## Abstract

Metabolic flux analysis using ^{13}C-tracer experiments is an important tool in metabolic engineering since intracellular fluxes are non-measurable quantities *in vivo*. Current metabolic flux analysis approaches are fully based on stoichiometric constraints and carbon atom balances, where the over-determined system is iteratively solved by a parameter estimation approach. However, the unavoidable measurement noises involved in the fractional enrichment data obtained by ^{13}C-enrichment experiment and the possible existence of unknown pathways prevent a simple parameter estimation method for intracellular flux quantification. The MCMC (Markov chain–Monte Carlo) method, which obtains intracellular flux distributions through delicately constructed Markov chains, is shown to be an effective approach for deep understanding of the intracellular metabolic network. Its application is illustrated through the simulation of an example metabolic network.

^{13}C tracer- intracellular flux
- least squares estimation (LSE)
- Markov chain–Monte Carlo (MCMC) method
- metabolic flux distribution
- stochastic behaviour

## Introduction

Metabolic engineering is the targeted and purposeful alteration of metabolic pathways found in an organism in order to better understand and use cellular pathways for chemical transformation, energy transduction and supramolecular assembly. Hence, it is necessary to study the behaviour of fluxes, i.e. reaction rates, among the metabolic systems. The quantification of all intracellular metabolic fluxes in a given model of cellular metabolism, i.e. MFA (metabolic flux analysis), has long been recognized as an important approach in metabolic network analysis [1].

The common approach for MFA using ^{13}C-tracer experiments is by feeding a substrate with a known labelling state into the system during stationary state. The labelling states of intracellular metabolites can be measured by using NMR and/or GC–MS. When steady state is reached, the stoichiometry balance and carbon atom balances often present an over-determined system whose solution is expected through iterative algorithms. However, the stochastic behaviour from gene expression levels, evident through experimental studies in recent years [2], implies the stochastic behaviour underlying intracellular fluxes. Therefore a stochastic model is more appropriate for flux distribution analysis. In the present paper, we propose to use the MCMC (Markov chain–Monte Carlo) method for flux distribution analysis.

## MCMC method for MFA

In a stationary state, if we use v and ṽ to denote the system extracellular and intracellular fluxes respectively, the stoichiometric equations can then be built upon the structure of a metabolic system based on the knowledge that the fluxes flowing out of a metabolic pool equal the fluxes entering it, so that (1) Similarly, if the labelling states of the extracellular and intracellular metabolites are denoted by x and respectively, the carbon balance equations can be given by (2) or (3) Note that the matrix Γ is normally a non-singular square matrix, except in the circumstance where the carbon atom network becomes disconnected by virtue of vanishing fluxes [3]. If the labelling measurements are y, then (4) The matrix H is the correlating matrix between the measurements and the system intracellular fractional enrichments, denoting the available measurements from experiments.

The above flux quantification problem can be solved by the LSE [LS (least squares) estimation] approach, if we replace x with its measurement y when all labelling measurements are available. However, in practical systems, there are only partial measurements available which prevents the simple LS approach. Considering the uncertainties and possible disturbances in the system, the above equations are transformed to the formats given below:
(5)
and the output model, eqn (4), is changed to a more general format:
(8)
where w, u, m and n are the noise models in an examined system. Assuming that w, u, m and n are all truncated Gaussian noise models with mean 0 and variances Σ_{w}, Σ_{u}, Σ_{m} and Σ_{n} respectively, then the full conditional distribution of v and x will be:
(9)
Here *N* represents the truncated Gaussian distribution. Based on the full conditional distribution of both x and v, we utilize the Metropolis and Gibbs algorithm in MCMC [4] for the flux distribution and quantification analysis. The procedure of MCMC for MFA is shown in Table 1, where U(0,1) denotes uniform distribution between 0 and 1. The sampled x and v are used for their distribution analysis by discarding the initial samples in the ‘burn-in’ period.

## Simulation results

The example metabolic system, namely the cyclic pentose phosphate pathway, as in [5], was used to test the proposed model. The system is set to be identifiable by fixing the flux Gly1 with its underlying quantity. The MSE (mean squared error) and variance of MSE of the 50 simulation results between 2 and 12% noise levels are shown in Table 2, where LS represents the simulation results from LSE, and MCMC denotes the results from the proposed MCMC approach. It is found that the MSEs of the MCMC method in all noise levels are substantially lower than those of the LS approach. However, the variances of MSE of the MCMC method are much bigger in all noise levels.

## Conclusions

In the present study, the MCMC method is applied to a flux distribution analysis problem. It is illustrated that the proposed algorithm is an effective approach for MFA. Future work will be on the flux distribution analysis under various uncertainties.

## Footnotes

Large-Scale Screening: A Focus Topic at BioScience2005, held at SECC Glasgow, U.K., 17–21 July 2005. Edited by B. Baum (Ludwig Institute, London, U.K.), K. Brindle (Cambridge, U.K.), S. Eaton (Institute of Child Health, London, U.K.) and I. Johnstone (Glasgow, U.K.).

**Abbreviations:**
LS, least squares;
LSE, LS estimation;
MCMC, Markov chain–Monte Carlo;
MFA, metabolic flux analysis;
MSE, mean squared error

- © 2005 The Biochemical Society