Analysis of stimulus-response chains using nonlinear dynamics

Stimulus chains represent those series of coupled biochemical events through which receptor occupation is transformed into physiological or pharmacological responses. These events are commonly modeled as sets of sequential hyperbolic responses that may be treated mathematically as an iterative process. Methods used in the analysis of iterated map functions such as bifurcation analysis and graphical iteration are applied to the stimulus sequence to understand the dynamics of the process. Examples are given to illustrate determination of fixed points of the map as well as techniques for determining their stability. When present, a central repelling fixed point reveals the existence and location of a stimulus threshold, while the upper attracting fixed point determines the size of the terminal stimulus (S omega) in the signaling pathway. Estimates of S omega are used to develop equations for predicting changes in agonist EC50S at each point in the stimulus chain. Similarly, Lyapunov exponents are found to provide a link between changes in cell or tissue parameters and changes in the rate of stimulus amplification and agonist maximal response. These findings demonstrate that techniques of nonlinear dynamics when combined with biochemical studies of cellular signaling, provide useful new approaches for quantitative and qualitative analysis of drug action.