Self-sustained oscillations in the Jacob-monod mode of gene regulation

Abstract

A plausible macroscopic model of genetic regulation by feedback repression, evolved from the original formulation of Goodwin, is considered.This model has the form of a second-order non-linear dynamic system, ~ = fl(x,y), # = f2(x,y), where x(t) ~ 0 describes the mRNA which codes for the protein y(t) ~ 0, the latter acting as a repressor. Published analytical, analog, and digital studies of this model do not report the existence of any periodic solutions.A stochastic analysis has produced some irregular undamped oscillations, but these appear to be due to the stochastic elements so introduced.

Global asymptotic stability of certain models for protein synthesis and repression

“…Thus, models based on concentrations of mRNA, genes, repressors, etc., should be interpreted so that the numbers and parameters involved represent "averaged" values over an aggregate.) Often in the literature [7,8,9,24] one finds a (perhaps only implicit) simplifying assumption that x3 = Xx2, so that in place of (3.1) one considers a two-dimensional version (the two-dimensional Goodwin model)…”

“…This may be due to a number of phenomena, including diffusion, packaging and transport across membranes, etc. Some authors [1,26,27,32] have used partial differential equation models to investigate this aspect of protein synthesis in eukaryotes while others [5,9,16,17,24,28,29] have tried to modify the models of Goodwin to include some type of time delay directly. Our interest in this paper is in the latter alternative.…”

“…Various special cases of (3.4) or (3.5) have been partially analyzed in the literature [5,9,16,17,24,28,29] (as Goodwin [6, p. 8] correctly observes, rigorous analysis of delay equations poses nontrivial difficulties; it is not surprising that much of the analysis of delay system protein models has involved linearization, approximation, numerical simulation, or a combination of these). In view of a number of incorrect conclusions ai.J claims based on non-rigorous analysis and/or computer simulations of other protein synthesis models (e.g., some authors carry out analysis of a linearized system, obtain local stability results and incorrectly conclude that this rules out oscillations; for remarks on other erroneous claims see [31]), it is exceedingly important in trying to argue validity or lack thereof of these models that one have completely rigorous discussions to support any claims for a certain qualitative behavior that a particular model might or might not possess.…”