metabol
This test system models a metabolic pathway considered by Arkin and Ross (1995) and in Gennemark and Wedelin (2007).
The system has two input variables X1 and X2 and five dependent variables X3-X7. The reactions v1-v6 follow Michaelis-Menten kinetics (non-competitive inhibition) and are catalyzed by different enzymes which also are present at constant levels. The system can be described by the following equations:
X3'(t) = - v1 - v2 + v3 + v4 X4'(t) = v1 - v3 X5'(t) = v2 - v4 X6'(t) = -X7'(t) = v5 - v6
The kinetic equations are specified according to
v1 = X3(t) Vmax1 / ( (X3(t) + KD1) (1 + X1(t)/KI1) )
v2 = X3(t) Vmax2 / ( (X3(t) + KD2) (1 + X2(t)/KI2) )
v3 = X4(t) Vmax3 / (X4(t) + KD3)
v4 = X5(t) Vmax4 / (X5(t) + KD4)
v5 = X7(t) Vmax5 / ( (X7(t) + KD5) (1 + X3(t)/KI3) )
v6 = X6(t) Vmax6 / (X6(t) + KD6)
The following rate constants, taken from Arkin and Ross (1995), were used: Vmax1-2 = 5, Vmax3-4 = Vmax6 = 1, Vmax5 = 10, KD1-6 = 5 and KI1-3 = 1.
Note that the general problem, in which conservation of mass (in this case X3+X4+X5=const and X6+X7=const) is not assumed as prior knowledge.
The system specification in the same format as the problem: metabol.
The system specification in SBML format: metabol.xml.
A simple Matlab script for simulating the system is given in metabol.m.
About the problems
The problems metabol1-metabol3 are presenetd in Gennemark and Wedelin (2007).
problem metabol1 considers perfect data.
Problem metabol2 considers the same initial conditions as in metabol1. 21 uniformly distributed data-points are sampled per variable and experiment between t=0 and t=150. Gaussian noise with 10% standard deviation relative to the particular experimental value was added. At t=0, the mean of five repetitions was used.
Problem metabol3 is similar to metabol2 but with 20% Gaussian noise instead of 10%.
References
Arkin,A.P., Ross,J. (1995) Statistical construction of chemical reaction mechanisms from measured time-series. J. Phys. Chem. 99, 970-979. J. Phys. Chem.
Gennemark,P.,Wedelin,D. (2007) Efficient algorithms for ordinary differential equation model identification of biological systems. IET Syst Biol., 1, 120-9. PMID:17441553