# Realizations of kinetic differential equations

## TL;DR

This paper explores whether polynomial differential equations can be realized as reaction networks with specific properties, aiding in modeling biochemical systems and solving mathematical problems.

## Contribution

It provides constructive methods to determine if a polynomial differential equation can be induced by a reaction network with desired properties.

## Key findings

- Conditions for realizing polynomial equations as reaction networks.
- Methods to construct networks with properties like reversibility and mass conservation.
- Applications in fitting models to data and mathematical problem solving.

## Abstract

The induced kinetic differential equation of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a polynomial differential equation, is it possible to find a network which induces the equation? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc. The constructive answers presented to a series of questions of the above type are useful when fitting a differential equation to measurements, or when trying to find out the dynamic behavior of the solutions of a differential equation. It turns out that some of the results can be applied when trying to solve purely mathematical problems, like the existence of positive solutions to polynomial equation.