Experimental observations suggest that proteins follow different pathways under different environmental conditions. We perform molecular dynamics simulations of a model of the SH3 domain over a broad range of temperatures, and identify distinct pathways in the folding transition. We determine the kinetic partition temperature --the temperature for which the SH3 domain undergoes a rapid folding transition with minimal kinetic barriers-- and observe that below this temperature the model protein may undergo a folding transition via multiple folding pathways. The folding kinetics is characterized by slow and fast pathways and the presence of only one or two intermediates. Our findings suggest the hypothesis that the SH3 domain, a protein for which only two-state folding kinetics was observed in previous experiments, may exhibit intermediates states under extreme experimental conditions, such as very low temperatures. A very recent report (Viguera et al., Proc. Natl. Acad. Sci. USA, 100:5730--5735, 2003) of an intermediate in the folding transition of the Bergerac mutant of the alpha-spectrin SH3 domain protein supports this hypothesis.
A small model polypeptide represented in atomic detail is folded using Monte Carlo dynamics. The polypeptide is designed to have a native conformation similar to the central part of the helix-turn-helix protein ROP. Starting from a beta-strand conformation or two different loop conformations of the protein glutamine synthetase, six trajectories are generated using the so-called window move in dihedral angle space. This move changes conformations locally and leads to realistic, quasi-continuously evolving trajectories. Four of the six trajectories end in stable native-like conformations. Their …
We introduce a method for calculating the extent to which chain non-crossing is important in the most efficient, optimal trajectories or pathways for a protein to fold. This involves recording all unphysical crossing events of a ghost chain, and calculating the minimal uncrossing cost that would have been required to avoid such events. A depth-first tree search algorithm is applied to find minimal transformations to fold $α$, $β$, $α/β$, and knotted proteins. In all cases, the extra uncrossing/non-crossing distance is a small fraction of the total distance travelled by a ghost chain. Different…
The functions of RNA pseudoknots (PKs), which are minimal tertiary structural motifs and an integral part of several ribozymes and ribonucleoprotein complexes, are determined by their structure, stability and dynamics. Therefore, it is important to elucidate the general principles governing their thermodynamics/folding mechanisms. Here, we combine experiments and simulations to examine the folding/unfolding pathways of the VPK pseudoknot, a variant of the Mouse Mammary Tumor Virus (MMTV) PK involved in ribosomal frameshifting. Fluorescent nucleotide analogs (2-aminopurine and pyrrolocytidine) …
We study the folding process in the shallowly knotted protein MJ0366 within two variants of a structure-based model. We observe that the resulting topological pathways are much richer than identified in previous studies. In addition to the single knot-loop events, we find novel, and dominant, two-loop mechanisms. We demonstrate that folding takes place in a range of temperatures and the conditions of most successful folding are at temperatures which are higher than those required for the fastest folding. We also demonstrate that nascent conditions are more favorable to knotting than off-riboso…
We study the protein folding problem on the base of the quantum approach we proposed recently by considering the model of protein chain with nine amino-acid residues. We introduced the concept of distance space and its projections on a $XY$-plane, and two characteristic quantities, one is called compactness of protein structure and another is called probability ratio involving shortest path. Our results not only confirmed the fast quantum folding time but also unveiled the existence of quantum intelligence hidden behind in choosing protein folding pathways.
Two proteins, one belonging to the mainly alpha class and the other belonging to the alpha/beta class, are selected to test a kinetic mechanism for protein folding. Targeted molecular dynamics is applied to generate folding pathways for those two proteins, starting from two well defined initial conformations: a fully extended and a alpha-helical conformation. The results show that for both proteins the alpha-helical initial conformation provides overall lower energy pathways to the native state. For the alpha/beta protein, 30 % (40%) of the pathways from an initial alpha-helix (fully extended)…
We report on atomistic simulation of the folding of a natively-knotted protein, MJ0366, based on a realistic force field. To the best of our knowledge this is the first reported effort where a realistic force field is used to investigate the folding pathways of a protein with complex native topology. By using the dominant-reaction pathway scheme we collected about 30 successful folding trajectories for the 82-amino acid long trefoil-knotted protein. Despite the dissimilarity of their initial unfolded configuration, these trajectories reach the natively-knotted state through a remarkably simila…
We introduce a method for predicting RNA folding pathways, with an application to the most important RNA tetraloops. The method is based on the idea that ensembles of three-dimensional fragments extracted from high-resolution crystal structures are heterogeneous enough to describe metastable as well as intermediate states. These ensembles are first validated by performing a quantitative comparison against available solution NMR data of a set of RNA tetranucleotides. Notably, the agreement is better with respect to the one obtained by comparing NMR with extensive all-atom molecular dynamics sim…
We propose a general theory to describe the distribution of protein-folding transition paths. We show that transition paths follow a predictable sequence of high-free-energy transient states that are separated by free-energy barriers. Each transient state corresponds to the assembly of one or more discrete, cooperative units, which are determined directly from the native structure. We show that the transition state on a folding pathway is reached when a small number of critical contacts are formed between a specific set of substructures, after which folding proceeds downhill in free energy. Th…
We introduce a powerful iterative algorithm to compute protein folding pathways, with realistic all-atom force fields. Using the path integral formalism, we explicitly derive a modified Langevin equation which samples directly the ensemble of reactive pathways, exponentially reducing the cost of simulating thermally activated transitions. The algorithm also yields a rigorous stochastic estimate of the reaction coordinate. After illustrating this approach on a simple toy model, we successfully validate it against the results of ultra-long plain MD protein folding simulations for a fast folding …
Three-dimensional interphase organization of metazoan genomes has been linked to cellular identity. However, the principles governing 3D interphase genome architecture and its faithful transmission through disruptive events of cell-cycle, like mitosis, are not fully understood. By using Brownian dynamics simulations of Drosophila chromosome 3R up to time-scales of minutes, we show that chromatin binding profile of Polycomb-repressive-complex-1 robustly predicts a sub-set of topologically associated domains (TADs), and inclusion of other factors recapitulates the profile of all TADs, as observe…
Recent advances in computational power and simulation programs finally delivered the first examples of reversible folding for small proteins with an all-atom description. But having at hand the atomistic details of the process did not lead to a straightforward interpretation of the mechanism. For the case of the Fip35 WW-domain where multiple long trajectories of 100 μs are available from D. E. Shaw Research, different interpretations emerged. Some of those are in clear contradiction with each other while others are in qualitative agreement. Here, we present a network-based analysis of the sam…
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…
Theoretical studies of protein folding on lattice models relie on the assumption that water close to amino-acids is always in thermal equilibrium all along the folding pathway. Within this framework, it has always been considered that out-of-equilibrium properties, such as folding time, could be evaluated equivalently from an averaging over a collection of trajectories of the protein with water described eitherexplicitly or through a mean-field approach. To critically assess this hypothesis, we built a two-dimensional lattice model of a protein in interaction with water molecules that can adop…
Newly-translated glycoproteins in the endoplasmic reticulum (ER) often undergo cycles of chaperone binding and release in order to assist in folding. Quality control is required to distinguish between proteins that have completed native folding, those that have yet to fold, and those that have misfolded. Using quantitative modeling, we explore how the design of the quality-control pathway modulates its efficiency. Our results show that an energy-consuming cyclic quality-control process, similar to the observed physiological system, outperforms alternative designs. The kinetic parameters that o…
The process of protein folding from an unfolded state to a biologically active, folded conformation is governed by many parameters e.g the sequence of amino acids, intermolecular interactions, the solvent, temperature and chaperon molecules. Our study, based on random matrix modeling of the interactions, shows however that the evolution of the statistical measures e.g Gibbs free energy, heat capacity, entropy is single parametric. The information can explain the selection of specific folding pathways from an infinite number of possible ways as well as other folding characteristics observed in …
A kinetic model for the nucleation mechanism of protein folding is proposed. A protein is modeled as a heteropolymer consisting of hydrophobic and hydrophilic beads with equal constant bond lengths and bond angles. The total energy of the heteropolymer is determined by the repulsive/attractive interactions of non-linked beads and the contribution from the dihedral angles involved. Their parameters can be rigorously defined, unlike the ill defined surface tension of a cluster of protein residues which is the basis of the previous model. As a crucial idea of the model, the dihedral potential in …
A major challenge in molecular simulations is to describe denaturant-dependent folding of proteins order to make direct comparisons with {\it in vitro} experiments. We use the molecular transfer model, which is currently the only method that accomplishes this goal albeit phenomenologically, to quantitatively describe urea-dependent folding of PDZ domain, which plays a significant role in molecular recognition and signaling. Experiments show that urea-dependent unfolding rates of the PDZ2 domain exhibit a downward curvature at high urea concentrations, which has been interpreted by invoking the…
We investigate the interplay between post-translational folding and escape of two small single-domain proteins at the ribosomal exit tunnel by using Langevin dynamics with coarse-grained models. It is shown that at temperatures lower or near the temperature of the fastest folding, folding proceeds concomitantly with the escape process, resulting in vectorial folding and enhancement of foldability of nascent proteins. The concomitance between the two processes, however, deteriorates as temperature increases. Our folding simulations as well as free energy calculation by using umbrella sampling s…
A model to describe the mechanism of conformational dynamics in secondary protein based on matter interactions is proposed. The approach deploys the lagrangian method by imposing certain symmetry breaking. The protein backbone is initially assumed to be nonlinear and represented by the Sine-Gordon equation, while the nonlinear external bosonic sources is represented by $ϕ^4$ interaction. It is argued that the nonlinear source induces the folding pathway in a different way than the previous work with initially linear backbone. Also, the nonlinearity of protein backbone decreases the folding spe…
Self-assembly of polypeptides into fibrillar structures can be initiated by planar surfaces that interact favorably with certain residues. Using a coarse grained model, we systematically studied the folding and adsorption behavior of a $β$-roll forming polypeptide. We find that there are two different folding pathways depending on the temperature: (i) at low temperature, the polypeptide folds in solution into a $β$-roll before adsorbing onto the attractive surface, (ii) at higher temperature, the polypeptide first adsorbs in a disordered state, and folds while on the surface. The folding tempe…
Atomically detailed simulations of RNA folding have proven very challenging in view of the difficulties of developing realistic force fields and the intrinsic computational complexity of sampling rare conformational transitions. To tackle both these issues, we extend to RNA an enhanced path sampling method previously successfully applied to proteins. In this scheme, the information about the RNA's native structure is harnessed by a soft history-dependent biasing force, which is added to the atomistic force field, thus promoting the generation of productive folding trajectories. Here, we report…
We review theoretical approaches, experiments and numerical simulations that have been recently proposed to investigate the folding problem in single-domain proteins. From a theoretical point of view, we emphasize the energy landscape approach. As far as experiments are concerned, we focus on the recent development of single-molecule techniques. In particular, we compare the results obtained with two main techniques: single protein force measurements with optical tweezers and single-molecule fluorescence in studies on the same protein (RNase H). This allows us to point out some controversial i…
We present a new computational approach, Action-CSA, to sample multiple reaction pathways with fixed initial and final states through global optimization of the Onsager-Machlup action using the conformational space annealing method. This approach successfully samples not only the most dominant pathway but also many other possible paths without initial guesses on reaction pathways. Pathway space is efficiently sampled by crossover operations of a set of paths and preserving the diversity of sampled pathways. The sampling ability of the approach is assessed by finding pathways for the conformati…
A model to describe the mechanism of conformational dynamics in protein based on matter interactions using lagrangian approach and imposing certain symmetry breaking is proposed. Both conformation changes of proteins and the injected non-linear sources are represented by the bosonic lagrangian with an additional ϕ^4 interaction for the sources. In the model the spring tension of protein representing the internal hydrogen bonds is realized as the interactions between individual amino acids and nonlinear sources. The folding pathway is determined by the strength of nonlinear sources that propaga…
We explore the consequences of very high dimensionality in the dynamical landscape of protein folding. Consideration of both typical range of stabilising interactions, and folding rates themselves, leads to a model of the energy hypersurface that is characterised by the structure of diffusive "hypergutters" as well as the familiar "funnels". Several general predictions result: (1) intermediate subspaces of configurations will always be visited; (2) specific but non-native interactions are important in stabilising these low-dimensional diffusive searches on the folding pathway; (3) sequential b…
The enterobacteria lambda phage is a paradigm temperate bacteriophage. Its lysogenic and lytic life cycles echo competition between the DNA binding $λ$-repressor (CI) and CRO proteins. Here we scrutinize the structure, stability and folding pathways of the $λ$-repressor protein, that controls the transition from the lysogenic to the lytic state. We first investigate the super-secondary helix-loop-helix composition of its backbone. We use a discrete Frenet framing to resolve the backbone spectrum in terms of bond and torsion angles. Instead of four, there appears to be seven individual loops. W…
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…
Collapse of the polypeptide backbone is an integral part of protein folding. Using polyglycine as a probe, we explore the nonequilibrium pathways of protein collapse in water. We find that the collapse depends on the competition between hydration effects and intra-peptide interactions. Once intra-peptide van der Waal interactions dominate, the chain collapses along a nonequilibrium pathway characterized by formation of pearl-necklace-like local clusters as intermediates that eventually coagulate into a single globule. By describing this coarsening through the contact probability as a function …
We discuss the ``soft-ratcheting'' algorithm which generates targeted stochastic trajectories in molecular systems with scores corresponding to their probabilities. The procedure, which requires no initial pathway guess, is capable of rapidly determining multiple pathways between known states. Monotonic progress toward the target state is not required. The soft-ratcheting algorithm is applied to an all-atom model of alanine dipeptide, whose unbiased trajectories are assumed to follow overdamped Langevin dynamics. All possible pathways on the two-dimensional dihedral surface are determined. The…
The thesis examines in detail the folding and unfolding processes of a number of proteins including hbSBD, DDLNF4, single and multi Ubiquitin. Using simplified coarse-grained off-lattice Go model and CD experiments we have shown the two-state behavior of hbSBD protein. It was shown that refolding pathways of single Ubiquitin depend on what end is anchored to the surface. Namely, the fixation of the N-terminal changes refolding pathways but anchoring the C-terminal leaves them unchanged. Interestingly, the end fixation has no effect on multi-domain Ubiquitin. Using the Go modeling and all-atom …
The influence of the salts KCl, NaCl, and NaI at molar concentrations on the α-helical folding kinetics of the alanine-based oligopeptide Ace-AEAAAKEAAAKA-Nme is investigated by means of (explicit-water) molecular dynamics simulations and a diffusional analysis. The mean first passage times for folding and unfolding are found to be highly salt-specific. In particular, the folding times increase about one order of magnitude for the sodium salts. The drastic slowing down can be traced back to long-lived, compact configurations of the partially folded peptide, in which sodium ions are tightly bou…
Neither of the two prevalent theories, namely thermodynamic stability and kinetic stability, provides a comprehensive understanding of protein folding. The thermodynamic theory is misleading because it assumes that free energy is the exclusive dominant mechanism of protein folding, and attributes the structural transition from one characteristic state to another to energy barriers. Conversely, the concept of kinetic stability overemphasizes dominant mechanisms that are related to kinetic factors. This article explores the stability condition of protein structures from the viewpoint of meso-sci…
Searching for reaction pathways describing rare events in large systems presents a long-standing challenge in chemistry and physics. Incorrectly computed reaction pathways result in the degeneracy of microscopic configurations and inability to sample hidden energy barriers. To this aim, we present a general enhanced sampling method to find multiple diverse reaction pathways of ligand unbinding through non-convex optimization of a loss function describing ligand-protein interactions. The method successfully overcomes large energy barriers using an adaptive bias potential, and constructs possibl…
The weighted ensemble (WE) simulation strategy provides unbiased sampling of non-equilibrium processes, such as molecular folding or binding, but the extraction of rate constants relies on characterizing steady state behavior. Unfortunately, WE simulations of sufficiently complex systems will not relax to steady state on observed simulation times. Here we show that a post-simulation clustering of molecular configurations into "microbins" using methods developed in the Markov State Model (MSM) community, can yield unbiased kinetics from WE data before steady-state convergence of the WE simulati…
One of the most important problems in development is how epigenetic domains can be first established, and then maintained, within cells. To address this question, we propose a framework which couples 3D chromatin folding dynamics, to a "recolouring" process modelling the writing of epigenetic marks. Because many intra-chromatin interactions are mediated by bridging proteins, we consider a "two-state" model with self-attractive interactions between two epigenetic marks which are alike (either active or inactive). This model displays a first-order-like transition between a swollen, epigeneticall…
Glycosylation is a highly complex process to produce a diverse repertoire of cellular glycans that are attached to proteins and lipids. Glycans are involved in fundamental biological processes, including protein folding and clearance, cell proliferation and apoptosis, development, immune responses, and pathogenesis. One of the major types of glycans, N-linked glycans, is formed by sequential attachments of monosaccharides to proteins by a limited number of enzymes. Many of these enzymes can accept multiple N-linked glycans as substrates, thereby generating a large number of glycan intermediate…
The energy for protein folding arises from multiple sources and is not large in total. In spite of the many specific successes of energy landscape and other approaches, there still seems to be some missing guiding factor that explains how energy from diverse small sources can drive a complex molecule to a unique state. We explore the possibility that the missing factor is in the geometry. A comparison of folding with other physical phenomena, together with analytic modeling of a molecule, led us to analyze the physics of optical caustic formation and of folding behavior side-by-side. The physi…
We propose a tight-binding model to investigate electronic transport properties of single helical protein molecules incorporating both the helical symmetry and the possibility of multiple charge transfer pathways. Our study reveals that due to existence of both the multiple charge transfer pathways and helical symmetry, the transport properties are quite rigid under influence of envi- ronmental fluctuations which indicates that these biomolecules can serve as better alternatives in nanoelectronic devices than its other biological counterparts e.g., single-stranded DNA.
The steep climbing of victims caused by the new coronavirus disease 2019 (COVID-19) throughout the planet is sparking an unprecedented effort to identify effective therapeutic regimens to tackle the pandemic. The SARS-CoV-2 virus is known to gain entry into various cell types through the binding of one of its surface proteins (spike) to the host Angiotensin-Converting Enzyme 2 (ACE2). Thus, spike-ACE2 interaction represents a major target for vaccines and antiviral drugs. A novel method has been recently described by some of the authors to pharmacologically downregulate the expression of targe…
We propose that the first stage of membrane fusion need not be the formation of a single stalk. Instead, we consider a scenario for defect-induced membrane fusion that proceeds cooperatively via multiple stalk formation. The defects (stalks or pores) attract each other via membrane-mediated capillary interactions that result in a condensation transition of the defects. The resulting dense phase of stalks corresponds to the so-called fusion intermediate.
Fragment-based assembly has been widely used in Ab initio protein folding simulation which can effectively reduce the conformational space and thus accelerate sampling. The efficiency of fragment-based movement as well as the quality of fragment library determine whether the folding process can lead the free energy landscape to the global minimum and help the protein to reach near-native folded state. We designed an improved fragment-based movement, "fragmove", which substituted multiple backbone dihedral angles in every simulation step. This movement strategy was derived from the fragment lib…
We theoretically investigate exciton relaxation dynamics in molecular aggregates based on model photosynthetic complexes under various conditions of incoherent excitation. We show that noise-induced quantum coherence is generated between spatially-separated exciton states which belong to the same or different energy transfer pathways, coupled via real and virtual transfer processes. Such quantum coherence effects may be used to improve light-harvesting efficiency and to reveal quantum phenomena in biology.
Recent single-molecule force measurements on single-domain proteins have highlighted a three-state folding mechanism where a stabilized intermediate state (I) is observed on the folding trajectory between the stretched state and the native state. Here we investigate on-lattice protein-like heteropolymer models that lead to a three-state mechanism and show that force experiments can be useful to determine the structure of I. We have mostly found that I is composed of a core stabilized by a high number of native contacts, plus an unstructured extended chain. The lifetime of I is shown to be sens…
Multiple dynamic pathways always exist in biological networks, but their robustness against internal fluctuations and relative stability have not been well recognized and carefully analyzed yet. Here we try to address these issues through an illustrative example, namely the Siah-1/beta-catenin/p14/19 ARF loop of protein p53 dynamics. Its deterministic Boolean network model predicts that two parallel pathways with comparable magnitudes of attractive basins should exist after the protein p53 is activated when a cell becomes harmfully disturbed. Once the low but non-neglectable intrinsic fluctuat…
