constants and the numbers of native contacts from “kinetic viewpoint”. The protein folding speed limit and stability in this model are consistent with experimental observations.

Biomolecular folding, at least in simple systems, can be described as a two state transition in a free energy landscape with two deep wells separated by a high barrier. Transition paths are the short part of the trajectories that cross the barrier. Average transition path times and, recently, their full probability distribution have been measured for several biomolecular systems, e.g. in the folding of nucleic acids or proteins. Motivated by these experiments, we have calculated the full transition path time distribution for a single stochastic particle crossing a parabolic barrier, focusing on the underdamped regime. Our analysis thus includes inertial terms, which were neglected in previous studies. These terms influence the short time scale dynamics of a stochastic system, and can be of experimental relevance in view of the short duration of transition paths. We derive the full transition path time distribution in the underdamped case and discuss the similarities and differences with the high friction (overdamped) limit., Comment: 16 pages, 5 figures

Characterizing the equilibrium ensemble of folding pathways, including their relative probability, is one of the major challenges in protein folding theory today. Although this information is in principle accessible via all-atom molecular dynamics simulations, it is difficult to compute in practice because protein folding is a rare event and the affordable simulation length is typically not sufficient to observe an appreciable number of folding events, unless very simplified protein models are used. Here we present an approach that allows for the reconstruction of the full ensemble of folding pathways from simulations that are much shorter than the folding time. This approach can be applied to all-atom protein simulations in explicit solvent. It does not use a predefined reaction coordinate but is based on partitioning the state space into small conformational states and constructing a Markov model between them. A theory is presented that allows for the extraction of the full ensemble of transition pathways from the unfolded to the folded configurations. The approach is applied to the folding of a PinWW domain in explicit solvent where the folding time is two orders of magnitude larger than the length of individual simulations. The results are in good agreement with kinetic experimental data and give detailed insights about the nature of the folding process which is shown to be surprisingly complex and parallel. The analysis reveals the existence of misfolded trap states outside the network of efficient folding intermediates that significantly reduce the folding speed.

The minimal folding pathway or trajectory for a biopolymer can be defined as the transformation that minimizes the total distance traveled between a folded and an unfolded structure. This involves generalizing the usual Euclidean distance from points to one-dimensional objects such as a polymer. We apply this distance here to find minimal folding pathways for several candidate protein fragments, including the helix, the β-hairpin, and a nonplanar structure where chain noncrossing is important. Comparing the distances traveled with root mean-squared distance and mean root-squared distance, we show that chain noncrossing can have large effects on the kinetic proximity of apparently similar conformations. Structures that are aligned to the β-hairpin by minimizing mean root-squared distance, a quantity that closely approximates the true distance for long chains, show globally different orientation than structures aligned by minimizing root mean-squared distance.

Protein structure prediction without using templates (i.e., ab initio folding) is one of the most challenging problems in structural biology. In particular, conformation sampling poses as a major bottleneck of ab initio folding. This article presents CRFSampler, an extensible protein conformation sampler, built on a probabilistic graphical model Conditional Random Fields (CRFs). Using a discriminative learning method, CRFSampler can automatically learn more than ten thousand parameters quantifying the relationship among primary sequence, secondary structure, and (pseudo) backbone angles. Using only compactness and self-avoiding constraints, CRFSampler can efficiently generate protein-like conformations from primary sequence and predicted secondary structure. CRFSampler is also very flexible in that a variety of model topologies and feature sets can be defined to model the sequence-structure relationship without worrying about parameter estimation. Our experimental results demonstrate that using a simple set of features, CRFSampler can generate decoys with much higher quality than the most recent HMM model.

We present a new computational methodology aimed to calculate the thermodynamics and kinetics of peptide folding. We focus in particular on temperature jump experiments of folding rates and show how a combination of replica exchange molecular dynamics (REMD) followed by multiplexed molecular dynamics starting from structures taken from the REMD runs can be used to extract properties in line with experiments. A model system, alanine20, is studied in this article as a proof of principle and used to describe the methodology.

We report rate constant calculations and a reaction coordinate analysis of the rate-limiting folding and unfolding process of the Trp-cage mini-protein in explicit solvent using transition interface sampling. Previous transition path sampling simulations revealed that in this (un)folding process the protein maintains its compact configuration, while a (de)increase of secondary structure is observed. The calculated folding rate agrees reasonably with experiment, while the unfolding rate is 10 times higher. We discuss possible origins for this mismatch. We recomputed the rates with the forward flux sampling method, and found a discrepancy of four orders of magnitude, probably caused by the method's higher sensitivity to the choice of order parameter with respect to transition interface sampling. Finally, we used the previously computed transition path-sampling ensemble to screen combinations of many order parameters for the best model of the reaction coordinate by employing likelihood maximization. We found that a combination of the root mean-square deviation of the helix and of the entire protein was, of the set of tried order parameters, the one that best describes the reaction coordination.

By suitably extending a recent approach [Bussi, G.; et al. J. Am. Chem. Soc. 2006, 128, 13435] we introduce a powerful methodology that allows the parallel reconstruction of the free energy of a system in a virtually unlimited number of variables. Multiple metadynamics simulations of the same system at the same temperature are performed, biasing each replica with a time-dependent potential constructed in a different set of collective variables. Exchanges between the bias potentials in the different variables are periodically allowed according to a replica exchange scheme. Due to the efficaciously multidimensional nature of the bias the method allows exploring complex free energy landscapes with high efficiency. The usefulness of the method is demonstrated by performing an atomistic simulation in explicit solvent of the folding of a Triptophane cage miniprotein. It is shown that the folding free energy landscape can be fully characterized starting from an extended conformation with use of only 40 ns of simulation on 8 replicas.

Principal component analysis is a powerful method for projecting multidimensional conformational space of peptides or proteins onto lower dimensional subspaces in which the main conformations are present, making it easier to reveal the structures of molecules from e.g. molecular dynamics simulation trajectories. However, the identification of all conformational states is still difficult if the subspaces consist of more than two dimensions. This is mainly due to the fact that the principal components are not independent with each other, and states in the subspaces cannot be visualized. In this work, we propose a simple and fast scheme that allows one to obtain all conformational states in the subspaces. The basic idea is that instead of directly identifying the states in the subspace spanned by principal components, we first transform this subspace into another subspace formed by components that are independent of one other. These independent components are obtained from the principal components by employing the independent component analysis method. Because of independence between components, all states in this new subspace are defined as all possible combinations of the states obtained from each single independent component. This makes the conformational analysis much simpler. We test the performance of the method by analyzing the conformations of the glycine tripeptide and the alanine hexapeptide. The analyses show that our method is simple and quickly reveal all conformational states in the subspaces. The folding pathways between the identified states of the alanine hexapeptide are analyzed and discussed in some detail. Proteins 2007. © 2007 Wiley-Liss, Inc.

Using time-resolved IR spectroscopy, we monitored the kinetics of folding and unfolding processes of a photoswitchable 16-residue alanine-based α-helical peptide on a timescale from few picoseconds to almost 40 μs and over a large temperature range (279–318 K). The folding and unfolding processes were triggered by an ultrafast laser pulse that isomerized the cross linker within a few picoseconds. The main folding and unfolding times (700 ns and 150 ns, respectively, at room temperature) are in line with previous T-jump experiments obtained from similar peptides. However, both processes show complex, strongly temperature-dependent spectral kinetics that deviate clearly from a single-exponential behavior. Whereas in the unfolding experiment the ensemble starts from a well defined folded state, the starting ensemble in the folding experiment is more heterogeneous, which leads to distinctly different kinetics of the experiments, because they are sensitive to different regions of the energy surface. A qualitative agreement with the experimental data-set can be obtained by a model where the unfolded states act as a hub connected to several separated “misfolded” states with a distribution of rates. We conclude that a rather large spread of rates ( k 1 : k n ≈ 9) is needed to explain the experimentally observed stretched exponential response with stretching factor β = 0.8 at 279 K.

Molecular dynamics simulations have been used extensively to model the folding and unfolding of proteins. The rates of folding and unfolding should follow the Arrhenius equation over a limited range of temperatures. This study shows that molecular dynamic simulations of the unfolding of crambin between 500K and 560K do follow the Arrhenius equation. They also show that while there is a large amount of variation between the simulations the average values for the rate show a very high degree of correlation.

The dynamics of proteins in the unfolded state can be quantified in computer simulations by calculating a spectrum of relaxation times which describes the time scales over which the population fluctuations decay to equilibrium. If the unfolded state space is discretized, we can evaluate the relaxation time of each state. We derive a simple relation that shows the mean first passage time to any state is equal to the relaxation time of that state divided by the equilibrium population. This explains why mean first passage times from state to state within the unfolded ensemble can be very long but the energy landscape can still be smooth (minimally frustrated). In fact, when the folding kinetics is two-state, all of the unfolded state relaxation times within the unfolded free energy basin are faster than the folding time. This result supports the well-established funnel energy landscape picture and resolves an apparent contradiction between this model and the recently proposed kinetic hub model of protein folding. We validate these concepts by analyzing a Markov state model of the kinetics in the unfolded state and folding of the miniprotein NTL9 (where NTL9 is the N-terminal domain of the ribosomal protein L9), constructed from a 2.9 ms simulation provided by D. E. Shaw Research.

The free energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. The profile together with the optimal coordinate allows one to directly determine such basic properties of folding dynamics as the configurations of the minima and transition states, the heights of the barriers, the value of the pre-exponential factor and its relation to the transition path times. In the present study, we review the framework, in particular, the approach to determine such an optimal coordinate, and its application to the analysis of simulated protein folding dynamics.

Biased simulations have great potential for the study of slow processes, including protein folding. Atomic motions in molecules are nonlinear, which suggests that simulations with enhanced sampling of collective motions traced by nonlinear dimensionality reduction methods may perform better than linear ones. In this study, we compare an unbiased folding simulation of the Trp-cage miniprotein with metadynamics simulations using both linear (principle component analysis) and nonlinear (Isomap) low dimensional embeddings as collective variables. Folding of the mini-protein was successfully simulated in 200 ns simulation with linear biasing and non-linear motion biasing. The folded state was correctly predicted as the free energy minimum in both simulations. We found that the advantage of linear motion biasing is that it can sample a larger conformational space, whereas the advantage of nonlinear motion biasing lies in slightly better resolution of the resulting free energy surface. In terms of sampling efficiency, both methods are comparable.

It is important to extract reaction coordinates or order parameters from protein simulations in order to investigate the local minimum-energy states and the transitions between them. The most popular method to obtain such data is principal component analysis, which extracts modes of large conformational fluctuations around an average structure. We recently applied relaxation mode analysis for protein systems, which approximately estimates the slow relaxation modes and times from a simulation and enables investigations of the dynamic properties underlying the structural fluctuations of proteins. In this study, we apply this relaxation mode analysis to extract reaction coordinates for a system in which there are large conformational changes such as those commonly observed in protein folding/unfolding. We performed a 750-ns simulation of chignolin protein near its folding transition temperature, and observed many transitions between the most stable, misfolded, intermediate, and unfolded states. We then applied principal component analysis and relaxation mode analysis to the system. In the relaxation mode analysis, we could automatically extract good reaction coordinates. The free-energy surfaces provide a clearer understanding of the transitions not only between local minimum-energy states but also between the folded and unfolded states, even though the simulation involved large conformational changes. Moreover, we propose a new analysis method called Markov state relaxation mode analysis. We applied the new method to states with slow relaxation, which are defined by the free-energy surface obtained in the relaxation mode analysis. Finally, the relaxation times of the states obtained with a simple Markov state model and the proposed Markov state relaxation mode analysis are compared and discussed., Comment: 27 pages

The conformational samplings are indispensible for obtaining reliable canonical ensembles, which provide statistical averages of physical quantities such as free energies. However, the samplings of vast conformational space of biomacromolecules by conventional molecular dynamics (MD) simulations might be insufficient, due to their inadequate accessible time-scales for investigating biological functions. Therefore, the development of methodologies for enhancing the conformational sampling of biomacromolecules still remains as a challenging issue in computational biology. To tackle this problem, we newly propose an efficient conformational search method, which is referred as TaBoo SeArch (TBSA) algorithm. In TBSA, an inverse energy histogram is used to select seeds for the conformational resampling so that states with high frequencies are inhibited, while states with low frequencies are efficiently sampled to explore the unvisited conformational space. As a demonstration, TBSA was applied to the folding of a mini-protein, chignolin, and automatically sampled the native structure (Cα root mean square deviation

Predicting the spacial folding structure of a protein, given its sequence of amino acids, is one of the central problems in computational biology field. This paper studies the AB off-lattice model with two species of monomers, called hydrophobic (A) and hydrophilic (B). Based on this simplified model, the low energy configurations are searched by using the GAPSO. A kind of optimization about the mutation mechanism and the Euclidean interference mechanism are presented, where a novel local adjustment strategy is also used to enhance the searching ability of the global minimum within the AB off-lattice model. Starting from random conformations, the GAPSO method can find the low-energy conformation of the Fibonacci sequences and the real protein sequences. Compared with other optimization methods, the proposed novel method could converge to the lower energy folds. It appears that the proposed method can used for solving protein folding problem, which is based on the thermodynamic hypothesis.