inhosc
This simple inhibitory oscillator presents a basic case for constant nonlinear oscillations.
The system has a constant input flow for X1 and a constant output flow for X2. As can be seen in the figure, X1 exerts an inhibitory influence to the input of the X2, and the X2 produces the same effect on the output of the X1. The system can be modelled as follows:
X1'(t) = In - k1/(X2+k2) X2'(t) = k3/(X1+k4) - Out
The following rate constants were used: k1=k2=0.5, k3=k4=0.7,
A simple Matlab script for simulating the system is given in inhosc.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:
inhosc1 is designed to be a simple test problem with perfect data.
inhosc2 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=20.
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.