Stoichiometry matrix

1. Definition

Stoichiometry matrix (SM) is a systematic arrangement of stoichiometric information from the reactions comprising the model. In a system with m species and n reactions the dimensions of the matrix are mxn. Chemical species are represented by rows and reactions – by columns. The elements of the matrix are corresponding stoichiometric coefficients. The selection of the system boundaries defines the complexity of the SM. When the concentration of a specie is considered fixed, the reaction is removed from the matrix.

The set of equations represented in the matrix together expresses the dynamics of the metabolite concentrations as

dS/dt = N*v,

where N is the matrix, v is the vector of fluxes and S is the vector of metabolite concentrations.

2. Applications

SM implies a steady state assuming that at any given time the concentration of the specie is constant. By using SM it is possible to enumerate all possible steady state flux solutions of a given network.

Personally, I like the fact that the SM is a crossroads of mathematics and biology, equally making sense for a person with a background in biology or information technology or mathematics.

2.1. Network reconstruction

The whole table of reactions encoded in the genome may be represented as SM. If the genes that encode for enzymes and reactions that each enzyme carries out are listed, the resulting table can be converted into the SM.

2.2. Mass conservation analysis

SM contains all information about the reaction network, therefore all necessary data to analyse mass conservation. Such relations can be retrieved from the SM as linear combinations of other rows. The result of removing all rows that are linear combinations of other rows is the reduced matrix which is used by software packages such as COPASI.

2.3. Stoichiometric modelling

In stoichiometric modelling, there are three major approaches are metabolic flux analysis (MFA), flux balance analysis (FBA) and metabolic pathway analysis (MPA). All three work by defining a high-dimension solution space of possible metabolic flux distributions based on the SM specifying system conservation relationships. The difference between the three approaches lies in how metabolic flux distributions are selected from the solution space.

MFA is a traditional approach which relies on extensive experimental data and computes a metabolic flux vector for a particular condition. Experimental data is used to simplify the SM.

FBA identifies only one optimal solution while alternative optimal solutions may exist. It very much depends on the validity of the model.

MPA, unlike the other two methods, can identify all metabolic flux vectors in a network. A finite set of solutions is achieved by additional constraints on the flux space.

References

Smolke C, The Metabolic Pathway Engineering Handbook: Fundamentals, CRC Press, 2009

Trinh T, Wlaschin A, Srienc F, Elementary Mode Analysis: A Useful Metabolic Pathway Analysis Tool for Characterizing Cellular Metabolism, Appl Microbiol Biotechnol, 2009, 81(5), pp 813-826

Wang X, Chen J, Quinn P, Reprogramming Microbial Metabolic Pathways, Springer, 2012

by Evgeny. Also posted on my website