# Stoichiometric Analysis¶

## Preliminaries¶

A network of chemical species and reactions can be described by the by stochiometry matrix . is the net number of species produced or consumed in reaction . The dynamics of the network are described by

where is the vector of species concentrations, is a vector of time independent parameters, and is time.

Each structural conservation, or interchangably, *conserved sum* (e.g. conserved moiety) in
the network coresponds to a lineraly dependent row in the stoichiometry matrix .

If there are conserved sums, then the row rank, of is , and
the stochiometry matrix may first be re-ordered such that the first are linearly
independent, and the remaining rows are linear combinations of the first
rows. The *reduced stochiometry matrix* is then formed from the first
rows of . Finally, may be expressed as a product of the
*link matrix* and the matrix:

The link matrix has the form

where is the identity matrix and is a matrix.

## Full vs. Extended Stoichiometry Matrix¶

The “full” stoichiometric matrix includes any conserved quantities (as opposed to the reduced stoichiometric matrix,
which does not). The *extended* stoichiometric matrix is equal to the full stoichiometric matrix plus additional
rows representing boundary species and sources / sinks. For example, consider the following reaction system:

```
reaction1: => C
reaction2: C =>
reaction3: C =>
reaction4: MI => M
reaction5: M => MI
reaction6: XI => X
reaction7: X => XI
```

The extended stoichiometry matrix for this system is:

```
>>> rr.getExtendedStoichiometryMatrix()
reaction1, reaction2, reaction3, reaction4, reaction5, reaction6, reaction7
C [[ 1, -1, -1, 0, 0, 0, 0],
M [ 0, 0, 0, 1, -1, 0, 0],
X [ 0, 0, 0, 0, 0, 1, -1],
MI [ 0, 0, 0, -1, 1, 0, 0],
XI [ 0, 0, 0, 0, 0, -1, 1],
reaction1_source [ -1, 0, 0, 0, 0, 0, 0],
reaction2_sink [ 0, 1, 0, 0, 0, 0, 0],
reaction3_sink [ 0, 0, 1, 0, 0, 0, 0]]
```

## Methods¶

The following methods are related to the analysis of the stoichiometric matrix.