Stoichiometry#Stoichiometry matrix

The stoichiometric matrix is ​​a matrix representing the stoichiometry of a chemical system in a compact form. It is usually abbreviated with. In general, the columns of the reactions of the system while the rows correspond to the corresponding chemical species. Species of a reaction, which are consumed in total, received a record with a negative sign and species, which are produced in total, received a positive entry at the position of which corresponds to the given reaction and species. The change of the species according to the time is given by where the vector of the reaction rates (also called " flux vector " ) represents. Therefore applicable in a stationary state, i.e., the system is in a dynamic equilibrium. The stoichiometric matrix allows to draw conclusions on the reaction rates of the stationary states. In steady state, then at least apply, ie the set of all such lie in the null space of. This condition is independent of the kinetics of which is the given chemical system is based.

Example

The four reactions

Can be encoded as a matrix as follows:

Wherein the columns of the array after the reaction and the rows correspond to the species. Reaction consumes one unit of species and produces one unit of species (first column). Likewise, a unit of species produced during net a unit is consumed by species (second column ) in response net. In the example, is then obtained, and the system is in a steady state.

As you can see in this example, the original chemical system can be reconstructed not only through knowledge of the stoichiometric matrix. For reaction exist an infinite number of options that produce the same column in:

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

Other properties

Be the set of all real numbers greater than zero and the set of all real numbers greater than or equal to zero. Continue to be the number of responses at the given chemical system. The following amounts of vectors represent stationary states of chemical systems with different boundary conditions:

  • Only reversible reactions, ie reactions which can run in forward and reverse direction :;
  • Only irreversible reactions, ie reactions which can take place only in the forward direction :;
  • A mixture of reversible and irreversible reactions: , wherein the set of indices of the irreversible reactions referred to.

Application

The stoichiometric matrix is a central tool in systems biology. It allows a systematic analysis of the flow vectors of stationary states of a chemical or biological system. In general, in this case the size of the system to be analyzed is almost no limit as their use requires only methods from linear algebra. Methods which have recourse to the stoichiometric matrix, 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 ).

Chemistry Chemical kinetics Index#Mathematics
750137
de