# Statistical mechanics for metabolic networks during steady-state growth

## TL;DR

This paper extends flux balance analysis to a maximum entropy framework at the single-cell level, accurately modeling E. coli metabolism and predicting flux variability, correlations, and growth rate distributions validated by microfluidic data.

## Contribution

It introduces a maximum entropy model for metabolic fluxes that generalizes FBA, providing more accurate predictions and insights into single-cell metabolic variability and regulation.

## Key findings

- E. coli metabolism is near, but not at, FBA optimality.
- Model accurately predicts flux variability and correlations.
- Scaling relations for growth rate distributions are validated by experiments.

## Abstract

Which properties of metabolic networks can be derived solely from stoichiometric information about the network's constituent reactions? Predictive results have been obtained by Flux Balance Analysis (FBA), by postulating that cells set metabolic fluxes within the allowed stoichiometry so as to maximize their growth. Here, we generalize this framework to single cell level using maximum entropy models from statistical physics. We define and compute, for the core metabolism of Escherichia coli, a joint distribution over all fluxes that yields the experimentally observed growth rate. This solution, containing FBA as a limiting case, provides a better match to the measured fluxes in the wild type and several mutants. We find that E. coli metabolism is close to, but not at, the optimality assumed by FBA. Moreover, our model makes a wide range of predictions: (i) on flux variability, its regulation, and flux correlations across individual cells; (ii) on the relative importance of stoichiometric constraints vs. growth rate optimization; (iii) on quantitative scaling relations for singe-cell growth rate distributions. We validate these scaling predictions using data from individual bacterial cells grown in a microfluidic device at different sub-inhibitory antibiotic concentrations. Under mild dynamical assumptions, fluctuation-response relations further predict the autocorrelation timescale in growth data and growth rate adaptation times following an environmental perturbation.