Applying a Rigorous Quasi-Steady State Approximation Method for Proving the Absence of Oscillations in Models of Genetic Circuits

Abstract

Abstract. In this paper, we apply a rigorous quasi-steady state approximation method on a family of models describing a gene regulated by a polymer of its own protein. We study the absence of oscillations for this family of models and prove that Poincaré-Andronov-Hopf bifurcations arise if and only if the number of polymerizations is greater than 8. A result presented in a former paper at Algebraic Biology 2007 is thereby generalized. The rigorous method is illustrated over the basic enzymatic reaction.

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

“…Using the subtropical approach from Sect. 4 we discovered a suitable zero in 144 cases (30%), where the inequalities were always satisfied. We could exclude the existence of a zero in 338 cases (67%).…”

“…21 The detailed transformation can be found in [3, Section 2.1]. In a later publication Boulier et al revised their quasi-steady state approximation to obtain another biologically more appropriate model [4]. For the applicability of our method and the quality of our results that improvement makes no difference.…”

On Muldowney’s Criteria for Polynomial Vector Fields with Constraints

“…If all variables and parameters are known to be positive, the technique of positive quantifier elimination [14,15] can be used, which was first used to solve semi-algebraic criteria for the existence of Hopf bifurcation fixed points [16,17] arising in the context of chemistry and algebraic biology [18][19][20][21].…”

Chemical Reaction Systems, Computer Algebra and Systems Biology

“…See [29] for a closely related work. Our algorithm was successfully applied over a more complicated model [6,5], featuring two genes, arising from the modeling problem of the circadian clock of the green algae Ostreococcus tauri [12].…”

“…By differentiating the probability generating function (5) and evaluating it at z 1 = z 2 = · · · = z n = 1, one gets formulae which bind φ and the moments of the random variables which count the numbers of molecules. A mere evaluation yields: φ(t, z) | z=1 = 1.…”