Islands of stability in motif distributions of random networks.

We consider random nondirected networks subject to dynamics conserving vertex degrees and study, analytically and numerically, equilibrium three-vertex motif distributions in the presence of an external field h coupled to one of the motifs. For small h, the numerics is well described by the "chemical kinetics" for the concentrations of motifs based on the law of mass action. For larger h, a transition into some trapped motif state occurs in Erdos-Rényi networks. We explain the existence of the transition by employing the notion of the entropy of the motif distribution and describe it in terms of a phenomenological Landau-type theory with a nonzero cubic term. A localization transition should always occur if the entropy function is nonconvex. We conjecture that this phenomenon is the origin of the motifs' pattern formation in real evolutionary networks.

Planar diagrams from optimization for concave potentials.

We propose a toy model of a heteropolymer chain capable of forming planar secondary structures typical for RNA molecules. In this model, the sequential intervals between neighboring monomers along a chain are considered as quenched random variables, and energies of nonlocal bonds are assumed to be concave functions of those intervals. A few factors are neglected: the contribution of loop factors to the partition function, the variation in energies of different types of complementary nucleotides, the stacking interactions, and constraints on the minimal size of loops. However, the model captures well the formation of folded structures without pseudoknots in an arbitrary sequence of nucleotides. Using the optimization procedure for a special class of concave-type potentials, borrowed from optimal transport analysis, we derive the local difference equation for the ground state free energy of the chain with the planar (RNA-like) architecture of paired links. We consider various distribution functions of intervals between neighboring monomers (truncated Gaussian and scale free) and demonstrate the existence of a topological crossover from sequential to essentially nested configurations of paired links.

New alphabet-dependent morphological transition in random RNA alignment.

We study the fraction f of nucleotides involved in the formation of a cactuslike secondary structure of random heteropolymer RNA-like molecules. In the low-temperature limit, we study this fraction as a function of the number c of different nucleotide species. We show, that with changing c, the secondary structures of random RNAs undergo a morphological transition: f(c)→1 for c≤c(cr) as the chain length n goes to infinity, signaling the formation of a virtually perfect gapless secondary structure; while f(c)<1 for c>c(cr), which means that a nonperfect structure with gaps is formed. The strict upper and lower bounds 2≤c(cr)≤4 are proven, and the numerical evidence for c(cr) is presented. The relevance of the transition from the evolutional point of view is discussed.

Chaotic Hamiltonian systems: survival probability.

We consider the dynamical system described by the area-preserving standard mapping. It is known for this system that P(t), the normalized number of recurrences staying in some given domain of the phase space at time t (so-called "survival probability") has the power-law asymptotics, P(t) approximately t{-nu}. We present new semiphenomenological arguments which enable us to map the dynamical system near the chaos border onto the effective "ultrametric diffusion" on the boundary of a treelike space with hierarchically organized transition rates. In the framework of our approach we have estimated the exponent nu as nu=ln 2/ln(1+r{g}) approximately 1.44, where rg=([square root] 5-1)/2 is the critical rotation number.

Unzipping of two random heteropolymers: ground-state energy and finite-size effects.

We consider a pair of random heteropolymer chains with quenched primary sequences. For this system we have analyzed the dependence of average ground state energy per monomer E on chain length n in the ensemble of chains with uniform distribution of primary sequences of monomers. Every monomer of the first (second) chain is randomly and independently chosen with the uniform probability distribution p=1/c from a set of c different types A , B , C , D ,... (A', B', C', D',...) . Monomers of the first chain could form saturating reversible bonds with monomers of the second chain. The bonds between similar monomer types (such as A-A', B-B', C-C', etc.) have the attraction energy u , while the bonds between different monomer types (such as A-B', A-D', B-D', etc.) have the attraction energy v . The main attention is paid to the computation of the normalized free energy E(n) for intermediate chain lengths n and different ratios a=v/u at sufficiently low temperatures, when the entropic contribution of the loop formation is negligible compared to direct energetic interactions between chain monomers, and when the partition function of the chains is dominated by the ground state. The performed analysis allows one to derive the force f(x) which is necessary to apply for unzipping of two random heteropolymers of equal lengths whose ends are separated by the distance x , averaged over all equally distributed primary structures at low temperatures for fixed values a and c .

Necklace-cloverleaf transition in associating RNA-like diblock copolymers.

We consider a A{m}B{n} diblock copolymer, whose links are capable of forming local reversible bonds with each other. We assume that the resulting structure of the bonds is RNA like--i.e., topologically isomorphic to a tree. We show that, depending on the relative strengths of A-A , A-B , and B-B contacts, such a polymer can be in one of two different states. Namely, if a self-association is preferable (i.e., A-A and B-B bonds are comparatively stronger than A-B contacts), then the polymer forms a typical randomly branched cloverleaf structure with the so-called roughness exponent gamma = 1/2 . On the contrary, if alternating association is preferable (i.e., A-B bonds are stronger than A-A and B-B contacts), then the polymer tends to form a generally linear necklace structure with gamma = 1 . The transition between cloverleaf and necklace states is studied in detail, and it is shown that it is a second-order phase transition.

Statistics of ideal randomly branched polymers in a semi-space.

We investigate the statistical properties of a randomly branched 3-functional N-link polymer chain without excluded volume, whose one point is fixed at the distance d from the impenetrable surface in a 3-dimensional space. Exactly solving the Dyson-type equation for the partition function Z(N, d )=N(-theta)e(gamma N) in 3D, we find the "surface" critical exponent theta=[Formula: see text], as well as the density profiles of 3-functional units and of dead ends. Our approach enables to compute also the pairwise correlation function of a randomly branched polymer in a 3D semi-space.

[Analysis of mechanisms underlying adaptation processes]. / Analiz mekhanizmov adaptatsionnykh protsessov.

A system of elementary adaptation mechanisms is presented. The adaptations are considered as transients to a new homeostatic condition induced by an environmental change. We propose to distinguish adaptation mechanisms not directly related to gene expression (changes in the rate of synthesis and degradation of proteins, protein-ligand interactions, changes in viscosity of the membrane lipids) and the mechanisms relying on gene expression (changes in expression of already functioning genes, expression of new genes, mutations). Most of these mechanisms have phenotypic nature and just one (mutations) is genotypic. By the nature and time pattern of the environmental influence the adaptation processes can be divided into four types: phenotypic adaptations under rapidly (A) or gradually (B) alternating environmental factors, genotypic adaptations induced by an instant change (mutation) (C), and step adaptational changes (several or many mutations) (D). We propose a model based on a (second-order) linear differential equation qualitatively describing all four types of the adaptation processes.

Limiting probability distribution for a random walk with topological constraints.

The joint limiting probability distribution is studied for the two-dimensional random walk with topological constraints, omega(2ns), on Z(2) lattice, where 2n is its total length and (0

[The role of topological limitations in the kinetics of homopolymer collapse and self-assembly of biopolymers]. / Rol' topologicheskikh ogranichenii v kinetike kollapsa gomopolimera i v samoorganizatsii biopolimerov.

Fast collapse of a linear homopolymer after a spasmodic temperature decrease or deterioration of the solvent quality results in the initiation of folded nonequilibrium globule. The chain line in it is a fractal with the dimension mu less than or equal to 2 at small ranges and 3 at the large ones. This is provided by non-selfintersection of the chain and initiated therefore spatial segregation from each other at all the ranges of the chain regions, globulized each in itself. Further relaxation of the folded globule proceeds very slowly only at the expense of reptation (diffusion crawling) of the chain along itself, and results in the polymer knotting. The model of the folded globule permits explanation from a single viewpoint of a number of globular proteins properties. Predictions are formulated whose checking in a real or computer experiment should reveal the adequacy of our results.