Theoretical Exploration of Stability and Steepness of Algebraic Steady State Equations Describing Kinetics of Elementary Cellular Reaction Cycles

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Cellular stimulus/response kinetics is described using ordinary differential equations (ODEs). In a steady state system, time-dependent ODEs can be converted to algebraic equations. The solutions of the equations reveal the relationships between the strength of an extracellular signal and the magnitude of the cellular response (the stimulus/response curves). In the present study, we investigated the dependence between the number of solutions (multistability) of the equations and the gradient of the inflection point (steepness) of the stimulus/response curve on a kinetic parameter space, defined by the ratios of the catalytic efficiency (referred to as \lambda_{1} and \lambda_{2}) of bi-connected elementary cellular reaction cycles (cycles 1 and 2, respectively), composed of a substrate and two enzymes. When \lambda_{2} was initially small and then increased (\lambda_{1} was fixed), the steepness also increased, leading to the monostability to bistability transition. This was the case with the \lambda_{1} axis, although when \lambda_{1} increased (\lambda_{2} was fixed), the steepness decreased, resulting in the bistability to monostability transition. This is the first report to systematically show that the steepness is closely correlated with the stability of elementary cellular reaction systems.
2011-04-15