Rate Equation - Stoichiometric Reaction Networks

Stoichiometric Reaction Networks

The most general description of a chemical reaction network considers a number of distinct chemical species reacting via reactions. The chemical equation of the -th reaction can then be written in the generic form

s_{1j} X_1 + s_{2j} X_2 \ldots + s_{Nj} X_{N} \xrightarrow{k_j} \ r_{1j} X_{1} + \ r_{2j} X_{2} + \ldots + r_{Nj} X_{N},

which is often written in the equivalent form

\sum_{i=1}^{N} s_{ij} X_i \xrightarrow{k_j} \sum_{i=1}^{N}\ r_{ij} X_{i}.

Here

is the reaction index running from 1 to ,
denotes the -th chemical species,
is the rate constant of the -th reaction and
and are the stoichiometric coefficients of reactants and products, respectively.

The rate of such reaction can be inferred by the law of mass action

f_j= k_j \prod_{z=1}^N ^{s_{zj}}

which denotes the flux of molecules per unit time and unit volume. Here is the vector of concentrations. Note that this definition includes the elementary reactions:

zero-order reactions
for which for all ,
first-order reactions
for which for a single ,
second-order reactions
for which for exactly two, i.e, a bimolecular reaction, or for a single, i.e., a dimerization reaction.

Each of which are discussed in detail below. One can define the stoichiometric matrix

denoting the net extend of molecules of in reaction . The reaction rate equations can then be written in the general form

\frac{d }{dt} =\sum_{j=1}^{R} S_{ij} f_j.

Note that this is the product of the stochiometric matrix and the vector of reaction rate functions. Particular simple solutions exist in equilibrium, for systems composed of merely reversible reactions. In this case the rate of the forward and backward reactions are equal, a principle called detailed balance. Note that detailed balance is a property of the stoichiometric matrix alone and does not depend on the particular form of the rate functions . All other cases where detailed balance is violated are commonly studied by flux balance analysis which has been developed to understand metabolic pathways.

Read more about this topic:  Rate Equation

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