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

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.

[1]  V. S. Vladimirov,et al.  Equations of mathematical physics , 1972 .

[2]  A. Scott,et al.  Perturbation analysis of fluxon dynamics , 1978 .

[3]  M A Fuentes,et al.  Nonlocal interaction effects on pattern formation in population dynamics. , 2003, Physical review letters.

[4]  V. Zarubaev,et al.  Activity of ultra-low doses of antibodies to gamma-interferon against lethal influenza A(H1N1)2009 virus infection in mice. , 2012, Antiviral research.

[5]  A. Davydov,et al.  The theory of contraction of proteins under their excitation. , 1973, Journal of theoretical biology.

[6]  Fordyce A. Davidson,et al.  Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation , 2001 .

[7]  A. Jamiołkowski.  Applications of Lie groups to differential equations , 1989 .

[8]  S. Takeno,et al.  Topological Solitons and Modulated Structure of Bases in DNA Double Helices A Dynamic Plane Base-Rotator Model , 1983 .

[9]  Zhang,et al.  Soliton excitations in deoxyribonucleic acid (DNA) double helices. , 1987, Physical review. A, General physics.

[10]  J. Cole,et al.  Similarity methods for differential equations , 1974 .

[11]  Pascual,et al.  Sine-Gordon solitons under weak stochastic perturbations. , 1985, Physical review. B, Condensed matter.

[12]  S. Yomosa.  Solitary Excitations in DNA Double Helices , 1985 .

[13]  T. Frank.  Stochastic systems with delay: Perturbation theory for second order statistics , 2016 .

[14]  Vlatko Vedral,et al.  Quantum physics meets biology , 2009, HFSP journal.

[15]  Bishop,et al.  Statistical mechanics of a nonlinear model for DNA denaturation. , 1989, Physical review letters.

[16]  Influence of electromagnetic field on soliton-mediated charge transport in biological systems* , 2015, Electromagnetic biology and medicine.

[17]  Leon A. Takhtajan,et al.  Hamiltonian methods in the theory of solitons , 1987 .

[18]  Peter V Coveney,et al.  Modelling biological complexity: a physical scientist's perspective , 2005, Journal of The Royal Society Interface.

[19]  Prohofsky.  Solitons hiding in DNA and their possible significance in RNA transcription. , 1988, Physical review. A, General physics.

[20]  L Brizhik,et al.  Solitons in alpha-helical proteins. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  E. Elizalde,et al.  Fermionic condensate and Casimir densities in the presence of compact dimensions with applications to nanotubes , 2011, 1102.2202.

[22]  A. Foletti,et al.  Nonlinear quantum phenomena and biophysical aspects of complexity related to health and disease. , 2014, Journal of biological regulators and homeostatic agents.

[23]  Irving R Epstein,et al.  Pattern formation mechanisms in reaction-diffusion systems. , 2009, The International journal of developmental biology.

[24]  O. Epstein,et al.  Novel Approach to Activity Evaluation for Release-Active Forms of Anti-Interferon-Gamma Antibodies Based on Enzyme-Linked Immunoassay , 2014, PloS one.

[25]  L. Yakushevich.  Is DNA a nonlinear dynamical system where solitary conformational waves are possible? , 2001, Journal of Biosciences.

[26]  A. Bishop,et al.  Classical particle-like behavior of Sine--Gordon solitons in scattering potentials and applied fields , 1976 .

[27]  C. Tsallis,et al.  Curl forces and the nonlinear Fokker-Planck equation. , 2016, Physical review. E.

[28]  A. Foletti,et al.  Nonlinearity, coherence and complexity: Biophysical aspects related to health and disease , 2017, Electromagnetic biology and medicine.

[29]  O. Epstein.  [The phenomenon of release activity and the hypothesis of "spatial" homeostasis]. , 2013, Uspekhi fiziologicheskikh nauk.

[30]  Chirality as an Instrument of Stratification of Hierarchical Systems in Animate and Inanimate Nature , 2012, 1212.1677.

[31]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[32]  Philip K. Maini,et al.  Experimental and Theoretical Advances in Biological Pattern Formation , 1993, NATO ASI Series.

[33]  A. Marciniak-Czochra,et al.  Unstable patterns in reaction–diffusion model of early carcinogenesis , 2011, 1104.3592.

[34]  E. Elizalde,et al.  Zeta Regularization Techniques with Applications , 1994 .

[35]  Mo-Lin Ge,et al.  Theory of Solitons , 2016 .

[36]  Y. Maruvka,et al.  Nonlocal competition and logistic growth: patterns, defects, and fronts. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Yuri S. Kivshar,et al.  Dynamics of Solitons in Nearly Integrable Systems , 1989 .

[38]  Alwyn C. Scott,et al.  Davydov's soliton revisited : self-trapping of vibrational energy in protein , 1990 .

[39]  R. G. Medhurst,et al.  Topics in the Theory of Random Noise , 1969 .

[40]  A. Davydov.  The lifetime of molecular (Davydov) solitons , 1991 .

[41]  Kimball A. Milton,et al.  The Reality of Casimir Friction , 2016, Symmetry.

[42]  A. Davydov,et al.  Solitons in molecular systems , 1979 .

[43]  Larissa Brizhik.  Nonlinear mechanism for weak photon emission from biosystems. , 2008, Indian journal of experimental biology.

[44]  O. Epstein,et al.  Subetta Treatment Increases Adiponectin Secretion by Mature Human Adipocytes In Vitro , 2013, International journal of endocrinology.

[45]  V. M. Tikhomirov,et al.  A Study of the Diffusion Equation with Increase in the Amount of Substance, and its Application to a Biological Problem , 1991 .

[46]  A. Heeger,et al.  Nature of the open state in long polynucleotide double helices: possibility of soliton excitations. , 1980, Proceedings of the National Academy of Sciences of the United States of America.

[47]  W. Miller,et al.  Group analysis of differential equations , 1982 .

[48]  Ervin K. Lenzi,et al.  Nonlinear Fokker–Planck equations, H – theorem, and entropies , 2017 .

[49]  A. Trifonov,et al.  Pattern formation in terms of semiclassically limited distribution on lower dimensional manifolds for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation , 2013, 1306.3765.

[50]  J Umbanhowar,et al.  Non-local concepts and models in biology. , 2001, Journal of theoretical biology.

[51]  Shigeru Kondo,et al.  Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation , 2010, Science.

[52]  J. McCauley.  A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck equations” by T.D. Frank , 2007, cond-mat/0701589.

[53]  A. Mogilner,et al.  A non-local model for a swarm , 1999 .

[54]  L. V. I︠A︡kushevich.  Nonlinear Physics of DNA , 1998 .

[55]  Angel Sánchez,et al.  Collective Coordinates and Length-Scale Competition in Spatially Inhomogeneous Soliton-Bearing Equations , 1998, SIAM Rev..

[56]  S. Takeno,et al.  Self-Localized Anharmonic Rotational Modes of Bases in DNA , 1990 .

[57]  A. Bishop,et al.  Dynamics of sine-Gordon solitons in the presence of perturbations , 1977 .

[58]  H. Meinhardt,et al.  Biological pattern formation: fmm basic mechanisms ta complex structures , 1994 .

[59]  TRANSFORMATION GROUPS APPLIED TO MATHEMATICAL PHYSICS (Mathematics and Its Applications (Soviet Series)) , 1986 .

[60]  C. Gardiner.  Stochastic Methods: A Handbook for the Natural and Social Sciences , 2009 .

[61]  On the modeling of the motion of a transcription bubble under constant torque , 2016 .

[62]  Yoh Iwasa,et al.  Mathematics for Life Science and Medicine , 2009 .

[63]  Y. Maruvka,et al.  Nonlocal competition and front propagation in branching-coalescence systems. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[64]  S. Cocco,et al.  Helicoidal model for DNA opening , 1999, chao-dyn/9904032.

[65]  G. Fleming,et al.  Quantum effects in biology , 2011 .

[66]  Kink dynamics in the medium with a random force and dissipation in the sine-Gordon model , 2008 .

[67]  L. A. Krasnobaeva,et al.  Kink velocity in nonstationary external fields for the sine-Gordon model with allowance for dissipation effects , 2008 .