feedf

Test problem that considers two Michaelis-Menten reactions affecting the production of a single variable. Constant rate flow inputs (In1 and In2) specify the steady state flow rate of the system.

The system has two input rates In1 and In2 and four dependent variables X1-X4. It can be described by the following equations: 



X1'(t) = In1 -  k1X1/(X1+k2)  

X2'(t) = In2 -  k3X2/(X2+k4)  

X3'(t) = k5X1/(X1+k6) + k7X2/(X2+k8) - k9X3/(X3+k10)

X4'(t) = k11X3/(X3+k12) - k13X4/(X4+k14)

The following rate constants were used: k1=k3=k5=k7=k9=k11=k13=1, k2=k6=0.5, k4=k8=0.4, k10=k12=k14=0.3

A simple Matlab script for simulating the system is given in feedf.m.

About the model spaces

The model space is defined differently for various variants of this problem:

1. Chemical rate equations is the basic case.

2. S-system (Savageau 1976, Voit 2000). This formalism 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.

3. Generalized Mass Action (GMA; Savageau 1976, Voit 2000). The GMA form uses a generic form of equation i as


Xi'(t) =  ∑j=1..Ni αij ∏k=1..n Xk(t)gijk

where Ni is the number of terms in equation i.

About the problems

Twelve different problems are considered. The following tags define the type of problem, the model space, and how data was generated from the source system.

Type of problem:

(no tag) full system identification

pe only parameter estimation considered (the structure is already defined)

Model space:

gma uses a GMA model space

ss uses a S-System model space

(no tag) uses a model space of linear rates

Data generation:

feedf1 is designed to be a simple test problem with perfect data.

feedf2 adds Gaussian noise with 5% standard deviation relative to each data point

Each problem contains data from several experiments, each with different initial conditions for the variables, and sampled time course data for all variables. In each experiment, each variable was assigned an initial value with a deviation from the steady-state of up to 75% of the steady state value.

For each variable in each experiment, 51 data-points were uniformly sampled between t=0 and t=5.0.

References

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.