The regulation of the antigen-activated T cell receptor (TCR) Erk signaling pathway serves a critical role in maintaining and coordinating immune responses. Extensive biochemical studies have identified many of the major molecular components and interactions of this pathway; however, this molecular-level knowledge conveys little information on how the entire signaling network operates as a functional unit. The embedded regulatory structures can be effectively discerned through a control analysis of a mathematical model describing the signal transduction pathway. Construction of the mathematical model utilizes many rate parameters and initial conditions even for small sections of the signaling pathway. Unfortunately, many of these rate parameters and initial conditions are not known. Furthermore, rate constants are typically defined in vitro and it is not well-established how in vitro rate constants compare to those within the cell. In this study, laboratory experiments, mathematical modeling and simulations were integrated to determine the principle regulatory structures of the pathway that give rise to the observed dynamics of the TCR and the protein tyrosine kinases Zap70 and Erk. Numerous approaches to fit a comprehensive mathematical model of the Erk signaling pathway to the time series data were unsuccessful due to the large number of uncertain parameters with considerable physiological ranges in potential values. Using an alternative approach loosely based upon a simple-to-general reconstruction modeling strategy [1], this work deduces the embedded regulatory structure from this limited time course data through a constrained step-wise evolution of a minimal model. Although the simple-to-general approaches have potential limitations associated with the existence of multiple local minima, the constraints imposed in our approach prevent divergent branching and corral the solution set to feasible solutions that are consistent with the existing knowledge of the specific pathway participants and their general roles. Briefly, the process is initiated with a rudimentary signaling diagram that contains only the elements for which time-course data exists. A mathematical model of the signaling diagram is constructed from ordinary differential equations and parameters are identified to best fit the data. Discrepancies are noted and an exogenous input is temporarily utilized to mediate intermediate reactions to fit the available data. Once satisfactory, the exogenous input is replaced by the addition of signaling elements or interconnections that achieve similar functional roles to the exogenous input. This process is repeated with each iteration using an exogenous input and subsequently replacing it with known signaling elements and or interconnections. In the final iteration, known signaling elements or interconnections were not capable of recreating the action of the exogenous input, thereby suggesting the participation of an unknown phosphatase. The result captures the primary dynamics of the experimental data with the simplest model and reveals the embedded regulatory structure. This work describes an alternative approach that utilizes a step-wise evolution of minimal models with temporary exogenous inputs to reveal the dominant regulatory structures employed by the T cell in achieving observed TCR, Zap70, and Erk time course dynamics. This approach integrates knowledge of established pathway elements to interpret experimental time course data within the mathematical framework.

1. Crampin, E.J., S. Schnell, and P.E. McSharry, Mathematical and computational techniques to deduce complex biochemical reaction mechanisms. Progress in Biophysics & Molecular Biology, 2004. 86(1): p. 77-112.