OptMetStrategy: Optimal metabolic strategies for microbial growth.

This is a Matlab implementation of the microbial metabolic model in arXiv.2210.11167 addressing optimal growth strategies under several environmental conditions.

A brief introduction to the theoretical framework

We assume a minimal microbial model suitable for a population of E.Coli in a carbon-limited media, whose full description is provided by two quantities, the expenditure $ϵ(x)$ and the specific uptake $q(x)$ governed by the internal degree of freedom $x$. Under this assumption, the growth rate is given by (see npj Sys Biol Appl 5, 1-9)

$\mu (x,s)=\frac{\varphi }{w+sq(x)+ϵ(x)}$

where $s$ is a stress variable modeling the environment, $\varphi$ and $w$ are the fraction of proteome devoted to constitutively expressed proteins and the proteome share to be allocated to ribosome-affiliated proteins per unit of growth rate respectively. There exists an optimal value $\hat{x}\left(s\right)$ such that the growth rate is maximized. However, being $s$ a random variable distributed according to $p(s)$, cells face a certain degree of uncertainty about the environment and, therefore, the strategy to adopt. We denote as $p(x|s)$ the stochastic rule to choose $x$ for a given $s$ which at optimality is given by

$p(x|s)\propto p(x)e^{\beta \mu (x,s)}$

for any given $\beta$. See arXiv.2210.11167 for a detailed derivation.

Documentation

Static environments

In main_script.m it is possible to select each of the four environmental conditions studied in arXiv.2210.11167 and characterized by a set of stress distribution $p(s)$. By adjusting the variable type_stress one can select one of the following stress distributions:

• uniform

$p(s)\propto \mathbb{I}[s_{min}<s<s_{max}]$

• exp

$p(s)\propto \exp (-s/s_0)\mathbb{I}[s_{min}<s<s_{max}]$

• power_law

$p(s)\propto \frac{1}{s}\mathbb{I}[s_{min}<s<s_{max}]$

• two_states

$p(s)\propto \rho \mathcal{N}(s_0,\sigma )+(1-\rho )\mathcal{N}(s_1,\sigma )$

The script returns the optimal strategies $p(x|s)$, the marginal distribution $p(x)$ as well as the distribution of the growth rate $p\left(\mu \right)$ for several values of the hyper-parameters $\beta \in [0,3\times {10}^2]$ (see text for more details).