Analytical Equilibrium Solutions of Biochemical Systems with Synthesis and Degradation

TL;DR

This paper introduces a graph-based method to analytically determine steady state solutions of biochemical networks with synthesis and degradation, extending previous work to include nonlinear systems and providing new theoretical insights.

Contribution

We develop a graph-theoretical framework for calculating steady states in biochemical systems with synthesis and degradation, including conditions for stability and methods for matrix inversion.

Findings

Framework applies to nonlinear systems via edge labels

Provides necessary and sufficient conditions for stability

Demonstrates utility with insulin secretion model

Abstract

Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph $G$ and the associated system of linear non-homogeneous differential equations with first order degradation and zeroth order synthesis. We also present a theorem which provides necessary and sufficient conditions for the dynamics to engender a unique stable steady state. Although the dynamics are linear, one can apply this framework to nonlinear systems by encoding nonlinearity into the edge labels. We answer open question from our previous work concerning the non-positiveness of the elements in the inverse of a perturbed Laplacian matrix. Moreover, we provide a graph theoretical framework for the computation of the inverse of a such matrix. This also completes our previous framework and makes it purely graph theoretical. Lately, we demonstrate the utility of this framework by applying it to a mathematical model of insulin secretion through ion channels and glucose metabolism in pancreatic $\beta$-cells.