The Extracellularly Regulated Kinase/Mitogen Activated Protein Kinase (ERK/MAPK) signaling pathway is a critical regulator of cellular processes in developing and adult tissues.^{1}^{–}^{4} At the core of the pathway is a three-stage cascade of phosphorylation-dephosphorylation reactions that culminates in a double phosphorylation of MAPK, the key serine/threonine kinase at the last stage of the cascade (). By acting on cytoplasmic and nuclear substrates, phosphorylated MAPK controls cellular processes, such as cell division and differentiation. The MAPK pathway is one of the most extensively modeled signaling systems.^{5}^{–}^{7} The goals of mathematical models of MAPK and other signaling pathways are to integrate the results of genetic, biochemical and cellular studies, and to predict the dynamics of signaling and its effects on cellular processes.

The first model of the MAPK cascade was developed by Huang and Ferrell, who used their model as a basis for linking the structure of the cascade and its dynamics (). Their model represents an idealized situation where the enzymes are placed in a well-mixed system with nonlimiting amounts of ATP and other co-factors. Based on the mass-action kinetics, the model describes the dynamics of 22 species participating in ten reactions.^{8} Huang and Ferrell hypothesized that the three-tiered structure of the cascade controls its steady-state input-output behavior. Since double phosphorylation of MAPK is necessary for its catalytic activity, the fraction of MAPK in its double phosphorylated form was considered the output of the cascade. To compute the input-output maps, the top-layer of the cascade was activated to some constant level and the downstream components were allowed to reach a steady state level of activation. Clearly, to perform such an analysis, one has to specify model parameters, such as the starting amounts of kinases and phosphatases and the rate constants of enzymatic reactions.

All in all, the model had 36 parameters, each of which was given within a 25-fold range, and calculation of input-output maps was based on sampling of their values from these ranges. Briefly, one generates a 36-dimensional vector of model parameters and then uses it to find the values of the output of the cascade for multiple values of inputs. Every one of such calculations generates a single input-output map. Based on calculations with hundreds of randomly generated parameter sets, Huang and Ferrell found that all of the input-output maps in their model are ultrasensitive: there is a single output for every input, but the output can switch from “off” to “on” within a very narrow range of inputs (). Recently, however, we have demonstrated that, within a very wide range of model parameters, the same model can behave as an irreversible switch and as an oscillator ().^{9}

Given the fact that even the simplest models of this important pathway can lead to complex dynamics, it is essential to experimentally determine what types of dynamics are actually realized in vivo. We are using the terminal patterning system in the early Drosophila embryo as an experimental model for monitoring the dynamics of MAPK signaling under a wide range of genetic and environmental conditions. Our long-term goal is to explore the regulation of MAPK signaling by multiple cellular and biochemical processes and to develop increasingly mechanistic models of this pathway. Here we review the first results of our approach that is based on a combination of previously developed genetic tools, quantitative imaging and biophysical models.