Abstract DNA-type systems

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Published 31 January 2006 2006 IOP Publishing Ltd and London Mathematical Society
, , Citation Diederik Aerts and Marek Czachor 2006 Nonlinearity 19 575DOI 10.1088/0951-7715/19/3/003

0951-7715/19/3/575

Abstract

An abstract DNA-type system is defined by a set of nonlinear kinetic equations with polynomial nonlinearities that admit soliton solutions associated with helical geometry. The set of equations allows for two different Lax representations: a von Neumann form and a Darboux-covariant Lax pair. We explain why non-Kolmogorovian probability models occurring in soliton kinetics are naturally associated with chemical reactions. The most general known characterization of soliton kinetic equations is given and a class of explicit soliton solutions is discussed. Switching between open and closed states is generic behaviour of the helices. The effect does not crucially depend on the order of nonlinearity (i.e. types of reactions), a fact that may explain why simplified models possess properties occurring in realistic systems. We explain also why fluctuations based on Darboux transformations will not destroy the dynamics but only switch between a finite number of helical structures.

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10.1088/0951-7715/19/3/003
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