# Description of chemical systems by means of response functions

## TL;DR

This paper introduces a new way to describe how parts of biochemical systems respond using mathematical tools called renewal equations.

## Contribution

The paper introduces a formalism to represent biochemical system interactions using renewal equations derived from linear ODEs.

## Key findings

- The formalism can represent interactions in biochemical systems using renewal equations under specific conditions.
- The approach applies to both linear and non-linear ODEs modeling biochemical systems like kinetic proofreading.
- Kernels from systems with reactions satisfying detail balance are completely monotone functions.

## Abstract

In this paper we introduce a formalism that allows to describe the response of a part of a biochemical system in terms of renewal equations. In particular, we examine under which conditions the interactions between the different parts of a chemical system, described by means of linear ODEs, can be represented in terms of renewal equations. We show also how to apply the formalism developed in this paper to some particular types of linear and non-linear ODEs, modelling some biochemical systems of interest in biology (for instance, some time-dependent versions of the classical Hopfield model of kinetic proofreading). We also analyse some of the properties of the renewal equations that we are interested in, as the long-time behaviour of their solution. Furthermore, we prove that the kernels characterising the renewal equations derived by biochemical system with reactions that satisfy the detail balance condition belong to the class of completely monotone functions.