August 26, 2009 – Herbert Sauro
Stoichiometry
Stoichiometry refers to the molar proportions of reactants and products in a chemical
reaction. Given a hypothetical reaction such as:
3A + 4B −→ 2C + D
with reactants A and B and products C and D, the stoichiometry is indicated by the
number of participating reactant and product molecules. Thus the stoichiometry for A is
three, for B, four, for C, two and for D, one.
When describing multiple reactions in a system, it is convenient to represent the stoichiometries in a compact form called the stoichiometry matrix, N . This matrix is a m row
by n column matrix where m is the number reactants and n the number reactions.
For example, consider the simple chain of reactions which has five molecular species and
four reactions. The four reactions are labeled, v1 to v4 .
S1 −→ S2 −→ S3 −→ S4 −→ S5
The stoichiometry matrix for this simple system is given by:


N= 



v1 v2 v3 v4
−1
0
0
0
1 −1
0
0
0
1 −1
0
0
0
1 −1
0
0
0
1

S1
 S2

 S3

 S4
S5
Entries in the stoichiometry matrix are computed as follows. Given a species Si and reaction vj , the corresponding entry in the stoichiometry matrix is given by the total stoichiometry for Si on the product side minus the total stoichiometry of Si on the reactant
side. Thus, considering species A in reaction v1 , we note that the total stoichiometry on
the product size is zero (no A molecules are formed on the product side) and the total
stoichiometry on the reactant side is +1. Subtracting one from the other (0 − 1) we obtain
−1 which is entered into the stoichiometry matrix.
For a more complex system such as:
3A −→ 2A + B
the stoichiometry entry for A is (2 − 3) that is −1, since the stoichiometry for A on the
product side is 2 and on the reactant side is 3.
Constructing a Model
To construct a model that describes the time evolution of a biochemical network, we need
one more piece of information in addition to the stoichiometries. This piece of information includes the rates of reaction. For the simple linear biochemical chain shown in the
previous section, the reaction rates are given by the symbols, v1 , v2 , v3 and v4 . We combine
these into a rate vector, v. We then combine the rate vector and stoichiometry matrix to
give us the differential equations that describe the rates of change of the pathway species.
dS
= Nv
dt
For example:






dS1 /dt
dS2 /dt
dS3 /dt
dS4 /dt
dS5 /dt


 
 
=
 
 
−1
0
0
0
1 −1
0
0
0
1 −1
0
0
0
1 −1
0
0
0
1
yields
dS1
dt
dS2
dt
dS3
dt
dS4
dt
dS5
dt
= −v1
= v1 − v2
= v2 − v3
= v3 − v4
= v4



 v1
  v2 


  v3 

v4