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Homoclinic solutions in mechanical systems with small dissipation. Application to DNA dynamics.

B Kazmierczak1, T Lipniacki

  • 1Institute of Fundamental Technological Research, Polish Academy of Sciences, Swietokrzyska St. 21, 00-049 Warsaw, Poland. bkazmier@ippt.gov.pl

Journal of Mathematical Biology
|May 2, 2002
PubMed
Summary

This study shows that homoclinic solutions persist in dissipative systems, even with energy input. This finding is applied to demonstrate the existence of torsional pulse-like traveling waves in DNA molecules.

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Area of Science:

  • Mathematical Physics
  • Dynamical Systems Theory
  • Biophysics

Background:

  • Perturbed systems of second-order ordinary differential equations (ODEs) are often derived from partial differential equations (PDEs) when analyzing traveling wave solutions.
  • Understanding the persistence of homoclinic solutions in such systems is crucial for modeling wave phenomena.

Purpose of the Study:

  • To analyze the persistence of homoclinic solutions in perturbed second-order ODE systems, particularly in the presence of dissipation.
  • To apply these findings to the specific case of torsional pulse-like traveling waves in nonelastic DNA molecules.

Main Methods:

  • Analysis of perturbed second-order ODEs.
  • Investigating the impact of dissipation and nonautonomous terms on homoclinic solutions.

Related Experiment Videos

  • Application of theoretical results to a biophysical model.

    • Main Results:

    • Homoclinic solutions are shown to persist even when dissipation is present in the system.
    • Dissipation can be effectively balanced by nonautonomous terms controlled by a single parameter.
    • The existence of torsional pulse-like traveling waves in nonelastic DNA is proven.

    Conclusions:

    • The persistence of homoclinic solutions is established in dissipative systems, offering a robust framework for wave analysis.
    • The study provides a theoretical basis for understanding energy transfer mechanisms, such as those involving RNA polymerase, in DNA dynamics.