Mathematical analysis of activation thresholds in enzyme-catalyzed positive feedbacks: application to the feedbacks of blood coagulation.

A hierarchy of enzyme-catalyzed positive feedback loops is examined by mathematical and numerical analysis. Four systems are described, from the simplest, in which an enzyme catalyzes its own formation from an inactive precursor, to the most complex, in which two sequential feedback loops act in a cascade. In the latter we also examine the function of a long-range feedback, in which the final enzyme produced in the second loop activates the initial step in the first loop. When the enzymes generated are subject to inhibition or inactivation, all four systems exhibit threshold properties akin to excitable systems like neuron firing. For those that are amenable to mathematical analysis, expressions are derived that relate the excitation threshold to the kinetics of enzyme generation and inhibition and the initial conditions. For the most complex system, it was expedient to employ numerical simulation to demonstrate threshold behavior, and in this case long-range feedback was seen to have two distinct effects. At sufficiently high catalytic rates, this feedback is capable of exciting an otherwise subthreshold system. At lower catalytic rates, where the long-range feedback does not significantly affect the threshold, it nonetheless has a major effect in potentiating the response above the threshold. In particular, oscillatory behavior observed in simulations of sequential feedback loops is abolished when a long-range feedback is present.

The Wsh and Ph mutations affect the c-kit expression profile: c-kit misexpression in embryogenesis impairs melanogenesis in Wsh and Ph mutant mice.

The receptor tyrosine kinases (RTKs) c-kit and platelet-derived growth factor receptor alpha chain (PDG-FRa) are encoded at the white spotting (W) and patch (Ph) loci on mouse chromosome 5. While W mutations affect melanogenesis, gametogenesis, and hematopoiesis, the Ph mutation affects melanogenesis and causes early lethality in homozygotes. W-sash (Wsh) is an expression mutation and blocks c-kit expression in certain cell types and enhances c-kit expression in others, including at sites important for early melanogenesis. We have determined the effect of Ph on c-kit expression during embryogenesis in Ph heterozygotes. Immunohistochemical analysis revealed enhanced c-kit expression in several cell types, including sites important for early melanogenesis. We propose that in both Wsh and Ph mutant mice c-kit misexpression affects early melanogenesis and is responsible for the pigment deficiency. Moreover, we have defined the organization of the RTKs in the W/Ph region on chromosome 5 and characterized the Wsh mutation by using pulsed-field gel electrophoresis. Whereas the order of the RTK genes was determined as Pdgfra-c-kit-flk1, analysis of the Wsh mutation revealed that the c-kit and Pdgfra genes are unlinked in Wsh, presumably because of an inversion of a small segment of chromosome 5. The Ph mutation consists of a deletion including Pdgfra and the 3' deletion endpoint of Ph lies between Pdgfra and c-kit. Therefore, positive 5' upstream elements controlling c-kit expression in mast cells and some other cell types are affected by the Wsh mutation and negative elements are affected by both the Wsh and the Ph mutation.