Stoichiometric matrix

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The stoichiometric matrix is a matrix which represents the stoichiometry of a reaction network in a compact form.

It is usually abbreviated with . Usually the columns correspond to the reactions of the system while the rows correspond to the chemical species . Species of a reaction that are consumed in total receive an entry with a negative sign and species that are produced in total receive a positive entry at the position of which corresponds to the given reaction and species. The change in species with time is then given by where represents the vector of the reaction rates (also called “flow vector”). In a steady state , therefore, applies , d. H. the system is in a steady state .

The stoichiometric matrix allows conclusions to be drawn about the reaction rates of the stationary states. In the steady state, at least the following must apply, i. H. the set of all such lies in the null space of . This condition is independent of the kinetics on which the given chemical system is based.

The rank of the stoichiometric matrix indicates the number of linearly independent reactions.

example

The four reactions

can be coded as a matrix as follows:

the columns corresponding to the reactions and the rows to the species . Reaction consumes a unit of species and produces a unit of species (first column). Likewise, a net unit of species is produced in reaction while a net unit of species is consumed (second column). If is in the example , then the result is and the system is in a stationary state.

As you can see from this example, the original chemical system cannot be reconstructed simply by knowing the stoichiometric matrix. There are infinite possibilities for reaction which produce the same column in :

Thus the stoichiometric matrix contains less information than the original set of reactions.

Other properties

Let the set of all real numbers be greater than zero and the set of all real numbers greater than or equal to zero. Furthermore, let the number of reactions in the given chemical system be. The following sets of vectors represent stationary states of chemical systems with different boundary conditions:

  • only reversible reactions, d. H. Reactions which may proceed in forward and reverse directions: ;
  • only irreversible reactions, d. H. Reactions which can proceed only in the forward direction: ;
  • a mixture of reversible and irreversible reactions:, where the set denotes the indices of the irreversible reactions.

application

The stoichiometric matrix is ​​a central tool in systems biology . It enables a systematic analysis of the flow vectors of stationary states of a chemical or biological system. In general, there is almost no limit to the size of the system to be analyzed, as their use only requires methods from linear algebra. Methods which use the stoichiometric matrix are e.g. For example: FBA ( Flux Balance Analysis ), FCA ( Flux Coupling Analysis ), FVA ( Flux Variability Analysis ), the concept of EFMs ( Elementary Flux Modes ) and similar methods such as Extreme Currents and Extreme Pathways , DFBA ( Dynamic FBA ) and CRNT ( Chemical Reaction Network Theory ).

Individual evidence

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