arXiv subjects

S. K. Nechaev

Publications and source records attributed to S. K. Nechaev.

At least 19 recordsLinked to original sources

arXiv2021

Fractional Brownian motion meets topology: statistical and topological properties of globular macromolecules with volume interactions

In the paper we investigate statistical and topological properties of fractional Brownian polymer chains, equipped with the short-range volume interactions. The attention is paid to statistical properties of collapsed conformations with the fractal dimension $D_f\ge 2$ in the three-dimensional space, which are analyzed both numerically and \textit{via} the mean-field Flory approach. Our study is motivated by an attempt to mimic the conformational statistics of collapsed unknotted polymer rings, which are known to form compact hierarchical crumpled globules (CG) with $D_f=3$ at large scales. Replacing the topologically-stabilized CG state by a self-avoiding fractal path adjusted to the fractal dimension $D_f=3$ we tremendously simplify the problem of generating compact self-avoiding conformations since we wash out the topological constraints from the consideration. We make use of the Monte-Carlo simulations to prepare the equilibrium ensemble of swollen chains with various fractal dimensions. A combination of the Flory arguments with statistical analysis of the conformations from simulations allows one to infer the dependence of the critical exponent of the swollen chains on the fractal dimension of the seed chain. We show that with the increase of $D_f$, typical conformations become more territorial and less knotted. Distributions of the knot complexity, $P(χ)$ for various fractal dimensions of the swollen chains suggest a close relationship between statistical and topological properties of fractal paths with volume interactions.

cond-mat.soft↗

arXiv2020

Analysis of English free association network reveals mechanisms of efficient solution of Remote Association Tests

We study correlations between the structure and properties of a free association network of the English language, and solutions of psycholinguistic Remote Association Tests (RATs). We show that average hardness of individual RATs is largely determined by relative positions of test words (stimuli and response) on the free association network. We argue that the solution of RATs can be interpreted as a first passage search problem on a network whose vertices are words and links are associations between words. We propose different heuristic search algorithms and demonstrate that in "easily-solving" RATs (those that are solved in 15 seconds by more than 64\% subjects) the solution is governed by "strong" network links (i.e. strong associations) directly connecting stimuli and response, and thus the efficient strategy consist in activating such strong links. In turn, the most efficient mechanism of solving medium and hard RATs consists of preferentially following sequence of "moderately weak" associations.

cs.AI↗

arXiv2019

Statistical Properties of a Polymer Globule Formed during Collapse with the Irreversible Coalescence of Units

Collapse of the polymer chain upon the sharp decrease of solvent quality is studied. During collapse, any pair of polymer units appearing in a sufficiently close vicinity in space has the possibility with a certain probability to form an irreversible crosslink, thereby preventing the interpenetration of chain material between the forming clusters. Globular structures having different spatial chain packing at various scales are obtained by computer simulations. It is shown that the dependence of probability of contact between two monomers in space , where s is a distance between monomers along chain, reproduces a number of characteristic features observed previously in experiments on the analysis of three-dimensional chromatin packing. The cluster analysis of intramolecular contact maps makes it possible to express the hypothesis that there are characteristic discrete hierarchical levels in polymer packing associated with the number-theoretic origin of rare-event statistics and inherent to individual maps of intra- and interchromosomal contacts.

cond-mat.soft↗

arXiv2018

On statistical models on super trees

We consider a particular example of interplay between statistical models related to CFT on one hand, and to the spectral properties of ODE, known as ODE/IS correspondence, on the other hand. We focus at the representation of wave functions of Schrödinger operators in terms of spectral properties of associated transfer matrices on "super trees" (the trees whose vertex degree changes with the distance from the root point). Such trees with varying branchings encode the structure of the Fock space of the model. We discuss basic spectral properties of "averaged random matrix ensembles" in terms of Hermite polynomials for the transfer matrix of super trees. At small "branching velocities" we have related the problem of paths counting on super trees to the statistics of area-weighted one-dimensional Dyck paths. We also discuss the connection of the spectral statistics of random walks on super trees with the Kardar-Parisi-Zhang scaling.

math-ph↗

arXiv2017

Paths counting on simple graphs: from escape to localization

We study the asymptotic behavior of the number of paths of length $N$ on several classes of infinite graphs with a single special vertex. This vertex can work as an entropic trap for the path, i.e. under certain conditions the dominant part of long paths become localized in the vicinity of the special point instead of spreading to infinity. We study the conditions for such localization on decorated star graphs, regular trees and regular hyperbolic graphs as a function of the functionality of the special vertex. In all cases the localization occurs for large enough functionality. The particular value of transition point depends on the large-scale topology of the graph. The emergence of localization is supported by the analysis of the spectra of the adjacency matrices of corresponding finite graphs.

cond-mat.stat-mech↗

arXiv2016

Concepts of polymer statistical topology

I review few conceptual steps in analytic description of topological interactions, which constitute the basis of a new interdisciplinary branch in mathematical physics, "Statistical Topology", emerged at the edge of topology and statistical physics of fluctuating non-phantom rope-like objects. This new branch is called statistical (or probabilistic) topology.

cond-mat.stat-mech↗

arXiv2014

Lamplighter model of a random copolymer adsorption on a line

We present a model of an AB-diblock random copolymer sequential self-packaging with local quenched interactions on a one-dimensional infinite sticky substrate. It is assumed that the A-A and B-B contacts are favorable, while A-B are not. The position of a newly added monomer is selected in view of the local contact energy minimization. The model demonstrates a self-organization behavior with the nontrivial dependence of the total energy, $E$ (the number of unfavorable contacts), on the number of chain monomers, $N$: $E\sim N^{3/4}$ for quenched random equally probable distribution of A- and B-monomers along the chain. The model is treated by mapping it onto the "lamplighter" random walk and the diffusion-controlled chemical reaction of $X+X\to 0$ type with the subdiffusive motion of reagents.

cond-mat.stat-mech↗

arXiv2014

Two conjectures about spectral density of diluted sparse Bernoulli random matrices

We consider the ensemble of $N\times N$ ($N\gg 1$) symmetric random matrices with the bimodal independent distribution of matrix elements: each element could be either "1" with the probability $p$, or "0" otherwise. We pay attention to the "diluted" sparse regime, taking $p=1/N +ε$, where $0<ε\ll 1/N$. In this limit the eigenvalue density, $ρ(λ)$, is essentially singular, consisting of a hierarchical ultrametric set of peaks. We provide two conjectures concerning the structure of $ρ(λ)$: (i) we propose an equation for the position of sequential (in heights) peaks, and (ii) we give an expression for the shape of an outbound enveloping curve. We point out some similarities of $ρ(λ)$ with the shapes constructed on the basis of the Dedekind modular $η$-function.

cond-mat.stat-mech↗

arXiv2014

Islands of stability in motif distributions of random networks

We consider random non-directed 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 non-zero cubic term. A localization transition should always occur if the entropy function is non-convex. We conjecture that this phenomenon is the origin of the motifs' pattern formation in real evolutionary networks.

cond-mat.stat-mech↗

arXiv2013

Statistical model of intra-chromosome contact maps

The statistical properties of intra-chromosome maps obtained by a genome-wide chromosome conformation capture method (Hi-C) are described in the framework of the hierarchical crumpling model of heteropolymer chain with quenched disorder in the primary sequence. We conjecture that the observed Hi-C maps are statistical averages over many different ways of hierarchical genome folding, and show that the existence of quenched primary structure coupled with hierarchical folding can induce the observed fine structure of intra-chromosome contact maps.

q-bio.GN↗

arXiv2013

Fractal globule as an artificial molecular machine

The relaxation of an elastic network, constructed by a contact map of a fractal (crumpled) polymer globule is investigated. We found that: i) the slowest mode of the network is separated from the rest of the spectrum by a wide gap, and ii) the network quickly relaxes to a low--dimensional (one-dimensional, in our demonstration) manifold spanned by slowest degrees of freedom with a large basin of attraction, and then slowly approaches the equilibrium not escaping this manifold. By these dynamic properties, the fractal globule elastic network is similar to real biological molecular machines, like myosin. We have demonstrated that unfolding of a fractal globule can be described as a cascade of equilibrium phase transitions in a hierarchical system. Unfolding manifests itself in a sequential loss of stability of hierarchical levels with the temperature change.

cond-mat.stat-mech↗

arXiv2013

On prediction of regulatory genes by analysis of C.elegans functional networks

Connectivity networks have recently become widely used in biology due to increasing amounts of information on the physical and functional links between individual proteins. This connectivity data provides valuable material for expanding our knowledge far beyond the experimentally validated via mathematical analysis and theoretical predictions of new functional interactions. In this paper we demonstrate an application of several algorithms developed for the ranking of potential gene-expression regulators within the context of an associated network. We analyze how different types of connectivity between genes and proteins affect the topology of the integral C.elegans functional network and thereby validate algorithmic performance. We demonstrate the possible definition of co-expression gene clusters within a network context from their specific motif distribution signatures. We also show that the method based on the shortest path function (SPF) applied to gene interactions sub-network of the co-expression gene cluster, efficiently predicts novel regulatory transcription factors (TFs). Simultaneous application of other methods, including only interactions with neighborhood genes, allows rapid ranking of potential regulators that could be functionally linked with the group of co-expressed genes. Predicting functions of regulators for a cluster of ribosomal/mRNA metabolic genes we highlight a role of mRNA translation and decay in a longevity of organisms.

q-bio.GN↗

arXiv2012

Planar diagrams from optimization

We propose a new 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. 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 embedded (nested) configurations of paired links.

cond-mat.soft↗

arXiv2012

New alphabet-dependent morphological transition in a random RNA alignment

We study the fraction $f$ of nucleotides involved in the formation of a cactus--like 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: $f(c)\to 1$ for $c \le c_{\rm cr}$ as the chain length $n$ goes to infinity, signaling the formation of a virtually "perfect" gapless secondary structure; while $f(c)<1$ for $c>c_{\rm cr}$, what means that a non-perfect structure with gaps is formed. The strict upper and lower bounds $2 \le c_{\rm cr} \le 4$ are proven, and the numerical evidence for $c_{\rm cr}$ is presented. The relevance of the transition from the evolutional point of view is discussed.

q-bio.GN↗

arXiv2010

Sequence matching algorithms and pairing of noncoding RNAs

A new statistical method of alignment of two heteropolymers which can form hierarchical cloverleaf-like secondary structures is proposed. This offers a new constructive algorithm for quantitative determination of binding free energy of two noncoding RNAs with arbitrary primary sequences. The alignment of ncRNAs differs from the complete alignment of two RNA sequences: in ncRNA case we align only the sequences of nucleotides which constitute pairs between two different RNAs, while the secondary structure of each RNA comes into play only by the combinatorial factors affecting the entropc contribution of each molecule to the total cost function. The proposed algorithm is based on two observations: i) the standard alignment problem is considered as a zero-temperature limit of a more general statistical problem of binding of two associating heteropolymer chains; ii) this last problem is generalized onto the sequences with hierarchical cloverleaf-like structures (i.e. of RNA-type). Taking zero-temperature limit at the very end we arrive at the desired "cost function" of the system with account for entropy of side cactus-like loops. Moreover, we have demonstrated in detail how our algorithm enables to solve the "structure recovery" problem. Namely, we can predict in zero-temperature limit the cloverleaf-like (i.e. secondary) structure of interacting ncRNAs by knowing only their primary sequences.

q-bio.QM↗

arXiv2010

On the motifs distribution in random hierarchical networks

The distribution of motifs in random hierarchical networks defined by nonsymmetric random block--hierarchical adjacency matrices, is constructed for the first time. According to the classification of U. Alon et al of network superfamilies by their motifs distributions, our artificial directed random hierarchical networks falls into the superfamily of natural networks to which the class of neuron networks belongs. This is the first example of ``handmade'' networks with the motifs distribution as in a special class of natural networks of essential biological importance.

q-bio.QM↗

arXiv2010

Chaotic Hamiltonian systems revisited: 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-clled "survival probability") has the power--law asymptotics, $P(t)\sim t^{-ν}$. We present new semi--phenomenological arguments which enable us to map the dynamical system near the chaos border onto the effective "ultrametric diffusion" on the boundary of a tree--like space with hierarchically organized transition rates. In the frameworks of our approach we have estimated the exponent $ν$ as $ν=\ln 2/\ln (1+r_g)\approx 1.44$, where $r_g=(\sqrt{5}-1)/2$ is the critical rotation number.

nlin.CD↗

arXiv2009

On scale-free and poly-scale behaviors of random hierarchical network

In this paper the question about statistical properties of block--hierarchical random matrices is raised for the first time in connection with structural characteristics of random hierarchical networks obtained by mipmapping procedure. In particular, we compute numerically the spectral density of large random adjacency matrices defined by a hierarchy of the Bernoulli distributions $\{q_1,q_2,...\}$ on matrix elements, where $q_γ$ depends on hierarchy level $γ$ as $q_γ=p^{-μγ}$ ($μ>0$). For the spectral density we clearly see the free--scale behavior. We show also that for the Gaussian distributions on matrix elements with zero mean and variances $σ_γ=p^{-νγ}$, the tail of the spectral density, $ρ_G(λ)$, behaves as $ρ_G(λ) \sim |λ|^{-(2-ν)/(1-ν)}$ for $|λ|\to\infty$ and $0<ν<1$, while for $ν\ge 1$ the power--law behavior is terminated. We also find that the vertex degree distribution of such hierarchical networks has a poly--scale fractal behavior extended to a very broad range of scales.

cond-mat.dis-nn↗