The importance of torsion vibration in the transmission of life information is indicated. The localization of quantum torsion state is proved. Following these analyses a formalism on the quantum theory of conformation-electron system is proposed. The conformational-electronic transition is calculated by non-adiabatic operator method. The protein folding is viewed from conformational quantum transition and the folding rate is calculated. The time-scale of microsecond to millisecond for the fundamental folding event (nucleation, collapse, etc) is deduced. The dependence of transition rate W on N inertial moments is given. It indicates how W increases with the number N of torsion angles and decreases with the inertial moment I of atomic group in cooperative transition. The temperature dependence is also deduced which is different from chemical reaction in high-temperature region. It is demonstrated that the conformational dynamics gives deep insights into the folding mechanism and provides a useful tool for analyzing and explaining experimental facts on the rate of protein folding.
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…
Intrinsically disordered proteins (IDPs) and multidomain proteins with flexible linkers show a high level of structural heterogeneity and are best described by ensembles consisting of multiple conformations with associated thermodynamic weights. Determining conformational ensembles usually involves integration of biophysical experiments and computational models. In this review, we discuss current approaches to determining conformational ensembles of IDPs and multidomain proteins, including the choice of biophysical experiments, computational models used to sample protein conformations, 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.
The protein folding is regarded as a quantum transition between torsion states on polypeptide chain. The deduction of the folding rate formula in our previous studies is reviewed. The rate formula is generalized to the case of frequency variation in folding. Then the following problems about the application of the rate theory are discussed: 1) The unified theory on the two-state and multi-state protein folding is given based on the concept of quantum transition. 2) The relationship of folding and unfolding rates vs denaturant concentration is studied. 3) The temperature dependence of folding r…
Although the importance of protein dynamics in protein function is generally recognized, the role of protein fluctuations in allosteric effects scarcely has been considered. To address this gap, the Kullback-Leibler divergence (Dx) between protein conformational distributions before and after ligand binding was proposed as a means of quantifying allosteric effects in proteins. Here, previous applications of Dx to methods for analysis and simulation of proteins are first reviewed, and their implications for understanding aspects of protein function and protein evolution are discussed. Next, equ…
In protein folding the term plasticity refers to the number of alternative folding pathways encountered in response to free energy perturbations such as those induced by mutation. Here we explore the relation between folding plasticity and a gross, generic feature of the native geometry, namely, the relative number of local and non-local native contacts. The results from our study, which is based on Monte Carlo simulations of simple lattice proteins, show that folding to a structure that is rich in local contacts is considerably more plastic than folding to a native geometry characterized by h…
Protein folding is a problem of large interest since it concerns the mechanism by which the genetic information is translated into proteins with well defined three-dimensional (3D) structures and functions. Recently theoretical models have been developed to predict the protein folding rate considering the relationships of the process with tolopological parameters derived from the native (atomic-solved) protein structures. Previous works classified proteins in two different groups exhibiting either a single-exponential or a multi-exponential folding kinetics. It is well known that these two cla…
The tertiary structure of protein, as well as the local secondary structure organization are fully determined by the angles of the peptidic bound. The backbone dihedral angles not only determine the global fold of the protein, but also the details of the local chain organization. Although a wealth of structural information is available in different databases and numerous structural biology softwares have been developed, rapid conformational characterization remains challenging. We present here RamaDA, a program able to give a synthetic description of the conformation of a protein. The RamaDA p…
Quantum mechanical calculations are performed on 116 conformers of the protein ubiquitin (Lange et al., Science 2008, 320, 1471-1475). The results indicate that the heat of formation (HOF), dipole moment, energy of the frontier orbitals HOMO and LUMO, and HOMO-LUMO gap fluctuate within their corresponding ranges. This study thus provides a link between the conformational dynamics of a protein and its electronic structure.
We use plasmon rulers to follow the conformational dynamics of a single protein for up to 24 h at a video rate. The plasmon ruler consists of two gold nanospheres connected by a single protein linker. In our experiment, we follow the dynamics of the molecular chaperone heat shock protein 90, which is known to show open and closed conformations. Our measurements confirm the previously known conformational dynamics with transition times in the second to minute time scale and reveals new dynamics on the time scale of minutes to hours. Plasmon rulers thus extend the observation bandwidth 3/4 order…
CLeFAPS, a fast and flexible pairwise structural alignment algorithm based on a rigid-body framework, namely CLePAPS, is proposed. Instead of allowing twists (or bends), the flexible in CLeFAPS means: (a) flexibilization of the algorithm's parameters through self-adapting with the input structures' size, (b) flexibilization of adding the aligned fragment pairs (AFPs) into an one-to-multi correspondence set instead of checking their position conflict, (c) flexible fragment may be found through an elongation procedure rooted in a vector-based score instead of a distance-based score. We perform a…
We study the conformational freedom of a protein made by two rigid domains connected by a flexible linker. The conformational freedom is represented as an unknown probability distribution on the space of allowed states. A new algorithm for the calculation of the Maximum Allowable Probability is proposed, which can be extended to any type of measurements. In this paper we use Pseudo Contact Shifts and Residual Dipolar Coupling. We reconstruct a single central tendency in the distribution and discuss in depth the results.
Protein binding and function often involves conformational changes. Advanced NMR experiments indicate that these conformational changes can occur in the absence of ligand molecules (or with bound ligands), and that the ligands may 'select' protein conformations for binding (or unbinding). In this review, we argue that this conformational selection requires transition times for ligand binding and unbinding that are small compared to the dwell times of proteins in different conformations, which is plausible for small ligand molecules. Such a separation of timescales leads to a decoupling and tem…
Conformational changes upon protein-protein association are the key element of the binding mechanism. The study presents a systematic large-scale analysis of such conformational changes in the side chains. The results indicate that short and long side chains have different propensities for the conformational changes. Long side chains with three or more dihedral angles are often subject to large conformational transition. Shorter residues with one or two dihedral angles typically undergo local conformational changes not leading to a conformational transition. The relationship between the local …
The conformational change of biological macromolecule is investigated from the point of quantum transition. A quantum theory on protein folding is proposed. Compared with other dynamical variables such as mobile electrons, chemical bonds and stretching-bending vibrations the molecular torsion has the lowest energy and can be looked as the slow variable of the system. Simultaneously, from the multi-minima property of torsion potential the local conformational states are well defined. Following the idea that the slow variables slave the fast ones and using the nonadiabaticity operator method we …
Using the perturbation-response scanning (PRS) technique, we study a set of 23 proteins that display a variety of conformational motions upon ligand binding (e.g. shear, hinge, allosteric). In most cases, PRS determines residues that may be manipulated to achieve the resulting conformational change. PRS reveals that for some proteins, binding induced conformational change may be achieved through the perturbation of residues scattered throughout the protein, whereas in others, perturbation of specific residues confined to a highly specific region are necessary. Correlations between the experime…
Protein folding produces characteristic and functional three-dimensional structures from unfolded polypeptides or disordered coils. The emergence of extraordinary complexity in the protein folding process poses astonishing challenges to theoretical modeling and computer simulations. The present work introduces molecular nonlinear dynamics (MND), or molecular chaotic dynamics, as a theoretical framework for describing and analyzing protein folding. We unveil the existence of intrinsically low dimensional manifolds (ILDMs) in the chaotic dynamics of folded proteins. Additionally, we reveal that …
The SARS-CoV-2 spike (S) protein facilitates viral infection, and has been the focus of many structure determination efforts. Its flexible loop regions are known to be involved in protein binding and may adopt multiple conformations. This paper identifies the S protein loops and studies their conformational variability based on the available Protein Data Bank (PDB) structures. While most loops had essentially one stable conformation, 17 of 44 loop regions were observed to be structurally variable with multiple substantively distinct conformations based on a cluster analysis. Loop modeling meth…
An approach that combines Self-Organizing maps, hierarchical clustering and network components is presented, aimed at comparing protein conformational ensembles obtained from multiple Molecular Dynamic simulations. As a first result the original ensembles can be summarized by using only the representative conformations of the clusters obtained. In addition the network components analysis allows to discover and interpret the dynamic behavior of the conformations won by each neuron. The results showed the ability of this approach to efficiently derive a functional interpretation of the protein d…
The Electron Microscopy Data Bank (EMDB) is a rapidly growing repository for the dissemination of structural data from single-particle reconstructions of supramolecular protein assemblies including motors, chaperones, cytoskeletal assemblies, and viral capsids. While the static structure of these assemblies provides essential insight into their biological function, their conformational dynamics and mechanics provide additional important information regarding the mechanism of their biological function. Here, we present an unsupervised computational framework to analyze and store for public acce…
A protein undergoes conformational dynamics with multiple time scales, which results in fluctuating enzyme activities. Recent studies in single molecule enzymology have observe this "age-old" dynamic disorder phenomenon directly. However, the single molecule technique has its limitation. To be able to observe this molecular effect with real biochemical functions {\it in situ}, we propose to couple the fluctuations in enzymatic activity to noise propagations in small protein interaction networks such as zeroth order ultra-sensitive phosphorylation-dephosphorylation cycle. We showed that enzyme …
The rates of protein folding with photon absorption or emission and the cross section of photon -protein inelastic scattering are calculated from the quantum folding theory by use of standard field-theoretical method. All these protein photo-folding processes are compared with common protein folding without interaction of photons (nonradiative folding). It is demonstrated that there exists a common factor (thermo-averaged overlap integral of vibration wave function, TAOI) for protein folding and protein photo-folding. Based on this finding it is predicted that: 1) the stimulated photo-folding …
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…
A mathematico-physically valid formulation is required to infer properties of disordered protein conformations from single-molecule Förster resonance energy transfer (smFRET). Conformational dimensions inferred by conventional approaches that presume a homogeneous conformational ensemble can be unphysical. When all possible---heterogeneous as well as homogeneous---conformational distributions are taken into account without prejudgement, a single value of average transfer efficiency $\langle E\rangle$ between dyes at two chain ends is generally consistent with highly diverse, multiple values of…
Wayne State University（韦恩州立大学）；Univ. of Bari Aldo Moro（巴里大学阿尔多·莫罗分校）；National Research Council (CONICET)（国家研究委员会（CONICET））；Daruma Institute for Applied Intelligence（达摩应用智能研究所）
All-atom and coarse-grained molecular dynamics are two widely used computational tools to study the conformational states of proteins. Yet, these two simulation methods suffer from the fact that without access to supercomputing resources, the time and length scales at which these states become detectable are difficult to achieve. One alternative to such methods is based on encoding the atomistic trajectory of molecular dynamics as a shorthand version devoid of physical particles, and then learning to propagate the encoded trajectory through the use of artificial intelligence. Here we show that…
The assumption that similar structures have similar folding probabilities ($p_{fold}$) leads naturally to a procedure to evaluate $p_{fold}$ for every snapshot saved along an equilibrium folding-unfolding trajectory of a structured peptide or protein. The procedure utilizes a structurally homogeneous clustering and does not require any additional simulation. It can be used to detect multiple folding pathways as shown for a three-stranded antiparallel $β$-sheet peptide investigated by implicit solvent molecular dynamics simulations.
Many proteins carry out their biological functions by forming the characteristic tertiary structures. Therefore, the search of the stable states of proteins by molecular simulations is important to understand their functions and stabilities. However, getting the stable state by conformational search is difficult, because the energy landscape of the system is characterized by many local minima separated by high energy barriers. In order to overcome this difficulty, various sampling and optimization methods for conformations of proteins have been proposed. In this study, we propose a new conform…
Kinetics of folding of a protein held in a force-clamp are compared to an unconstrained folding. The comparison is made within a simple topology-based dynamical model of ubiquitin. We demonstrate that the experimentally observed variations in the end-to-end distance reflect microscopic events during folding. However, the folding scenarios in and out of the force-clamp are distinct.
As protein folding is a NP-complete problem, artificial intelligence tools like neural networks and genetic algorithms are used to attempt to predict the 3D shape of an amino acids sequence. Underlying these attempts, it is supposed that this folding process is predictable. However, to the best of our knowledge, this important assumption has been neither proven, nor studied. In this paper the topological dynamic of protein folding is evaluated. It is mathematically established that protein folding in 2D hydrophobic-hydrophilic (HP) square lattice model is chaotic as defined by Devaney. Consequ…
Understanding protein folding has been one of the great challenges in biochemistry and molecular biophysics. Over the past 50 years, many thermodynamic and kinetic studies have been performed addressing the stability of globular proteins. In comparison, advances in the membrane protein folding field lag far behind. Although membrane proteins constitute about a third of the proteins encoded in known genomes, stability studies on membrane proteins have been impaired due to experimental limitations. Furthermore, no systematic experimental strategies are available for folding these biomolecules in…
We combined the genetic crossover, which is one of the operations of genetic algorithm, and replica-exchange method in parallel molecular dynamics simulations. The genetic crossover and replica-exchange method can search the global conformational space by exchanging the corresponding parts between a pair of conformations of a protein. In this study, we applied this method to an $α$-helical protein, Trp-cage mini protein, which has 20 amino-acid residues. The conformations obtained from the simulations are in good agreement with the experimental results.
In this conceptual paper we propose to explore the analogy between ontic/epistemic description of quantum phenomena and interrelation between dynamics of conformational and functional states of proteins. Another new idea is to apply theory of automata to model the latter dynamics. In our model protein's behavior is modeled with the aid of two dynamical systems, ontic and epistemic, which describe evolution of conformational and functional states of proteins, respectively. The epistemic automaton is constructed from the ontic automaton on the basis of functional (observational) equivalence rela…
Simple theoretical concepts and models have been helpful to understand the folding rates and routes of single-domain proteins. As reviewed in this article, a physical principle that appears to underly these models is loop closure.
We perform molecular dynamics simulations for a simple coarse-grained model of crambin placed inside of a softly repulsive sphere of radius R. The confinement makes folding at the optimal temperature slower and affects the folding scenarios, but both effects are not dramatic. The influence of crowding on folding are studied by placing several identical proteins within the sphere, denaturing them, and then by monitoring refolding. If the interactions between the proteins are dominated by the excluded volume effects, the net folding times are essentially like for a single protein. An introductio…
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…
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…
Protein binding often involves conformational changes. Important questions are whether a conformational change occurs prior to a binding event ('conformational selection') or after a binding event ('induced fit'), and how conformational transition rates can be obtained from experiments. In this article, we present general results for the chemical relaxation rates of conformational-selection and induced-fit binding processes that hold for all concentrations of proteins and ligands and, thus, go beyond the standard pseudo-first-order approximation of large ligand concentration. These results all…
Motivation: Proteins are known to undergo conformational changes in the course of their functions. The changes in conformation are often attributable to a small fraction of residues within the protein. Therefore identification of these variable regions is important for an understanding of protein function. Results: We propose a novel method for identification of local conformational changes in proteins. In our method, backbone conformations are encoded into a sequence of letters from a 16-letter alphabet (called D2 codes) to perform structural comparison. Since we do not use clustering analysi…
For the vast majority of naturally occurring, small, single domain proteins folding is often described as a two-state process that lacks detectable intermediates. This observation has often been rationalized on the basis of a nucleation mechanism for protein folding whose basic premise is the idea that after completion of a specific set of contacts forming the so-called folding nucleus the native state is achieved promptly. Here we propose a methodology to identify folding nuclei in small lattice polymers and apply it to the study of protein molecules with chain length N=48. To investigate the…
A fundamental question in protein folding is whether the coil to globule collapse transition occurs during the initial stages of folding (burst-phase) or simultaneously with the protein folding transition. Single molecule fluorescence resonance energy transfer (FRET) and small angle X-ray scattering (SAXS) experiments disagree on whether Protein L collapse transition occurs during the burst-phase of folding. We study Protein L folding using a coarse-grained model and molecular dynamics simulations. The collapse transition in Protein L is found to be concomitant with the folding transition. In …
We investigate the folding behavior of protein sequences by numerically studying all sequences with maximally compact lattice model through exhaustive enumeration. We get the prion-like behavior of protein folding. Individual proteins remaining stable in the isolated native state may change their conformations when they aggregate. We observe the folding properties as the interfacial interaction strength changes, and find that the strength must be strong enough before the propagation of the most stable structures happens.
The binding of a ligand molecule to a protein is often accompanied by conformational changes of the protein. A central question is whether the ligand induces the conformational change (induced-fit), or rather selects and stabilizes a complementary conformation from a pre-existing equilibrium of ground and excited states of the protein (selected-fit). We consider here the binding kinetics in a simple four-state model of ligand-protein binding. In this model, the protein has two conformations, which can both bind the ligand. The first conformation is the ground state of the protein when the liga…
A quantum theory on conformation-electron system is presented. Protein folding is regarded as the quantum transition between torsion states on polypeptide chain, and the folding rate is calculated by nonadiabatic operator method. The theory is used to study the temperature dependences of folding rate of 15 proteins and their non-Arrhenius behavior can all be deduced in a natural way. A general formula on the rate-temperature dependence has been deduced which is in good accordance with experimental data. These temperature dependences are further analyzed in terms of torsion potential parameters…
Exploring and understanding the protein-folding problem has been a long-standing challenge in molecular biology. Here, using molecular dynamics simulation, we reveal how parallel distributed adjacent planar peptide groups of unfolded proteins fold reproducibly following explicit physical folding codes in aqueous environments due to electrostatic attractions. Superfast folding of protein is found to be powered by the contribution of the formation of hydrogen bonds. Temperature-induced torsional waves propagating along unfolded proteins break the parallel distributed state of specific amino acid…
The current capacity of computers makes it possible to perform simulations of small systems with portable, explicit-solvent potentials achieving high degree of accuracy. However, simplified models must be employed to exploit the behaviour of large systems or to perform systematic scans of smaller systems. While powerful algorithms are available to facilitate the sampling of the conformational space, successful applications of such models are hindered by the availability of simple enough potentials able to satisfactorily reproduce known properties of the system. We develop an interatomic potent…
A growing number of experimental evidence shows that it is general for a ligand binding protein to have a potential for allosteric regulation and for further evolution. In addition, such proteins generically change their conformation upon binding. O. Rivoire has recently proposed an evolutionary scenario that explains these properties as a generic byproduct of selection for exquisite discrimination between very similar ligands. The initial claim was supported by two classes of basic examples: continuous protein models with small numbers of degrees of freedom, on which the development of a conf…
Nonnative residual interactions have attracted increasing attention in recent protein folding researches. Experimental and theoretical investigations had been set out to catch nonnative contacts that might dominate key events in protein folding. However, energetic frustrations caused by nonnative inter-residue interactions are not systematically characterized, due to the complicated folding mechanism. Recently, we studied the folding of a set of homologous all-α proteins and found that nonnative-contact-based energetic frustrations are closely related to protein native-contact networks. In thi…
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…
We investigated the impact of hydrodynamic interactions (HI) on protein folding using a coarse-grained model. The extent of the impact of hydrodynamic interactions, whether it accelerates, retards, or has no effect on protein folding, has been controversial. Together with a theoretical framework of the energy landscape theory (ELT) for protein folding that describes the dynamics of the collective motion with a single reaction coordinate across a folding barrier, we compared the kinetic effects of HI on the folding rates of two protein models that use a chain of single beads with distinctive to…
The fundamental law for protein folding is the Thermodynamic Principle: the amino acid sequence of a protein determines its native structure and the native structure has the minimum Gibbs free energy. If all chemical problems can be answered by quantum mechanics, there should be a quantum mechanics derivation of Gibbs free energy formula G(X) for every possible conformation X of the protein. We apply quantum statistics to derive such a formula. For simplicity, only monomeric self folding globular proteins are covered. We point out some immediate applications of the formula. We show that the fo…
Statistical thermodynamics basis of energy and residue position fluctuations is explained for native proteins. The protein and its surroundings are treated as a canonical system with emphasis on the effects of energy exchange between the two. Fluctuations of the energy are related to fluctuations of residue positions, which in turn are related to the connectivity matrix of the protein, thus establishing a connection between energy fluctuation pathways and protein architecture. The model gives the locations of hotspots for ligand binding, and identifies the pathways of energy conduction within …
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 …
This chapter describes the application of constrained geometric simulations for prediction of antibody structural dynamics. We utilize constrained geometric simulations method FRODAN, which is a low computational complexity alternative to Molecular Dynamics (MD) simulations that can rapidly explore flexible motions in protein structures. FRODAN is highly suited for conformational dynamics analysis of large proteins, complexes, intrinsically disordered proteins and dynamics that occurs on longer biologically relevant time scales which are normally inaccessible to classical MD simulations. This …
We review some of our recent results obtained within the scope of simple lattice models and Monte Carlo simulations that illustrate the role of native geometry in the folding kinetics of two state folders.
