Systems-Level Analysis of Multisite Protein Phosphorylation: Mathematical Induction, Geometric Series, and Entropy

Multisite protein phosphorylation plays a pivotal role in regulating cellular signaling and decision-making processes. In this study, we focus on the mathematical underpinnings and informational aspects of sequential, distributive phosphorylation systems. We first provide rigorous steady-state solutions derived using geometric series arguments and formal mathematical induction, demonstrating that the distribution of phosphorylation states follows a geometric progression determined by the kinase-to-phosphatase activity ratio. We then extend the analysis with entropy-based insights, quantifying uncertainty in phosphorylation states and examining the mutual information between kinase activity and phosphorylation levels through a truncated Poisson model. These results highlight how phosphorylation dynamics introduce both structured patterns and inherent signal variability. By combining exact mathematical proofs with entropy analysis, this work clarifies key quantitative features of multisite phosphorylation from a systems-level perspective.