Solution of a model of self-avoiding walks with multiple monomers per site on the Bethe lattice

Collapse transition in polymer models with multiple monomers per site and multiple bonds per edge

We present results from extensive Monte Carlo simulations of polymer models where each lattice site can be visited by up to K monomers and no restriction is imposed on the number of bonds on each lattice edge. These multiple monomer per site (MMS) models are investigated on the square and cubic lattices, for K=2 and 3, by associating Boltzmann weights ω_{0}=1, ω_{1}=e^{β_{1}}, and ω_{2}=e^{β_{2}} to sites visited by 1, 2, and 3 monomers, respectively. Two versions of the MMS models are considered for which immediate reversals of the walks are allowed (RA) or forbidden (RF). In contrast to previous simulations of these models, we find the same thermodynamic behavior for both RA and RF versions. In three dimensions, the phase diagrams, in space β_{2}×β_{1}, are featured by coil and globule phases separated by a line of Θ points, as thoroughly demonstrated by the metric ν_{t}, crossover ϕ_{t}, and entropic γ_{t} exponents. The existence of the Θ lines is also confirmed by the second virial coefficient. This shows that no discontinuous collapse transition exists in these models, in contrast to previous claims based on a weak bimodality observed in some distributions, which indeed exists in a narrow region very close to the Θ line when β_{1}<0. Interestingly, in two dimensions, only a crossover is found between the coil and globule phases.

Polymer models with competing collapse interactions on Husimi and Bethe lattices

In the framework of Husimi and Bethe lattices, we investigate a generalized polymer model that incorporates as special cases different models previously studied in the literature, namely, the standard interacting self-avoiding walk, the interacting self-avoiding trail, and the vertex-interacting self-avoiding walk. These models are characterized by different microscopic interactions, giving rise, in the two-dimensional case, to collapse transitions of an apparently different nature. We expect that our results, even though of a mean-field type, could provide some useful information to elucidate the role of such different θ points in the polymer phase diagram. These issues are at the core of a long-standing unresolved debate.

Nature of the collapse transition in interacting self-avoiding trails

We study the interacting self-avoiding trail (ISAT) model on a Bethe lattice of general coordination q and on a Husimi lattice built with squares and coordination q=4. The exact grand-canonical solutions of the model are obtained, considering that up to K monomers can be placed on a site and associating a weight ω_{i} with an i-fold visited site. Very rich phase diagrams are found with nonpolymerized, regular polymerized, and dense polymerized phases separated by lines (or surfaces) of continuous and discontinuous transitions. For a Bethe lattice with q=4 and K=2, the collapse transition is identified with a bicritical point and the collapsed phase is associated with the dense polymerized (solidlike) phase instead of the regular polymerized (liquidlike) phase. A similar result is found for the Husimi lattice, which may explain the difference between the collapse transition for ISATs and for interacting self-avoiding walks on the square lattice. For q=6 and K=3 (studied on the Bethe lattice only), a more complex phase diagram is found, with two critical planes and two coexistence surfaces, separated by two tricritical and two critical end-point lines meeting at a multicritical point. The mapping of the phase diagrams in the canonical ensemble is discussed and compared with simulational results for regular lattices.

Chemically controlled unfolding of a RNA-like polymer model

We consider a lattice polymer model of the two-tolerant type (i.e., a random walk allowed to visit lattice bonds at most twice), in which doubly visited bonds yield an attractive energy term (pairing energy). Such a model has been previously proposed as a rough, nonspecific description of the RNA folding mechanism. Indeed, the model predicts, besides the usual theta collapse, an extra transition to a low-temperature fully paired state. In the current work, we propose an extension of the model, in which a "micromolecular" chemical species can bind the polymer and locally forbid segment pairing. We investigate equilibrium thermodynamics in the grand-canonical picture, at the level of a Bethe approximation, which is, a refined mean-field technique, equivalent to the exact solution on a random-regular graph. The general trend we observe is that expected from the mechanism implemented in the model (increasing micromolecule concentration favors unfolding and lowers the transition temperature), but the resulting phase diagram turns out to be remarkably interesting and rich.

Walks with up to two monomers per site on the Bethe lattice: interpolation between models with immediate reversals allowed and immediate reversals forbidden

We generalize earlier calculations of the RA (immediate reversals allowed) and RF (immediate reversals forbidden) model with multiple monomers per site, including a parameter in the model, so that both cases studied before are particular cases of the model considered here. Also, we calculate the bulk free energy of the model using Gujrati's prescription, which leads to corrections in the localization of coexistence lines and qualitative changes in the phase diagram of the model. Thus, this calculation provides an example in which the use of the method of iterating recursion relations starting with natural initial conditions to locate coexistence loci is not trustworthy and may lead to qualitatively different results. A continuous collapse transition appears in all cases as a tricritical point.

Grand-canonical and canonical solution of self-avoiding walks with up to three monomers per site on the Bethe lattice

We solve a model of polymers represented by self-avoiding walks on a lattice, which may visit the same site up to three times in the grand-canonical formalism on the Bethe lattice. This may be a model for the collapse transition of polymers where only interactions between monomers at the same site are considered. The phase diagram of the model is very rich, displaying coexistence and critical surfaces, critical, critical end point, and tricritical lines, as well as a multicritical point. From the grand-canonical results, we present an argument to obtain the properties of the model in the canonical ensemble, and compare our results with simulations in the literature. We do actually find extended and collapsed phases, but the transition between them, composed by a line of critical end points and a line of tricritical points, separated by the multicritical point, is always continuous. This result is at variance with the simulations for the model, which suggest that part of the line should be a discontinuous transition. Finally, we discuss the connection of the present model with the standard model for the collapse of polymers (self-avoiding, self-attracting walks), where the transition between the extended and collapsed phases is a tricritical point.

Bethe approximation for a DNA-like self-avoiding walk model with variable solvent quality

The phase diagram and critical behavior of a simple toy model for DNA zipping/unzipping are examined in the framework of the Bethe approximation. The effects of solvent quality are included and found to lead to a variety of different thermodynamic behaviors.

Solution of a model of self-avoiding walks with multiple monomers per site on the Husimi lattice

We solve a model of self-avoiding walks which allows for a site to be visited up to two times by the walk on the Husimi lattice. This model is inspired in the Domb-Joyce model and was proposed to describe the collapse transition of polymers with one-site interactions only. We consider the version in which immediate self-reversals of the walk are forbidden. The phase diagram we obtain for the grand-canonical version of the model is similar to the one found in the solution of the Bethe lattice, with two distinct polymerized phases: a tricritical point and a critical endpoint.

Exact solution of a RNA-like polymer model on the Husimi lattice

We investigate a two-tolerant polymer model on the square Husimi lattice, which aims at describing the properties of RNA-like macromolecules. We solve the model in a numerically exact way, working out the grand-canonical phase diagram, both with and without taking into account the stacking effect. Besides a nonpolymerized phase, we observe two different polymerized phases characterized by a lower or higher density of doubly visited lattice bonds. The system exhibits three qualitatively different regimes, as a function of the monomer chemical potential. Below some T1 temperature and above some T2 temperature, the transition to the nonpolymerized phase is continuous, whereas, in the (T1,T2) temperature range, the transition is first order. In the dilute-solution limit, the high temperature regime corresponds to a swollen ("coil") state, the intermediate regime to a moderately collapsed ("molten") state, with a small fraction of paired segments, and the low temperature regime to an almost fully paired ("native") state. The molten state ends in a tricritical (Theta-like) transition at high temperature and in a critical end point at low temperature. Upon increasing the stacking energy parameter, the temperature range of the molten state turns out to be progressively reduced but never completely removed.