Interacting Elastic Lattice Polymers: a Study of the Free-Energy of Globular Rings

We introduce and implement a Monte Carlo scheme to study the equilibrium statistics of polymers in the globular phase. It is based on a model of "interacting elastic lattice polymers" and allows a sufficiently good sampling of long and compact configurations, an essential prerequisite to study the scaling behaviour of free energies. By simulating interacting self-avoiding rings at several temperatures in the collapsed phase, we estimate both the bulk and the surface free energy. Moreover from the corresponding estimate of the entropic exponent $\alpha-2$ we provide evidence that, unlike for swollen and $\Theta$-point rings, the hyperscaling relation is not satisfied for globular rings.Comment: 8 pages; v2: typos removed, published versio

Collapse transitions of a periodic hydrophilic hydrophobic chain

We study a single self avoiding hydrophilic hydrophobic polymer chain, through Monte Carlo lattice simulations. The affinity of monomer $i$ for water is characterized by a (scalar) charge $\lambda_{i}$, and the monomer-water interaction is short-ranged. Assuming incompressibility yields an effective short ranged interaction between monomer pairs $(i,j)$, proportional to $(\lambda_i+\lambda_j)$. In this article, we take $\lambda_i=+1$ (resp. ($\lambda_i=- 1$)) for hydrophilic (resp. hydrophobic) monomers and consider a chain with (i) an equal number of hydro-philic and -phobic monomers (ii) a periodic distribution of the $\lambda_{i}$ along the chain, with periodicity $2p$. The simulations are done for various chain lengths $N$, in $d=2$ (square lattice) and $d=3$ (cubic lattice). There is a critical value $p_c(d,N)$ of the periodicity, which distinguishes between different low temperature structures. For $p >p_c$, the ground state corresponds to a macroscopic phase separation between a dense hydrophobic core and hydrophilic loops. For $p <p_c$ (but not too small), one gets a microscopic (finite scale) phase separation, and the ground state corresponds to a chain or network of hydrophobic droplets, coated by hydrophilic monomers. We restrict our study to two extreme cases, $p \sim O(N)$ and $p\sim O(1)$ to illustrate the physics of the various phase transitions. A tentative variational approach is also presented.Comment: 21 pages, 17 eps figures, accepted for publication in Eur. Phys. J.

Supercoil formation in DNA denaturation

We generalize the Poland-Scheraga (PS) model to the case of a circular DNA, taking into account the twisting of the two strains around each other. Guided by recent single-molecule experiments on DNA strands, we assume that the torsional stress induced by denaturation enforces formation of supercoils whose writhe absorbs the linking number expelled by the loops. Our model predicts that, when the entropy parameter of a loop satisfies $c \le 2$, denaturation transition does not take place. On the other hand for $c>2$ a first-order denaturation transition is consistent with our model and may take place in the actual system, as in the case with no supercoils. These results are in contrast with other treatments of circular DNA melting where denaturation is assumed to be accompanied by an increase in twist rather than writhe on the bound segments.Comment: 4 pages, 3 figures, accepted for publication in PRE Rapid Com

Phase diagram of magnetic polymers

We consider polymers made of magnetic monomers (Ising or Heisenberg-like) in a good solvent. These polymers are modeled as self-avoiding walks on a cubic lattice, and the ferromagnetic interaction between the spins carried by the monomers is short-ranged in space. At low temperature, these polymers undergo a magnetic induced first order collapse transition, that we study at the mean field level. Contrasting with an ordinary $\Theta$ point, there is a strong jump in the polymer density, as well as in its magnetization. In the presence of a magnetic field, the collapse temperature increases, while the discontinuities decrease. Beyond a multicritical point, the transition becomes second order and $\Theta$-like. Monte Carlo simulations for the Ising case are in qualitative agreement with these results.Comment: 29 pages, 15 eps figures (one color figure). Submitted for publication to Eur.Phys.J.

Facilitated diffusion on confined DNA

In living cells, proteins combine 3D bulk diffusion and 1D sliding along the DNA to reach a target faster. This process is known as facilitated diffusion, and we investigate its dynamics in the physiologically relevant case of confined DNA. The confining geometry and DNA elasticity are key parameters: we find that facilitated diffusion is most efficient inside an isotropic volume, and on a flexible polymer. By considering the typical copy numbers of proteins in vivo, we show that the speedup due to sliding becomes insensitive to fine tuning of parameters, rendering facilitated diffusion a robust mechanism to speed up intracellular diffusion-limited reactions. The parameter range we focus on is relevant for in vitro systems and for facilitated diffusion on yeast chromatin

Topological and geometrical entanglement in a model of circular DNA undergoing denaturation

The linking number (topological entanglement) and the writhe (geometrical entanglement) of a model of circular double stranded DNA undergoing a thermal denaturation transition are investigated by Monte Carlo simulations. By allowing the linking number to fluctuate freely in equilibrium we see that the linking probability undergoes an abrupt variation (first-order) at the denaturation transition, and stays close to 1 in the whole native phase. The average linking number is almost zero in the denatured phase and grows as the square root of the chain length, N, in the native phase. The writhe of the two strands grows as the square root of N in both phases.Comment: 7 pages, 11 figures, revte

Ranking knots of random, globular polymer rings

An analysis of extensive simulations of interacting self-avoiding polygons on cubic lattice shows that the frequencies of different knots realized in a random, collapsed polymer ring decrease as a negative power of the ranking order, and suggests that the total number of different knots realized grows exponentially with the chain length. Relative frequencies of specific knots converge to definite values because the free energy per monomer, and its leading finite size corrections, do not depend on the ring topology, while a subleading correction only depends on the crossing number of the knots.Comment: 4 pages, 5 figure

Nonequilibrium Kinetics of One-Dimensional Bose Gases

We study cold dilute gases made of bosonic atoms, showing that in the mean-field one-dimensional regime they support stable out-of-equilibrium states. Starting from the 3D Boltzmann-Vlasov equation with contact interaction, we derive an effective 1D Landau-Vlasov equation under the condition of a strong transverse harmonic confinement. We investigate the existence of out-of-equilibrium states, obtaining stability criteria similar to those of classical plasmas.Comment: 16 pages, 6 figures, accepted for publication in Journal of Statistical Mechanics: Theory and Experimen

Switching dynamics in cholesteric blue phases

Blue phases are networks of disclination lines, which occur in cholesteric liquid crystals near the transition to the isotropic phase. They have recently been used for the new generation of fast switching liquid crystal displays. Here we study numerically the steady states and switching hydrodynamics of blue phase I (BPI) and blue phase II (BPII) cells subjected to an electric field. When the field is on, there are three regimes: for very weak fields (and strong anchoring at the boundaries) the blue phases are almost unaffected, for intermediate fields the disclinations twist (for BPI) and unzip (for BPII), whereas for very large voltages the network dissolves in the bulk of the cell. Interestingly, we find that a BPII cell can recover its original structure when the field is switched off, whereas a BPI cell is found to be trapped more easily into metastable configurations. The kinetic pathways followed during switching on and off entails dramatic reorganisation of the disclination networks. We also discuss the effect of changing the director field anchoring at the boundary planes and of varying the direction of the applied field.Comment: 17 pages, 11 figure