Mathematical modeling of biochemical signal propagation in many-stage enzymatic pathways

TL;DR

This paper develops a mathematical framework to understand how signals propagate through complex biochemical pathways, accounting for variability and heterogeneity in enzyme kinetics, and introduces a rescaling technique to stabilize wave transmission.

Contribution

It introduces a novel reciprocal-velocity spatial rescaling method that compensates for kinetic variations, enabling predictable signal propagation in heterogeneous biochemical networks.

Findings

Wave existence depends on activation bias as a bifurcation parameter.

Parameter gradients and kinetic variations distort wavefronts and affect propagation speed.

The rescaling technique stabilizes wave velocities and preserves wavefronts despite heterogeneity.

Abstract

Biochemical signalling cascades transduce extracellular stimuli into cellular responses through sequences of discrete, node-to-node activations. While signal fidelity depends critically on local interaction kinetics, the mechanisms governing information propagation in realistic, highly variable kinetic contexts remain poorly understood. In this paper, we develop a mathematical framework for travelling waves in canonical feed-forward pathways governed by nonlinear Michaelis-Menten-type kinetics. For uniform pathways, we characterise the complete steady-state landscape and demonstrate that activation bias (the contribution of the binary states of each node to downstream activation) between connected nodes acts as a key bifurcation parameter dictating wave existence. Extending this framework to heterogeneous networks, we show how parameter gradients and random kinetic variations distort wavefronts and induce heavy fluctuations in propagation speed. To recover predictable signal transmission, we introduce a novel reciprocal-velocity spatial rescaling technique. We demonstrate that this coordinate transformation inherently absorbs local kinetic variations, effectively smoothing wave velocities and preserving wavefront profiles without requiring bespoke parameter tuning or continuous limits. Finally, by testing the framework's limits against extreme parameter variability, we reveal how severe kinetic bottlenecks lead to functional pathway fragmentation, offering a mathematically justified basis for rational model reduction in complex biochemical networks.