We have developed a new simulation method to estimate the distance between the native state and the first transition state, and the distance between the intermediate state and the second transition state of a protein which mechanically unfolds via intermediates. Assuming that the end-to-end extension $ΔR$ is a good reaction coordinate to describe the free energy landscape of proteins subjected to an external force, we define the midpoint extension $ΔR^*$ between two transition states from either constant-force or constant loading rate pulling simulations. In the former case, $ΔR^*$ is defined as a middle point between two plateaus in the time-dependent curve of $ΔR$, while, in the latter one, it is a middle point between two peaks in the force-extension curve. Having determined $ΔR^*$, one can compute times needed to cross two transition state barriers starting from the native state. With the help of the Bell and microscopic kinetic theory, force dependencies of these unfolding times can be used to locate the intermediate state and to extract unfolding barriers. We have applied our method to the titin domain I27 and the fourth domain of {\em Dictyostelium discoideum} filamin (DDFLN4), and obtained reasonable agreement with experiments, using the C$_α$-Go model.
The novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a major worldwide public health emergency that has infected over $1.5$ million people. The partially open state of S1 subunit in spike glycoprotein is considered vital for its infection with host cell and is represented as a key target for neutralizing antibodies. However, the mechanism elucidating the transition from the closed state to the partially open state still remains unclear. Here, we applied a combination of Markov state model, transition path theory and random forest to analyze the S1 motion. Our results explo…
Inherent structure theory is used to discover strong connections between simple characteristics of protein structure and the energy landscape of a Go model. The potential energies and vibrational free energies of inherent structures are highly correlated, and both reflect simple measures of networks of native contacts. These connections have important consequences for models of protein dynamics and thermodynamics.
Water molecules and molecular chaperones efficiently help the protein folding process. Here we describe their action in the context of the energy and topological networks of proteins. In energy terms water and chaperones were suggested to decrease the activation energy between various local energy minima smoothing the energy landscape, rescuing misfolded proteins from conformational traps and stabilizing their native structure. In kinetic terms water and chaperones may make the punctuated equilibrium of conformational changes less punctuated and help protein relaxation. Finally, water and chap…
A simple approach is proposed to investigate the protein structure. Using a low complexity model, a simple pairwise interaction and the concept of global optimization, we are able to calculate ground states of proteins, which are in agreement with experimental data. All possible model structures of small proteins are available below a certain energy threshold. The exact lowenergy landscapes for the trp cage protein (1L2Y) is presented showing the connectivity of all states and energy barriers.
Deciphering the unique dynamic activation pathway in a G protein-coupled receptor enables unveiling biased signaling and identifying cryptic allosteric sites in conformational intermediates
Shanghai Jiao Tong University School of Medicine（上海交通大学医学院）；Ningxia Medical University（宁夏医科大学）；Peking University（北京大学）
Considering all the PDB annotated allosteric proteins (from ASD - AlloSteric Database) belonging to four different classes (kinases, nuclear receptors, peptidases and transcription factors), this work has attempted to decipher certain consistent patterns present in the residues constituting the allosteric communication sub-system (ACSS). The thermal fluctuations of hydrophobic residues in ACSSs were found to be significantly higher than those present in the non-ACSS part of the same proteins, while polar residues showed the opposite trend. The basic residues and hydroxyl residues were found to…
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…
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…
The dynamical characterization of proteins is crucial to understand protein function. From a microscopic point of view, protein dynamics is governed by the local atomic interactions that, in turn, trigger the functional conformational changes. Unfortunately, the relationship between local atomic fluctuations and global protein rearrangements is still elusive. Here, atomistic molecular dynamics simulations in conjunction with complex network analysis show that fast peptide relaxations effectively build the backbone of the global free-energy landscape, providing a connection between local and gl…
The interaction cutoff contribution to the ruggedness of protein-protein energy landscape (the artificial ruggedness) is studied in terms of relative energy fluctuations for 1/r^n potentials based on a simplistic model of a protein complex. Contradicting the principle of minimal frustration, the artificial ruggedness exists for short cutoffs and gradually disappears with the cutoff increase. The critical values of the cutoff were calculated for each of eleven popular power-type potentials with n=0-9, 12 and for two thresholds of 5% and 10%. The artificial ruggedness decreases to tolerable thre…
Biological forces govern essential cellular and molecular processes in all living organisms. Many cellular forces, e.g. those generated in cyclic conformational changes of biological machines, have repetitive components. However, little is known about how proteins process repetitive mechanical stresses. To obtain first insights into dynamic protein mechanics, we probed the mechanical stability of single and multimeric ubiquitins perturbed by periodic forces. Using coarse-grained molecular dynamics simulations, we were able to model repetitive forces with periods about two orders of magnitude l…
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 …
The energy landscape of DNA-binding proteins along the genome
Istituto Europeo di Oncologia（欧洲肿瘤研究所）；Heidelberg Institute for Theoretical Studies (HITS)（海德堡理论研究所）；Heidelberg University（海德堡大学）；Università degli Studi di Milano（米兰大学）；INFN（意大利国家核物理研究所）
We develop a simple but rigorous model of protein-protein association kinetics based on diffusional association on free energy landscapes obtained by sampling configurations within and surrounding the native complex binding funnels. Guided by results obtained on exactly solvable model problems, we transform the problem of diffusion in a potential into free diffusion in the presence of an absorbing zone spanning the entrance to the binding funnel. The free diffusion problem is solved using a recently derived analytic expression for the rate of association of asymmetrically oriented molecules. D…
Nested sampling is a Bayesian sampling technique developed to explore probability distributions lo- calised 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 algo- rithm 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, inc…
Energy landscape theory describes how a full-length protein can attain its native fold by sampling only a tiny fraction of all possible structures. Although protein folding is now understood to be concomitant with synthesis on the ribosome, there have been few attempts to modify energy landscape theory by accounting for cotranslational folding. Here we provide a model for cotranslational folding that leads to a natural definition of a nested energy landscape. By applying concepts drawn from submanifold differential geometry, the physics of protein folding on the ribosome can be explored in a q…
The notion of energy landscapes provides conceptual tools for understanding the complexities of protein folding and function. Energy Landscape Theory indicates that it is much easier to find sequences that satisfy the "Principle of Minimal Frustration" when the folded structure is symmetric (Wolynes, P. G. Symmetry and the Energy Landscapes of Biomolecules. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 14249-14255). Similarly, repeats and structural mosaics may be fundamentally related to landscapes with multiple embedded funnels. Here we present analytical tools to detect and compare structural rep…
Metadynamics is a powerful computational tool to obtain the free energy landscape of complex systems. The Monte Carlo algorithm has proven useful to calculate thermodynamic quantities associated with simplified models of proteins, and thus to gain an ever-increasing understanding on the general principles underlying the mechanism of protein folding. We show that it is possible to couple metadynamics and Monte Carlo algorithms to obtain the free energy of model proteins in a way which is computationally very economical.
Energy landscape theory describes how a full-length protein can attain its native fold after sampling only a tiny fraction of all possible structures. Although protein folding is now understood to be concomitant with synthesis on the ribosome there have been few attempts to modify energy landscape theory by accounting for cotranslational folding. This paper introduces a model for cotranslational folding that leads to a natural definition of a nested energy landscape. By applying concepts drawn from submanifold differential geometry the dynamics of protein folding on the ribosome can be explore…
Theoretical studies of stretching proteins with slipknots reveal a surprising growth of their unfolding times when the stretching force crosses an intermediate threshold. This behavior arises as a consequence of the existence of alternative unfolding routes that are dominant at different force ranges. Responsible for longer unfolding times at higher forces is the existence of an intermediate, metastable configuration where the slipknot is jammed. Simulations are performed with a coarsed grained model with further quantification using a refined description of the geometry of the slipknots. The …
Over the last 10-15 years a general understanding of the chemical reaction of protein folding has emerged from statistical mechanics. The lessons learned from protein folding kinetics based on energy landscape ideas have benefited protein structure prediction, in particular the development of coarse grained models. We survey results from blind structure prediction. We explore how second generation prediction energy functions can be developed by introducing information from an ensemble of previously simulated structures. This procedure relies on the assumption of a funnelled energy landscape ke…
The equilibrium free energy landscape of an off-lattice model protein as a function of an internal (reaction) coordinate is reconstructed from out-of-equilibrium mechanical unfolding manipulations. This task is accomplished via two independent methods: by employing an extended version of the Jarzynski equality (EJE) and the protein inherent structures (ISs). In a range of temperatures around the ``folding transition'' we find a good quantitative agreement between the free energies obtained via EJE and IS approaches. This indicates that the two methodologies are consistent and able to reproduce…
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…
Characterization of protein energy landscape and conformational ensembles is important for understanding mechanisms of protein folding and function. We studied ensembles of bound and unbound conformations of six proteins to explore their binding mechanisms and characterize the energy landscapes in implicit solvent. First, results show that bound and unbound spectra often significantly overlap. Moreover, the larger the overlap the smaller the RMSD between bound and unbound conformational ensembles. Second, the analysis of the unbound-to-bound changes points to conformational selection as the bi…
Recent studies attracted the attention on the inherent structure landscape (ISL) approach as a reduced description of proteins allowing to map their full thermodynamic properties. However, the analysis has been so far limited to a single topology of a two-state folding protein, and the simplifying assumptions of the method have not been examined. In this work, we construct the thermodynamics of four two-state folding proteins of different sizes and secondary structure by MD simulations using the ISL method, and critically examine possible limitations of the method. Our results show that the IS…
A geometric analysis of the global properties of the energy landscape of a minimalistic model of a polypeptide is presented, which is based on the relation between dynamical trajectories and geodesics of a suitable manifold, whose metric is completely determined by the potential energy. We consider different sequences, some with a definite protein-like behavior, a unique native state and a folding transition, and the others undergoing a hydrophobic collapse with no tendency to a unique native state. The global geometry of the energy landscape appears to contain relevant information on the beha…
This review is a tutorial for scientists interested in the problem of protein structure prediction, particularly those interested in using coarse-grained molecular dynamics models that are optimized using lessons learned from the energy landscape theory of protein folding. We also present a review of the results of the AMH/AMC/AMW/AWSEM family of coarse-grained molecular dynamics protein folding models to illustrate the points covered in the first part of the article. Accurate coarse-grained structure prediction models can be used to investigate a wide range of conceptual and mechanistic issue…
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.
Recently, there has been great interest in learning how to best represent proteins, specifically with fixed-length embeddings. Deep learning has become a popular tool for protein representation learning as a model's hidden layers produce potentially useful vector embeddings. TAPE introduced a number of benchmark tasks and showed that semi-supervised learning, via pretraining language models on a large protein corpus, improved performance on downstream tasks. Two of the tasks (fluorescence prediction and stability prediction) involve learning fitness landscapes. In this paper, we show that CNN …
The near-native free energy landscape of protein G is investigated through 0.4 microseconds-long atomistic molecular dynamics simulations in explicit solvent. A theoretical and computational framework is used to assess the time-dependence of salient thermodynamical features. While the quasi-harmonic character of the free energy is found to degrade in a few ns, the slow modes display a very mild dependence on the trajectory duration. This property originates from a striking self-similarity of the free energy landscape embodied by the consistency of the principal directions of the local minima, …
We study the dynamics of protein folding via statistical energy-landscape theory. In particular, we concentrate on the local-connectivity case with the folding progress described by the fraction of native conformations. We obtain information for the first passage-time (FPT) distribution and its moments. The results show a dynamic transition temperature below which the FPT distribution develops a power-law tail, a signature of the intermittency phenomena of the folding dynamics. We also discuss the possible application of the results to single-molecule dynamics experiments.
Simulations of protein folding and protein association happen on timescales that are orders of magnitude larger than what can typically be covered in all-atom molecular dynamics simulations. Use of low-resolution models alleviates this problem but may reduce the accuracy of the simulations. We introduce a replica-exchange-based multiscale sampling technique that combines the faster sampling in coarse-grained simulations with the potentially higher accuracy of all-atom simulations. After testing the efficiency of our Resolution Exchange with Tunneling (ResET) in simulations of the Trp-cage prot…
In modern biology, one of the most important research problems is to understand how protein sequences fold into their native 3D structures. To investigate this problem at a high level, one wishes to analyze the protein landscapes, i.e., the structures of the space of all protein sequences and their native 3D structures. Perhaps the most basic computational problem at this level is to take a target 3D structure as input and design a fittest protein sequence with respect to one or more fitness functions of the target 3D structure. We develop a toolbox of combinatorial techniques for protein land…
The folding of naturally occurring, single domain proteins is usually well-described as a simple, single exponential process lacking significant trapped states. Here we further explore the hypothesis that the smooth energy landscape this implies, and the rapid kinetics it engenders, arises due to the extraordinary thermodynamic cooperativity of protein folding. Studying Miyazawa-Jernigan lattice polymers we find that, even under conditions where the folding energy landscape is relatively optimized (designed sequences folding at their temperature of maximum folding rate), the folding of protein…
Experimental studies on enzyme evolution show that only a small fraction of all possible mutation trajectories are accessible to evolution. However, these experiments deal with individual enzymes and explore a tiny part of the fitness landscape. We report an exhaustive analysis of fitness landscapes constructed with an off-lattice model of protein folding where fitness is equated with robustness to misfolding. This model mimics the essential features of the interactions between amino acids, is consistent with the key paradigms of protein folding and reproduces the universal distribution of evo…
Acylphosphatase (AcP) is a small protein with 98 amino acid residues that catalyzes the hydrolysis of carboxyl-phosphate bonds. AcP is a typical two-state protein with slow folding rate due to its relatively large contact order in the native structure. The mechanical properties and unfolding behavior of AcP has been studied by atomic force microscope. But the folding and unfolding dynamics at low forces has not been reported. Here using stable magnetic tweezers, we measured the force-dependent folding rates within a force range from 1 pN to 3 pN, and unfolding rates from 15 pN to 40 pN. The ob…
Binding of ligands is often crucial for function yet the effects of ligand binding on the mechanical stability and energy landscape of proteins are incompletely understood. Here we use a combination of single-molecule optical tweezers and MD simulations to investigate the effect of ligand binding on the energy landscape of acyl-coenzyme A (CoA) binding protein (ACBP). ACBP is a topologically simple and highly conserved four-alpha-helix bundle protein that acts as an intracellular transporter and buffer for fatty-acyl CoA and is active in membrane assembly. We have previously described the beha…
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 study the geometric properties of the energy landscape of coarse-grained, off-lattice models of polymers by endowing the configuration space with a suitable metric, depending on the potential energy function, such that the dynamical trajectories are the geodesics of the metric. Using numerical simulations, we show that the fluctuations of the curvature clearly mark the folding transition, and that this quantity allows to distinguish between polymers having a protein-like behavior (i.e., that fold to a unique configuration) and polymers which undergo a hydrophobic collapse but do not have a …
A method to reconstruct the energy landscape of small peptides is presented with reference to a 2d off--lattice model. The starting point is a statistical analysis of the configurational distances between generic minima and directly connected pairs (DCP). As the mutual distance of DCP is typically much smaller than that of generic pairs, a metric criterion can be established to identify the great majority of DCP. Advantages and limits of this approach are thoroughly analyzed for three different heteropolymeric chains. A funnel--like structure of the energy landscape is found in all of the thre…
The folding pathway and rate coefficients of the folding of a knotted protein are calculated for a potential energy function with minimal energetic frustration. A kinetic transition network is constructed using the discrete path sampling approach, and the resulting potential energy surface is visualized by constructing disconnectivity graphs. Owing to topological constraints, the low-lying portion of the landscape consists of three distinct regions, corresponding to the native knotted state and to configurations where either the N- or C-terminus is not yet folded into the knot. The fastest fol…
Free energy landscapes decisively determine the progress of enzymatically catalyzed reactions[1]. Time-resolved macromolecular crystallography unifies transient-state kinetics with structure determination [2-4] because both can be determined from the same set of X-ray data. We demonstrate here how barriers of activation can be determined solely from five-dimensional crystallography [5]. Directly linking molecular structures with barriers of activation between them allows for gaining insight into the structural nature of the barrier. We analyze comprehensive time series of crystal-lographic dat…
We study folding dynamics of protein-like sequences on square lattice using physical move set that exhausts all possible conformational changes. By analytically solving the master equation, we follow the time-dependent probabilities of occupancy of all 802,075 conformations of 16-mers over 7-orders of time span. We find that (i) folding rates of these protein-like sequences of same length can differ by 4-orders of magnitude, (ii) folding rates of sequences of the same conformation can differ by a factor of 190, and (iii) parameters of the native structures, designability, and thermodynamic pro…
Associative memory Hamiltonian structure prediction potentials are not overly rugged, thereby suggesting their landscapes are like those of actual proteins. In the present contribution we show how basin-hopping global optimization can identify low-lying minima for the corresponding mildly frustrated energy landscapes. For small systems the basin-hopping algorithm succeeds in locating both lower minima and conformations closer to the experimental structure than does molecular dynamics with simulated annealing. For large systems the efficiency of basin-hopping decreases for our initial implement…
Local minima and the saddle points separating them in the energy landscape are known to dominate the dynamics of biopolymer folding. Here we introduce a notion of a "folding funnel" that is concisely defined in terms of energy minima and saddle points, while at the same time conforming to a notion of a "folding funnel" as it is discussed in the protein folding literature.
One of the key limitations of Molecular Dynamics simulations is the computational intractability of sampling protein conformational landscapes associated with either large system size or long timescales. To overcome this bottleneck, we present the REinforcement learning based Adaptive samPling (REAP) algorithm that aims to efficiently sample conformational space by learning the relative importance of each reaction coordinate as it samples the landscape. To achieve this, the algorithm uses concepts from the field of reinforcement learning, a subset of machine learning, which rewards sampling al…
Improving Protein Optimization with Smoothed Fitness Landscapes
Massachusetts Institute of Technology（麻省理工学院）；Czech Academy of Sciences（捷克科学院）；Czech Technical University in Prague（捷克布拉格技术大学）；University of Chemistry and Technology, Prague（布拉格化学与技术大学）
Chaperonins are biological nanomachines that help newly translated proteins to fold by rescuing them from kinetically trapped misfolded states. Protein folding assistance by the chaperonin machinery is obligatory in vivo for a subset of proteins in the bacterial proteome. Chaperonins are large oligomeric complexes, with unusual seven fold symmetry (group I) or eight/nine fold symmetry (group II), that form double-ring constructs, enclosing a central folding chamber. Dramatic large-scale conformational changes, that take place during ATP-driven cycles, allow chaperonins to bind misfolded protei…
A simple lattice model, recently introduced as a generalization of the Wako--Saitô model of protein folding, is used to investigate the properties of widely studied molecules under external forces. The equilibrium properties of the model proteins, together with their energy landscape, are studied on the basis of the exact solution of the model. Afterwards, the kinetic response of the molecules to a force is considered, discussing both force clamp and dynamic loading protocols and showing that theoretical expectations are verified. The kinetic parameters characterizing the protein unfolding are…
Robust Optimization in Protein Fitness Landscapes Using Reinforcement Learning in Latent Space
School of Computing, KAIST（韩国科学技术院计算机学院）；Center for Mathematical and Computational Sciences, Institute for Basic Science（基础科学研究院数学与计算科学中心）；Center for Biomolecular and Cellular Structure, Institute for Basic Science（基础科学研究院生物分子与细胞结构中心）；Department of Biological Sciences, KAIST（韩国科学技术院生物科学系）
Within the frame of an effective, coarse-grained hydrophobic-polar protein model, we employ multicanonical Monte Carlo simulations to investigate free-energy landscapes and folding channels of exemplified heteropolymer sequences, which are permutations of each other. Despite the simplicity of the model, the knowledge of the free-energy landscape in dependence of a suitable system order parameter enables us to reveal complex folding characteristics known from real bioproteins and synthetic peptides, such as two-state folding, folding through weakly stable intermediates, and glassy metastability…
Epistatic interactions between residues determine a protein's adaptability and shape its evolutionary trajectory. When a protein experiences a changed environment, it is under strong selection to find a peak in the new fitness landscape. It has been shown that strong selection increases epistatic interactions as well as the ruggedness of the fitness landscape, but little is known about how the epistatic interactions change under selection in the long-term evolution of a protein. Here we analyze the evolution of epistasis in the protease of the human immunodeficiency virus type 1 (HIV-1) using …
Dynamical connectivity graphs, which describe dynamical transition rates between local energy minima of a system, can be displayed against the background of a disconnectivity graph which represents the energy landscape of the system. The resulting supergraph describes both dynamics and statics of the system in a unified coarse-grained sense. We give examples of the supergraphs for several two dimensional spin and protein-related systems. We demonstrate that disordered ferromagnets have supergraphs akin to those of model proteins whereas spin glasses behave like random sequences of aminoacids w…
