Benchmarks for identification of ODEs from time series data

ss_cadBA

The model space is defined as a so called S-system model. The S-system formalism (Savageau 1976, Voit 2000) is based on approximating kinetic laws with multivariate power-law functions. A model consists of n non-linear ODEs and the generic form of equation i reads


X_i'(t) = α_i ∏_j=1..n X_j(t)^g_ij - β_i ∏_j=1..n X_j(t)^h_ij

where X is a vector (length n) of dependent variables, α and β are vectors (length n) of non-negative rate constants and g and h are matrices (n*n) of kinetic orders, that can be negative as well as positive.

We use data on the cadBA network in E. coli presented in Gonzalez et al. (2007). They variables are CadA (X₁), cadBA (X₂), Cadaverine (X₃), Lysine (X₄) and the pH (X₅). In Gonzalez et al. (2007), pH (X₅) is considered an input variable and values are interpolated (by some splining method) from data in the simulations. Here, in order to unambiguously define the benchmark problems, we let pH (X₅) be a dependent variable for which the ODE is fixed as


X₅'(t) = α₅  - β₅ X₅(t)^h₅₅

which allows for a good fit to the considered time-series. Standard deviation is assumed to be 5% of measured values for cadBA, Cadaverine and Lysine and pH, and 20% for CadA.

About the problems

ss_cadBA1 is presented in Gonzalez et al. (2007). No prior structure is assumed. In Gonzalez et al. (2007), the kinetic orders are said to be bound in [0,8], although the solution includes small negative kinetic orders. In the benchmark problem, we let the kinteic orders be bound in [-1,8] in order to allow similar models.

ss_cadBA2 is similar to ss_cadBA1 but with a smaller model space.

References

Gonzalez,O.R., Küper,C., Jung,K., Naval,P.C.Jr., Mendoza,E. (2007) Parameter estimation using Simulated Annealing for S-system models of biochemical networks. Bioinformatics, 23, 480-6. PMID:17038344

Savageau,M.A. (1976) Biochemical systems analysis: a study of function and design in molecular biology (Addison-Wesley, Reading, Mass).

Voit,E.O. (2000) Computational analysis of biochemical systems. A practical guide for biochemists and molecular biologists. Cambridge University Press, Cambridge, 176-184.