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 equation is expressed in the following form:
formula_1
The notation for the dependent variable x varies among authors. For example, some authors use s, indicating species. x is used here to match the state space notation used in control theory but either notation is acceptable.
formula_2 is the stoichiometry matrix which is an formula_3 by formula_4 matrix of stoichiometry coefficient. formula_3 is the number of species and formula_4 the number of biochemical reactions. The notation for formula_2 is also variable. In constraint-based modeling the symbol formula_2 tends to be used to indicate 'stoichiometry'. However in biochemical dynamic modeling and sensitivity analysis, formula_2 tends to be in more common use to indicate 'number'. 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. 
formula_10 is an n-dimensional column vector of reaction rates, and formula_11 is a p-dimensional column vector of parameters.
Example.
Given the biochemical network:
formula_12
where formula_13 and formula_14 are fixed species to ensure the system is open. The system equation can be written as:
 formula_15 formula_16
So that:
formula_17 formula_18
The elements of the rate vector will be rate equations that are functions of one or more species formula_19 and parameters, p. In the example, these might be simple mass-action rate laws such as formula_20 where formula_21 is the rate constant parameter. 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:
formula_17 formula_23
Analysis.
The system equation can be analyzed by looking at the linear response of the equation around the steady-state with respect to the parameter formula_24. At steady-state, the system equation is set to zero and given by:
formula_25
Differentiating the equation with respect to formula_26 and rearranging gives:
formula_27
This derivation assumes that the stoichiometry matrix has full rank. If this is not the case, then the inverse won't exist.
Example.
For example, consider the same problem from the previous section of a linear chain. The matrix formula_28 is the unscaled elasticity matrix:
 formula_29
In this specific problem there are 3 species (formula_30) and 4 reaction steps (formula_31), the elasticity matrix is therefore a formula_32 matrix. However, a number of entries in the matrix will be zero. For example formula_33 will be zero since formula_34 has no effect on formula_35. The matrix, therefore, will contain the following entries:
 formula_36
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. We also assume that each enzyme, formula_37, only can affect its own step and no other. The matrix formula_38 is the unscaled elasticity matrix with respect to the parameters. Since there are 4 reaction steps and 4 corresponding parameters, the matrix will be a 4 by 4 matrix. Since each parameter only affects one reaction, the matrix will be a diagonal matrix:
 formula_39
Since there are 3 species and 4 reactions, the resulting matrix formula_40 will be a 3 by 4 matrix
formula_41
formula_42
formula_43
formula_44
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.
Assumptions.
The biochemical systems equation makes two key assumptions:
References.
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