We develop a theoretical description of the critical zipping dynamics of a self-folding polymer. We use tension propagation theory and the formalism of the generalized Langevin equation applied to a polymer that contains two complementary parts which can bind to each other. At the critical temperature, the (un)zipping is unbiased and the two strands open and close as a zipper. The number of closed base pairs $n(t)$ displays a subdiffusive motion characterized by a variance growing as $\langle Δn^2(t) \rangle \sim t^α$ with $α< 1$ at long times. Our theory provides an estimate of both the asymptotic anomalous exponent $α$ and of the subleading correction term, which are both in excellent agreement with numerical simulations. The results indicate that the tension propagation theory captures the relevant features of the dynamics and shed some new insights on related polymer problems characterized by anomalous dynamical behavior.
The folding transition of single, long semiflexible polymers was studied with special emphasis on the chain length effect using Monte Carlo simulations. While a relatively short chain (10-25 Kuhn segments) undergoes a large discrete transiton between swollen coil and compact toroid conformations, a long chain (50 Kuhn segments) exhibits an intrachain segregated state between the disordered coil and ordered toroid.
We study the search process of a target on a rapidly folding polymer (`DNA') by an ensemble of particles (`proteins'), whose search combines 1D diffusion along the chain, Levy type diffusion mediated by chain looping, and volume exchange. A rich behavior of the search process is obtained with respect to the physical parameters, in particular, for the optimal search.
The effect of different move sets on the folding kinetics of the Monte Carlo simulations is analysed based on the conformation-network and the temperature-dependent folding kinetics. A new scheme of implementing Metropolis algorithm is given. The new method is shown to satisfy the detailed balance and converge efficiently towards thermal equilibrium. A new quantity, employed from the continuous time Monte Carlo method, is introduced to identify effectively the kinetic traps of foldings.
We theoretically investigate the kinetics of the folding transition of a single semiflexible polymer. In the folding transition, the growth rate decrease with an increase in the number of monomers in a collapsed domain, suggesting that the main contribution to dissipation is from the motion of the domain. In the unfolding transition, dynamic scaling exponents, 1/8 and 1/4, were determined for disentanglement and relaxation steps, respectively. We performed Langevin dynamics simulations to test our theory. It is found that our theory is in good agreement with simulations. We also propose the ki…
By simulating the first order globule-crystal transition of a flexible homopolymer chain, both by collision dynamics and Monte Carlo with non-kinetic moves, we show that the effective and the thermodynamic transition temperatures are different and we propose a way of quantifying the kinetic hindering. We then also observe that the top eigenvalue in the spectrum of the dynamical (contact or adjacency) matrix provides insight into the ensembles of folding and unfolding trajectories, and may be a suitable additional reaction coordinate for the folding transition of chain molecules.
Kinetics of conformational change of a semiflexible polymer under mechanical external field were investigated with Langevin dynamics simulations. It is found that a semiflexible polymer exhibits large hysteresis in mechanical folding/unfolding cycle even with a slow operation, whereas in a flexible polymer, the hysteresis almost disappears at a sufficiently slow operation. This suggests that the essential features of the structural transition of a semiflexible polymer should be interpreted at least on a two-dimensional phase space. The appearance of such large hysteresis is discussed in relati…
I develop a kinetic mechanism to explain chain folding in polymer crystallization which is based on the competition between the formation of stems, which is due to frequent occupations of trans states along the chains in the supercooled polymer melt, and the random coil structure of the polymer chains. Setting equal the average formation time of stems of length $% d_l$ with the Rouse time of a piece of polymer of the same arc length $d_l$ yields a lower bound for the thickness of stems and bundles. The estimated lamellar thickness is inversely proportional to the supercooling. The present appr…
In our original article (Phys. Chem. Chem. Phys., 2012, 14, 60446053) a convergence problem resulted in an averaging error in computing the entropy from a set of Wang-Landau Monte-Carlo simulations. Here we report corrected results for the freezing temperature of the homopolymer chain as a function of the range of the non-bonded interaction. We find that the previously reported forward-flux sampling (FFS) and brute-force (BF) simulation results are in agreement with the revised Wang-Landau (WL) calculations. This confirms the utility of FFS for computing crystallisation rates in systems of thi…
In suitable environments, proteins, nucleic acids and certain synthetic polymers fold into unique conformations. This work shows that it is possible to construct lattice models of foldable heteropolymers by expressing the energy only in terms of individual properties of monomers, such as the exposure to the solvent and the steric factor.
The non-Markovian nature of polymer motions is accounted for in folding kinetics, using frequency-dependent friction. Folding, like many other problems in the physics of disordered systems, involves barrier crossing on a correlated energy landscape. A variational transition state theory (VTST) that reduces to the usual Bryngelson-Wolynes Kramers approach when the non-Markovian aspects are neglected is used to obtain the rate, without making any assumptions regarding the size of the barrier, or the memory time of the friction. The transformation to collective variables dependent on the dynamics…
The extent of coupling between the folding of a protein and its binding to a substrate varies from protein to protein. Some proteins have highly structured native states in solution, while others are natively disordered and only fold fully upon binding. In this Letter, we use Monte Carlo simulations to investigate how disordered polymer chains grafted around a binding site affect the folding and binding of three model proteins. The protein that approaches the substrate fully folded is more hindered during the binding process than those whose folding and binding are cooperative. The polymer cha…
Different aspects of protein folding are illustrated by simplified polymer models. Stressing the diversity of side chains (residues) leads one to view folding as the freezing transition of an heteropolymer. Technically, the most common approach to diversity is randomness, which is usually implemented in two body interactions (charges, polar character,..). On the other hand, the (almost) universal character of the protein backbone suggests that folding may also be viewed as the crystallization transition of an homopolymeric chain, the main ingredients of which are the peptide bond and chirality…
We perform, with the help of cloud computing resources, extensive Langevin simulations which provide compelling evidence in favor of a general markovian framework for unbiased polymer translocation. Our statistical analysis consists of careful evaluations of (i) two-point correlation functions of the translocation coordinate and (ii) the empirical probabilities of complete polymer translocation (taken as a function of the initial number of monomers on a given side of the membrane). We find good agreement with predictions derived from the Markov chain approach recently addressed in the literatu…
Epigenetics is a driving force of important and ubiquitous phenomena in nature such as cell differentiation or even metamorphosis. Oppositely to its widespread role, understanding the biophysical principles that allow epigenetics to control and rewire gene regulatory networks remains an open challenge. In this work we study the effects of epigenetic modifications on the spatial folding of chromosomes -- and hence on the expression of the underlying genes -- by mapping the problem to a class of models known as magnetic polymers. In this work we show that a first-order phase transition underlies…
We study the kinetics of protein folding via statistical energy landscape theory. We concentrate on the local-connectivity case, where the configurational changes can only occur among neighboring states, with the folding progress described in terms of an order parameter given by the fraction of native conformations. The non-Markovian diffusion dynamics is analyzed in detail and an expression for the mean first-passage time (MFPT) from non-native unfolded states to native folded state is obtained. It was found that the MFPT has a V-shaped dependence on the temperature. We also find that the MFP…
Configurational entropy of randomly double-folding ring polymers
SISSA - Scuola Internazionale Superiore di Studi Avanzati（国际高等研究院）；ENS de Lyon, CNRS, Laboratoire de Physique (LPENSL UMR5672) et Centre Blaise Pascal（里昂高等师范学院，法国国家科学研究中心，物理实验室（LPENSL UMR5672）和布莱斯·帕斯卡中心）
A fascinating and open question challenging biochemistry, physics and even geometry is the presence of highly regular motifs such as alpha-helices in the folded state of biopolymers and proteins. Stimulating explanations ranging from chemical propensity to simple geometrical reasoning have been invoked to rationalize the existence of such secondary structures. We formulate a dynamical variational principle for selection in conformation space based on the requirement that the backbone of the native state of biologically viable polymers be rapidly accessible from the denatured state. The variati…
We discuss a problem of cyclization of a polymer molecule, which is an important example of reaction in a system showing strongly non-Markovian behavior on the timescales of interest. We show that the knowledge of the joint three-time probability distribution of the end-to-end distance is sufficient for the full description of the cyclization kinetics, so that the survival probability follows rigorously as a solution of the Volterra integral equation. The corresponding kinetics can easily be evaluated numerically. We moreover discuss how do some well-known approximations appear from this exact…
The hydrophobic effect is the dominant force which drives a protein towards its native state, but its physics has not been thoroughly understood yet. We introduce an exactly solvable model of the solvation of non-polar molecules in water, which shows that the reduced number of allowed configurations of water molecules when the solute is present is enough to give rise to hydrophobic behaviour. We apply our model to a non-polar homopolymer in aqueous solution, obtaining a clear evidence of both ``cold'' and ``warm'' collapse transitions that recall those of proteins. Finally we show how the mode…
Monte Carlo simulations of a Miyazawa-Jernigan lattice-polymer model indicate that, depending on the native's structure geometry, the model exhibits two broad classes of folding mechanisms for two-state folders. Folding to native structures of low contact order is driven by contact range and is characterised by progressive accumulation of structure towards the native fold. By contrast, folding to high contact order targets is largely dominated by intermediate stage contacts which are not present in the native fold, yielding a more cooperative folding process.
Molecular dynamics simulation methods are used to study the folding of polymer chains into packed cubic states. The polymer model, based on a chain of linked sites moving in the continuum, includes both excluded volume and torsional interactions. Different native-state packing arrangements and chain lengths are explored; the organization of the native state is found to affect both the ability of the chain to fold successfully and the nature of the folding pathway as the system is gradually cooled. An order parameter based on contact counts is used to provide information about the folding proce…
The Replica Exchange Wang-Landau Method is used to estimate the energy landscape of a polymer composed of a simple hydrophobic and polar sequence using the HP protein model. Calculations of state transitions between the energy levels of the derived energy landscape are made using an equation of motion from the steepest-entropy-ascent quantum thermodynamic (SEAQT) framework. The SEAQT framework makes it possible to determine the unique kinetic paths from an arbitrary quasi-equilibrium or non-equilibrium initial state to stable equilibrium. Calculations performed with SEAQT require significantly…
Any first course on polymer physics teaches that the dynamics of a tagged monomer of a polymer is anomalously subdiffusive, i.e., the mean-square displacement of a tagged monomer increases as $t^α$ for some $α<1$ until the terminal relaxation time $τ$ of the polymer. Beyond time $τ$ the motion of the tagged monomer becomes diffusive. Classical examples of anomalous dynamics in polymer physics are single polymeric systems, such as phantom Rouse, self-avoiding Rouse, self-avoiding Zimm, reptation, translocation through a narrow pore in a membrane, and many-polymeric systems such as polymer melts…
We study the geometric properties of the energy landscape of coarse-grained, off-lattice models of polymers by endowing the configuration space with a suitable metric, depending on the potential energy function, such that the dynamical trajectories are the geodesics of the metric. Using numerical simulations, we show that the fluctuations of the curvature clearly mark the folding transition, and that this quantity allows to distinguish between polymers having a protein-like behavior (i.e., that fold to a unique configuration) and polymers which undergo a hydrophobic collapse but do not have a …
At the cutting edge of materials science, matter is designed to self-organize into structures that perform a wide range of functions. The past two decades have witnessed major innovations in the versatility of building blocks, ranging from DNA on the nanoscale to handshaking materials on the macroscale. Like a jigsaw puzzle, one can reliably self-assemble arbitrary structures if all the pieces are distinct, but systems with fewer flavors of building blocks have so far been limited to the assembly of exotic crystals. Inspired by Nature's strategy of folding biopolymers into specific RNA and pro…
We investigate theoretically a diffusion-limited reaction between a reactant attached to a Rouse polymer and an external fixed reactive site in confinement. The present work completes and goes beyond a previous study [T. Guérin, O. Bénichou and R. Voituriez, Nat. Chem., 4, 268 (2012)] that showed that the distribution of the polymer conformations at the very instant of reaction plays a key role in the reaction kinetics, and that its determination enables the inclusion of non-Markovian effects in the theory. Here, we describe in detail this non-Markovian theory and we compare it with numerical …
Describing the kinetics of polymer reactions, such as the formation of loops and hairpins in nucleic acids or polypeptides, is complicated by the structural dynamics of their chains. Although both intramolecular reactions, such as cyclization, and intermolecular reactions have been studied extensively, both experimentally and theoretically, there is to date no exact explicit analytical treatment of transport-limited polymer reaction kinetics, even in the case of the simplest (Rouse) model of monomers connected by linear springs. We introduce a new analytical approach to calculate the mean reac…
Monte Carlo simulations of proteins are hindered by the system's high density which often makes local moves ineffective. Here we devise and test a set of long range moves that work well even when all sites of a lattice simulation are filled. We demonstrate that for a 27-mer cube, the ground state of random heteropolymers can quickly be reached. We discuss results for 48-mer systems where the ground state is known exactly. For ten sequences that were examined, the introduction of long range moves speeds up the search for the ground state by about one order of magnitude. The method is compared t…
We investigate the crystallization of a single, flexible homopolymer chain using transition path sampling (TPS). The chain consists of N identical spherical monomers evolved according to Langevin dynamics. While neighboring monomers are coupled via harmonic springs, the non-neighboring monomers interact via a hard core and a short-ranged attractive potential. For a sufficiently small interaction range λ, the system undergoes a first-order freezing transition from an expanded, disordered phase to a compact crystalline state. Using a new shooting move tailored to polymers combined with a committ…
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…
We investigate theoretically the physics of diffusion-limited intramolecular polymer reactions. The present work completes and goes beyond a previous study [Nat. Chem. 4, 268 (2012)] that showed that the distribution of the polymer conformations at the very instant of reaction plays a key role in the cyclization kinetics, and takes explicitly into account the non-Markovian nature of the reactant motion. Here, we present in detail this non-Markovian theory, and compare it explicitly with existing Markovian theories and with numerical stochastic simulations. A large focus is made on the descript…
We present results from extensive molecular dynamics simulations of collapse transitions of hydrophobic polymers in explicit water focused on understanding effects of lengthscale of the hydrophobic surface and of attractive interactions on folding. Hydrophobic polymers display parabolic, protein-like, temperature-dependent free energy of unfolding. Folded states of small attractive polymers are marginally stable at 300 K, and can be unfolded by heating or cooling. Increasing the lengthscale or decreasing the polymer-water attractions stabilizes folded states significantly, the former dominated…
How much time does it take for a fluctuating system, such as a polymer chain, to reach a target configuration that is rarely visited -- typically because of a high energy cost ? This question generally amounts to the determination of the first-passage time statistics to a target zone in phase space with lower occupation probability. Here, we present an analytical method to determine the mean first-passage time of a generic non-Markovian random walker to a rarely visited threshold, which goes beyond existing weak-noise theories. We apply our method to polymer systems, to determine (i) the first…
Coarse-grained (lattice-) models have a long tradition in aiding efforts to decipher the physical or biological complexity of proteins. Despite the simplicity of these models, however, numerical simulations are often computationally very demanding and the quest for efficient algorithms is as old as the models themselves. Expanding on our previous work [T. Wüst and D. P. Landau, Phys. Rev. Lett. 102, 178101 (2009)], we present a complete picture of a Monte Carlo method based on Wang-Landau sampling in combination with efficient trial moves (pull, bond-rebridging and pivot moves) which is partic…
While the structure of chromatin has been studied in great detail on length scales below 30 nm, amazingly little is known about the higher-order folding motifs of chromatin in interphase. Recent experiments give evidence that the folding may depend locally on gene density and transcriptional activity and show a leveling-off at long distances where approximately $<R^2> \sim O(1)$. We propose a new model that can explain this leveling-off by the formation of random loops. We derive an analytical expression for the mean square displacement between two beads where the average is taken over the the…
We investigate the crystallization mechanism of a single, flexible homopolymer chain with short range attractions. For a sufficiently narrow attractive well, the system undergoes a first-order like freezing transition from an expanded disordered coil to a compact crystalline state. Based on a maximum likelihood analysis of committor values computed for configurations obtained by Wang-Landau sampling, we construct a non-linear string reaction coordinate for the coil-to-crystal transition. In contrast to a linear reaction coordinate, the string reaction coordinate captures the effect of differen…
We investigate the folding and forced-unbinding transitions of adsorbed semiflexible polymer chains using theory and simulations. These processes describe biologically relevant phenomena that include adhesive interactions between proteins and tethering of receptors to cell walls. The binding interface is modeled as a solid surface and the worm-like chain is used for the semiflexible chain. Using Langevin simulations we examine the ordering kinetics of racquet-like and toroidal structures in the presence of attractive interaction between the surface and the polymer chain. For a range of interac…
We review pro and contra of the hypothesis that generic polymer properties of topological constraints are behind many aspects of chromatin folding in eukaryotic cells. For that purpose, we review, first, recent theoretical and computational findings in polymer physics related to concentrated, topologically-simple (unknotted and unlinked) chains or a system of chains. Second, we review recent experimental discoveries related to genome folding. Understanding in these fields is far from complete, but we show how looking at them in parallel sheds new light on both.
The problem of the helix-coil transition of biopolymers in explicit solvents, like water, with the ability for hydrogen bonding with solvent is addressed analytically using a suitably modified version of the Generalized Model of Polypeptide Chains. Besides the regular helix-coil transition, an additional coil-helix or reentrant transition is also found at lower temperatures. The reentrant transition arises due to competition between polymer-polymer and polymer-water hydrogen bonds. The balance between the two types of hydrogen bonding can be shifted to either direction through changes not only…
We describe a class of growth algorithms for finding low energy states of heteropolymers. These polymers form toy models for proteins, and the hope is that similar methods will ultimately be useful for finding native states of real proteins from heuristic or a priori determined force fields. These algorithms share with standard Markov chain Monte Carlo methods that they generate Gibbs-Boltzmann distributions, but they are not based on the strategy that this distribution is obtained as stationary state of a suitably constructed Markov chain. Rather, they are based on growing the polymer by succ…
Single two dimensional polymers confined to a strip are studied by Monte Carlo simulations. They are described by N-step self-avoiding random walks on a square lattice between two parallel hard walls with distance 1 << D << N^ν(ν= 3/4 is the Flory exponent). For the simulations we employ the pruned-enriched-Rosenbluth method (PERM) with Markovian anticipation. We measure the densities of monomers and of end points as functions of the distance from the walls, the longitudinal extent of the chain, and the forces exerted on the walls. Their scaling with D and the universal ratio between force and…
The looping of polymers such as DNA is a fundamental process in the molecular biology of living cells, whose interior is characterised by a high degree of molecular crowding. We here investigate in detail the looping dynamics of flexible polymer chains in the presence of different degrees of crowding. From the analysis of the looping-unlooping rates and the looping probabilities of the chain ends we show that the presence of small crowders typically slow down the chain dynamics but larger crowders may in fact facilitate the looping. We rationalise these non-trivial and often counterintuitive e…
The threading of a polymer chain through a small pore is a classic problem in polymer dynamics and underlies nanopore sensing technology. However important experimental aspects of the polymer motion in a solid-state nanopore, such as an accurate measurement of the velocity variation during translocation, have remained elusive. In this work we analysed the translocation through conical quartz nanopores of a 7 kbp DNA double-strand labelled with six markers equally spaced along its contour. These markers, constructed from DNA hairpins, give direct experimental access to the translocation dynamic…
The present research is dedicated to provide deeper understanding of the impact of complex architecture of branched polymers on their behaviour in solvents. The folding dynamics of macromolecules and hydrodynamics of polymer fluids are strongly dependent on size and shape measures of single macromolecules, which in turn are determined by their topology. For this aim, we use combination of analytical theory, based on path integration method, and molecular dynamics simulations to study structural properties of complex Gaussian polymers containing $f^c$ linear branches and $f^r$ closed loops graf…
The motion involved in barrier crossing for protein folding are investigated in terms of the chain dynamics of the polymer backbone, completing the microscopic description of protein folding presented in the previous paper. Local reaction coordinates are identified as collective growth modes of the unstable fluctuations about the saddle-points in the free energy surface. The description of the chain dynamics incorporates internal friction (independent of the solvent viscosity) arising from the elementary isomerizations of the backbone dihedral angles. We find that the folding rate depends line…
Compact polymers are self-avoiding random walks which visit every site on a lattice. This polymer model is used widely for studying statistical problems inspired by protein folding. One difficulty with using compact polymers to perform numerical calculations is generating a sufficiently large number of randomly sampled configurations. We present a Monte-Carlo algorithm which uniformly samples compact polymer configurations in an efficient manner allowing investigations of chains much longer than previously studied. Chain configurations generated by the algorithm are used to compute statistics …
The calculation of Euclidean distance between points is generalized to one-dimensional objects such as strings or polymers. Necessary and sufficient conditions for the minimal transformation between two polymer configurations are derived. Transformations consist of piecewise rotations and translations subject to Weierstrass-Erdmann corner conditions. Numerous examples are given for the special cases of one and two links. The transition to a large number of links is investigated, where the distance converges to the polymer length times the mean root square distance (MRSD) between polymer config…
The chromosome is a key player of cell physiology, and its dynamics provides valuable information about its physical organization. In both prokaryotes and eukaryotes, the short-time motion of chromosomal loci has been described as a Rouse model in a simple or viscoelastic medium. However, little emphasis has been put on the role played by the folded organization of chromosomes on the local dynamics. Clearly, stress-propagation, and thus dynamics, must be affected by such organization, but a theory allowing to extract such information from data, e.g.\ of two-point correlations, is lacking. Here…
We address the problem of inverse polymer swelling. This phenomenon, in which a collapsed polymer chain swells upon decreasing temperature, can be observed experimentally in so-called thermoreversible homopolymers in aqueous solution, and is believed to be related to the role of hydrophobicity in protein folding. We consider a lattice-fluid model of water, defined on a body-centered cubic lattice, which has been previously shown to account for most thermodynamic anomalies of water and of hydrophobic solvation for monomeric solutes. We represent the polymer as a self-avoiding walk on the same l…
Many small proteins fold via a first-order "all-or-none" transition directly from an expanded coil to a compact native state. Here we study an analogous direct freezing transition from an expanded coil to a compact crystallite for a simple flexible homopolymer. Wang-Landau sampling is used to construct the 1D density of states for square-well chains of length 128. Analysis within both the micro-canonical and canonical ensembles shows that, for a chain with sufficiently short-range interactions, the usual polymer collapse transition is preempted by a direct freezing or "folding" transition. A 2…
We introduce an exactly solvable statistical-mechanical model of the hydration of non-polar compounds, based on grouping water molecules in clusters where hydrogen bonds and isotropic interactions occur; interactions between clusters are neglected. Analytical results show that an effective strengthening of hydrogen bonds in the presence of the solute, together with a geometric reorganization of water molecules, are enough to yield hydrophobic behavior. We extend our model to describe a non-polar homopolymer in aqueous solution, obtaining a clear evidence of both ``cold'' and ``warm'' swelling …
Biopolymer looping is a dynamic process that occurs ubiquitously in cells for gene regulation, protein folding, etc. In cellular environments, biopolymers are often subject to tensions which are either static, or temporally fluctuating far away from equilibrium. We study the dynamics of semiflexible polymer looping in the presence of such tensions by using Brownian dynamics simulation combined with an analytical theory. We show a minute tension dramatically changes the looping time, especially for long chains. Considering a dichotomically flipping noise as a simple example of the nonequilibriu…
