arXiv subjects

A. Kh. Bikulov

Publications and source records attributed to A. Kh. Bikulov.

12 recordsLinked to original sources

arXiv2026

A toy model of a protein prototype reveals nontrivial ultrametricity of the energy landscape

A model for studying the ultrametricity of the energy landscape in a disordered heteropolymer is presented. It is treated as a simplified model of a protein molecule in which amino acid residues are modeled as point masses. Pairwise interactions include universal repulsion, the Lennard-Jones potential, the Coulomb potential with screening, and the elastic potential for bonds between adjacent residues. An analogy with spin glass models is used, allowing the application of replica theory methods. Unlike the standard approach to disordered systems, averaging over disorder is not performed. The overlap between replicas is defined as the Pearson correlation coefficient between the vectors of average pairwise energies, which corresponds to a comparison of thermodynamic averages in the spirit of spin glass theory. The results of a computational experiment conducted using the developed algorithm on a graphics processing unit (GPU) are presented. The simulations were performed using a 128-residue-long sequence, with 50 independent disorder realizations and 50 replicas for each sequence at a temperature of T = 1.0. It was found that for 90.0% of the sequences, the distance matrix between replicas contains more than half of the ultrametric triangles, and nontrivial ultrametricity predominates in 97.8% of them, indicating a hierarchical organization of the energy landscape. A repeated computational experiment for selected sequences confirms the reliability of the observations: 95.5% of them again demonstrated ultrametricity, of which 97.7% showed a predominance of the nontrivial type of ultrametricity. The obtained results confirm Frauenfelder's hypothesis of protein ultrametricity and pave the way for a systematic study of ultrametric properties in more realistic protein models.

cond-mat.dis-nn↗

arXiv2025

Classical first passage problems for $p$-adic stochastic processes

In this paper we present a comprehensive analysis of the solution of the classical problem of finding the distribution density of a random variable - the first passage time to a given domain by the trajectory of a $p$-adic Markov stochastic process with probability density function satisfying the solution of the Cauchy problem for the Vladimirov equation (a $p$-adic analog of the Kolmogorov-Feller equation with the kernel of the Vladimirov operator) with uniform initial distribution in the unit ball. We consider three equivalent approaches to obtain equations for the distribution density of a random variable - the first passage time to a given domain by a stochastic trajectory. We find a solution to these equations for the distribution density, analyze its properties, and compare them with the properties of the distribution density of a random variable - the first return timeof a stochastic trajectory to the support of the initial distribution. We also solve the problem of finding the number of hittings a given domain and analyze the solution obtained. In conclusion, we discuss a class of problems related to the study of the distribution density of the passage time to a given domain and the return time to the initial domain for other types of $p$-adic Markov stochastic processes.

math-ph↗

arXiv2024

Power laws and logarithmic oscillations in diffusion processes on infinite countable ultrametric spaces

It is shown that if the initial condition of the Cauchy problem for the diffusion equation on a general infinite countable ultrametric space is spherically symmetric with respect to some point, then this problem has an exact analytical solution. A general solution of this problem is presented for pure ultrametric diffusion, as well as for ultrametric diffusion with a reaction sink concentrated at the center of spherical symmetry. Conditions on the ultrametric and the distribution of the number of states in ultrametric spheres are found that lead at large times to the asymptotic behavior of the solutions obtained in the form of a power law modulated by a bounded function that is log-periodic under some additional conditions.

math-ph↗

arXiv2023

Oscillations in p-adic diffusion processes and simulation of the conformational dynamics of protein

Logarithmic oscillations superimposed on a power-law trend appear in the behavior of various complex hierarchical systems. In this paper, we study the logarithmic oscillations of relaxation curves in p-adic diffusion models that are used to describe the conformational dynamics of protein. We consider the case of a purely p-adic diffusion, as well as the case of p-adic diffusion with a reaction sink. We show that, relaxation curves for large times in these two cases are described by a power law on which logarithmic oscillations are superimposed whose period and amplitude are determined by the parameters of the model. We also provide a physical explanation of the emergence of oscillations in relaxation curves and discuss the relation of the results to the experiments on relaxation dynamics of protein.

q-bio.BM↗

arXiv2022

The sojourn time problem for a $p$-adic random walk and its applications to the spectral diffusion of proteins

We consider the problem of the distribution of the sojourn time in a compact set $\mathbb{Z}_{p}$ in the case of a $p$-adic random walk. We rely on the results of our previous studies of the distribution of the first return time for a $p$-adic random walk and the results of Takacs on the study of the sojourn time problem for a wide class of random processes. For a $p$-adic random walk we find the mean sojourn time of the trajectory in $\mathbb{Z}_{p}$ and the asymptotics as $t\rightarrow\infty$ of arbitrary moments of the distribution of the sojourn time in $\mathbb{Z}_{p}$. We also discuss some possible applications of our results to the modeling of relaxation processes related to the conformational dynamics of protein.

math.PR↗

arXiv2020

Ultrametric theory of conformational dynamics of protein molecules in a functional state and the description of experiments on the kinetics of CO binding to myoglobin

The paper is devoted to a systematic account of the theory of conformational dynamics of protein molecules. As an example of application of this theory, we provide a complete analytical description of experiments on the kinetics of CO binding to myoglobin, which were carried out by the group of Frauenfelder more than 30 years ago and acquired the status of base experiments for studying the properties of the fluctuation dynamic mobility of protein molecules. As early as 2001, the authors could demonstrate that, within the model of ultrametric random walk with a reaction sink, the experimental curves of CO binding to myoglobin can be reproduced in the high-temperature region. Later, in 2010, the authors proposed a modified model and, based on its numerical analysis, demonstrated that this model can reproduces the experimental results over the whole temperature range covered in the experiment. In the present study, based on the previously proposed model, we formulate a rigorous mathematical theory of conformational dynamics of protein molecules. We demonstrate that the proposed theory provides not only a complete description of the experiment over the whole temperature range of $\left(60\div300\right)$ K and in the observation time window of $\left(10^{-7}\div10^{2}\right)$ s but also a unified picture of the conformational mobility of a protein molecule, as well as allows one to realize the fact that the mobility changes in a self-similar way. This specific feature of protein molecules, which has remained hidden to date, significantly expands the ideas of dynamic symmetry that proteins apparently possess. In addition, we show that the model provides a prediction for the behavior of the kinetic curves of the experiment in the low-temperature range of $\left(60\div180\right)$ K at times not covered by the experiment (more than $10^{2}$ s).

cond-mat.stat-mech↗

arXiv2015

Model of the $p$-Adic Random Walk in a Potential

We consider the $p$-adic random walk model in a potential, which can be viewed as a generalization of $p$-adic random walk models used for description of protein conformational dynamics. This model is based on the Kolmogorov--Feller equations for the distribution function defined on the field of $p$-adic numbers in which the probability of transitions per unit time depends on ultrametric distance between the transition points as well as on function of potential violating the spatial homogeneity of a random process. This equation, which will be called the equation of $p$-adic random walk in a potential, is equivalent to the equation of $p$-adic random walk with modified measure and reaction source. With a special choice of a power-law potential the last equation is shown to have an exact analytic solution. We find the analytic solution of the Cauchy problem for such equation with an initial condition, whose support lies in the ring of integer $p$-adic numbers. We also examine the asymptotic behaviour of the distribution function for large times. It is shown that in the limit $t\rightarrow\infty$ the distribution function tends to the equilibrium solution according to the law, which is bounded from above and below by power laws with the same exponent. Our principal conclusion is that the introduction of a potential in the ultrametric model of conformational dynamics of protein conserves the power-law behaviour of relaxation curves for large times.

cond-mat.dis-nn↗

arXiv2015

On one real basis for $L^2(Q_p)$

We construct new bases of real functions from $L^{2}\left(B_{r}\right)$ and from $L^{2}\left(\mathbb{Q}_{p}\right)$. These functions are eigenfunctions of the $p$-adic pseudo-differential Vladimirov operator, which is defined on a compact set $B_{r}\subset\mathbb{Q}_{p}$ of the field of $p$-adic numbers $\mathbb{Q}_{p}$ or, respectively, on the entire field $\mathbb{Q}_{p}$. A relation between the basis of functions from $L^{2}\left(\mathbb{Q}_{p}\right)$ and the basis of $p$-adic wavelets from $L^{2}\left(\mathbb{Q}_{p}\right)$ is found. As an application, we consider the solution of the Cauchy problem with the initial condition on a compact set for a pseudo-differential equation with a general pseudo-differential operator, which is diagonal in the basis constructed.

math-ph↗

arXiv2015

Application of $p$-adic analysis methods in describing Markov processes on ultrametric spaces isometrically embeddable into $\mathbb{Q}_{p}$

We propose a method for describing stationary Markov processes on the class of ultrametric spaces $\mathbb{U}$ isometrically embeddable in the field $\mathbb{Q}_{p}$ of $p$-adic numbers. This method is capable of reducing the study of such processes to the investigation of processes on $\mathbb{Q}_{p}$. Thereby the traditional machinery of $p$-adic mathematical physics can be applied to calculate the characteristics of stationary Markov processes on such spaces. The Cauchy problem for the Kolmogorov-Feller equation of a~stationary Markov process on such spaces is shown as being reducible to the Cauchy problem for a pseudo-differential equation on $\mathbb{Q}_{p}$ with non-translation-invariant measure $m\left(x\right)d_{p}x$. The spectrum of the pseudo-differential operator of the Kolmogorov-Feller equation on $\mathbb{Q}_{p}$ with measure $m\left(x\right)d_{p}x$ is found. Orthonormal basis of real valued functions for $L^{2}\left(\mathbb{Q}_{p},m\left(x\right)d_{p}x\right)$ is constructed from the eigenfunctions of this operator.

math-ph↗

arXiv2008

Random Hierarchical Matrices: Spectral Properties and Relation to Polymers on Disordered Trees

We study the statistical and dynamic properties of the systems characterized by an ultrametric space of states and translationary non-invariant symmetric transition matrices of the Parisi type subjected to "locally constant" randomization. Using the explicit expression for eigenvalues of such matrices, we compute the spectral density for the Gaussian distribution of matrix elements. We also compute the averaged "survival probability" (SP) having sense of the probability to find a system in the initial state by time $t$. Using the similarity between the averaged SP for locally constant randomized Parisi matrices and the partition function of directed polymers on disordered trees, we show that for times $t>t_{\rm cr}$ (where $t_{\rm cr}$ is some critical time) a "lacunary" structure of the ultrametric space occurs with the probability $1-{\rm const}/t$. This means that the escape from some bounded areas of the ultrametric space of states is locked and the kinetics is confined in these areas for infinitely long time.

cond-mat.dis-nn↗

arXiv2008

First Passage Time Distribution and Number of Returns for Ultrametric Random Walk

In this paper, we consider a homogeneous Markov process ξ(t;ω) on an ultrametric space Q_p, with distribution density f(x,t), x in Q_p, t in R_+, satisfying the ultrametric diffusion equation df(x,t)/dt =-Df(x,t). We construct and examine a random variable τ(ω) that has the meaning the first passage times. Also, we obtain a formula for the mean number of returns on the interval (0,t] and give its asymptotic estimates for large t.

math-ph↗

arXiv2002

p-Adic description of characteristic relaxation in complex systems

This work is a further development of an approach to the description of relaxation processes in complex systems on the basis of the p-adic analysis. We show that three types of relaxation fitted into the Kohlrausch-Williams-Watts law, the power decay law, or the logarithmic decay law, are similar random processes. Inherently, these processes are ultrametric and are described by the p-adic master equation. The physical meaning of this equation is explained in terms of a random walk constrained by a hierarchical energy landscape. We also discuss relations between the relaxation kinetics and the energy landscapes.

cond-mat.dis-nn↗