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.

Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets

An integrated understanding of molecular and developmental biology must consider the large number of molecular species involved and the low concentrations of many species in vivo. Quantitative stochastic models of molecular interaction networks can be expressed as stochastic Petri nets (SPNs), a mathematical formalism developed in computer science. Existing software can be used to define molecular interaction networks as SPNs and solve such models for the probability distributions of molecular species. This approach allows biologists to focus on the content of models and their interpretation, rather than their implementation. The standardized format of SPNs also facilitates the replication, extension, and transfer of models between researchers. A simple chemical system is presented to demonstrate the link between stochastic models of molecular interactions and SPNs. The approach is illustrated with examples of models of genetic and biochemical phenomena where the UltraSAN package is used to present results from numerical analysis and the outcome of simulations.

Functional domains are specified to single-cell resolution in a Drosophila epithelium

Specification of pattern is fundamental to the development of a multicellular organism. The Malpighian (renal) tubule of Drosophila melanogaster is a simple epithelium that proliferates under the direction of a single tip cell into three morphologically distinct domains. However, systematic analysis of a panel of over 700 P{GAL4} enhancer trap lines reveals unexpected richness for such an apparently simple tissue. Using numerical analysis, it was possible formally to reconcile apparently similar or complementary expression domains and thus to define at least five genetically defined domains and multiple cell types. Remarkably, the positions of domain boundaries and the numbers of both principal and secondary (“stellate”) cell types within each domain are reproducible to near single-cell precision between individual animals. Domains of physiological function were also mapped using transport or expression assays. Invariably, they respect the boundaries defined by enhancer activity. These genetic domains can also be visualized in vivo, both in transgenic and wild-type flies, providing an “identified cell” system for epithelial physiology. Building upon recent advances in Drosophila Malpighian tubule physiology, the present study confirms this tissue as a singular model for integrative physiology.