Mathematical modeling and analysis of the pathway network consisting of symmetrical complexes with N monomers, like the activation of MMP2

TL;DR

This paper develops a mathematical model for symmetrical pathway networks involving N monomers, generalizing previous work on a 3-monomer system, and analyzes their stability and equilibrium properties.

Contribution

It introduces a generalized mathematical framework for symmetrical pathway networks with N monomers and derives their equilibrium concentrations and stability properties.

Findings

Complex concentrations converge to equilibrium values.

The pathway network with symmetrical complexes is asymptotically stable.

Identifies regulator parameters influencing the target complex.

Abstract

The activation of matrix metalloproteinase 2 (MMP2) is a crucial event during tumor metastasis and invasion, and this pathway network consists of 3 monomers. The pathway network of the activation obeys to a set of specified reaction rules. According to the rules, the individual molecules localize in a particular order and symmetrically around a homodimer following the formation of that dimer. We generalized the homodimer pathway network obeying to similar reaction rules, by changing the number of monomers involved in this pathway from 3 to N. At the previous work, we found the molecules in the pathway network are classified to some reaction groups. We derived the law of mass conservation between the groups. Each group concentration converges to its equilibrium solution. Using these results, we derive the concentrations of the complexes theoretically and reveal that each complex concentration converges to its equilibrium value. We can say the pathway network with homodimer symmetric form complexes is asymptotic stable and identify the regulator parameter of the target complex in the network. Our mathematical approach may help us understand the mechanism of this type pathway network by knowing the background mathematical laws which govern this type pathway network.