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 having a very large number of long-range contacts (i.e., contacts between amino acids that are separated by more than 12 units of backbone distance). The smaller folding plasticity of `non-local' native geometries is probably a direct consequence of their higher folding cooperativity that renders the folding reaction more robust against single- and multiple-point mutations.
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 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…
Most known proteins in nature consist of multiple domains. Interactions between domains may lead to unexpected folding and misfolding phenomena. This study of human γD-crystallin, a two-domain protein in the eye lens, revealed one such surprise: conformational catalysis of misfolding via intermolecular domain interface ''stealing''. An intermolecular interface between the more stable domains outcompetes the native intramolecular domain interface. Loss of the native interface in turn promotes misfolding and subsequent aggregation, especially in cataract-related γD-crystallin variants. This phen…
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…
We analytically derive the lower bound of the total conformational energy of a protein structure by assuming that the total conformational energy is well approximated by the sum of sequence-dependent pairwise contact energies. The condition for the native structure achieving the lower bound leads to the contact energy matrix that is a scalar multiple of the native contact matrix, i.e., the so-called Go potential. We also derive spectral relations between contact matrix and energy matrix, and approximations related to one-dimensional protein structures. Implications for protein structure predic…
The relevance of various residue positions for the stability and the folding characteristics of the prion protein are investigated by using molecular dynamics simulations of models exploiting the topology of the native state. Highly significant correlations are found between the most relevant sites in our analysis and the single point mutations known to be associated with the arousal of the genetic forms of prion disease (caused by the conformational change from the cellular to the scrapie isoform). Considerable insight into the conformational change is provided by comparing the folding proces…
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…
Natively unstructured proteins defy the classical "one sequence-one structure" paradigm of protein science. Monomers of these proteins in pathological conditions can aggregate in the cell, a process that underlies socially relevant neurodegenerative diseases such as Alzheimer and Parkinson. A full comprehension of the formation and structure of the so-called misfolded intermediates from which the aggregated states ensue is still lacking. We characterized the folding and the conformational diversity of alpha-synuclein (aSyn), a natively unstructured protein involved in Parkinson disease, by mec…
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 …
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…
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…
The elastic network (EN) is a prime model that describes the long-time dynamics of biomolecules. However, the use of harmonic potentials renders this model insufficient for studying large conformational changes of proteins (e.g. stretching of proteins, folding and thermal unfolding). Here, we extend the capabilities of the EN model by using a harmonic approximation described by Lennard-Jones (LJ) interactions for far contacts and native contacts obtained from the standard overlap criterion as in the case of Go-like models. While our model is validated against the EN model by reproducing the eq…
Unavailability of experimental 3D structural data on protein folding dynamics and necessity for a new generation of structure prediction methods in this context
The total conformational energy is assumed to consist of pairwise interaction energies between atoms or residues, each of which is expressed as a product of a conformation-dependent function (an element of a contact matrix, C-matrix) and a sequence-dependent energy parameter (an element of a contact energy matrix, E-matrix). Such pairwise interactions in proteins force native C-matrices to be in a relationship as if the interactions are a Go-like potential [N. Go, Annu. Rev. Biophys. Bioeng. 12. 183 (1983)] for the native C-matrix, because the lowest bound of the total energy function is equal…
In this study, we explore nucleation and the transition state ensemble of the ribosomal protein S6 using a Monte Carlo Go model in conjunction with restraints from experiment. The results are analyzed in the context of extensive experimental and evolutionary data. The roles of individual residues in the folding nucleus are identified and the order of events in the S6 folding mechanism is explored in detail. Interpretation of our results agrees with, and extends the utility of, experiments that shift f-values by modulating denaturant concentration and presents strong evidence for the realism of…
Non-native conformations drive protein misfolding diseases, complicate bioengineering efforts, and fuel molecular evolution. No current experimental technique is well-suited for elucidating them and their phenotypic effects. Especially intractable are the transient conformations populated by intrinsically disordered proteins. We describe an approach to systematically discover, stabilize, and purify native and non-native conformations, generated in vitro or in vivo, and directly link conformations to molecular, organismal, or evolutionary phenotypes. This approach involves high-throughput disul…
Novel numerical techniques, validated by an analysis of barnase and chymotrypsin inhibitor, are used to elucidate the paramount role played by the geometry of the protein backbone in steering the folding to the correct native state. It is found that, irrespective of the sequence, the native state of a protein has exceedingly large number of conformations with a given amount of structural overlap compared to other compact artificial backbones; moreover the conformational entropies of unrelated proteins of the same length are nearly equal at any given stage of folding. These results are suggesti…
We develop a theoretical approach to the protein folding problem based on out-of-equilibrium stochastic dynamics. Within this framework, the computational difficulties related to the existence of large time scale gaps in the protein folding problem are removed and simulating the entire reaction in atomistic details using existing computers becomes feasible. In addition, this formalism provides a natural framework to investigate the relationships between thermodynamical and kinetic aspects of the folding. For example, it is possible to show that, in order to have a large probability to remain u…
An effective potential function is critical for protein structure prediction and folding simulation. Simplified protein models such as those requiring only $C_α$ or backbone atoms are attractive because they enable efficient search of the conformational space. We show residue specific reduced discrete state models can represent the backbone conformations of proteins with small RMSD values. However, no potential functions exist that are designed for such simplified protein models. In this study, we develop optimal potential functions by combining contact interaction descriptors and local sequen…
A global optimization method, conformational space annealing (CSA), is applied to study a 46-residue protein with the sequence B_9N_3(LB)_4N_3B_9N_3(LB)_5L, where B, L and N designate hydrophobic, hydrophilic, and neutral residues, respectively. The 46-residue BLN protein is folded into the native state of a four-stranded beta-barrel. It has been a challenging problem to locate the global minimum of the 46-residue BLN protein since the system is highly frustrated and consequently its energy landscape is quite rugged. The CSA successfully located the global minimum of the 46-mer for all 100 ind…
We apply the general protocol of parameter optimization (Lee, J. et al. Phys. Chem. B 2001, 105, 7291) to the UNRES potential. In contrast to the earlier works where only the relative weights of various interaction terms were optimized, we optimize all linear parameters of the potential. The method exploits the high efficiency of the conformal space annealing method in finding distinct low energy conformations. For a given training set of proteins, the parameters are modified to make the native-like conformations energetically more favorable than the non-native ones. Linear approximation is us…
A new statistical mechanics formulation of characterizing the structural fluctuation of protein correlated with that of water is presented based on the generalized Langevin equation and the 3D-RISM/RISM theory of molecular liquids. The displacement vector of atom positions and their conjugated momentum, are chosen for the dynamic variables for protein, while the density fields of atoms and their momentum fields are chosen for water. Projection of other degrees of freedom onto those dynamic variables using the standard projection operator method produces essentially two equations which describe…
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…
The prediction of the biologically active native conformation of a protein is one of the fundamental challenges of structural biology. This problem remains yet unsolved mainly due to three factors: the partial knowledge of the effective free energy function that governs the folding process, the enormous size of the conformational space of a protein and, finally, the relatively small differences of energy between conformations, in particular, between the native one and the ones that make up the unfolded state. Herein, we recall the importance of taking into account, in a detailed manner, the ma…
A protein is a linear chain containing a set of amino acids, which folds on itself to create a specific native structure, also called the minimum energy conformation. It is the native structure that determines the functionality of each protein. The protein folding problem (PFP) remains one of the more difficult problems in computational and chemical biology. The principal challenge of PFP is to predict the optimal conformation of a given protein by considering only its amino acid sequence. As the conformational space contains a very large number of conformations, even when addressing short seq…
We assume that the protein folding process follows two autonomous steps: the conformational search for the native, mainly ruled by the hydrophobic effect; and, the final adjustment stage, which eventually gives stability to the native. Our main tool of investigation is a 3D lattice model provided with a ten-letter alphabet, the stereochemical model. This model was conceived for Monte Carlo (MC) simulations when one keeps in mind the kinetic behavior of protein-like chains in solution. In order to characterize the folding characteristic time (τ) by two distinct sampling methods, first we presen…
Despite the spontaneity of some in vitro protein folding reactions, native folding in vivo often requires the participation of barrel-shaped multimeric complexes known as chaperonins. Although it has long been known that chaperonin substrates fold upon sequestration inside the chaperonin barrel, the precise mechanism by which confinement within this space facilitates folding remains unknown. In this study, we examine the possibility that the chaperonin mediates a favorable reorganization of the solvent for the folding reaction. We begin by discussing the effect of electrostatic charge on solve…
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…
Intrinsically disordered proteins participate in many biological processes by folding upon binding with other proteins. However, coupled folding and binding processes are not well understood from an atomistic point of view. One of the main questions is whether folding occurs prior to or after binding. Here we use a novel unbiased high-throughput adaptive sampling approach to reconstruct the binding and folding between the disordered transactivation domain of \mbox{c-Myb} and the KIX domain of the CREB-binding protein. The reconstructed long-term dynamical process highlights the binding of a sh…
The prediction of the three-dimensional native structure of proteins from the knowledge of their amino acid sequence, known as the protein folding problem, is one of the most important yet unsolved issues of modern science. Since the conformational behaviour of flexible molecules is nothing more than a complex physical problem, increasingly more physicists are moving into the study of protein systems, bringing with them powerful mathematical and computational tools, as well as the sharp intuition and deep images inherent to the physics discipline. This work attempts to facilitate the first ste…
First shells of hydration and bulk solvent plays a crucial role in the folding of proteins. Here, the role of water in the dynamics of proteins has been investigated using a theoretical protein-solvent model and a statistical physics approach. We formulate a hydration model where the hydrogen bonds between water molecules pertaining to the first shell of the protein conformation may be either mainly formed or broken. At thermal equilibrium, hydrogen bonds are formed at low temperature and are broken at high temperature. To explore the solvent effect, we follow the folding of a large sampling o…
