Proteins experience a wide variety of conformational dynamics that can be crucial for facilitating their diverse functions. How is the intrinsic flexibility required for these motions encoded in their three-dimensional structures? Here, the overall flexibility of a protein is demonstrated to be tightly coupled to the total amount of surface area buried within its fold. A simple proxy for this, the relative solvent accessible surface area (Arel), therefore shows excellent agreement with independent measures of global protein flexibility derived from various experimental and computational methods. Application of Arel on a large scale demonstrates its utility by revealing unique sequence and structural properties associated with intrinsic flexibility. In particular, flexibility as measured by Arel shows little correspondence with intrinsic disorder, but instead tends to be associated with multiple domains and increased α- helical structure. Furthermore, the apparent flexibility of monomeric proteins is found to be useful for identifying quaternary structure errors in published crystal structures. There is also a strong tendency for the crystal structures of more flexible proteins to be solved to lower resolutions. Finally, local solvent accessibility is shown to be a primary determinant of local residue flexibility. Overall this work provides both fundamental mechanistic insight into the origin of protein flexibility and a simple, practical method for predicting flexibility from protein structures.
Several recent works have shown that protein structure can predict site-specific evolutionary sequence variation. In particular, sites that are buried and/or have many contacts with other sites in a structure have been shown to evolve more slowly, on average, than surface sites with few contacts. Here, we present a comprehensive study of the extent to which numerous structural properties can predict sequence variation. The quantities we considered include buriedness (as measured by relative solvent accessibility), packing density (as measured by contact number), structural flexibility (as meas…
We formulate a simple solvation potential based on a coarsed-grain representation of amino acids with two spheres modeling the $C_α$ atom and an effective side-chain centroid. The potential relies on a new method for estimating the buried area of residues, based on counting the effective number of burying neighbours in a suitable way. This latter quantity shows a good correlation with the buried area of residues computed from all atom crystallographic structures. We check the discriminatory power of the solvation potential alone to identify the native fold of a protein from a set of decoys and…
Protein-nucleic acid complexes are important for many cellular processes including the most essential function such as transcription and translation. For many protein-nucleic acid complexes, flexibility of both macromolecules has been shown to be critical for specificity and/or function. Flexibility-rigidity index (FRI) has been proposed as an accurate and efficient approach for protein flexibility analysis. In this work, we introduce FRI for the flexibility analysis of protein-nucleic acid complexes. We demonstrate that a multiscale strategy, which incorporates multiple kernels to capture var…
Computational docking methods can provide structural models of protein-protein complexes, but protein backbone flexibility upon association often thwarts accurate predictions. In recent blind challenges, medium or high accuracy models were submitted in less than 20% of the "difficult" targets (with significant backbone change or uncertainty). Here, we describe recent developments in protein-protein docking and highlight advances that tackle backbone flexibility. In molecular dynamics and Monte Carlo approaches, enhanced sampling techniques have reduced time-scale limitations. Internal coordina…
Binding of a ligand on a protein changes the flexibility of certain parts of the protein, which directly affects its function. These changes are not the same at each point, some parts become more flexible and some others become stiffer. Here, an equation is derived that gives the stiffness map for proteins. The model is based on correlations of fluctuations of pairs of points that need to be evaluated by molecular dynamics simulations. The model is also cast in terms of the Gaussian Network Model and changes of stiffness upon dimerization of AKT1 are evaluated as an example.
Classical simulations of protein flexibility remain computationally expensive, especially for large proteins. A few years ago, we developed a fast method for predicting protein structure fluctuations that uses a single protein model as the input. The method has been made available as the CABS-flex web server and applied in numerous studies of protein structure-function relationships. Here, we present a major update of the CABS-flex web server to version 2.0. The new features include: extension of the method to significantly larger and multimeric proteins, customizable distance restraints and s…
A simple way to get insights about the possible functional motions of a protein is to perform a normal mode analysis (NMA). Indeed, it has been shown that low-frequency modes thus obtained are often closely related to domain motions involved in protein function. Moreover, because protein low-frequency modes are known to be robust, NMA can be performed using coarse-grained models. As a consequence, it can be done for large ensembles of conformations as well as for large systems, like the ribosome, whole virus capsids, etc. Unexpectedly, on the high-frequency side, modes obtained with cutoff-bas…
Motivation: Identification of flexible regions of protein structures is important for understanding of their biological functions. Recently, we have developed a fast approach for predicting protein structure fluctuations from a single protein model: the CABS-flex. CABS-flex was shown to be an efficient alternative to conventional all-atom molecular dynamics (MD). In this work, we evaluate CABS-flex and MD predictions by comparison with protein structural variations within NMR ensembles. Results: Based on a benchmark set of 140 proteins, we show that the relative fluctuations of protein residue…
Proteins are the most important biomolecules for living organisms. The understanding of protein structure, function, dynamics and transport is one of most challenging tasks in biological science. In the present work, persistent homology is, for the first time, introduced for extracting molecular topological fingerprints (MTFs) based on the persistence of molecular topological invariants. MTFs are utilized for protein characterization, identification and classification. The method of slicing is proposed to track the geometric origin of protein topological invariants. Both all-atom and coarse-gr…
The flexibility-rigidity index (FRI) is a newly proposed method for the construction of atomic rigidity functions. The FRI method analyzes protein rigidity and flexibility and is capable of predicting protein B-factors without resorting to matrix diagonalization. A fundamental assumption used in the FRI is that protein structures are uniquely determined by various internal and external interactions, while the protein functions, such as stability and flexibility, are solely determined by the structure. As such, one can predict protein flexibility without resorting to the protein interaction Ham…
Protein function frequently involves conformational changes with large amplitude on timescales which are difficult and computationally expensive to access using molecular dynamics. In this paper, we report on the combination of three computationally inexpensive simulation methods-normal mode analysis using the elastic network model, rigidity analysis using the pebble game algorithm, and geometric simulation of protein motion-to explore conformational change along normal mode eigenvectors. Using a combination of ELNEMO and FIRST/FRODA software, large-amplitude motions in proteins with hundreds …
We have studied the mobility of the folding catalyst, protein disulphide-isomerase (PDI), by molecular dynamics and by a rapid approach based on flexibility. We analysed our simulations using measures of backbone movement, relative positions and orientations of domains, and distances between functional sites. Despite their different assumptions, the two methods are surprisingly consistent. Both methods agree that motion of domains is dominated by hinge and rotation motion of the a and a' domains relative to the central b-b' domain core. We identify the a' domain as showing the greatest intra-d…
Upon studying the B-Factors of all the atoms of all non-redundant proteins belonging to 76 most commonly found structural domains of all four major structural classes, it was found that the residue mobility has decreased during the course of evolution. Though increased residue-flexibility was preferred in the early stages of protein structure evolution, less flexibility is preferred in the medieval and recent stages. GLU is found to be the most flexible residue while VAL recorded to have the least flexibility. General trends in decrement of B-Factors conformed to the general trend in the order…
DBB and SciLifeLab at Stockholm University（斯德哥尔摩大学生物生物学与系统生物学中心及科学生命实验室）；Max Planck Institute for Polymer Research（马克斯·普朗克聚合物研究所）；Heidelberg Institute for Theoretical Studies（海德尔堡理论研究所）；IWR, Heidelberg University（海德尔堡大学计算与数学研究所）
Most widely used ligand docking methods assume a rigid protein structure. This leads to problems when the structure of the target protein deforms upon ligand binding. In particular, the ligand's true binding pose is often scored very unfavorably due to apparent clashes between ligand and protein atoms, which lead to extremely high values of the calculated van der Waals energy term. Traditionally, this problem has been addressed by explicitly searching for receptor conformations to account for the flexibility of the receptor in ligand binding. Here we present a deep learning model trained to ta…
The rigidity and flexibility of homologous psychrophilic(P), mesophilic(M) and thermophilic(T) proteins have been investigated at the global and local levels in terms of packing factor and atomic fluctuations obtained from B-factors. For comparison of atomic fluctuations, correction of errors by considering errors in B-factors from all sources in a consolidated manner and conversion of the fluctuations to the same temperature have been suggested and validated. Results indicate no differences in the global values like average packing factor among the three classes of protein homologs but at loc…
Debye-Waller factor, a measure of X-ray attenuation, can be experimentally observed in protein X-ray crystallography. Previous theoretical models have made strong inroads in the analysis of B-factors by linearly fitting protein B-factors from experimental data. However, the blind prediction of B-factors for unknown proteins is an unsolved problem. This work integrates machine learning and advanced graph theory, namely, multiscale weighted colored graphs (MWCGs), to blindly predict B-factors of unknown proteins. MWCGs are local features that measure the intrinsic flexibility due to a protein st…
In simple models side chains are often represented implicitly (e.g., by spin-states) or simplified as one atom. We study side chain effects using square lattice and tetrahedral lattice models, with explicitly side chains of two atoms. We distinguish effects due to chirality and effects due to side chain flexibilities, since residues in proteins are L-residues, and their side chains adopt different rotameric states. Short chains are enumerated exhaustively. For long chains, we sample effectively rare events (eg, compact conformations) and obtain complete pictures of ensemble properties of these…
$α$-helices stand out as common and relatively invariant secondary structural elements of proteins. However, $α$-helices are not rigid bodies and their deformations can be significant in protein function ({\it e.g.} coiled coils). To quantify the flexibility of $α$-helices we have performed a structural principal-component analysis of helices of different lengths from a representative set of protein folds in the Protein Data Bank. We find three dominant modes of flexibility: two degenerate bend modes and one twist mode. The data are consistent with independent Gaussian distributions for each m…
An intrinsically disordered protein (IDP) lacks a stable three-dimensional structure, while it folds into a specific structure when it binds to a target molecule. In some IDP-target complexes, not all target binding surfaces are exposed on the outside, and intermediate states are observed in their binding processes. We consider that stepwise target recognition via intermediate states is a characteristic of IDP binding to targets with "hidden" binding sites. To investigate IDP binding to hidden target binding sites, we constructed an IDP lattice model based on the HP model. In our model, the ID…
Flexibility is an intrinsic essential feature of protein structures, directly linked to their functions. To this day, most of the prediction methods use the crystallographic data (namely B-factors) as the only indicator of protein's inner flexibility and predicts them as rigid or flexible.PredyFlexy stands differently from other approaches as it relies on the definition of protein flexibility (i) not only taken from crystallographic data, but also (ii) from Root Mean Square Fluctuation (RMSFs) observed in Molecular Dynamics simulations. It also uses a specific representation of protein structu…
Elastic network models (ENM) and constraint-based, topological rigidity analysis are two distinct, coarse-grained approaches to study conformational flexibility of macromolecules. In the two decades since their introduction, both have contributed significantly to insights into protein molecular mechanisms and function. However, despite a shared purpose of these approaches, the topological nature of rigidity analysis, and thereby the absence of motion modes, has impeded a direct comparison. Here, we present an alternative, kinematic approach to rigidity analysis, which circumvents these drawbac…
Side chain flexibility is an important factor in ligand binding. In order to determine the extent to which side chain flexibility is involved in ligand binding, a knowledge-based approach was taken. A database composed of examples of protein structures in the presence or absence of a given ligand is used to analyze which side chains undergo side chain conformational changes. Such an analysis has determined that up to 40% of binding site do not present side chain conformational changes. A total of three residues undergoing side chain conformational changes encompass approximately 85% of the bin…
Flexibility-rigidity index (FRI) has been developed as a robust, accurate and efficient method for macromolecular thermal fluctuation analysis and B-factor prediction. The performance of FRI depends on its formulations of rigidity index and flexibility index. In this work, we introduce alternative rigidity and flexibility formulations. The structure of the classic Gaussian surface is utilized to construct a new type of rigidity index, which leads to a new class of rigidity densities with the classic Gaussian surface as a special case. Additionally, we introduce a new type of flexibility index …
Despite considerable efforts, structural prediction of protein-peptide complexes is still a very challenging task, mainly due to two reasons: high flexibility of the peptides and transient character of their interactions with proteins. Recently we have developed an automated web server CABS-dock (http://biocomp.chem.uw.edu.pl/CABSdock), which conducts flexible protein-peptide docking without any knowledge about the binding site. Our method allows for full flexibility of the peptide, whereas the flexibility of the receptor is restricted to near native conformations considering the main chain, a…
Protein data bank entries obtain distinct, reproducible flexibility characteristics determined by normal mode analyses of their three dimensional coordinate files. We study the effectiveness and sensitivity of this technique by analyzing the results on one class of glycosidases: family 10 xylanases. A conserved tryptophan that appears to affect access to the active site can be in one of two conformations according to X-ray crystallographic electron density data. The two alternate orientations of this active site tryptophan lead to distinct flexibility spectra, with one orientation thwarting th…
The amino acid sequences of proteins provide rich information for inferring distant phylogenetic relationships and for predicting protein functions. Estimating the rate matrix of residue substitutions from amino acid sequences is also important because the rate matrix can be used to develop scoring matrices for sequence alignment. Here we use a continuous time Markov process to model the substitution rates of residues and develop a Bayesian Markov chain Monte Carlo method for rate estimation. We validate our method using simulated artificial protein sequences. Because different local regions s…
Protein folds are built primarily from the packing together of two types of structures: alpha-helices and beta-sheets. Neither structure is rigid, and the flexibility of helices and sheets is often important in determining the final fold ({\it e.g.}, coiled coils and beta-barrels). Recent work has quantified the flexibility of alpha-helices using a principal-component analysis (PCA) of database helical structures (Emberly, 2003). Here, we extend the analysis to beta-sheet flexibility using PCA on a database of beta-sheet structures. For sheets of varying dimension and geometry, we find two dom…
Flexibility-Conditioned Protein Structure Design with Flow Matching
Heidelberg Institute for Theoretical Studies, Heidelberg, Germany（海德堡理论研究所）；IWR, Heidelberg University, Heidelberg, Germany（海德堡大学生物医学研究所）；Max Planck Institute for Polymer Research, Mainz, Germany（马克斯·普朗克聚合物研究所）；Biophysics at Stockholm University（斯德哥尔摩大学生物物理学系）；Science for Life Laboratory, Stockholm, Sweden（生命科学实验室）；IAR, Karlsruhe Institute of Technology, Karlsruhe, Germany（卡尔斯鲁厄理工学院）
Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of topological representations of the molecule. However, persistent homology is rarely {employed} for the analysis of atomic properties, such as biomolecular flexibility analysis or B factor prediction. This work introduces atom-specific persistent homology to provide a local atomic level representation of a molecule via a global topological tool. This is achie…
We explore how inherent flexibility of a protein molecule influences the mechanism controlling the kinetics of allosteric transitions using a variational model inspired from work in protein folding. The striking differences in the predicted transition mechanism for the opening of the two domains of calmodulin (CaM) emphasizes that inherent flexibility is key to understanding the complex conformational changes that occur in proteins. In particular, the C-terminal domain of CaM (cCaM) which is inherently less flexible than its N-terminal domain (nCaM) reveals "cracking" or local partial unfoldin…
Recent work has shown that the hydrophobic protein surfaces in aqueous solution sit near a drying transition. The tendency for these surfaces to expel water from their vicinity leads to self assembly of macromolecular complexes. In this article we show with a realistic model for a biologically pertinent system how this phenomenon appears at the molecular level. We focus on the association of the C-terminal domain (CA-C) of the human immunodeficiency virus (HIV) capsid protein. By combining all-atom simulations with specialized sampling techniques we measure the water density distribution durin…
The ability to precisely visualize the atomic geometry of the interactions between a drug and its protein target in structural models is critical in predicting the correct modifications in previously identified inhibitors to create more effective next generation drugs. It is currently common practice among medicinal chemists while attempting the above to access the information contained in three-dimensional structures by using two-dimensional projections, which can preclude disclosure of useful features. A more precise visualization of the three-dimensional configuration of the atomic geometry…
Proteins can exhibit dynamic structural flexibility as they carry out their functions, especially in binding regions that interact with other molecules. For the key SARS-CoV-2 spike protein that facilitates COVID-19 infection, studies have previously identified several such highly flexible regions with therapeutic importance. However, protein structures available from the Protein Data Bank are presented as static snapshots that may not adequately depict this flexibility, and furthermore these cannot keep pace with new mutations and variants. In this paper we present a sequential Monte Carlo me…
Many biologically important ligands of proteins are large, flexible, and often charged molecules that bind to extended regions on the protein surface. It is infeasible or expensive to locate such ligands on proteins with standard methods such as docking or molecular dynamics (MD) simulation. The alternative approach proposed here is the scanning of a spatial and angular grid around the protein with smaller fragments of the large ligand. Energy values for complete grids can be computed efficiently with a well-known Fast Fourier Transform accelerated algorithm and a physically meaningful interac…
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…
Protein rigidity and flexibility can be analyzed accurately and efficiently using the program FIRST. Previous studies using FIRST were designed to analyze the rigidity and flexibility of proteins using a single static (snapshot) structure. It is however well known that proteins can undergo spontaneous sub-molecular unfolding and refolding, or conformational dynamics, even under conditions that strongly favour a well-defined native structure. These (local) unfolding events result in a large number of conformers that differ from each other very slightly. In this context, proteins are better repr…
The predominant structural protein in vertebrates is collagen, which plays a key role in extracellular matrix and connective tissue mechanics. Despite its prevalence and physical importance in biology, the mechanical properties of molecular collagen are far from established. The flexibility of its triple helix is unresolved, with descriptions from different experimental techniques ranging from flexible to semirigid. Furthermore, it is unknown how collagen type (homo- vs. heterotrimeric) and source (tissue-derived vs. recombinant) influence flexibility. Using SmarTrace, a chain tracing algorith…
We study four citrate synthase homodimeric proteins within a structure-based coarse-grained model. Two of these proteins come from thermophilic bacteria, one from a cryophilic bacterium and one from a mesophilic organism; three are in the closed and two in the open conformations. Even though the proteins belong to the same fold, the model distinguishes the properties of these proteins in a way which is consistent with experiments. For instance, the thermophilic proteins are more stable thermodynamically than their mesophilic and cryophilic homologues, which we observe both in the magnitude of …
The growing interest for comparing protein internal dynamics owes much to the realization that protein function can be accompanied or assisted by structural fluctuations and conformational changes. Analogously to the case of functional structural elements, those aspects of protein flexibility and dynamics that are functionally oriented should be subject to evolutionary conservation. Accordingly, dynamics-based protein comparisons or alignments could be used to detect protein relationships that are more elusive to sequence and structural alignments. Here we provide an account of the progress th…
We explored the Protein DataBank (PDB) to collect protein-ssDNA structures and create a multiconformational docking benchmark including both bound and unbound protein structures. Due to ssDNA high flexibility when not bound, no ssDNA unbound structure is included in the benchmark. For the 91 sequence-identity groups identified as bound-unbound structures of the same protein, we studied the conformational changes in the protein induced by the ssDNA binding. Moreover, based on several bound or unbound protein structures in some groups, we also assessed the intrinsic conformational variability in…
A simple force-probe setup is employed to study the mechanical properties of transcription activator-like effector (TALE) proteins in computer experiments. It is shown that their spring-like arrangement benefits superelastic behaviour which is manifested by large-scale global conformational changes along the helical axis, thus linking structure and dynamics in TALE proteins. As evidenced from the measured force-extension curves the dHax3 and PthXo1 TALEs behave like linear springs, obeying Hooke's law, for moderate global structural changes. For larger deformations, however, the proteins exhib…
Geometric and structural constraints greatly restrict the selection of folds adapted by protein backbones, and yet, folded proteins show an astounding diversity in functionality. For structure to have any bearing on function, it is thus imperative that, apart from the protein backbone, other tunable degrees of freedom be accountable. Here, we focus on side-chain interactions, which non-covalently link amino acids in folded proteins to form a network structure. At a coarse-grained level, we show that the network conforms remarkably well to realizations of random graphs and displays associated p…
ProtAgents: Protein discovery via large language model multi-agent collaborations combining physics and machine learning
Massachusetts Institute of Technology（麻省理工学院）；Laboratory for Atomistic and Molecular Mechanics (LAMM)（原子与分子力学实验室）；Center for Computational Science and Engineering（计算科学与工程中心）；Schwarzman College of Computing（施瓦茨曼计算学院）
