Exact Analysis of Intrinsic Qualitative Features of Phosphorelays using Mathematical Models

TL;DR

This paper provides an analytical framework for understanding the intrinsic features of phosphorelays, revealing how their responses are shaped by biological parameters and how ultrasensitivity can be modulated within these signaling systems.

Contribution

The authors develop a comprehensive mathematical model of phosphorelays, proving the existence of a unique stable steady-state and deriving explicit formulas for stimulus-response relationships.

Findings

Phosphorelays have a unique stable steady-state under mass-action kinetics.

Limited ultrasensitivity in the bottom layer is intrinsic and independent of reaction rates.

Adjusting reaction rates can enhance ultrasensitivity in intermediate layers.

Abstract

Phosphorelays are a class of signaling mechanisms used by cells to respond to changes in their environment. Phosphorelays (of which two-component systems constitute a special case) are particularly abundant in prokaryotes and have been shown to be involved in many fundamental processes such as stress response, osmotic regulation, virulence, and chemotaxis. We develop a general model of phosphorelays extending existing models of phosphorelays and two-component systems. We analyze the model analytically under the assumption of mass-action kinetics and prove that a phosphorelay has a unique stable steady-state. Furthermore, we derive explicit functions relating stimulus to the response in any layer of a phosphorelay and show that a limited degree of ultrasensitivity (the ability to respond to changes in stimulus in a switch-like manner) in the bottom layer of a phosphorelay is an intrinsic feature which does not depend on any reaction rates or substrate amounts. On the other hand, we show how adjusting reaction rates and substrate amounts may lead to higher degrees of ultrasensitivity in intermediate layers. The explicit formulas also enable us to prove how the response changes with alterations in stimulus, kinetic parameters, and substrate amounts. Aside from providing biological insight, the formulas may also be used to avoid time-consuming simulations in numerical analyses and simulations.