The statistical mechanics of random copolymers

Decoding Interaction Patterns from the Chemical Sequence of Polymers Using Neural Networks

The relation between chemical sequences and the properties of polymers is considered using artificial neural networks with a low-dimensional bottleneck layer of neurons. These encoder-decoder architectures may compress the input information into a meaningful set of physical variables that describe the correlation between distinct types of data. In this work, neural networks were trained to translate a sequence of hydrophilic and hydrophobic segments into the effective free energy landscape of a copolymer interacting with a lipid membrane. The training data were obtained by the sampling of coarse-grained polymer conformations in a given membrane density field. Neural networks that were split into separate channels have learned to decompose the free energy into independent components that are explainable by known concepts from polymer physics. The semantic information in the hidden layers was employed to predict polymer translocation events through a membrane for a more detailed dynamic model via a transfer learning procedure. The search for minimal translocation times in the compressed chemical space underlined that nontrivial sequence motifs may lead to optimal properties.

Polymers undergoing inhomogeneous adsorption: Order parameters for a partially directed walk model

We consider partially directed walk models of polymers undergoing inhomogeneous adsorption. The inhomogeneity can be in the polymer, in the surface, or in both. For the cases where the polymer is either a homopolymer or a strictly alternating copolymer and where the surface is either homogeneous or has stripes of width 1, we calculate detailed order parameters and show that these provide important information about the ways in which the polymer adsorbs.

Pulling alternating copolymers adsorbed on a striped surface

We consider a partially directed walk model of a strictly alternating copolymer adsorbing on a striped surface where the energy is associated with the numbers of the two types of monomers adsorbed on the two types of surface sites. A force is applied to the last monomer and the polymer responds to this force, sometimes by desorbing. The force can be applied at various angles, with the surface component parallel or perpendicular (or at some other angle) to the stripe direction. The desorption behavior is strongly dependent on the force direction and the response gives information about the shape and direction of the polymer adsorbed on the surface, especially at low temperatures. In some cases the ground state is degenerate and this also has an important effect on the temperature dependence of the critical force needed for desorption. We give a complete solution of the problem using generating function techniques and an approximate treatment that is especially useful at low temperatures and helps in our physical understanding of the situation.

Graph theoretical analysis of the energy landscape of model polymers

In systems characterized by a rough potential-energy landscape, local energetic minima and saddles define a network of metastable states whose topology strongly influences the dynamics. Changes in temperature, causing the merging and splitting of metastable states, have nontrivial effects on such networks and must be taken into account. We do this by means of a recently proposed renormalization procedure. This method is applied to analyze the topology of the network of metastable states for different polypeptidic sequences in a minimalistic polymer model. A smaller spectral dimension emerges as a hallmark of stability of the global energy minimum and highlights a nonobvious link between dynamic and thermodynamic properties.

Critical behavior of the particle mass spectra in a family of gelling systems

The critical and post-critical behavior of coagulating systems, where the coagulation efficiency grows with the masses of colliding particles g and l as K(g,l)=0.5(g(alpha)(l)(beta)+g(beta)(l)(alpha), lambda=alpha+beta>1, mu=/alpha - beta/ <1 is studied. The instantaneous sink that removes the particles with masses exceeding G is introduced which allows one to describe the coagulation kinetics by a finite set of equations and define the gel as the deposit of particles with masses between G+1 and 2G . This system displays the critical behavior (the sol-gel transition) in the limit G-->infinity if lambda=alpha+beta>1. The exact post-critical particle mass spectrum is shown to be an algebraic function of g times a growing exponent. All critical parameters of the systems are determined as the functions of alpha and beta .

Polymer-population mapping and localization in the space of phenotypes

We present a mapping between the thermodynamics of an ideal heteropolymer in an external field and the dynamics of structured populations in fluctuating environments. We employ a population model in which individuals may adopt different phenotypes, each of which may be optimal in a different environment. Using this mapping, we develop a path integral formulation for populations and predict the existence of a biological counterpart for the well-known heteropolymer localization phase transition.

Exact kinetics of sol-gel transition in a coagulating mixture

The formation of a gel in a two component disperse system wherein binary coagulation governs the temporal changes to particle composition spectra is studied under the assumption that the coagulation kernel is proportional to m1n2+m2n1, with m,n being the numbers of monomers of the first and the second component in the coalescing pair of particles. This model is shown to reveal the sol-gel transition, i.e., the formation of one giant cluster with the mass comparable to the total mass of the whole system. This paper reports on the exact solution of this model within the Marcus-Lushnikov stochastic scheme. The evolution equation for the generating functional of the probability to find in the system a given set of occupation numbers (the numbers of particles containing m and n monomers of each component) at time t is formulated and solved exactly. The expression for the particle composition spectrum is derived and analyzed in the thermodynamic limit. It is shown that after a critical time a giant single particle (the gel) appears. The time evolution of its composition is found. Special attention is given to the transition point, where the gel is appearing. The time dependencies of the gel composition, the number concentration, and the second moments of the particle composition spectrum are found.

Smoothing of depinning transitions for directed polymers with quenched disorder

We consider disordered models of pinning of directed polymers on a defect line, including (1 + 1)-dimensional interface wetting models, disordered Poland-Scheraga models of DNA denaturation, and other (1 + d)-dimensional polymers in interaction with columnar defects. We consider also random copolymers at a selective interface. These models are known to have a (de)pinning transition at some critical line in the phase diagram. In this work we prove that, as soon as disorder is present, the transition is at least of second order: the free energy is differentiable at the critical line, and the order parameter (contact fraction) vanishes continuously at the transition. On the other hand, it is known that the corresponding nondisordered models can have a first order (de)pinning transition, with a jump in the order parameter. Our results confirm predictions based on the Harris criterion.