Piecewise linear kinetics

These assumptions are really quite mild the / need only be increasing and sufficiently smooth. It is not even required that f, be bounded on IR. From a biological perspective, of course, the model loses relevance for really large values of S. In addition to the Monod functions, other functions that have been suggested include the exponential kinetics AM(1—exp(—S log 2/a)), hyperbolic kinetics m tanh(S log 3/2a), and piece-wise linear kinetics given by mS/2a for S<2a and by w for S > 2a. The piecewise linear kinetics fails to satisfy the strict monotonocity of (iii) and fails to satisfy (iv) at one point, but these assumptions could be weakened so as to include this case. 

Dynamical modeling requires detailed kinetic information on the mechanisms involved that is often not available. To cope with this problem, the formalism of piecewise-linear differential equations abstracts from the biochemical details of the regulatory mechanisms and exploits the switchlike nature of gene expression [de Jong 2002],... 

To calculate the dependence of wave speed, c, on Fe concentration, z, TYSON and FIFE replaced the nonlinear Oregonator kinetics (1) by a piecewise linear approximation. They found that c=c(z) is given implicitly by the transcendental equation... 

For numerical calculations, three specific models have been used extensively. These are the Oregonator model of the BZ reaction [12, 14, 24], the FitzHugh-Nagumo model [25, 26], and the (piecewise linear) Pushchino kinetics (also called Kinetics B ) [24, 27, 28]. These are all two-variable models with a diagonal diffusion matrix. 

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