Coil-globule transition of a homopolymer chain in a square-well potential: Comparison between Monte Carlo canonical replica exchange and Wang-Landau sampling
Artem Badasyan, Trinh Xuan Hoang, Rudolf Podgornik, Achille Giacometti
We study the equilibrium properties of a flexible homopolymer where consecutive monomers are represented by impenetrable hard spheres that are tangent to each other, and non-consecutive monomers interact via a square-well potential. To this aim, we use both replica exchange canonical simulations and micro-canonical Wang-Landau techniques for relatively short chains, and perform a close comparative analysis of the corresponding results. These investigations are then further exploited to reproduce, at a much shorter scale and, hence, computational effort, the phase diagram previously studied with much longer chains. This opens up the possibility of improving the model and introduce specificities typical, among other examples, of protein folding.
We propose the quantitative mean-field theory of mechanical unfolding of a globule formed by long flexible homopolymer chain collapsed in poor solvent and subjected to an extensional force We show that with an increase in the applied force the globule unfolds as a whole without formation of an intermediate state. The value of the threshold force and the corresponding jump in the distance between chain ends increase with a deterioration of the solvent quality and / or with an increase in the degree of polymerization. This way of globule unfolding is compared with that in the D-ensemble, when th…
A simple model of a polymer with long rigid segments which interact via excluded volume repulsions and short ranged attractions is proposed. The coil-globule transition of this model polymer is strongly first order, the globule is crystalline and the coil which coexists with the globule is swollen. A virial expansion truncated at low order is shown to provide a very poor approximation to the free energy and so a cell theory is used to calculate the free energy of the globule.
We propose the quantitative mean-field theory of mechanical unfolding of a globule formed by long flexible homopolymer chain collapsed in poor solvent and subjected to extensional deformation. We demonstrate that depending on the degree of polymerization and solvent quality (quantified by the Flory-Huggins $χ$ parameter) the mechanical unfolding of the collapsed chain may either occur continuously (by passing a sequence of uniformly elongated configurations) or involves intra-molecular micro-phase coexistence of a collapsed and a stretched segment followed by an abrupt unraveling transition. T…
Homopolymer chain with beads forming pairwise reversible bonds is a well-known model in polymer physics. We studied kinetics of homopolymer chain collapse, which was induced by pairwise reversible bonds formation. We compared kinetic mechanism of this coil-globule transition with the mechanism of collapse in a poor solvent. We discovered, that coil-globule transition occurs sufficiently more homogeneously on different scales, if collapse is induced by pairwise reversible bonds formation. This effect leads to formation of transient structures, which are not similar to the classical pearl-neckla…
The determination of the coil-globule transition of a polymer is generally based on the reconstruction of scaling laws, implying the need for samples from a rather wide range of different polymer lengths $N$. The spectral point of view developed in this work allows for a very parsimonious description of all the aspects of the finite-size coil-globule transition on the basis of the first two Rouse (cosine) modes only, shedding new light on polymer theory. Capturing the relevant configuration path features, the proposed approach enables to determine the state of a polymer without the need of any…
Polymers adopt extended coil and compact globule states according to the balance between entropy and interaction energies. The transition of a polymer between an extended coil state and compact globule state can be induced by changing thermodynamic force such as temperature to alter the energy/entropy balance. Previously, this transition was theoretically studied by taking into account the excluded-volume interaction between monomers of a polymer chain using the partition function. For binary mixtures of a long polymer and short polymers, the coil-globule transition can be induced by changing …
We present results of numerical self-consistent field (SCF) calculations for the equilibrium mechanical unfolding of a globule formed by a single flexible polymer chain collapsed in a poor solvent. In accordance with earlier scaling theory and stochastic dynamics simulations findings we have identified three regimes of extensional deformation: (i) a linear response regime characterized by a weakly elongated (ellipsoidal) shape of the globule at small deformations, (ii) a tadpole structure with a globular "head" coexisting with a stretched "tail" at intermediate ranges of deformations, and (iii…
Equilibrium mechanical unfolding of a globule formed by long flexible homopolymer chain collapsed in a poor solvent and subjected to an extensional force f (force-clamp mode) or extensional deformation D (position-clamp mode) is studied theoretically. Our analysis, like all previous analysis of this problem, shows that the globule behaves essentially differently in two modes of extension. In the force-clamp mode, mechanical unfolding of the globule with increasing applied force occurs without intramolecular microphase segregation, and at certain threshold value of the pulling force the globule…
Poly-N-isopropylacrylamide (PNIPAM) phase diagram is explored in a wide range of temperature and pressure using extensive all-atom molecular dynamics simulations. By exploiting a simple model of an atactic linear polymer chain, we provide novel insights into PNIPAM coil-to-globule transition addressing the roles played by both temperature and pressure. We find that a coil-to-globule transition exists up to large pressures, undergoing an intriguing reentrant behavior of the lower critical solution temperature with increasing pressure in agreement with experimental observations. Furthermore, we …
We study the nonequilibrium kinetics during the coil-globule transition of a flexible polymer chain with active beads after a quench from good to poor solvent condition using molecular dynamics simulation. Activity for each bead is introduced via the well-known Vicsek-like alignment rule due to which the velocity of a bead tries to align towards the average direction of its neighbors. We investigate the role of quenching temperature with varying activity during collapse of this polymer. We find that although for lower activities the kinetics remains qualitatively similar for different temperat…
We present an off-lattice statistical model of a single polymer chain in mixed solvent media. Taking into account a polymer conformational entropy, renormalization of solvent composition near the polymer backbone, the universal intermolecular excluded volume and Van-der-Waals interactions within the self-consistent field theory the reentrant coil-to-globule-to-coil transition (co-nonsolvency) has been described in this paper. For convenience we split the system volume in two parts: the volume occupied by the polymer chain and the volume of bulk solution. Considering the equilibrium between two…
In this article, a novel cluster analysis algorithm was employed in the study of polymer coil to globule transition via single chain Monte Carlo simulations. The algorithm, which has been recently applied in Molecular Dynamics simulations of atomistic systems that tend to phase separate [arXiv: 1307.7366 [cond-mat.soft]], provides us with a convenient means to map out the dynamics of "pearls" formation along the backbone chain together with extracting meaningful quantitative information about their shape and size distribution. Preliminary findings tend to favour a two-stage model of collapse k…
We study the conformational behavior of polarizable polymer chain under an external homogeneous electric field within the Flory type self-consistent field theory. We consider the influence of electric field on the polymer coil as well as on the polymer globule. We show that when the polymer chain conformation is a coil, application of external electric field leads to its additional swelling. However, when the polymer conformation is a globule, a sufficiently strong field can induce a globule-coil transition. We show that such $"$field-induced$"$ globule-coil transition at the sufficiently smal…
We explore the coupling between the helix-coil and coil-globule transitions of a helical polymer using Monte Carlo simulations. A very rich state diagram is found. Each state is characterized by a specific configuration of the chain which could be a helix, a random coil, an amorphous globule or one of various other globular states which carry residual helical strands. We study the boundaries between states and provide further insight into the physics of the system with a detailed analysis of the order parameter and other properties.
We investigate a model of self-avoiding walk exhibiting a first-order coil-globule transition. This first-order nature, unravelled through the coexistence of distinct coil and globule populations, has observable consequences on the scaling properties. A thorough analysis of the size dependence of the mean radius of gyration evidences a breakdown of the plain tricritical scaling behaviour. In some regimes, anomalous exponents are observed in the transition region and logarithmic corrections arise along the coexistence curve.
In this work, we report our results on the phase transition of a flexible homopolymer from a stretched chain to a compact globule. The Wang-landau method is used to study the thermodynamic properties of a the chain up to 512 monomers. We believe that the peak in the specific heat at low temperature $T\approx 0.05$ for small chain sizes $N<100$ is a clear evidence of the existence of metastable helical structures observed in a previous study.
We consider a free energy functional on the monomer density function that is suitable for the study of coil-globule transition. We demonstrate, with explicitly stated assumptions, why the entropic contribution is in the form of the Kullback-Leibler distance, and that the energy contribution is given by two-body and three-body terms. We then solve for the free energy analytically on a set of trial density functions, and reproduce de Gennes' classical theory on polymer coil-globule transition. We then discuss how our formalism can be applied to study polymer dynamics from the perspective of dyna…
We study flexible polymer macromolecules in a crowded (porous) environment, modelling them as self-attracting self-avoiding walks (SASAW) on site-diluted percolative lattices in space dimensions d=2, 3. The influence of stretching force on the polymer folding and properties of globule-coil transitions are analyzed. Applying the pruned-enriched Rosenbluth chain-growth method (PERM), we estimate the transition temperature T_Θ between collapsed and extended polymer configurations and construct the phase diagrams of the globule-coil coexistence when varying temperature and stretching force. The tr…
By means of atomistic molecular dynamics simulations we investigate the behaviour of poly(N-isopropylacrylamide), PNIPAM, in water at temperatures below and above the lower critical solution temperature (LCST), including the undercooled regime. The transition between water soluble and insoluble states at the LCST is described as a cooperative process involving an intramolecular coil-to-globule transition preceding the aggregation of chains and the polymer precipitation. In this work we investigate the molecular origin of such cooperativity and the evolution of the hydration pattern in the unde…
We have studied the thermodynamics of isolated homopolymer chains of varying stiffness using a lattice model. A complex phase behaviour is found; phases include chain-folded `crystalline' structures, the disordered globule and the coil. It is found, in agreement with recent theoretical calculations, that the temperature at which the solid-globule transition occurs increases with chain stiffness, whilst the $θ$-point has only a weak dependence on stiffness. Therefore, for sufficiently stiff chains there is no globular phase and the polymer passes directly from the solid to the coil. This effect…
We explore the mechanical response of a single polyelectrolyte chain under tension in good and poor solvents using a combination of simulation and theory. In poor solvents, where the equilibrium state of the chain is a collapsed globule, we find that the chain undergoes a globule-coil transition, as the magnitude of the force is increased beyond a critical value. This transition, where the polymer size changes discontinuously from a small to a large value, is accompanied by release of bound counterions from the chain. We explain these results by adhering to a statistical mechanical theory of c…
Poly(N-isopropylacrylamide), PNIPAM, is a widely studied polymer, which serves as a key constituent of nanostructured soft materials with responsive properties. Upon increasing temperature the PNIPAM polymer chain undergoes a reversible coil-to-globule transition at ~305K, which is reflected by a volume phase transition in cross-linked architectures, such as microgels, valuable for many practical applications. The addition of a cosolvent is a simple method to tune the transition temperature according to the specific purpose. In this study, we use atomistic molecular dynamics simulations to exp…
The phase behaviour of a single large semiflexible polymer immersed in a suspension of spherical particles is studied. All interactions are simple excluded volume interactions and the diameter of the spherical particles is an order of magnitude larger than the diameter of the polymer. The spherical particles induce a quite long ranged depletion attraction between the segments of the polymer and this induces a continuous coil-globule transition in the polymer. This behaviour gives an indication of the condensing effect of macromolecular crowding on DNA.
Understanding the role of self-propulsion on the conformational properties of active filamentous objects has relevance in biology. In this context, we consider a flexible bead-spring model polymer for which along with both attractive and repulsive interactions among the non-bonded monomers, activity for each bead works along its intrinsic direction of self-propulsion. We study its kinetics in the overdamped limit, following a quench from good to poor solvent condition. We observe that with low activities, though the kinetic pathways remain similar, the scaling exponent for the relaxation time …
The microphase separated morphology of a melt of a specific class of comb-coil diblock copolymers, consisting of an AB comb block and a linear homopolymer A block, is analyzed in the weak segregation limit. On increasing the length of the homopolymer A block, the systems go through a characteristic series of structural transitions. Starting from the pure comb copolymer the first series of structures involve the short length scale followed by structures involving the large length scale. A maximum of two critical points exists. Furthermore, in the two parameter space, characterizing the comb-coi…
We study the role of topological restrictions for the conformational structure of a nonphantom homopolymer chain in a lattice Monte Carlo model. In the athermal regime we find that the standard Metropolis algorithm violates the detailed balance condition if both local monomer and global reptational moves are included. However, if about 1 reptation is performed per N local moves the balance is recovered and we obtain the Flory exponent value close to nu=0.588, where N is the degree of polymerisation. We also find that the structure of the collapsed globule is different at equilibrium from that …
A self-consistent field theory was developed in the grand-canonical ensemble formulation to study transitions in a helix-coil multiblock globule. Helical and coil parts are treated as stiff rods and self-avoiding walks of variable lengths correspondingly. The resulting field-theory takes, in addition to the conventional Zimm-Bragg (B.H. Zimm, I.K. Bragg, J. Chem. Phys. 31, 526 (1959)) parameters, also three-dimensional interaction terms into account. The appropriate differential equations which determine the self-consistent fields were solved numerically with finite element method. Three diffe…
The coil-globule transition of hetero-polymer chains is studied here. By means of extensive Molecular Dynamics simulations, we show that the transition is directly linked to the complexity of the chain, which depends on the number of chemical species defined in the environment and the location of the binding sites along the polymer. In addition, when the number of species increases, we find that the distribution of binding sites plays an important role in triggering the transition, beyond the standard control parameters of the polymer model, i.e. binders concentration and binding affinity. Ove…
We study long polymer chains in a poor solvent, confined to the space between two parallel hard walls. The walls are energetically neutral and pose only a geometric constraint which changes the properties of the coil-globule (or "$θ$-") transition. We find that the $θ$ temperature increases monotonically with the width $D$ between the walls, in contrast to recent claims in the literature. Put in a wider context, the problem can be seen as a dimensional cross over in a tricritical point of a $ϕ^4$ model. We roughly verify the main scaling properties expected for such a phenomenon, but we find a…
We present a scaling theory describing the collapse of a homopolymer chain in poor solvent. At time t after the beginning of the collapse, the original Gaussian chain of length N is streamlined to form N/g segments of length R(t), each containing g ~ t monomers. These segments are statistical quantities representing cylinders of length R ~ t^{1/2} and diameter d ~ t^{1/4}, but structured out of stretched arrays of spherical globules. This prescription incorporates the capillary instability. We compare the time-dependent structure factor derived for our theory with that obtained from ultra-larg…
We develop a theory to probe the effect of non-equilibrium fluctuation-induced forces on the size of a polymer confined between two horizontal thermally conductive plates subject to a constant temperature gradient, $\nabla T$. We assume that (a) the solvent is good and (b) the distance between the plates is large so that in the absence of a thermal gradient the polymer is a coil whose size scales with the number of monomers as $N^ν$, with $ν\approx 0.6$. We predict that above a critical temperature gradient, $\nabla T_c \sim N^{-\frac{5}{4}}$, favorable attractive monomer-monomer interaction d…
We elucidate the structural transitions in a helical off-lattice homopolymer induced by crowding agents, as function of the number of monomers ($N$) and volume fraction ($ϕ_c$) of crowding particles. At $ϕ_c = 0$, the homopolymer undergoes transitions from a random coil to a helix, helical hairpin \textbf{HH}, and helix bundle \textbf{HB} structures depending on $N$, and temperature. Crowding induces chain compaction that can promote \textbf{HH} or \textbf{HB} formation depending on $ϕ_c$. Typically, the helical content decreases which is reflected in the decrease in the transition temperature…
The coil-globule transition of an energy polydisperse chain, a model heteropolymer system where the number of monomer species is as large as the total number of monomers, is studied by means of computer simulations. In this study, we systematically explore the consequences of having different functional form and variance of the energy distribution on the coil-globule transition in general. In particular, considering Gaussian (G) and uniform (U) distributions, the effect of varying polydispersity index, $δ$, on the transition temperature $θ^\ast$, chain size, internal structure and spatial orga…
It has been revealed by mean-field theories and computer simulations that the nature of the collapse transition of a polymer is influenced by its bending stiffness $ε_{\rm b}$. In two dimensions, a recent analytical work demonstrated that the collapse transition of a partially directed lattice polymer is always first-order as long as $ε_{\rm b}$ is positive [H. Zhou {\em et al.}, Phys. Rev. Lett. {\bf 97}, 158302 (2006)]. Here we employ Monte Carlo simulation to investigate systematically the effect of bending stiffness on the static properties of a 2D lattice polymer. The system's phase-diagr…
We perform numerical simulations of an active fully flexible self-avoiding polymer as a function of the quality of the embedding solvent described in terms of an effective monomer-monomer interaction. Specifically, by extracting the Flory exponent of the active polymer under different conditions, we are able to pin down the location of the coil-globule transition for different strength of the active forces. Remarkably, we find that a simple rescaling of the temperature is capable of qualitatively capture the dependence of the $Θ$-point of the polymer with the amplitude of the active fluctuatio…
We have analyzed the equilibrium response of chain molecules to stretching. For a homogeneous sequence of monomers, the induced transition from compact globule to extended coil below the $θ$-temperature is predicted to be sharp. For random sequences, however, the transition may be smoothed by a prevalence of necklace-like structures, in which globular regions and coil regions coexist in a single chain. As we show in the context of a random copolymer, preferential solvation of one monomer type lends stability to such structures. The range of stretching forces over which necklaces are stable is …
Detailed thermodynamic analysis of complex systems with multiple stable configurational states allows for insight into the cooperativity of each individual transition. In this work we derive a heat capacity decomposition comprising contributions from each individual configurational state, which together sum to a baseline heat capacity, and contributions from each state-to-state transition. We apply this analysis framework to a series of replica exchange molecular dynamics simulations of linear and 1-1 coarse-grained homo-oligomer models which fold into stable, configurationally well-defined mo…
A self-consistent-field theory (SCFT) in the grand canonical ensemble formulation is used to study transitions in a helix-coil multiblock copolymer globule. The helices are modeled as stiff rods. In addition to the established coil-globule transition we show for the first time that, even without explicit rod-rod alignment interaction, the system undergoes a transition to a nematic liquid-crystalline (LC) globular state. The LC-globule formation is driven by the hydrophobic helical segment attraction and the anisotropy of the globule surface energy. The full phase diagram of the copolymer was c…
By means of continuous space Monte Carlo simulation we study conformational structures formed by star and comb heteropolymers during kinetics of folding from the coil to the globule, as well as the corresponding equilibrium states on going from the good to the poor solution. Particular examples of combs with hydrophobic backbone and hydrophilic side-groups (and vice versa), as well as stars with flexible and semi-stiff arms are studied. It is interesting to note that star-like conformations naturally appear for a comb polymer with a strongly hydrophobic backbone. We emphasise the crucial diffe…
We derive an analytical pair potential of mean force for Brownian molecules in the liquid-state. Our approach accounts for many-particle correlations of crowding particles of the liquid, and for diffusive transport across the spatially modulated local density of crowders in the dense environment. Specializing on the limit of equal-size particles, we show that this diffusive transport leads to additional density- and structure-dependent terms in the interaction potential, and to a much stronger attraction (by a factor ~4 at average volume fraction of crowders 0.25) than in the standard depletio…
We investigate by the use of the Martin - Siggia - Rose generating functional technique and the self - consistent Hartree approximation, the dynamics of the ring homopolymer collapse (swelling) following an instantaneous change into a poor (good) solvent conditions.The equation of motion for the time dependent monomer - to - monomer correlation function is systematically derived. It is argued that for describing of the coarse - graining process (which neglects the capillary instability and the coalescence of ``pearls'') the Rouse mode representation is very helpful, so that the resulting equat…
The coil-globule transition has been studied for A-B copolymer chains both by means of lattice Monte Carlo simulations using bond fluctuation algorithm and by a numerical self-consistent field method. Copolymer chains of fixed length with A and B monomeric units with regular, random and specially designed (protein-like) primary sequences have been investigated. The dependence of the transition temperature on the AB sequence has been analyzed. A protein-like copolymer is more stable than a copolymer with statistically random sequence. The transition is more sharpe for random copolymers. It is f…
We study the behavior of a flexible polymer chain in the presence of a low-molecular weight solvent in the vicinity of a liquid-gas critical point within the framework of a self-consistent field theory. The total free energy of the dilute polymer solution is expressed as a function of the radius of gyration of the polymer and the average solvent number density within the gyration volume at the level of the mean-field approximation. Varying the strength of attraction between polymer and solvent we show that two qualitatively different regimes occur at the liquid-gas critical point. In case of w…
We present results of Monte Carlo study of the monomer-monomer correlation functions, static structure factor and asphericity characteristics of a single homopolymer in the coil and globular states for three distinct architectures of the chain: ring, open and star. To rationalise the results we introduce the dimensionless correlation functions rescaled via the corresponding mean-squared distances between monomers. For flexible chains with some architectures these functions exhibit a large degree of universality by falling onto a single or several distinct master curves. In the repulsive regime…
We present a simple analytical theory of flexible polymer chain dissolved in a good solvent, carrying permanent freely oriented dipoles on the monomers. We take into account the dipole correlations within the random phase approximation (RPA), as well as a dielectric heterogeneity in the internal polymer volume relative to the bulk solution. We demonstrate that the dipole correlations of monomers can be taken into account as pairwise ones only when the polymer chain is in a coil conformation. In this case the dipole correlations manifest themselves through the Keesom interactions of the permane…
We study the dynamics and conformation of polymers composed by active monomers. By means of Brownian dynamics simulations we show that when the direction of the self-propulsion of each monomer is aligned with the backbone, the polymer undergoes a coil-to-globule-like transition, highlighted by a marked change of the scaling exponent of the gyration radius. Concurrently, the diffusion coefficient of the center of mass of the polymer becomes essentially independent of the polymer size for sufficiently long polymers or large magnitudes of the self-propulsion. These effects are reduced when the se…
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…
In this paper we present a method to study the folding structure of a simple model consisting of two kinds of monomers, hydrophobic and hydrophilic. This method has three main steps: an efficient simulation method to bring an open sequence of homopolymer to a folded state, the application of a painting method called (regular hull) to the folded globule and the refolding process of the obtained copolymer sequence. This study allows us to suggest a theoretical function of disorder distribution for copolymer sequences that give rise to a compacted and well micro-phase separated globule.
A version of the Gaussian self-consistent (GSC) method, which avoids the use of the Edwards' virial expansion, is presented. Instead, the mean energy is evaluated directly via a convolution of the attractive part of the pair-wise non-bonded potential with the Gaussian trial radial distribution function. The hard sphere repulsion is taken into account via a suitably generalised Carnahan-Starling term. Comparison of the mean-squared inter-monomer distances and radius of gyration, as well as of the mean energy, between the results from the GSC calculations and Monte Carlo (MC) simulation in conti…
The coil-globule transition of an isolated polymer has been well established to be a second-order phase transition described by a standard tricritical O(0) field theory. We present Monte-Carlo simulations of interacting self-avoiding walks and interacting self-avoiding trails in four dimensions which provide compelling evidence that the approach to this (tri)critical point is dominated by the build-up of first-order-like singularities masking the second-order nature of the coil-globule transition.
Recently, it has been demonstrated [Magee et al., Phys. Rev. Lett. 96, 207802 (2006)] that isolated, square-well homopolymers can spontaneously break chiral symmetry and freeze into helical structures at sufficiently low temperatures. This behavior is interesting because the square-well homopolymer is itself achiral. In this work, we use event-driven molecular dynamics, combined with an optimized parallel tempering scheme, to study this polymer model over a wide range of parameters. We examine the conditions where the helix structure is stable and determine how the interaction parameters of th…
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.
In this paper we extend the Gaussian self-consistent method to permit study of the equilibrium and kinetics of conformational transitions for heteropolymers with any given primary sequence. The kinetic equations earlier derived by us are transformed to a form containing only the mean squared distances between pairs of monomers. These equations are further expressed in terms of instantaneous gradients of the variational free energy. The method allowed us to study exhaustively the stability and conformational structure of some periodic and random aperiodic sequences. A typical phase diagram of a…
