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.

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.

Handedness, origin of life and evolution.

Biological polymers have a preferred chirality ond can replicate themselves. Physical arguments provide insight into which of these unique and apparently related properties evolved first, and by what mechanism.

Chiral purity of nucleotides as a necessary condition of complementarity.

This work discusses the question about the role of chiral purity (homochirality) of nucleotides in the formation of complementary replicas. A qualitative answer to this question can be obtained from molecular models constructed to simulate the chiral defect in the polynucleotidic chain. It shows the necessity of homochirality of nucleotides for the complementarity preservation. The necessity of the strong mirror-symmetry breaking in the abiogenic formation of the self-replicating oligonucleotide structures is discussed in the context of prebiological evolution.