Nearest-neighbor directed random hyperbolic graphs.

Undirected hyperbolic graph models have been extensively used as models of scale-free small-world networks with high clustering coefficient. Here we presented a simple directed hyperbolic model where nodes randomly distributed on a hyperbolic disk are connected to a fixed number m of their nearest spatial neighbors. We introduce also a canonical version of this network (which we call "network with varied connection radius"), where maximal length of outgoing bond is space dependent and is determined by fixing the average out-degree to m. We study local bond length, in-degree, and reciprocity in these networks as a function of spacial coordinates of the nodes and show that the network has a distinct core-periphery structure. We show that for small densities of nodes the overall in-degree has a truncated power-law distribution. We demonstrate that reciprocity of the network can be regulated by adjusting an additional temperature-like parameter without changing other global properties of the network.

Many-body contacts in fractal polymer chains and fractional Brownian trajectories.

We calculate the probabilities that a trajectory of a fractional Brownian motion with arbitrary fractal dimension d_{f} visits the same spot n≥3 times, at given moments t_{1},...,t_{n}, and obtain a determinant expression for these probabilities in terms of a displacement-displacement covariance matrix. Except for the standard Brownian trajectories with d_{f}=2, the resulting many-body contact probabilities cannot be factorized into a product of single-loop contributions. Within a Gaussian network model of a self-interacting polymer chain, which we suggested recently [K. Polovnikov et al., Soft Matter 14, 6561 (2018)1744-683X10.1039/C8SM00785C], the probabilities we calculate here can be interpreted as probabilities of multibody contacts in a fractal polymer conformation with the same fractal dimension d_{f}. This Gaussian approach, which implies a mapping from fractional Brownian motion trajectories to polymer conformations, can be used as a semiquantitative model of polymer chains in topologically stabilized conformations, e.g., in melts of unconcatenated rings or in the chromatin fiber, which is the material medium containing genetic information. The model presented here can be used, therefore, as a benchmark for interpretation of the data of many-body contacts in genomes, which we expect to be available soon in, e.g., Hi-C experiments.

Effective Hamiltonian of topologically stabilized polymer states.

Topologically stabilized polymer conformations in melts of nonconcatenated polymer rings and crumpled globules are considered to be a good candidate for the description of the spatial structure of mitotic chromosomes. Despite significant efforts, the microscopic Hamiltonian capable of describing such systems still remains unknown. We describe a polymer conformation by a Gaussian network - a system with a Hamiltonian quadratic in all coordinates - and show that by tuning interaction constants, one can obtain equilibrium conformations with any fractal dimension between 2 (an ideal polymer chain) and 3 (a crumpled globule). Monomer-to-monomer distances in topologically stabilized states, according to available numerical data, fit very well the Gaussian distribution, giving an additional argument in support of the quadratic Hamiltonian model. Mathematically, the polymer conformations are mapped onto the trajectories of a subdiffusive fractal Brownian particle. Moreover, we explicitly show that the quadratic Hamiltonian with a hierarchical set of coupling constants provides the microscopic background for the description of the path integral of the fractional Brownian motion with an algebraically decaying kernel.

Fractal Folding and Medium Viscoelasticity Contribute Jointly to Chromosome Dynamics.

Chromosomes are key players of cell physiology, their dynamics provides valuable information about its physical organization. In both prokaryotes and eukaryotes, the short-time motion of chromosomal loci has been described with a Rouse model in a simple or viscoelastic medium. However, little emphasis has been put on the influence of the folded organization of chromosomes on the local dynamics. Clearly, stress propagation, and thus dynamics, must be affected by such organization, but a theory allowing us to extract such information from data, e.g., on two-point correlations, is lacking. Here, we describe a theoretical framework able to answer this general polymer dynamics question. We provide a scaling analysis of the stress-propagation time between two loci at a given arclength distance along the chromosomal coordinate. The results suggest a precise way to assess folding information from the dynamical coupling of chromosome segments. Additionally, we realize this framework in a specific model of a polymer whose long-range interactions are designed to make it fold in a fractal way and immersed in a medium characterized by subdiffusive fractional Langevin motion with a tunable scaling exponent. This allows us to derive explicit analytical expressions for the correlation functions.

Anomalous diffusion in fractal globules.

The fractal globule state is a popular model for describing chromatin packing in eukaryotic nuclei. Here we provide a scaling theory and dissipative particle dynamics computer simulation for the thermal motion of monomers in the fractal globule state. Simulations starting from different entanglement-free initial states show good convergence which provides evidence supporting the existence of a unique metastable fractal globule state. We show monomer motion in this state to be subdiffusive described by ⟨X(2)(t)⟩∼t(αF) with αF close to 0.4. This result is in good agreement with existing experimental data on the chromatin dynamics, which makes an additional argument in support of the fractal globule model of chromatin packing.

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.

Overlap of two Brownian trajectories: exact results for scaling functions.

We consider two random walkers starting at the same time t=0 from different points in space separated by a given distance R. We compute the average volume of the space visited by both walkers up to time t as a function of R and t and dimensionality of space d. For d<4, this volume, after proper renormalization, is shown to be expressed through a scaling function of a single variable R/√t. We provide general integral formulas for scaling functions for arbitrary dimensionality d<4. In contrast, we show that no scaling function exists for higher dimensionalities d≥4.

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.

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 .

The global phase behavior of the two-component systems with intracomponent association: Flory approach.

Within the Flory approach we study the phase diagrams of two-component fluids, the molecules of each component A(f(A)), B(f(B)) bearing f(A) (f(B)) functional groups capable of forming thermoreversible A-A and B-B bonds. We develop a general procedure to classify these diagrams depending on the values of four governing parameters -- entropies and normalized energies of A-A and B-B bonds, and give full topological classification of phase diagrams with f(A,B)> or =3. We show that these phase diagrams can have immiscibility loops and up to four critical points.

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.