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.

Path Counting on Tree-like Graphs with a Single Entropic Trap: Critical Behavior and Finite Size Effects

It is known that maximal entropy random walks and partition functions that count long paths on graphs tend to become localized near nodes with a high degree. Here, we revisit the simplest toy model of such a localization: a regular tree of degree p with one special node ("root") that has a degree different from all the others. We present an in-depth study of the path-counting problem precisely at the localization transition. We study paths that start from the root in both infinite trees and finite, locally tree-like regular random graphs (RRGs). For the infinite tree, we prove that the probability distribution function of the endpoints of the path is a step function. The position of the step moves away from the root at a constant velocity v=(p-2)/p. We find the width and asymptotic shape of the distribution in the vicinity of the shock. For a finite RRG, we show that a critical slowdown takes place, and the trajectory length needed to reach the equilibrium distribution is on the order of N instead of logp-1N away from the transition. We calculate the exact values of the equilibrium distribution and relaxation length, as well as the shapes of slowly relaxing modes.

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.

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.

A statistical model of intra-chromosome contact maps

A statistical model describing a fine structure of the intra-chromosome maps obtained by a genome-wide chromosome conformation capture method (Hi-C) is proposed. The model combines hierarchical chain folding with a quenched heteropolymer structure of primary chromatin sequences. It is conjectured that the observed Hi-C maps are statistical averages over many different ways of hierarchical genome folding. It is shown that the existence of a quenched primary structure coupled with hierarchical folding induces a full range of features observed in experimental Hi-C maps: hierarchical elements, chess-board intermittency and large-scale compartmentalization.

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 Erdős-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.

How a viscoelastic solution of wormlike micelles transforms into a microemulsion upon absorption of hydrocarbon: new insight

In this article, we investigate the effect of hydrocarbon addition on the rheological properties and structure of wormlike micellar solutions of potassium oleate. We show that a viscoelastic solution of entangled micellar chains is extremely responsive to hydrocarbons-the addition of only 0.5 wt % n-dodecane results in a drastic drop in viscosity by up to 5 orders of magnitude, which is due to the complete disruption of micelles and the formation of microemulsion droplets. We study the whole range of the transition of wormlike micelles into microemulsion droplets and discover that it can be divided into three regions: (i) in the first region, the solutions retain a high viscosity (∼10-350 Pa·s), the micelles are entangled but their length is reduced by the solubilization of hydrocarbons; (ii) in the second region, the system transitions to the unentangled regime and the viscosity sharply decreases as a result of further micelle shortening and the appearance of microemulsion droplets; (iii) in the third region, the viscosity is low (∼0.001 Pa·s) and only microemulsion droplets remain in the solution. The experimental studies were accompanied by theoretical considerations, which allowed us to reveal for the first time that (i) one of the leading mechanisms of micelle shortening is the preferential accumulation of the solubilized hydrocarbon in the spherical end caps of wormlike micelles, which makes the end caps thermodynamically more favorable; (ii) the onset of the sharp drop in viscosity is correlated with the crossover from the entangled to unentangled regime of the wormlike micellar solution taking place upon the shortening of micellar chains; and (iii) in the unentangled regime short cylindrical micelles coexist with microemulsion droplets.

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.

Phase transition in random planar diagrams and RNA-type matching

We study the planar matching problem, defined by a symmetric random matrix with independent identically distributed entries, taking values zero and one. We show that the existence of a perfect planar matching structure is possible only above a certain critical density, p(c), of allowed contacts (i.e., of ones). Using a formulation of the problem in terms of Dyck paths and a matrix model of planar contact structures, we provide an analytical estimation for the value of the transition point, p(c), in the thermodynamic limit. This estimation is close to the critical value, p(c)≈0.379, obtained in numerical simulations based on an exact dynamical programming algorithm. We characterize the corresponding critical behavior of the model and discuss the relation of the perfect-imperfect matching transition to the known molten-glass transition in the context of random RNA secondary structure formation. In particular, we provide strong evidence supporting the conjecture that the molten-glass transition at T=0 occurs at p(c).

Number of common sites visited by N random walkers

We compute analytically the mean number of common sites, W(N)(t), visited by N independent random walkers each of length t and all starting at the origin at t = 0 in d dimensions. We show that in the (N-d) plane, there are three distinct regimes for the asymptotic large-t growth of W(N)(t). These three regimes are separated by two critical lines d = 2 and d = d(c)(N) = 2N/(N-1) in the (N-d) plane. For d<2, W(N) (t) ~ t(d/2) for large t (the N dependence is only in the prefactor). For 2 < d < d(c)(N), W(N)(t) ~ t(ν) where the exponent ν = N-d(N-1)/2 varies with N and d. For d > d(c)(N), W(N)(t) → const as t → ∞. Exactly at the critical dimensions there are logarithmic corrections: for d=2, we get W(N)(t) ~ t/[ln t](N), while for d = d(c)(N), W(N)(t) ~ ln t for large t. Our analytical predictions are verified in numerical simulations.

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.