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
Xi'(t) = αi ∏j=1..n Xj(t)gij - βi ∏j=1..n Xj(t)hij
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.
A gene expression time series data set of mice is presented in Barrett et al. (2005) and is available on http://www.ncbi.nlm.nih.gov/ (search for data set GDS404). The data consist of 12488 genes.
Seven genes form this data set is used as described in Daisuke and Horton 2006. The clone ID of the seven genes are 101458at, 92257at, 93619at, 93694at, 94420fat, 94421rat and 97724at. They are Wee1 (X1), Clock (X2), Per1 (X3), Per2 (X4), Cry1 (X5), Cry2 (X6) and Cry2 (X7), respectively. Wee1 regulates cell cycle (suppress procession from G2 to M), and the others regulate the circadian clock.
Data points are sampled every 4 hours (0, 8, 12, ..., 44 hours). Hence, the total number of time points in the benchmark problem is 12. At t=0, a replicate is sampled, all for all other points one repetition is considered.
Since there is mostly only one repetition available, standard deviations can not be estimated from data but are assumed to be proportional to the measured value. The proportional constant was chosen as the median proportional constant obtained at the initial time-point for the seven variables.
ss_clock1 is presented in Daisuke and Horton (2006). No prior structure is assumed.
ss_clock2 is similar to ss_clock1 but here a prior structure is assumed in form of non-zero hii's for all i.
Barrett,T.,Suzek,T.O., Troup,D.B., Wilhite,S.E., Ngau,W.C., Ledoux,P.,Rudnev,D., Lash,A.E., Fujibuchi,W., Edgar,R. (2005) NCBI GEO: mining millions of expression profiles--database and tools. Nucleic Acids Res., 33,D562-6. PMID:15608262
Daisuke,T.,Horton,P. (2006) Inference of scale-free networks from gene expression time series. J Bioinform Comput Biol., 4, 503-14. PMID:16819798
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.