ss_branch
The model is defined as a so called S-system. 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 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.
The model was introduced by Voit (2000) and represents a branched pathway with four dependent variables, X1...X4, and one input variable X5.
The parameter values are:
| i | αi | gi1 | gi2 | gi3 | gi4 | gi5 | βi | hi1 | hi2 | hi3 | hi4 | hi5 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 12 | -0.8 | 1 | 10 | 0.5 | |||||||
| 2 | 8 | 0.5 | 3 | 0.75 | ||||||||
| 3 | 3 | 0.75 | 5 | 0.5 | 0.2 | |||||||
| 4 | 2 | 0.5 | 6 | 0.8 | ||||||||
| 5 | Input | |||||||||||
An empty element corresponds to 0.0. Each row corresponds to one ODE according to Eq. 1, e.g. the first row gives X1'(t) = 12*X3(t)-0.8 - 10*X1(t))0.5.
In most of the test problems, the input variable X5 is not explicitly considered, and hence, a reduced S-system can be used:
| i | αi | gi1 | gi2 | gi3 | gi4 | βi | hi1 | hi2 | hi3 | hi4 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 12 | -0.8 | 10 | 0.5 | ||||||
| 2 | 8 | 0.5 | 3 | 0.75 | ||||||
| 3 | 3 | 0.75 | 5 | 0.5 | 0.2 | |||||
| 4 | 2 | 0.5 | 6 | 0.8 |
The system specification in the same format as the problem: ss_branch and ss_branch_5variables.
The system specification in SBML format: ss_branch.xml.
A simple Matlab script for simulating the system is given in ss_branch.m.
About the problems
ss_branch1 is presented in Voit and Almeida (2004).
ss_branch2 is preseneted in Marino and Voit (2006). Note that the hii's are non-zero initially, but that they are not constrained to positive values. They can hence be assigned zero along the identification procedure.
ss_branch3 is presented in Tucker and Moulton (2006).
ss_branch4 is presented in Kutalik et al.(2007).
ss_branch5 is presented in Kutalik et al. (2007). The same 4 experiments defined in ss_branch4 is used and Gaussian noise with a 2.5% standard deviation relative to the particular experimental value is added. Note that Kutalik et al. used 5% Gaussian noise but considered 4 replicates of each experiment. This corresponds to 2.5% noise and one replicate of each experiment.
ss_branch6 is presented in Gonzalez et al. (2007). Note that α1=20 in this problem.
pe_ss_branch4 is the same as ss_branch4 but with a fixed structure, i.e. a parameter estimation problem.
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
Kutalik,Z., Tucker,W., Moulton,V. (2007) S-system parameter estimation for noisy metabolic profiles using newton-flow analysis. IET Syst Biol., 1, 174-80. PMID:17591176
Marino,S., Voit,E.O. (2006) An automated procedure for the extraction of metabolic network information from time series data. J Bioinform Comput Biol., 4, 665-91. PMID:16960969
Savageau,M.A. (1976) Biochemical systems analysis: a study of function and design in molecular biology (Addison-Wesley, Reading, Mass).
Tucker,W.,Moulton,V. (2006) Parameter reconstruction for biochemical networks using interval analysis, Reliable Computing, 12, 389-402. http://www.springerlink.com/content/r5252637515v1qq3/
Voit,E.O. (2000) Computational analysis of biochemical systems. A practical guide for biochemists and molecular biologists. Cambridge University Press, Cambridge, 176-184.
Voit,E.O.,Almeida,J. (2004) Decoupling dynamical systems for pathway identification from metabolic profiles. Bioinformatics, 20, 1670-81. PMID:14988125