# Computing the projected reachable set of switched affine systems: an application to systems biology

## TL;DR

This paper introduces a novel method for approximating the reachable set of switched affine systems, enabling better analysis of biochemical networks and optimizing external signals to reach desired states.

## Contribution

It presents a new approach to approximate the reachable set of switched affine systems and demonstrates its application to systems biology for improved accuracy.

## Key findings

- More accurate estimates of protein mean and variance reachable sets.
- Method efficiently computes projections without full set enumeration.
- Validated results with experimental data.

## Abstract

A fundamental question in systems biology is what combinations of mean and variance of the species present in a stochastic biochemical reaction network are attainable by perturbing the system with an external signal. To address this question, we show that the moments evolution in any generic network can be either approximated or, under suitable assumptions, computed exactly as the solution of a switched affine system. Motivated by this application, we propose a new method to approximate the reachable set of switched affine systems. A remarkable feature of our approach is that it allows one to easily compute projections of the reachable set for pairs of moments of interest, without requiring the computation of the full reachable set, which can be prohibitive for large networks. As a second contribution, we also show how to select the external signal in order to maximize the probability of reaching a target set. To illustrate the method we study a renown model of controlled gene expression and we derive estimates of the reachable set, for the protein mean and variance, that are more accurate than those available in the literature and consistent with experimental data.