Current protein forcefields like the ones seen in CHARMM or Xplor-NIH have many terms that include bonded and non-bonded terms. Yet the forcefields do not take into account the use of hydrogen bonds which are important for secondary structure creation and stabilization of proteins. SCOPE is an open-source program that generates proteins from rotamer space. It then creates a forcefield that uses only non-bonded and hydrogen bond energy terms to create a profile for a given protein. The profiles can then be used in an artificial neural network to create a linear model that is funneled to the true protein conformation.
We report the reproducible first-principles folding of the 40 amino acid, three-helix headpiece of the HIV accessory protein in a recently developed all-atom free-energy forcefield. Six of twenty simulations using an adapted basin-hopping method converged to better than 3 Åbackbone RMS deviation to the experimental structure. Using over 60,000 low-energy conformations of this protein, we constructed a decoy tree that completely characterizes its folding funnel.
Use of a combination of statistical thermodynamics and the Gershgorin theorem enable us to guess, in the thermodynamic limit, a plausible value for the upper bound free-energy difference between native-like structures of monomeric globular proteins. Support to our result in light of both the observed free-energy change between the native and denatured states and the microstability free-energy values obtained from the observed micro-unfolding tendency of nine globular proteins, will be here discussed.
The analysis of coevolution of residues in homologous proteins is a powerful tool to predict their native conformation. The standard framework in which coevolutionary analysis is usually worked out is that of equilibrium Potts models, assuming that proteins have evolved for enough time to reach thermodynamic equilibrium in sequence space. Here we propose a model to describe correlations in sequences based on an explicit description of the evolutionary kinetics of proteins. We show that this procedure improves the correct prediction of native contacts with respect to equilibrium--based models.
By observing trends in the folding kinetics of experimental 2-state proteins at their transition midpoints, and by observing trends in the barrier heights of numerous simulations of coarse grained, C-alpha model, Go proteins, we show that folding rates correlate with the degree of heterogeneity in the formation of native contacts. Statistically significant correlations are observed between folding rates and measures of heterogeneity inherent in the native topology, as well as between rates and the variance in the distribution of either experimentally measured or simulated phi-values.
Proteins are minimally frustrated polymers. However, for realistic protein models non-native interactions must be taken into account. In this paper we analyze the effect of non-native interactions on the folding rate and on the folding free energy barrier. We present an analytic theory to account for the modification on the free energy landscape upon introduction of non-native contacts, added as a perturbation to the strong native interactions driving folding. Our theory predicts a rate-enhancement regime at fixed temperature, under the introduction of weak, non-native interactions. We have th…
Normal mode analysis offers an efficient way of modeling the conformational flexibility of protein structures. Simple models defined by contact topology, known as elastic network models, have been used to model a variety of systems, but the validation is typically limited to individual modes for a single protein. We use anisotropic displacement parameters from crystallography to test the quality of prediction of both the magnitude and directionality of conformational variance. Normal modes from four simple elastic network model potentials and from the CHARMM forcefield are calculated for a dat…
We used statistical thermodynamics of conformational fluctuations and the elements of algebraic graph theory together with data from 2000 protein crystal structures, and showed that folded native proteins with harmonic interactions exhibit distribution functions each of which appear to be universal across all proteins. The three universal distributions are: (i) the eigenvalue spectrum of the protein graph Laplacian, (ii) the B-factor distribution of residues, and (iii) the vibrational entropy difference per residue between the unfolded and the folded states. The three distributions, which look…
An effective potential function is critical for protein structure prediction and folding simulation. For simplified models of proteins where coordinates of only $C_α$ atoms need to be specified, an accurate potential function is important. Such a simplified model is essential for efficient search of conformational space. In this work, we present a formulation of potential function for simplified representations of protein structures. It is based on the combination of descriptors derived from residue-residue contact and sequence-dependent local geometry. The optimal weight coefficients for cont…
Using a beta-hairpin protein as a representative example of two-state folders, we studied how the exploration of native-like states affects the folding kinetics. It has been found that the first-passage time (FPT) distributions are essentially single-exponential not only for the times to overcome the free energy barrier that separates unfolded and native-like states but also for the times to find the native state among the native-like ones. If the protein explores native-like states for a time much longer than the time to overcome the free energy barrier, which was found to be characteristic o…
The presence of metamorphism in the protein's native state is not yet fully understood. In an attempt to throw light on this issue here we present an assessment, in terms of the amide hydrogen exchange protection factor, that aims to determine the likely existence of structural fluctuations in the native-state consistent with both the upper bound marginal stability of proteins and the metamorphism presence. The preliminary results enable us to conclude that the native-state metamorphism is, indeed, more probable than thought.
Using a variant of parallel tempering, we study the changes in sampling within a simulation when the all-atom model is coupled to a Go-like potential. We find that the native structure is not the lowest-energy configuration in the all-atom force field. Adding a Go-term deforms the energy landscape in a way that the native configuration becomes the global minimum.
A complex network approach to protein folding is proposed. The graph object is the network of shortcut edges present in a native-state protein (SCN0). Although SCN0s are found via an intuitive message passing algorithm (S. Milgram, Psychology Today, May 1967 pp. 61-67), they are meaningful enough that the logarithm form of their contact order (SCN0_lnCO) correlates significantly with protein kinetic rates, regardless of protein size. Further, the clustering coefficient of a SCN0 (CSCN0) can be used to combine protein segments iteratively within the Restricted Binary Collision model to form the…
In this paper we investigate the role of native geometry on the kinetics of protein folding based on simple lattice models and Monte Carlo simulations. Results obtained within the scope of the Miyazawa-Jernigan indicate the existence of two dynamical folding regimes depending on the protein chain length. For chains larger than 80 amino acids the folding performance is sensitive to the native state's conformation. Smaller chains, with less than 80 amino acids, fold via two-state kinetics and exhibit a significant correlation between the contact order parameter and the logarithmic folding times.…
Studying evolutionary correlations in alignments of homologous sequences by means of an inverse Potts model has proven useful to obtain residue-residue contact energies and to predict contacts in proteins. The quality of the results depend much on several choices of the detailed model and on the algorithms used. We built, in a very controlled way, synthetic alignments with statistical properties similar to those of real proteins, and used them to assess the performance of different inversion algorithms and of their variants. Realistic synthetic alignments display typical features of low--tempe…
The differing ability of polypeptide conformations to act as the native state of proteins has long been rationalized in terms of differing kinetic accessibility or thermodynamic stability. Building on the successful applications of physical concepts and sampling algorithms recently introduced in the study of disordered systems, in particular artificial neural networks, we quantitatively explore how well a quantity known as the local entropy describes the native state of model proteins. In lattice models and all-atom representations of proteins, we are able to efficiently sample high local entr…
Stochastic simulations of coarse-grained protein models are used to investigate the propensity to form knots in early stages of protein folding. The study is carried out comparatively for two homologous carbamoyltransferases, a natively-knotted N-acetylornithine carbamoyltransferase (AOTCase) and an unknotted ornithine carbamoyltransferase (OTCase). In addition, two different sets of pairwise amino acid interactions are considered: one promoting exclusively native interactions, and the other additionally including non-native quasi-chemical and electrostatic interactions. With the former model …
We recently introduced a physical model [Hoang et al., P. Natl. Acad. Sci. USA (2004), Banavar et al., Phys. Rev. E (2004)] for proteins which incorporates, in an approximate manner, several key features such as the inherent anisotropy of a chain molecule, the geometrical and energetic constraints placed by the hydrogen bonds and sterics, and the role played by hydrophobicity. Within this framework, marginally compact conformations resembling the native state folds of proteins emerge as broad competing minima in the free energy landscape even for a homopolymer. Here we show how the introductio…
Natural proteins fold to a unique, thermodynamically dominant state. Modeling of the folding process and prediction of the native fold of proteins are two major unsolved problems in biophysics. Here, we show successful all-atom ab initio folding of a representative diverse set of proteins, using a minimalist transferable energy model that consists of two-body atom-atom interactions, hydrogen-bonding, and a local sequence energy term that models sequence-specific chain stiffness. Starting from a random coil, the native-like structure was observed during replica exchange Monte Carlo (REMC) simul…
Though the problem of sequence-reversed protein folding is largely unexplored, one might speculate that reversed native protein sequences should be significantly more foldable than purely random heteropolymer sequences. In this article, we investigate how the reverse-sequences of native proteins might fold by examining a series of small proteins of increasing structural complexity (α-helix, \b{eta}-hairpin, α-helix bundle, and α/\b{eta}-protein). Employing a tandem protein structure prediction algorithmic and molecular dynamics simulation approach, we find that the ability of reverse sequences…
Predicting protein 3D structure from amino acid sequence remains as a challenge in the field of computational biology. If protein structure homologues are not found, one has to construct 3D structural conformations from the very beginning by the so-called ab initio approach, using some empirical energy functions. A successful algorithm in this category, Rosetta, creates an ensemble of decoy conformations by assembling selected best short fragments of known protein structures and then recognizes the native state as the highly populated one with a very low energy. Typically, an energy function i…
Mechanical unfolding of the fourth domain of Distyostelium discoideum filamin (DDFLN4) was studied by all-atom molecular dynamics simulations, using the GROMOS96 force field 43a1 and the simple point charge explicit water solvent. Our study reveals an important role of non-native interactions in the unfolding process. Namely, the existence of a peak centered at the end-to-end extension 22 nm in the force-extension curve, is associated with breaking of non-native hydrogen bonds. Such a peak has been observed in experiments but not in Go models, where non-native interactions are neglected. We pr…
The similarity in the three-dimensional structures of homologous proteins imposes strong constraints on their sequence variability. It has long been suggested that the resulting correlations among amino acid compositions at different sequence positions can be exploited to infer spatial contacts within the tertiary protein structure. Crucial to this inference is the ability to disentangle direct and indirect correlations, as accomplished by the recently introduced Direct Coupling Analysis (DCA) (Weigt et al. (2009) Proc Natl Acad Sci 106:67). Here we develop a computationally efficient implemen…
Cotranslational folding depends on the folding speed and stability of the nascent protein. It remains difficult, however, to predict which proteins cotranslationally fold. Here, we simulate evolution of model proteins to investigate how native structure influences evolution of cotranslational folding. We developed a model that connects protein folding during and after translation to cellular fitness. Model proteins evolved improved folding speed and stability, with proteins adopting one of two strategies for folding quickly. Low contact order proteins evolve to fold cotranslationally. Such pro…
We provide evidence that the energy landscapes of folded proteins do not shift with temperature, but the onset of functional dynamics is associated with its effective sampling. The motion of the backbone is described by three distinct regimes. One is associated with slow time scales of the activity along the envelope of the energy surface defining the folded protein. Another, with fast time scales, is due to activity along the pockets decorating the folded-state envelope. The intermediate regime emerges at temperatures where jumps between the pockets become possible, leading to an active prote…
Models of protein energetics which neglect interactions between amino acids that are not adjacent in the native state, such as the Go model, encode or underlie many influential ideas on protein folding. Implicit in this simplification is a crucial assumption that has never been critically evaluated in a broad context: Detailed mechanisms of protein folding are not biased by non-native contacts, typically imagined as a consequence of sequence design and/or topology. Here we present, using computer simulations of a well-studied lattice heteropolymer model, the first systematic test of this oft-a…
Understanding the biological function of knots in proteins and their folding process is an open and challenging question in biology. Recent studies classify the topology and geometry of knotted proteins by analysing the distribution of a protein's planar projections using topological objects called knotoids. We approach the analysis of proteins with the same topology by introducing a topologically inspired statistical metric between their knotoid distributions. We detect geometric differences between trefoil proteins by characterising their entanglement and we recover a clustering by sequence …
We study a large data set of protein structure ensembles of very diverse sizes determined by nuclear magnetic resonance. By examining the distance-dependent correlations in the displacement of residues pairs and conducting finite size scaling analysis it was found that the correlations and susceptibility behave as in systems near a critical point implying that, at the native state, the motion of each amino acid residue is felt by every other residue up to the size of the protein molecule. Furthermore certain protein's shapes corresponding to maximum susceptibility were found to be more probabl…
Native protein folds often have a high degree of symmetry. We study the relationship between the symmetries of native proteins, and their designabilities -- how many different sequences encode a given native structure. Using a two-dimensional lattice protein model based on hydrophobicity, we find that those native structures that are encoded by the largest number of different sequences have high symmetry. However only certain symmetries are enhanced, e.g. x/y-mirror symmetry and $180^o$ rotation, while others are suppressed. If it takes a large number of mutations to destabilize the native sta…
We propose an automated protocol for designing the energy landscape of a protein energy function by optimizing its parameters. The parameters are optimized so that not only the global minimum energy conformation becomes native-like, but also the conformations distinct from the native structure have higher energies than those close to the native one. We successfully apply our protocol to the parameter optimization of the UNRES potential energy, using the training set of betanova, 1fsd, the 36-residue subdomain of chicken villin headpiece (PDB ID 1vii), and the 10-55 residue fragment of staphylo…
We explore the consequences of very high dimensionality in the dynamical landscape of protein folding. Consideration of both typical range of stabilising interactions, and folding rates themselves, leads to a model of the energy hypersurface that is characterised by the structure of diffusive "hypergutters" as well as the familiar "funnels". Several general predictions result: (1) intermediate subspaces of configurations will always be visited; (2) specific but non-native interactions are important in stabilising these low-dimensional diffusive searches on the folding pathway; (3) sequential b…
The folding rates of two-state proteins have been found to correlate with simple measures of native-state topology. The most prominent among these measures is the relative contact order (CO), which is the average CO or 'localness' of all contacts in the native protein structure, divided by the chain length. Here, we test whether such measures can be generalized to capture the effect of chain crosslinks on the folding rate. Crosslinks change the chain connectivity and therefore also the localness of some of the the native contacts. These changes in localness can be taken into account by the gra…
While all the information required for the folding of a protein is contained in its amino acid sequence, one has not yet learned how to extract this information to predict the three--dimensional, biologically active, native conformation of a protein whose sequence is known. Using insight obtained from simple model simulations of the folding of proteins, in particular of the fact that this phenomenon is essentially controlled by conserved (native) contacts among (few) strongly interacting ("hot"), as a rule hydrophobic, amino acids, which also stabilize local elementary structures (LES, hidden,…
Residue-wise contact order (RWCO) is a new kind of one-dimensional protein structures which represents the extent of long-range contacts. We have recently shown that a set of three types of one-dimensional structures (secondary structure, contact number, and RWCO) contains sufficient information for reconstructing the three-dimensional structure of proteins. Currently, there exist prediction methods for secondary structure and contact number from amino acid sequence, but none exists for RWCO. Also, the properties of amino acids that affect RWCO is not clearly understood. Here, we present a lin…
Electrospray ion-beam deposition (ES-IBD) is a versatile tool to study structure and reactivity of molecules from small metal clusters to large protein assemblies. It brings molecules gently into the gas phase where they can be accurately manipulated and purified, followed by controlled deposition onto various substrates. In combination with imaging techniques, direct structural information of well-defined molecules can be obtained, which is essential to test and interpret results from indirect mass spectrometry techniques. To date, ion-beam deposition experiments are limited to a small number…
We incorporate hydrodynamic interactions (HI) in a coarse-grained and structure-based model of proteins by employing the Rotne-Prager hydrodynamic tensor. We study several small proteins and demonstrate that HI facilitate folding. We also study HIV-1 protease and show that HI make the flap closing dynamics faster. The HI are found to affect time correlation functions in the vicinity of the native state even though they have no impact on same time characteristics of the structure fluctuations around the native state.
A single protein molecule is regarded as a contact network of amino-acid residues. Some studies have indicated that this network is a small world network (SWN), while other results have implied that this is a fractal network (FN). However, SWN and FN are essentially different in the dependence of the shortest path length on the number of nodes. In this paper, we investigate this dependence in the residue contact networks of proteins in native structures, and show that the networks are not SWN but FN. FN is generally characterized by several dimensions. Among them, we focus on three dimensions;…
The native structures of proteins, except for notable exceptions of intrinsically disordered proteins, in general take their most stable conformation in the physiological condition to maintain their structural framework so that their biological function can be properly carried out. Experimentally, the stability of a protein can be measured by several means, among which the pulling experiment using the atomic force microscope (AFM) stands as a unique method. AFM directly measures the resistance from unfolding, which can be quantified from the observed force-extension profile. It has been shown …
Several ionic liquids (ILs) are known to revert aggregation processes and improve the in vitro refolding of denatured proteins. In this paper the capacity of a particular class of ammonium based ILs to act as refolding enhancers was tested using lysozyme as a model protein. Raman spectra of ILs treated fibrillar lysozyme as well as lysozyme in its native and fibrillar conformations were collected and carefully analyzed to characterize the refolding extent under the effect of the IL interaction. Results obtained confirm and largely extend the earlier knowledge on this class of protic ILs and in…
Motivation. Protein design aims to identify sequences compatible with a given protein fold but incompatible to any alternative folds. To select the correct sequences and to guide the search process, a design scoring function is critically important. Such a scoring function should be able to characterize the global fitness landscape of many proteins simultaneously. Results. To find optimal design scoring functions, we introduce two geometric views and propose a formulation using mixture of nonlinear Gaussian kernel functions. We aim to solve a simplified protein sequence design problem. Our goa…
We present a new method to extract distance and orientation dependent potentials between amino acid side chains using a database of protein structures and the standard Boltzmann device. The importance of orientation dependent interactions is first established by computing orientational order parameters for proteins with alpha-helical and beta-sheet architecture. Extraction of the anisotropic interactions requires defining local reference frames for each amino acid that uniquely determine the coordinates of the neighboring residues. Using the local reference frames and histograms of the radial …
Simplified Go models, where only native contacts interact favorably, have proven useful to characterize some aspects of the folding of small proteins. The success of these models is limited by the fact that all residues interact in the same way, so that the folding features of a protein are determined only by the geometry of its native conformation. We present an extended version of a C-alpha based Go model where different residues interact with different energies. The model is used to calculate the thermodynamics of three small proteins (Protein G, SrcSH3 and CI2) and the effect of mutations …
Unavailability of experimental 3D structural data on protein folding dynamics and necessity for a new generation of structure prediction methods in this context
Protein one-dimensional (1D) structures such as secondary structure and contact number provide intuitive pictures to understand how the native three-dimensional (3D) structure of a protein is encoded in the amino acid sequence. However, it has not been clear whether a given set of 1D structures contains sufficient information for recovering the underlying 3D structure. Here we show that the 3D structure of a protein can be recovered from a set of three types of 1D structures, namely, secondary structure, contact number and residue-wise contact order which is introduced here for the first time.…
Protein conformational transitions, which are essential for function, may be driven either by entropy or enthalpy when molecular systems comprising solute and solvent molecules are the focus. Revealing thermodynamic origin of a given molecular process is an important but difficult task, and general principles governing protein conformational distributions remain elusive. Here we demonstrate that when protein molecules are taken as thermodynamic systems and solvents being treated as the environment, conformational entropy is an excellent proxy for free energy and is sufficient to explain protei…
We present a simple model of protein folding dynamics that captures key qualitative elements recently seen in all-atom simulations. The goals of this theory are to serve as a simple formalism for gaining deeper insight into the physical properties seen in detailed simulations as well as to serve as a model to easily compare why these simulations suggest a different kinetic mechanism than previous simple models. Specifically, we find that non-native contacts play a key role in determining the mechanism, which can shift dramatically as the energetic strength of non-native interactions is changed…
In order to elucidate the role of the native state topology and the stability of subdomains in protein folding, we investigate free energy landscape of human lysozyme, which is composed of two subdomains, by Monte Carlo simulations. A realistic lattice model with Go-like interaction is used. We take the relative interaction strength (stability, in other word) of two subdomains as a variable parameter and study the folding process. A variety of folding process is observed and we obtained a phase diagram of folding in terms of temperature and the relative stability. Experimentally-observed diver…
We report on atomistic simulation of the folding of a natively-knotted protein, MJ0366, based on a realistic force field. To the best of our knowledge this is the first reported effort where a realistic force field is used to investigate the folding pathways of a protein with complex native topology. By using the dominant-reaction pathway scheme we collected about 30 successful folding trajectories for the 82-amino acid long trefoil-knotted protein. Despite the dissimilarity of their initial unfolded configuration, these trajectories reach the natively-knotted state through a remarkably simila…
We explore the dynamical large-deviations of a lattice heteropolymer model of a protein by means of path sampling of trajectories. We uncover the existence of non-equilibrium dynamical phase-transitions in ensembles of trajectories between active and inactive dynamical phases, whose nature depends on properties of the interaction potential. When the full heterogeneity of interactions due to the amino-acid sequence is preserved, as in a fully interacting model or in a heterogeneous version of the Gō model where only native interactions are considered, the transition is between the equilibrium n…
Native shortcut networks (SCN0) are sub-graphs of native protein residue networks (PRN0). In this paper, we propose the Network Dynamics (ND) model, which reconstructs a PRN0 by adding back its edges according to some recipe, while its nodes are fixed at native (PDB) locations. A PRN0 reconstruction will eventually reconstruct its SCN0, but only after exploring several non-native shortcut network (SCN) configurations. It is these other SCN configurations that are of interest to us, as they produce the statistics to evaluate the different recipes. The recipes vary from each other slightly to in…
The folding of a protein towards its native state is a rather complicated process. However there are empirical evidences that the folding time correlates with the contact order, a simple measure of the spatial organisation of the native state of the protein. Contact order is related to the average length of the main chain loops formed by amino acids which are in contact. Here we argue that folding kinetics can be influenced also by the entanglement that loops may undergo within the overall three dimensional protein structure. In order to explore such possibility, we introduce a novel descripto…
There are several simple criteria of folding to a native state in model proteins. One of them involves crossing of a threshold value of the RMSD distance away from the native state. Another checks whether all native contacts are established, i.e. whether the interacting amino acids come closer than some characteristic distance. We use Go-like models of proteins and show that such simple criteria may prompt one to declare folding even though fragments of the resulting conformations have a wrong sense of chirality. We propose that a better condition of folding should augment the simple criteria …
We present a simple physical model which demonstrates that the native state folds of proteins can emerge on the basis of considerations of geometry and symmetry. We show that the inherent anisotropy of a chain molecule, the geometrical and energetic constraints placed by the hydrogen bonds and sterics, and hydrophobicity are sufficient to yield a free energy landscape with broad minima even for a homopolymer. These minima correspond to marginally compact structures comprising the menu of folds that proteins choose from to house their native-states in. Our results provide a general framework fo…
AMix-2: Establishing Protein as a Native Modality in Large Language Models
Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）；Generative Symbolic Intelligence Lab (GenSI), Tsinghua University（生成符号智能实验室（GenSI），清华大学）；Institute for AI Industry Research (AIR), Tsinghua University（人工智能产业研究院（AIR），清华大学）
The primary structure of proteins, that is their sequence, represents one of the most abundant set of experimental data concerning biomolecules. The study of correlations in families of co--evolving proteins by means of an inverse Ising--model approach allows to obtain information on their native conformation. Following up on a recent development along this line, we optimize the algorithm to calculate effective energies between the residues, validating the approach both back-calculating interaction energies in a model system, and predicting the free energies associated to mutations in real sys…
The aim of this work is to elucidate how physical principles of protein design are reflected in natural sequences that evolved in response to the thermal conditions of the environment. Using an exactly solvable lattice model, we design sequences with selected thermal properties. Compositional analysis of designed model sequences and natural proteomes reveals a specific trend in amino acid compositions in response to the requirement of stability at elevated environmental temperature, i.e. the increase of fractions of hydrophobic and charged amino acid residues at the expense of polar ones. We s…
Proper folding of deeply knotted proteins has a very low success rate even in structure-based models which favor formation of the native contacts but have no topological bias. By employing a structure-based model, we demonstrate that cotranslational folding on a model ribosome may enhance the odds to form trefoil knots for protein YibK without any need to introduce any non-native contacts. The ribosome is represented by a repulsive wall that keeps elongating the protein. On-ribosome folding proceeds through a a slipknot conformation. We elucidate the mechanics and energetics of its formation. …
A theoretical framework is developed to study the dynamics of protein folding. The key insight is that the search for the native protein conformation is influenced by the rate r at which external parameters, such as temperature, chemical denaturant or pH, are adjusted to induce folding. A theory based on this insight predicts that (1) proteins with non-funneled energy landscapes can fold reliably to their native state, (2) reliable folding can occur as an equilibrium or out-of-equilibrium process, and (3) reliable folding only occurs when the rate r is below a limiting value, which can be calc…
