# Murray's law for discrete and continuum models of biological networks

## TL;DR

This paper validates Murray's law, a key scaling relation for biological network conductivities, across discrete and continuum models, including generalized and steady-state cases, enhancing understanding of biological transport systems.

## Contribution

It generalizes Murray's law for discrete networks with multiple branches and metabolic coefficients, and proves its validity for continuum models derived from discrete limits and phenomenological assumptions.

## Key findings

- Generalized 3/4-law for discrete networks with multiple branches
- Proved Murray's law for continuum models from discrete limits
- Validated Murray's law for stable steady states in phenomenological models

## Abstract

We demonstrate the validity of Murray's law, which represents a scaling relation for branch conductivities in a transportation network, for discrete and continuum models of biological networks. We first consider discrete networks with general metabolic coefficient and multiple branching nodes and derive a generalization of the classical 3/4-law. Next we prove an analogue of the discrete Murray's law for the continuum system obtained in the continuum limit of the discrete model on a rectangular mesh. Finally, we consider a continuum model derived from phenomenological considerations and show the validity of the Murray's law for its linearly stable steady states.