First-order phase transition in a stiff polymer chain

Influence of solvent quality on the swelling and deswelling and the shear modulus of semi-dilute solution cross-linked poly(vinyl acetate) gels

We systematically examine the influence of varying temperature (T) over a large range in model poly(vinyl acetate) gels swollen in isopropyl alcohol. The theta temperature Θ, at which the second virial coefficient A2 vanishes, is found to be equal to within numerical uncertainty to the corresponding high molecular mass polymer solution value without cross-links, and we quantify the swelling and deswelling of our model gels relative to their size at T = Θ, as customary for individual flexible polymer chains in solutions. We also quantify the "solvent quality" dependence of the shear modulus G relative to G(T = Θ) and compare to the hydrogel swelling factor, α. We find that all our network swelling and deswelling data can be reduced to a scaling equation of the same general form as derived from renormalization group theory for flexible linear polymer chains in solutions so that it is not necessary to invoke either the Flory-Huggins mean field theory or the Flory-Rehner hypothesis that the elastic and mixing contributions to the free energy of network swelling are separable to describe our data. We also find that changes of G relative to G(T = Θ) are directly related to α. At the same time, we find that classical rubber elasticity theory describes many aspects of these semi-dilute solution cross-linked networks, regardless of the solvent quality, although the prefactor clearly reflects the existence of network defects whose concentration depends on the initial polymer concentration of the polymer solution from which the networks were synthesized.

Protein structure predictions by enhanced conformational sampling methods

In this Special Festschrift Issue for the celebration of Professor Nobuhiro Gō's 80th birthday, we review enhanced conformational sampling methods for protein structure predictions. We present several generalized-ensemble algorithms such as multicanonical algorithm, replica-exchange method, etc. and parallel Monte Carlo or molecular dynamics method with genetic crossover. Examples of the results of these methods applied to the predictions of protein tertiary structures are also presented.

On the Pseudo Phase Diagram of Single Semi-Flexible Polymer Chains: A Flat-Histogram Monte Carlo Study

Local stiffness of polymer chains is instrumental in all structure formation processes of polymers, from crystallization of synthetic polymers to protein folding and DNA compactification. We present Stochastic Approximation Monte Carlo simulations-a type of flat-histogram Monte Carlo method-determining the density of states of a model class of single semi-flexible polymer chains, and, from this, their complete thermodynamic behavior. The chains possess a rich pseudo phase diagram as a function of stiffness and temperature, displaying non-trivial ground-state morphologies. This pseudo phase diagram also depends on chain length. Differences to existing pseudo phase diagrams of semi-flexible chains in the literature emphasize the fact that the mechanism of stiffness creation matters.

Knots as a Topological Order Parameter for Semiflexible Polymers

Using a combination of the multicanonical Monte Carlo algorithm and the replica-exchange method, we investigate the influence of bending stiffness on the conformational phases of a bead-stick homopolymer model and present the pseudophase diagram for the complete range of semiflexible polymers, from flexible to stiff. Although it is a simple model, we observe a rich variety of conformational phases, reminiscent of conformations observed for synthetic polymers or biopolymers. Depending on the bending stiffness, the model exhibits different pseudophases like bent, hairpin, or toroidal. In particular, we find thermodynamically stable knots and unusual transitions into these knotted phases with a clear phase coexistence, but almost constant mean total energy, and hence almost no latent heat.

Thermodynamics and structure of macromolecules from flat-histogram Monte Carlo simulations

Over the last decade flat-histogram Monte Carlo simulations, especially multi-canonical and Wang-Landau simulations, have emerged as a strong tool to study the statistical mechanics of polymer chains. These investigations have focused on coarse-grained models of polymers on the lattice and in the continuum. Phase diagrams of chains in bulk as well as chains attached to surfaces were studied, for homopolymers as well as for protein-like models. Also, aggregation behavior in solution of these models has been investigated. We will present here the theoretical background for these simulations, explain the algorithms used and discuss their performance and give an overview over the systems studied with these methods in the literature, where we will limit ourselves to studies of coarse-grained model systems. Implementations of these algorithms on parallel computers will be also briefly described. In parallel to the development of these simulation methods, the power of a micro-canonical analysis of such simulations has been recognized, and we present the current state of the art in applying the micro-canonical analysis to phase transitions in nanoscopic polymer systems.

Coil-globule transition of a single semiflexible chain in slitlike confinement

Single polymer chains undergo a phase transition from coiled conformations to globular conformations as the effective attraction between monomers becomes strong enough. In this work, we investigated the coil-globule transition of a semiflexible chain confined between two parallel plates, i.e. a slit, using the lattice model and Pruned-enriched Rosenbluth method (PERM) algorithm. We find that as the slit height decreases, the critical attraction for the coil-globule transition changes non-monotonically due to the competition of the confinement free energies of the coiled and globular states. In wide (narrow) slits, the coiled state experiences more (less) confinement free energy, and hence the transition becomes easier (more difficult). In addition, we find that the transition becomes less sharp with the decreasing slit height. Here, the sharpness refers to the sensitivity of thermodynamic quantities when varying the attraction around the critical value. The relevant experiments can be performed for DNA condensation in microfluidic devices.

Enhanced sampling algorithms

In biomolecular systems (especially all-atom models) with many degrees of freedom such as proteins and nucleic acids, there exist an astronomically large number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. Enhanced conformational sampling techniques are thus in great demand. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present various extensions of these three generalized-ensemble algorithms. The effectiveness of the methods is tested with short peptide and protein systems.

Topological phase transition in a RNA model in the de Gennes regime

We study a simplified model of the RNA molecule proposed by Vernizzi in the regime of strong concentration of positive ions in solution. The model considers a flexible chain of equal bases that can pairwise interact with any other one along the chain while preserving the property of saturation of the interactions. In the regime considered, we observe the emergence of a critical temperature Tc separating two phases that can be characterized by the topology of the predominant configurations: in the large temperature regime, the dominant configurations of the molecule have very large genera (on the order of the size of the molecule), corresponding to a complex topology, whereas in the opposite regime of low temperatures the dominant configurations are simple and have the topology of a sphere. We determine that this topological phase transition is of first order and provide an analytical expression for Tc. The regime studied for this model exhibits analogies with the dense polymer systems studied by de Gennes.

Coil-globule coexistence and compaction of DNA chains

In this work we discuss different factors governing coil-globule coexistence in the compaction process of DNA. We initially analyse the role played by fluctuations in the degree of binding of an external compacting agent in the conformational behavior of the chain backbone. The analysis relies both on Monte Carlo simulation results and simple statistical approaches. Compacting agents of various binding characteristics are taken into consideration and the degree of charge neutralization upon the chain is related to conformational indicators. Selected model systems comprising stiff chains in the presence of multivalent ions are employed to assess intrinsic single-chain conformational fluctuation, in the presence of external agents but not resulting from differences in binding. It is shown that trends found for a variety of compacting agents, including the extension of the coil-globule coexistence regions, can be rationalised on the basis of this analysis.

Stability of toroid and rodlike globular structures of a single stiff-chain macromolecule for different bending potentials

We study the effect of the bending potential on the stability of toroidal and rodlike globules which are typical collapsed conformations of a single stiff-chain macromolecule. We perform numerical calculations in the framework of the bead-stick model of a polymer chain. The intrinsic structure of globules is also analyzed. It was shown that the bending potential affects the packing geometry of bundles in a toroidal globule in the ground state. This potential also influences the bends at the ends of a rodlike globule: both the shape of the loops and the number of bonds in each loop have been investigated numerically as well as by Monte Carlo computer simulations performed for a separate loop. Our main results are (1) the shape of the bending potential could be possibly seen from the geometry of a globule; (2) toroidal globules are always more favorable than the rodlike ones.

Semiflexible amphiphilic polymers: Cylindrical-shaped, collagenlike, and toroidal structures

A coarse-grained model is used to study the conformational properties of semiflexible polymers with amphiphilic monomer units containing both hydrophilic and hydrophobic interaction sites. The hydrophobically driven conformational transitions are studied using molecular dynamics simulations for the chains of varying stiffness, as characterized by intrinsic Kuhn segment lengths that vary over a decade. It is shown that the energy of hydrophobic attraction required for the realization of the coil-to-globule transition increases with increasing chain stiffness. For rather stiff backbone, the coil-to-globule transition corresponds to a first order phase transition. We find that depending on the chain stiffness, a variety of thermodynamically stable anisometric chain morphologies are possible in a solvent selectively poor for hydrophobic sites of amphiphilic monomer units. For flexible chains, the amphiphilic polymer forms a cylindrical globule having blob structure with nearly spherical blobs. With increasing stiffness, the number of blobs composing the globule decreases and the shape of blobs transforms into elongated cylinder. Further increase in stiffness leads to compaction of macromolecules into a collagenlike structure when the chain folds itself several times and different strands wind round each other. In this state, the collagenlike structures coexist with toroidal globules, both conformations having approximately equal energies.

Single macromolecules: hierarchic thermodynamics, irreversibility and biological function

Single homo-polymers show two different paths in the folding transition; a liquid-like spherical globule is generated for flexible polymers, whereas a rich variety of ordered structures are formed for semi-flexible polymers. This unique characteristic is discussed in relation to its biological significance.

Generalized-ensemble algorithms for molecular simulations of biopolymers

In complex systems with many degrees of freedom such as peptides and proteins, there exists a huge number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present three new generalized-ensemble algorithms that combine the merits of the above methods. The effectiveness of the methods for molecular simulations in the protein folding problem is tested with short peptide systems.