An Algebraic Approach to Signaling Cascades with n Layers

TL;DR

This paper presents an algebraic framework for analyzing n-layer signaling cascades, revealing unique steady states, rational relationships among concentrations, and how responses shift along the cascade, with implications for enzyme competition and substrate dynamics.

Contribution

It introduces a novel algebraic method to analyze multi-layer signaling cascades, providing explicit solutions and insights into steady states and response behaviors.

Findings

Unique steady state for any total substrate and enzyme amounts

Stimulus-response curves are inverse rational functions

Response curves shift leftward down the cascade

Abstract

Posttranslational modification of proteins is key in transmission of signals in cells. Many signaling pathways contain several layers of modification cycles that mediate and change the signal through the pathway. Here, we study a simple signaling cascade consisting of n layers of modification cycles, such that the modified protein of one layer acts as modifier in the next layer. Assuming mass-action kinetics and taking the formation of intermediate complexes into account, we show that the steady states are solutions to a polynomial in one variable, and in fact that there is exactly one steady state for any given total amounts of substrates and enzymes. We demonstrate that many steady state concentrations are related through rational functions, which can be found recursively. For example, stimulus-response curves arise as inverse functions to explicit rational functions. We show that the stimulus-response curves of the modified substrates are shifted to the left as we move down the cascade. Further, our approach allows us to study enzyme competition, sequestration and how the steady state changes in response to changes in the total amount of substrates. Our approach is essentially algebraic and follows recent trends in the study of posttranslational modification systems.