Gene regulatory networks and the repressilator

Elowitz and Leibler modeled their system in terms of the concentrations of three repressor proteins, lacl, tetR, and cl, and their corresponding mRNA concentrations. If we use subscripts 1,2, and 3 to denote these three, their generic equations for the mRNA and protein concentrations m, and p, (i = 1, 2, 3) are 

There are six ODEs and eighteen parameters in the full model of Elowitz and Leibler. To simplify the analysis, Elowitz and Leibler assumed that all the parameters are independent of the i in their subscripts. That leaves six parameters, which can be further simplified by introducing unitless variables 

To obtain the steady state(s) of Equation (5.43), we have by repeated substitution  

The 1952 Hodgkin-Huxley model for membrane electrical potential is perhaps the oldest and the best known cellular kinetic model that exhibits temporal oscillations. The phenomenon of the nerve action potential, also known as excitability, has grown into a large interdisciplinary area between biophysics and neurophysiology, with quite sophisticated mathematical modeling. See [103] for a recent treatise. 

Derived from Hodgkin-Huxley s celebrated theory and inspired by the experimental observations, cellular calcium dynamics, either stimulated via inositol 1,4,5-trisphosphate (IP3) receptor in many non-muscle cells [69,139], or via the ryanodine receptor in muscle cells [108], is another extensively studied oscillatory system. Both receptors are themselves Ca2+ channels, and both can be activated by Ca2+, leading to calcium-induced calcium release from endoplasmic reticulum. 

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