Some Aspects of Nonlinearity and Self-Organization In Biosystems on Examples of Localized Excitations in the DNA Molecule and Generalized Fisher–KPP Model

V Obukhov,A Shapovalov

doi:10.3390/sym10030053

Abstract

This review deals with ideas and approaches to nonlinear phenomena, based on different branches of physics and related to biological systems, that focus on how small impacts can significantly change the state of the system at large spatial scales. This problem is very extensive, and it cannot be fully resolved in this paper. Instead, some selected physical effects are briefly reviewed. We consider sine-Gordon solitons and nonlinear Schrodinger solitons in some models of DNA as examples of self-organization at the molecular level, as well as examine features of their formation and dynamics under the influence of external influences. In addition, the formation of patterns in the generalized Fisher–KPP model is viewed as a simple example of self-organization in a system with nonlocal interaction at the cellular level. Symmetries of model equations are employed to analyze the considered nonlinear phenomena. In this context the possible relations between phenomena considered and released activity effect, which is assessed differently in the literature, are discussed.

Highlights

Interdisciplinary analysis of known phenomena and patterns in light of the emergence of new facts and challenges requires a deeper understanding of the basics of the studied field of science and its relationship with other fields
We consider how soliton-like states may arise in the BP model with the use of the inverse scattering transform (IST) method, which is a cornerstone in nonlinear mathematical physics
If we accept the hypothesis about the released activity (RA) effect on the cell population dynamics, one can explore the dependence between the change in the character of the population growth and the peculiarities of the corresponding small influences related to RA

Summary

Introduction

Interdisciplinary analysis of known phenomena and patterns in light of the emergence of new facts and challenges requires a deeper understanding of the basics of the studied field of science and its relationship with other fields. The formation of internal structures (spatially stable or space-time (dynamic)) occurs due to nonlinear interactions between the elements of a complex system, or its subsystems of the same level of complexity. This problem is extremely extensive and complex, and we do not set out the task of highlighting it in any finished form. We discuss pattern formation in the well-known generalized Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) model as a simple example of self-organization in a system with nonlocal interaction at the cellular and population levels. Solitons are important examples of solutions endowed with symmetry properties, and to explore the generalized Fisher–KPP model equation, we apply the semiclassical approximation approach, which incorporates the methods of symmetry analysis of model equations

The Concept of Solitons in Molecules

Kink Dynamics asExternal Factors for the Sine-Gordon DNA Model

Localized Energy Distributions in the Framework of The Peyrard–Bishop Model

Summary

Pattern Formation in Cell Populations

Conclusions