# jireva/mexmodel

A model of Methylobacterium extorquens central carbon and energy metabolism
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# Methylobacterium extorquens central carbon and energy metabolism

Last updated: 2018-10-19

author: José Rojas Echenique

## Introduction

This document describes a mathematical model of the central carbon and energy metabolism of the bacterium Methylobacterium extorquens.

The model takes, as inputs, the initial concentrations of metabolites and the kinetic parameters of each enzyme and predicts the overall rate at which the enzymes in the metabolic network convert substrate to biomass. We express the model as a system of ordinary differential equations:

\dot x = f(x | \vartheta )


where

• f is the set of coupled kinetic rate laws that represents the metabolic network,
• x is a vector of the concentrations of each internal metabolite, and
• \vartheta is the prior distribution of a set of unknown metabolic parameters.

For example, the coupled kinetic rate law that describes \dot {x}_\text{CH2O}, the change in formaldehyde over time:

\dot {x}_\text{CH2O} = { d [\text{CH}_2\text{OH}] \over dt} = v_\text{MDH} - v_\text{GFA}

v_\text{MDH} = { [\text{MDH}] V_\text{MDH} [\text{CH}_3\text{OH}] \over K_\text{MDH} + [\text{CH}_3\text{OH}]}


consist of the rate law for the methanol dehydrogenase reaction (MDH), minus the rate law for the glutathione dependent formaldehyde activation (GFA) reaction.

The system is open: there are input metabolites whose concentrations do not change, and output metabolites which increase in concentration indefinitely. We are interested in the steady state of this system where only the output metabolites have a non-zero rate of change.

Some of the inputs to the model are wholly unknown because they have not or cannot be measured. Others are merely uncertain. We attempt to account for this uncertainty and for the unavoidable uncertainty of the output by embeding our mechanistic description of metabolism in a Bayesian inference context. The paramters to the model are random variables who's posterior distributions can be infered from data.

The metabolic control data y that we can measure is a collection of either

• the concentrations of important metabolites like ATP and NADH under different physiological conditions,

x_\text{ATP}

• or the growth rate as a function enzyme expression

\vartheta _\text{MDH} \mapsto  \dot x_\text{biomass}


We assume that the data is a function of the steady state concentrations of metabolites in the model:

y = g(x^*)


To quantify the fit between the model and the data we use the sum of the squared differences between the true data and the simulated data \sum (y-g(x^*))². We say that the model matches the data when this metric is below a threshold value \varepsilon .

To sample the posterior distribution of parameters \vartheta |y we will use the algorithm of approximate Bayesian computation:

1. draw a candidate \vartheta ' from \vartheta
2. generate x^*_{\vartheta '} by integrating f|\vartheta ' to a steady state
3. if g(x^*_{\vartheta '}) matches y, keep \vartheta ' as a sample from the distribution \vartheta |y
4. repeat

One goal of the model is to use data collected from experiments to learn about the distributions of the enzyme kinetic parameters. A second, perhaps more important, goal for the model is to explore how flux through this metabolic network might evolve.