Kinetics and thermodynamics of first-order Markov chain copolymerization

We report a theoretical study of stochastic processes modeling the growth of first-order Markov copolymers, as well as the reversed reaction of depolymerization. These processes are ruled by kinetic equations describing both the attachment and detachment of monomers. Exact solutions are obtained for these kinetic equations in the steady regimes of multicomponent copolymerization and depolymerization. Thermodynamic equilibrium is identified as the state at which the growth velocity is vanishing on average and where detailed balance is satisfied. Away from equilibrium, the analytical expression of the thermodynamic entropy production is deduced in terms of the Shannon disorder per monomer in the copolymer sequence. The Mayo-Lewis equation is recovered in the fully irreversible growth regime. The theory also applies to Bernoullian chains in the case where the attachment and detachment rates only depend on the reacting monomer

Monte Carlo Algorithm for Simulating Reversible Aggregation of Multisite Particles

We present an efficient and exact Monte Carlo algorithm to simulate reversible aggregation of particles with dedicated binding sites. This method introduces a novel data structure of dynamic bond tree to record clusters and sequences of bond formations. The algorithm achieves a constant time cost for processing cluster association and a cost between $\mathcal{O}(\log M)$ and $\mathcal{O}(M)$ for processing bond dissociation in clusters with $M$ bonds. The algorithm is statistically exact and can reproduce results obtained by the standard method. We applied the method to simulate a trivalent ligand and a bivalent receptor clustering system and obtained an average scaling of $\mathcal{O}(M^{0.45})$ for processing bond dissociation in acyclic aggregation, compared to a linear scaling with the cluster size in standard methods. The algorithm also demands substantially less memory than the conventional method.Comment: 8 pages, 3 figure

Effects of Kinks on DNA Elasticity

We study the elastic response of a worm-like polymer chain with reversible kink-like structural defects. This is a generic model for (a) the double-stranded DNA with sharp bends induced by binding of certain proteins, and (b) effects of trans-gauche rotations in the backbone of the single-stranded DNA. The problem is solved both analytically and numerically by generalizing the well-known analogy to the Quantum Rotator. In the small stretching force regime, we find that the persistence length is renormalized due to the presence of the kinks. In the opposite regime, the response to the strong stretching is determined solely by the bare persistence length with exponential corrections due to the ``ideal gas of kinks''. This high-force behavior changes significantly in the limit of high bending rigidity of the chain. In that case, the leading corrections to the mechanical response are likely to be due to the formation of multi-kink structures, such as kink pairs.Comment: v1: 16 pages, 7 figures, LaTeX; submitted to Physical Review E; v2: a new subsection on soft kinks added to section Theory, sections Introduction and Conclusions expanded, references added, other minor changes; v3: a reference adde

Proteins and polymers

Proteins, chain molecules of amino acids, behave in ways which are similar to each other yet quite distinct from standard compact polymers. We demonstrate that the Flory theorem, derived for polymer melts, holds for compact protein native state structures and is not incompatible with the existence of structured building blocks such as $\alpha$-helices and $\beta$-strands. We present a discussion on how the notion of the thickness of a polymer chain, besides being useful in describing a chain molecule in the continuum limit, plays a vital role in interpolating between conventional polymer physics and the phase of matter associated with protein structures.Comment: 7 pages, 6 figure

Topological Solitons and Folded Proteins

We propose that protein loops can be interpreted as topological domain-wall solitons. They interpolate between ground states that are the secondary structures like alpha-helices and beta-strands. Entire proteins can then be folded simply by assembling the solitons together, one after another. We present a simple theoretical model that realizes our proposal and apply it to a number of biologically active proteins including 1VII, 2RB8, 3EBX (Protein Data Bank codes). In all the examples that we have considered we are able to construct solitons that reproduce secondary structural motifs such as alpha-helix-loop-alpha-helix and beta-sheet-loop-beta-sheet with an overall root-mean-square-distance accuracy of around 0.7 Angstrom or less for the central alpha-carbons, i.e. within the limits of current experimental accuracy.Comment: 4 pages, 4 figure

Birth and growth of cavitation bubbles within water under tension confined in a simple synthetic tree

Water under tension, as can be found in several systems including tree vessels, is metastable. Cavitation can spontaneously occur, nucleating a bubble. We investigate the dynamics of spon- taneous or triggered cavitation inside water filled microcavities of a hydrogel. Results show that a stable bubble is created in only a microsecond timescale, after transient oscillations. Then, a diffusion driven expansion leads to filling of the cavity. Analysis reveals that the nucleation of a bubble releases a tension of several tens of MPa, and a simple model captures the different time scales of the expansion process

Two-way coupling of FENE dumbbells with a turbulent shear flow

We present numerical studies for finitely extensible nonlinear elastic (FENE) dumbbells which are dispersed in a turbulent plane shear flow at moderate Reynolds number. The polymer ensemble is described on the mesoscopic level by a set of stochastic ordinary differential equations with Brownian noise. The dynamics of the Newtonian solvent is determined by the Navier-Stokes equations. Momentum transfer of the dumbbells with the solvent is implemented by an additional volume forcing term in the Navier-Stokes equations, such that both components of the resulting viscoelastic fluid are connected by a two-way coupling. The dynamics of the dumbbells is given then by Newton's second law of motion including small inertia effects. We investigate the dynamics of the flow for different degrees of dumbbell elasticity and inertia, as given by Weissenberg and Stokes numbers, respectively. For the parameters accessible in our study, the magnitude of the feedback of the polymers on the macroscopic properties of turbulence remains small as quantified by the global energy budget and the Reynolds stresses. A reduction of the turbulent drag by up to 20% is observed for the larger particle inertia. The angular statistics of the dumbbells shows an increasing alignment with the mean flow direction for both, increasing elasticity and inertia. This goes in line with a growing asymmetry of the probability density function of the transverse derivative of the streamwise turbulent velocity component. We find that dumbbells get stretched referentially in regions where vortex stretching or bi-axial strain dominate the local dynamics and topology of the velocity gradient tensor.Comment: 20 pages, 10 Postscript figures (Figures 5 and 10 in reduced quality

Self-Assembly of Patchy Particles into Polymer Chains: A Parameter-Free Comparison between Wertheim Theory and Monte Carlo Simulation

We numerically study a simple fluid composed of particles having a hard-core repulsion, complemented by two short-ranged attractive (sticky) spots at the particle poles, which provides a simple model for equilibrium polymerization of linear chains. The simplicity of the model allows for a close comparison, with no fitting parameters, between simulations and theoretical predictions based on the Wertheim perturbation theory, a unique framework for the analytic prediction of the properties of self-assembling particle systems in terms of molecular parameter and liquid state correlation functions. This theory has not been subjected to stringent tests against simulation data for ordering across the polymerization transition. We numerically determine many of the thermodynamic properties governing this basic form of self-assembly (energy per particle, order parameter or average fraction of particles in the associated state, average chain length, chain length distribution, average end-to-end distance of the chains, and the static structure factor) and find that predictions of the Wertheim theory accord remarkably well with the simulation results

Kinetics of bond formation in crosslinked gelatin gels

In chemical crosslinking of gelatin solutions, two different time scales affect the kinetics of the gel formation in the experiments. We complement the experimental study with Monte Carlo numerical simulations of a lattice model. This approach shows that the two characteristic time scales are related to the formation of single bonds crosslinker-chain and of bridges between chains. In particular their ratio turns out to control the kinetics of the gel formation. We discuss the effect of the concentration of chains. Finally our results suggest that, by varying the probability of forming bridges as an independent parameter, one can finely tune the kinetics of the gelation via the ratio of the two characteristic times.Comment: 8 pages, 9 figures, revised versio

Reentrant phase diagram and pH effects in cross-linked gelatin gels

Experimental results have shown that the kinetics of bond formation in chemical crosslinking of gelatin solutions is strongly affected not only by gelatin and reactant concentrations but also by the solution pH. We present an extended numerical investigation of the phase diagram and of the kinetics of bond formation as a function of the pH, via Monte Carlo simulations of a lattice model for gelatin chains and reactant agent in solution. We find a reentrant phase diagram, namely gelation can be hindered either by loop formation, at low reactant concentrations, or by saturation of active sites of the chains via formation of single bonds with crosslinkers, at high reactant concentrations. The ratio of the characteristic times for the formation of the first and of the second bond between the crosslinker and an active site of a chain is found to depend on the reactant reactivity, in good agreement with experimental data.Comment: 8 pages, 8 figure