This text was written by Karin Sasaki: http://intranet.embl.de/training/eicat/members/index.php?s_personId=CP-60020237
Chemical reaction network theory is an area of applied mathematics that attempts to model the behaviour of real world chemical systems. Mathematical modelling of chemical reaction networks usually focuses on what happens to the concentrations of the various chemicals involved, as time passes. (Other factors that affect the reactions, apart from concentration, are, for example, temperature, pressure and physical state of reactants.)
A chemical reaction network comprises a set of reactants, a set of products and a set of reactions. For example the pair of combustion reactions below forms a reaction network:
Let [H_2] represent the concentration of H2, [H2O] that of H2O, etc. Since all of these concentrations will not in general remain constant, they can be written as a function of time:
These variables can then be combined into a vector
and their evolution with time can be written
(Note: We will eventually drop the squared brackets that indicate concentration.)
The number of molecules of each reactant used up each time a reaction occurs is constant, as is the number of molecules produced of each product. These numbers are referred to as the stoichiometry of the reaction.
The reaction is assumed to happen a fixed number of times in unit time; that is, the rate of the reaction is constant and this reaction constant needs to be calculated, either experimentally or computationally (see Part IV).
It turns out that the equation representing the chemical reaction network can be rewritten as:
Where each column of the constant matrix Γ represents the net stoichiometry of each reaction (i.e. column 1 for reaction 1, column 2 for reaction 2, etc) and V is a column vector where each input represents the reaction rate of each reaction, which we calculate given certain assumptions, using kinetic laws. Below we see why this is the case, using the assumption of Mass Action kinetics.
Mass action kinetics assumes a linear dependence of the rate of the reaction on substrates concentrations. So for example, for the reaction A + B -> C, the rate of the reaction would be v = k*A*B, where parameter k is the reaction constant. The law of mass action makes sense in systems where the numbers of molecules are relatively similar to each other, in magnitude.
Now, assuming mass action for the pair of combustion reactions, we can calculate the rate of the reactions using a diagram such as the one below:
Mathematically, we can write this as:
and given the assumption of mass action we have that
If we do this calculation for both reactions, we will soon realise that we can write V as:
and we can write
as follows:
So the reaction rates of vector V are calculated depending on the kinetic law that is assumed, or on the system that you have. There are other kinetic laws, such as Michaelis-Menten and the Hill equation and there are systems where these kinetic laws actually do not apply. To learn about how to apply Michaelis-Menten and the Hill equation to mathematical modelling, we refer you to reference 4, section 3.8. If you would like to learn more in general about reaction kinetics, we recommend you look at a chemistry reference book, such as reference 5. chapter 3.
This text was written by Karin Sasaki: http://intranet.embl.de/training/eicat/members/index.php?s_personId=CP-60020237