Protein Ultrametricity and Spectral Diffusion in Deeply Frozen Proteins

135

p-Adic mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of p-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

Section: Applications To Proteins and Genomesmentioning

confidence: 99%

p-Adic mathematical physics: the first 30 years

Драгович

Khrennikov

Козырев

et al. 2017

135

“…In doing so it is assumed that the probabilities of transitions per unit time through the activation barriers (that is, transitions between conformational marostates) obey the Arrhenius relation. Such a model, which looks simple at first sight, has enabled one to adequately describe two principal experiments: the experiment on spectral diffusion in proteins [17] and the experiment on binding kinetics of CO by myoglobin [20,21]. We note that in these two experiments, the fluctuation dynamics of a protein is described in a unified way.…”

Section: Introductionmentioning

confidence: 99%

Model of p-Adic Random Walk in a Potential

Бикулов¹,

Зубарев

2018

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 → ∞ 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.

“…In this case states are associated with local minima of the potential energy landscape of a protein molecule, and the energy landscape is represented as a hierarchy of nested basins of free energy local minima. Ultrametric models of conformational dynamics of protein molecules have been developed in a series of papers [13,14,15,16,17,18,19].…”

Section: Introductionmentioning

confidence: 99%

On stochastic generation of ultrametrics in high-dimensional Euclidean spaces

Зубарев

2014

We present a proof of the theorem which states that a matrix of Euclidean distances on a set of specially distributed random points in the n-dimensional Euclidean space R n converges in probability to an ultrametric matrix as n → ∞. Values of the elements of an ultrametric distance matrix are completely determined by variances of coordinates of random points. Also we preset a probabilistic algorithm for generation of finite ultrametric structures of any topology in high-dimensional Euclidean space. Validity of the algorithm is demonstrated by explicit calculations of distance matrices and ultrametricity indexes for various dimensions n.