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In their classical work (Proc. Natl. Acad. Sci. USA, 1981, 78:6840-6844),Goldbeter and Koshland mathematically analyzed a reversible covalentmodification system which is highly sensitive to the concentration ofeffectors. Its signal-response curve appears sigmoidal, constituting abiochemical switch. However, the switch behavior only emerges in the"zero-order region", i.e. when the signal molecule concentration is much lowerthan that of the substrate it modifies. In this work we showed that theswitching behavior can also occur under comparable concentrations of signalsand substrates, provided that the signal molecules catalyze the modificationreaction in cooperation. We also studied the effect of dynamic disorders on theproposed biochemical switch, in which the enzymatic reaction rates, instead ofconstant, appear as stochastic functions of time. We showed that the system isrobust to dynamic disorder at bulk concentration. But if the dynamic disorderis quasi-static, large fluctuations of the switch response behavior may beobserved at low concentrations. Such fluctuation is relevant to many biologicalfunctions. It can be reduced by either increasing the conformationinterconversion rate of the protein, or correlating the enzymatic reactionrates in the network.

The Goldbeter-Koshland switch in the first-order region and its response to dynamic disorder