Traditional approaches for metabolic network exploration are mainly based on physicochemical kinetics laws and principles. However,the difficulty for monitoring nutritional changes in metabolic biosystem neccecitates a more reductionist form of information especially available for computational processing. In 1992, Savinell and Palsson[1,2]proposed flux balance analysis(FBA) to simulate microbial metabolism and in 1993 FBA was applied in exploration of E.coli System[3,4,5].Thereafer, Palsson and his parters extend the uses of FBA even to genetic level[6,7].

      Here are the mathematical description of FBA[8]. Firstly, FBA assumes that metabolic networks will reach homeostasis constrained by stoichiometry without knowing material changes all along the process. The values in this stoichiometric matrix are the stoichiometric coefficients of each reaction in the system as the set of constraints for the optimization. The mathematical description can be interpreted in the simple example with graphs following.

      In the simple biosystem illustrated above, the compounds served as the nodes are connected by the reactions for conversion among compounds. The increase of Compound A is due to the converstion from Compound C and A, while the decrease of Compound A is the result of the conversion from Compound A to C. The graph below shows the overall change of the amount per time unit of Compound A, B, C.

      Since the whole system will reach homeostasis, which means a state that compound in the system will be constant, in other words, the flux-in and flux-out of the node(the compound) will be equal. That is why we call this method ‘Flux Balance Analysis’.

      Secondly, the results above can be translated into a stoichiometric matrix. In Matrix S below.

    Each col of Matrix S represents stoichiometric coefficients the related to each compound and each row of Matrix S refers to coefficients of each reaction according to the three equations above. The plus and minus signs are determined by the direction of reations according to the compound itself.

      While in the vector above, v1 and v2 represents the reaction rate namely fluxes of the two reactions in this sample system.

So you can see the multiplication of Matrix S and the vector can return to those three original equations above. As has been explained, the multiplication will be the zero vector.

      In the general case we can write:

      Thirdly,with stoichiometry prepared and the objective function(the biomass function) determined, linear programming can be performed for optimization.

      In many cases, constraints are set upon the values of fluxes based on some thermodynamic conditions.

(All the graphs are from Wikipedia.)

Reference:

 1.Savinell JM, Palsson BO: Optimal selection of metabolic fluxes for in vivo measurement. I. Development of mathematical methods. J Theor Biol 1992, 155:201-214.

2. Savinell JM, Palsson BO: Optimal selection of metabolic fluxes for in vivo measurement. II. Application to Escherichia coli and hybridoma cell metabolism. J Theor Biol 1992,155:215-242.

3.Varma A, Palsson BO: Metabolic capabilities of Escherichia coli:I. Synthesis of biosynthetic precursors and cofactors. J TheorBiol 1993, 165:477-502.

4. Varma A, Palsson BO: Metabolic capabilities of Escherichia coli:II. Optimal growth patterns. J Theor Biol 1993,165:503-522.

5. Varma A, Palsson BO: Stoichiometric flux balance models quantitatively predict growth and metabolic by-product secretion in wild-type Escherichia coli W3110. Appl Environ Microbiol 1994, 60:3724-3731.

6. Edwards JS, Palsson BO: The Escherichia coli MG1655 in silico metabolic genotype: its definition, characteristics, and capabilities. Proc Natl Acad Sci USA 2000, 97:5528-5533.

7.Edwards JS, Palsson BO: Robustness analysis of the Escherichia coli metabolic network. Biotechnol Prog 2000, 16:927-939.

8. Kenneth J Kauffman, Purusharth Prakash and Jeremy S Edwards: Advances in flux balance analysis, Current Opinion in Biotechnology 2003, 14:491–496