Menu: Home :: go to Journal :: switch to Russian :: switch to English
You are here: all Journals and Issues→ Journal→ Issue→ Article

The Filippov theory and its application to gene regulatory networks

Annotation

We study some properties of piecewise-linear differential systems describing gene regulatory networks, where the dynamics is governed by sigmoid-type nonlinearities which are close to or coincide with the step functions. To overcome the difficulty of describing the dynamics of the system near singular stationary points (i.e. belonging to the discontinuity set of the system) we use the concept of Filippov solutions. It consists in replacing differential equations with discontinuous right-hand sides with differential inclusions with multi-valued functions. The global existence and some basic properties of Filippov solutions such as continuous dependence on parameters are studied. We also study uniqueness and non-uniqueness of the Fillipov solutions in singular domains. The concept of Filippov stationary point is extensively exploited in the paper. We compare two approaches in defining the singular stationary points: based on Fillipov theory and based on replacing step functions by steep sigmoids and investigating the smooth systems thus obtained.

Keywords

piecewise-linear differential system; singular stationary point; differential inclusion; Filippov solution

Full-text in one file

Download

UDC

517.958

Pages

760-762

References

1. Glass L., Kaufmann S.A. Co-operative components, spatial localization and oscillatory cellular dynamics // J. Theor. Biol. 1972. V. 34. P. 219-237. 2. Glass L., Kaufmann S.A. The logical analysis of continuous, non-linear biochemical control networks // J. Theor. Biol. 1973. V. 39. P. 103-129. 3. Mestl T., Plahte E., and Omholt S.W. A mathematical framework for describing and analysing gene regulatory networks // J. Theor. Biol. 1995. V. 176. P. 291--300. 4. Plahte E., Kjoglum S. Analysis and generic properties of gene regulatory networks with graded response functions // J. Physica D. 2005. V. 201. P. 150--176. 5. Plahte E., Mestl T., Omholt S.W. Global analysis of steady points for systems of differential equations with sigmoid interactions // J. Dynamics and Stability of Systems. 1994. V. 9. P. 275--291. 6. Plahte E., Mestl T., Omholt S.W. A methodological basis for the description and analysis of systems with complex switch-like interactions // J. Math. Biol. 1998. V. 36. P. 321--348. 7. Gouze J.-L. and Sari T. A class of piecewise linear differential equations arising in biological models // Dynamical Systems: An International Journal. 2002. V. 17. P. 299-316. 8. Jong H. de, Gouze J.-L., Hernandez C., Page M., Sari T., and Geiselmann J. Qualitative simulations of genetic regulatory networks using piecewise linear models // Bulletin of mathematical biology. 2004. V. 66 (2). P. 301-340.

Section of issue

Articles

Для корректной работы сайта используйте один из современных браузеров. Например, Firefox 55, Chrome 60 или более новые.