This paper presents a mathematical technique for determining reaction stoichiometries. This is achieved purely from analysis of experimental composition data, obtained from batch reactors operating away from steady state, and assumes no a priori information about the stoichiometries or species structures. The method is composed of two stages: initially conservation relations (reaction invariants) are obtained from the process data and secondly the reaction stoichiometry is estimated from the conservation relations.

For isothermal batch reactors conservation relations exist when there are dependencies in the set of species concentrations (i.e. the number of species exceeds the number of linearly independent reactions). These relations provide an alternative description of the network in terms of mass balance relationships and therefore provide information about the process chemistry. They define certain combinations of species concentrations that remain unchanged or combinations of the time derivatives of the species concentrations that add to zero. Thus, L(x-x₀) = c or Lx' = 0 (where L is a matrix of the conservation relations, x is vector of the species concentrations at a specified time instant, x₀ are the initial conditions, x' is a vector of the rate of change of species concentrations at a specified time instant and c is a vector of constants). Therefore the conservation relations may be obtained through calculation of the left null space of a matrix of the process data.

Further, as the matrix of conservation relations also lies in the left null space of the stoichiometric matrix, it is also true that the stoichiometric matrix lies in the right null space of the matrix of conservations. Therefore, unknown reaction stoichiometries may be obtained through calculation of the right null space of the matrix of the identified conservation relations. In this work we calculate the null spaces through Gauss-Jordan elimination which gives a sparse representation of the reaction stoichiometry, however, alternative approaches aimed at calculating a sparse basis for the null space could also be used (see Gilbert et al, 1987).

This work shows how the conservation relations and stoichiometries may be obtained through simple mathematical manipulation of a matrix of the experimental data. Furthermore the presented techniques are extended to handle the presence of measurement error and to exploit possible a priori stoichiometric information. The proposed method compliments many other techniques in the field (e.g. see Hamer, 1989; Bonvin and Rippin, 1990; Fotopoulos et al, 1994; Georgakis and Lin, 2005; Bernard and Bastin, 2005 and Bernard, 2007) and is consistent with the work completed in biochemical systems theory (Palsson, 2006). The method is demonstrated using data obtained from a simulated batch reactor, and it is shown how conservation relations and stoichiometries may be identified.

References

Bonvin, D. and Rippin, D. W. T. (1990) Target factor analysis for the identification of

stoichiometric models, Chemical Engineering Science, 44, 3417-3426.

Bernard, O. (2007) Use Of Modulating Functions For Reaction Network Identification, Proceedings of the 10th International Symposium on Computer Applications in Biotechnology, Cancun.

Bernard, O. and Bastin, G. (2005) Identification of reaction networks for bioprocesses: determination of a partially unknown pseudo-stoichiometric matrix, Bioprocess and Biosystems Engineering, 27, 5, 293 – 301.

Fotopoulos, J., Georgakis, C., and Stenger, H. G. (1994) Structured Target Factor Analysis for the Stoichiometric Modeling of Batch Reactors, Proceedings of the 1994 American Control Conference, 495-499.

Georgakis, C. and Lin, R. (2005) Stoichiometric Modeling of Complex Pharmaceutical Reactions, Proceedings of the Annual AIChE meeting, Cincinnati, OH, November.

Gilbert, J. R., and Heath, M.T. (1987) Computing a Sparse Basis for the Null Space, SIAM J. Algebraic Discrete Methods 8:446–459.

Hamer, J. W. (1989) Stoichiometric interpretation of multi-reaction data: application to fed-batch fermentation, Chemical Engineering Science, 44, 2363-2374.

Palsson, B. O. (2006) Systems Biology: Properties of Reconstructed Networks, Cambridge University Press, New York.