The biochemical systems equation is a compact equation of nonlinear differential equations for describing a kinetic model for any network of coupled biochemical reactions and transport processes.
The notation for the dependent variable x varies among authors.
For example, some authors use s, indicating species.
[2] x is used here to match the state space notation used in control theory but either notation is acceptable.
the number of biochemical reactions.
In constraint-based modeling the symbol
However in biochemical dynamic modeling[3] and sensitivity analysis,
In the chemistry domain, the symbol used for the stoichiometry matrix is highly variable though the symbols S and N have been used in the past.
is an n-dimensional column vector of reaction rates, and
is a p-dimensional column vector of parameters.
are fixed species to ensure the system is open.
The system equation can be written as:[1][6] So that:
and parameters, p. In the example, these might be simple mass-action rate laws such as
The particular laws chosen will depend on the specific system under study.
Assuming mass-action kinetics, the above equation can be written in complete form as:
The system equation can be analyzed by looking at the linear response of the equation around the steady-state with respect to the parameter
[7] At steady-state, the system equation is set to zero and given by:
Differentiating the equation with respect to
This derivation assumes that the stoichiometry matrix has full rank.
If this is not the case, then the inverse won't exist.
For example, consider the same problem from the previous section of a linear chain.
is the unscaled elasticity matrix: In this specific problem there are 3 species (
The matrix, therefore, will contain the following entries: The parameter matrix depends on which parameters are considered.
In Metabolic control analysis, a common set of parameters are the enzyme activities.
For the sake of argument, we can equate the rate constants with the enzyme activity parameters.
is the unscaled elasticity matrix with respect to the parameters.
Each expression in the matrix describes how a given parameter influences the steady-state concentration of a given species.
Note that this is the unscaled derivative.
It is often the case that the derivative is scaled by the parameter and concentration to eliminate units as well as turn the measure into a relative change.
The biochemical systems equation makes two key assumptions: