Four-state models and regulation of contraction of smooth muscle. I. Physical considerations, stability, and solutions.

Cyclic four-state models are frequently used in biology to represent a variety of molecular behaviors. A common experimental strategy to test such models is to follow the behavior of the real system after some of the rate constants are changed in a stepwise manner. We analyze the mathematical behavior of a simple example of such a model applicable to the regulation of contraction of smooth muscle, but our results apply in general to any linear, cyclic four-state model. We discuss detailed balance and requirements for linearity. We find that the only way to have sustained oscillations is for the rate constants of the model to be themselves oscillatory. We state conditions for decaying oscillations and find that in models that do not follow strictly first-order kinetics and do not satisfy detailed balance, these conditions can hold. We show analytically that the response of any state to step changes in the rate constants is the sum of three weighted exponentials plus a constant term, the steady-state value. We provide explicit expressions for the time dependence of all state variables. We discuss a simple way to use these results to obtain numerical solutions in cases where the rate constants change in an arbitrary way.