Convergence Properties of Posttranslationally Modified Protein-Protein Switching Networks with Fast Decay Rates.
A significant conceptual difficulty in the use of switching systems to model regulatory networks is the presence of so-called "black walls," co-dimension 1 regions of phase space with a vector field pointing inward on both sides of the hyperplane. Black walls result from the existence of direct negative self-regulation in the system. One biologically inspired way of removing black walls is the introduction of intermediate variables that mediate the negative self-regulation. In this paper, we study such a perturbation. We replace a switching system with a higher-dimensional switching system with rapidly decaying intermediate proteins, and compare the dynamics between the two systems. We find that the while the individual solutions of the original system can be approximated for a finite time by solutions of a sufficiently close perturbed system, there are always solutions that are not well approximated for any fixed perturbation. We also study a particular example, where global basins of attraction of the perturbed system have a strikingly different form than those of the original system. We perform this analysis using techniques that are adapted to dealing with non-smooth systems.
Department of Mathematics, University of Utah, Salt Lake City, UT, 84112-0090, USA.
Gaoyang
G
Fan
Department of Mathematical Sciences, Montana State University, Bozeman, MT, 59717, USA. cummins@math.montana.edu.
Bree
B
Cummins
Department of Mathematical Sciences, Montana State University, Bozeman, MT, 59717, USA.
Tomáš
T
Gedeon
D053263:Gene Regulatory Networks; D055641:Mathematical Concepts; D008954:Models, Biological; D008957:Models, Genetic; D060066:Protein Interaction Maps; D011499:Protein Processing, Post-Translational; D049490:Systems Biology / Q000706:statistics & numerical data
Gene regulation; Negative self-regulation; Piecewise-linear; Singular perturbation; Transcription/translation model