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) an uniformly stretched chain at strong extensions. The conformational transition from the tadpole to the stretched chain is accompanied by an abrupt unfolding of the depleted globular head and a corresponding jump-wise drop in the intrachain tension. The unfolding-refolding cycle demonstrates a hysteresis loop in the vicinity of the transition point. These three regimes of deformation, as well as the first-order like transition between the tadpole and the stretched chain conformations, can be experimentally observable provided that the number of monomer units in the chain is large and/or the solvent quality is sufficiently poor. For short chains, on the other hand, under moderately poor solvent strength conditions, the unfolding transition is continuous. Upon an increase in the imposed end-to-end distance the extended globule retains a longitudinally uniform shape at any degree of deformation. In all cases the system exhibits a negative extensional modulus in the intermediate range of deformations. We anticipate that predicted patterns in force-deformation curves for polymer molecules in poor solvent can be observed in single molecule atomic spectroscopy experiments.
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 wit…
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…
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…
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…
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 study the force-induced unfolding of a homopolymer on the three dimensional Sierpinski gasket. The polymer is subject to a contact energy between nearest neighbour sites not consecutive along the chain and to a stretching force. The hierarchical nature of the lattice we consider allows for an exact treatment which yields the phase diagram and the critical behaviour. We show that for this model mean field predictions are not correct, in particular in the exact phase diagram there is {\em not} a low temperature reentrance and we find that the force induced unfolding transition below the theta…
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 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 …
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…
Folding and unfolding of biopolymers are often manipulated in experiment by tuning pH, temperature, single-molecule force or shear field. Here we carry out Brownian dynamics simulations to explore the behavior of a single self-attracting chain in the suspension of self-propelling particles (SPPs). As the propelling force increases, globule-stretch (G-S) transition of the chain happens due to the enhanced disturbance from SPPs. Two distinct mechanisms of the transition in the limits of low and high rotational diffusion rates of SPPs have been observed: shear effect at low rate and collision-ind…
We describe a systematic approach to construct coarse-grained Markov state models from molecular dynamics data of systems driven into a non-equilibrium steady state. We apply this method to study the globule-stretch transition of a single tethered model polymer in shear flow. The folding/unfolding rates of the coarse-grained model agree with the original detailed model. We demonstrate that the folding/unfolding proceeds through the same narrow region of configuration space but along different cycles.
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…
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 …
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 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…
Force induced stretching of polymers is important in a variety of contexts. We have used theory and simulations to describe the response of homopolymers, with $N$ monomers, to force ($f$) in good and poor solvents. In good solvents and for {sufficiently large} $N$ we show, in accord with scaling predictions, that the mean extension along the $f$ axis $<Z>\sim f$ for small $f$, and $<Z>\sim f^{2/3}$ (the Pincus regime) for intermediate values of $f$. The theoretical predictions for $\la Z\ra$ as a function of $f$ are in excellent agreement with simulations for N=100 and 1600. However, even with…
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…
Understanding how proteins fold into their native structure is a fundamental problem in biophysics, crucial for protein design. It has been hypothesized that the formation of a molten globule intermediate precedes folding to the native conformation of globular proteins; however, its thermodynamic properties are poorly known. We perform single-molecule pulling experiments of protein barnase in the range of 7$^\circ$C to 37$^\circ$C using a temperature-jump optical trap. We derive the folding free energy, entropy and enthalpy, and heat capacity change ($ΔC_p = 1050\pm50$ cal/mol$\cdot$K) at low …
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.
Conformational transitions of a single macromolecule of finite size $N$ cannot be described within standard thermodynamic framework. Taking as a basis a simple model of homopolymer exhibiting a coil-globule transition, we show that a relevant approach is to describe the thermal equilibrium distribution $P_N^{(β)}(t)$ of some variable $t$ characterizing the conformation. Although the mean order parameter exhibits a second-order behaviour in the infinite-size limit, the $Θ$-point arises from the coexistence of two distinct populations, associated with two well-separated peaks of $P_N^{(β)}(t)$ a…
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…
The conformational behavior of a giant DNA mediated by condensing agents in the bulk solution has been investigated through experimental and theoretical approaches. Experimentally, a pronounced conformational hysteresis is observed for folding and unfolding processes, by increasing and decreasing the concentration of condensing agent PEG (Polyethylene glycol), respectively. To elucidate the observed hysteresis, a semiflexible chain model is studied by using Monte Carlo simulations for the coil-globule transition. In the simulations, the hysteresis loop emerges for stiff enough chains, indicati…
We investigate the behavior of single polyelectrolytes in multivalent salt solutions under the action of electric fields through computer simulations. The chain is unfolded in a strong electric field and aligned parallel to the field direction, and the chain size shows a sigmoidal transition. The unfolding electric field $E^*$ depends on the salt concentration and scales as $V^{-1/2}$ with $V$ being the ellipsoidal volume occupied by the chain. The magnitude of the electrophoretic mobility of chain drastically increases during the unfolding. The fact that $E^*$ depends on the chain length prov…
The phase structure of a homopolymer chain is investigated in terms of a universal theoretical model, designed to describe the infrared limit of slow spatial variations. The effects of chirality are studied and compared with the influence of a short-range attractive interaction between monomers, at various ambient temperature values. In the high temperature limit the homopolymer chain is in the self-avoiding random walk phase. But at low temperatures, two different phases are possible: When short-range attractive interactions dominate over chirality, the chain collapses into a space- filling c…
We consider a stiff polymer chain in poor solvent and apply a force at one end of the chain. We find that by varying the stiffness parameter, polymer undergoes a transition from the globule state to the folded like state. The conformation of folded state mimics the $β$-sheet as seen in titin molecule. Using exact enumeration technique, we study the extension-force and force-temperature diagrams of such a system. Force-temperature diagram shows the re-entrance behaviour for flexible chain. However, for stiff chain this re-entrance behaviour is absent and there is an enhancement in $θ$-temperatu…
We study numerically a lattice model of semiflexible homopolymers with nearest neighbor attraction and energetic preference for straight joints between bonded monomers. For this we use a new algorithm, the "Pruned-Enriched Rosenbluth Method" (PERM). It is very efficient both for relatively open configurations at high temperatures and for compact and frozen-in low-T states. This allows us to study in detail the phase diagram as a function of nn-attraction epsilon and stiffness x. It shows a theta-collapse line with a transition from open coils to molten compact globules (large epsilon) and a fr…
The unfolding of a polymer below the $θ$ point when pulled by an external force is studied both in d=2 on the lattice and in $d=3$ off lattice. A ground state analysis of finite length chains shows that the globule unfolds via multiple steps, corresponding to transitions between different minima, in both cases. In the infinite length limit, these intermediate minima have a qualitative effect only in $d=2$. The phase diagram in d=2 is determined using transfer matrix techniques. Energy-entropy and renormalization group arguments are given which predict a qualitatively correct phase diagram and …
