Abstract DNA-type systems

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.

[1]  Kenneth Steiglitz,et al.  State transformations of colliding optical solitons and possible application to computation in bulk media , 1998 .

[2]  L M Adleman,et al.  Molecular computation of solutions to combinatorial problems. , 1994, Science.

[3]  Kenneth Steiglitz,et al.  Information transfer between solitary waves in the saturable Schrödinger equation , 1997 .

[4]  S. B. Leble,et al.  Elementary and binary Darboux transformations at rings , 1998 .

[5]  Kenneth Steiglitz,et al.  Embedding Computation in One-Dimensional Automata by Phase Coding Solitons , 1988, IEEE Trans. Computers.

[6]  Maria Przybylska,et al.  On matrix differential equations and abstract FG algorithm , 2002 .

[7]  L. Gabora,et al.  Quantum morphogenesis: a variation on Thom's catastrophe theory. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Gernot Neugebauer,et al.  Einstein-Maxwell solitons , 1983 .

[9]  Diederik Aerts,et al.  LETTER TO THE EDITOR: Quantum aspects of semantic analysis and symbolic artificial intelligence , 2003, quant-ph/0309022.

[10]  M. Eigen.  Selforganization of matter and the evolution of biological macromolecules , 1971, Naturwissenschaften.

[11]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[12]  J. Cieśliński.  The Darboux-Bäcklund transformation without using a matrix representation , 2000 .

[13]  Marek Czachor,et al.  Darboux covariant equations of von Neumann type and their generalizations , 2002, nlin/0205061.

[14]  Marek Czachor,et al.  Darboux-integrable equations with non-Abelian nonlinearities , 2000 .

[15]  Marek Czachor,et al.  Darboux-integrable nonlinear Liouville-von Neumann equation , 1998 .

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

[17]  Orlando Ragnisco,et al.  Extension of the Zakharov-Shabat generalized inverse method to solve differential-difference and difference-difference equations , 1980 .

[18]  J. Cieśliński.  An effective method to compute N‐fold Darboux matrix and N‐soliton surfaces , 1991 .

[19]  Maria Przybylska,et al.  Isospectral-like flows and eigenvalue problem , 2003, Future Gener. Comput. Syst..

[20]  A. Mikhailov,et al.  The reduction problem and the inverse scattering method , 1981 .

[21]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

[22]  V. V. Gafiychuk,et al.  Replicator - Mutator Evolutionary Dynamics , 2004 .

[23]  N. V. Ustinov,et al.  Darboux integration of , 2000 .

[24]  Solitons and Deformed Lattices I , 2002, cond-mat/0211239.

[25]  N. V. Ustinov.  The reduced self-dual Yang–Mills equation, binary and infinitesimal Darboux transformations , 1998 .

[26]  P. Libby.  The Scientific American , 1881, Nature.

[27]  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.

[28]  W. Symes.  The QR algorithm and scattering for the finite nonperiodic Toda Lattice , 1982 .

[29]  B. M. Fulk.  MATH , 1992 .

[30]  J. Cieśliński,et al.  A compact form of the Darboux–Bäcklund transformation for some spectral problems in Clifford algebras , 2001 .

[31]  Marek Czachor,et al.  Nonlinear von Neumann-type equations: Darboux invariance and spectra , 1999 .

[32]  R. Landauer,et al.  The Fundamental Physical Limits of Computation. , 1985 .

[33]  Diederik Aerts,et al.  PROBING THE STRUCTURE OF QUANTUM MECHANICS , 2002, Probing the Structure of Quantum Mechanics.

[34]  Marek Czachor,et al.  NAMBU-TYPE GENERALIZATION OF THE DIRAC EQUATION , 1996, quant-ph/9601015.

[35]  J. M. Oshorn.  Proc. Nat. Acad. Sei , 1978 .

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

[37]  Jan Naudts,et al.  MICROSCOPIC FOUNDATION OF NONEXTENSIVE STATISTICS , 1999 .

[38]  Jan L. Cieśliński,et al.  An algebraic method to construct the Darboux matrix , 1995 .

[39]  Josef Hofbauer,et al.  The theory of evolution and dynamical systems , 1988 .