Optimal phosphorylation step number of intracellular signal-transduction pathway

Eukaryotic cells use signal-transduction network to respond in specific ways to external signals from their environment. Several signal-transduction pathway are composed of multi-step chemical reactions. We here theoretically study what determines the number of kinase phosphorylation steps composing of the intracellular signal-transduction cascade. We examine a simple mathematical model for the association and phosphorylation process of kinases in the signal-transduction cascade. We focus on the speed of signal transduction as the criterion for determining the optimal response. The present model first reveals that the initial expression level of kinase in each step of the cascade must be the same in the optimal response under the constraint of the constant total kinase concentration. The second conclusion is that the optimal step number of kinase cascade is primarily determined by the ratio of the target concentration of the final phosphorylated kinase in the cascade to that of input signal molecule, C/S. A multi-step cascade is optimum if the amplification of the final product concentration C relative to the input signal S is sufficiently large; whereas, a single-step reaction is optimum if C/S is small. This suggests that multi-step phosphorylation would have evolved to accelerate the speed of transduction of weak signals.