# A mathematical framework for analysing particle flow in a network with multiple pools

## TL;DR

This paper introduces a mathematical model to study how particles flow through networks with multiple pools, showing how changes in one part can affect others in unexpected ways.

## Contribution

The paper introduces a new network model of ribosome flow models with multiple pools and demonstrates non-trivial effects of resource sharing.

## Key findings

- The network model always reaches a steady state.
- Increasing transition rates can simultaneously increase and decrease output rates in different parts of the network.
- The model provides insights into particle sharing dynamics in multi-pool networks.

## Abstract

In many real-world systems, the entry rate of particles into a lane is affected by the occupancy of nearby pools. For instance, in biological networks, the concentration of molecules on the side of a membrane affects the entry of particles through the membrane. To understand the behaviour of such networks, we develop a network model of ribosome flow models (RFMs) having multiple pools where each RFM captures the dynamics of particle flow in a lane and competes for the finite resources present at the nearby pool. We study a ribosome flow model network with two pools (RFMNTP) and show that the network always admits a steady state. We then analyse the behaviour of the RFMNTP with respect to modifying the transition rate through a theoretical framework. Simulations of the RFMNTP demonstrate a counterintuitive result. For example, increasing any of the transition rates in the presence of a slow site in an RFM can increase the output rate of some RFMs and decrease the output rate of the other RFMs simultaneously. This suggests that the role of local sharing of particles incorporated is non-trivial. Finally, we illustrate how these results can provide insights into studying a network with multiple pools.