Equations, stoichiometric biochemical

A considerable improvement over purely graph-based approaches is the analysis of metabolic networks in terms of their stoichiometric matrix. Stoichiometric analysis has a long history in chemical and biochemical sciences [59 62], considerably pre-dating the recent interest in the topology of large-scale cellular networks. In particular, the stoichiometry of a metabolic network is often available, even when detailed information about kinetic parameters or rate equations is lacking. Exploiting the flux balance equation, stoichiometric analysis makes explicit use of the specific structural properties of metabolic networks and allows us to put constraints on the functional capabilities of metabolic networks [61,63 69]. 

This shows that the solution s to the system of linear equations represented by equation 6.1-1 is made up of the stoichiometric numbers s that give the number of times the various biochemical reactions have to occur to accomplish the net reaction. Equation 6.1-1 is conveniently written in matrix notation as... 

Glycolysis involves 10 biochemical reactions and 16 reactants. Water is not counted as a reactant in writing the stoichiometric number matrix or the conservation matrix for reasons described in Section 6.3. Thus there are six components because C = N — R = 16 — 10 = 6. From a chemical standpoint this is a surprise because the reactants involve only C, H, O, N, and P. Since H and O are not conserved at specified pH in dilute aqueous solution, there are only three conservation equations based on elements. Thus three additional conservation relations arise from the mechanisms of the enzyme-catalyzed reactions in glycolysis. Some of these conservation relations are discussed in Alberty (1992a). At specified pH in dilute aqueous solutions the reactions in glycolysis are... 

The Vj are the stoichiometric numbers of reactants in the biochemical equation (positive for reactants on the right side of the equation and negative for reactants on the left side). The prime is needed on the stoichiometric numbers to distinguish them from the stoichiomeric numbers in the underlying chemical reactions. The standard transformed enthalpy of reaction Ar H(heat of reaction) is related to the standard transformed enthalpies of formation Af //,of the reactants by... 

A is the apparent conservation matrix, and y is the stoichiometric number matrix for the biochemical reaction system. This equation makes it possible to calculate a basis for the stoichiometric number matrix from the apparent conservation matrix by use of NullSpace. A has the dimensions C xA where C is the apparent number of components (C - 1) and N is the number of reactants (sums of species), y has the dimensions N R Note that N = C + R where R is the number of independent biochemical reactions. Equation 7.2-5 makes it possible to obtain a basis for the apparent stoichiometric number matrix by use of NullSpace. 

A basis for the stoichiometric number matrix for the biochemical reaction in the system being discussed can be obtained by applying equation 7.2-5 to conmatS. 

Many biochemical reactions, perhaps most, have exactly this stoichiometric number matrix. A basis for the apparent conservation matrix can be obtained by use of equation 7.2-6. 

The row reduced form shows that this reaction system involves two biochemical reactions. But there is a second way to obtain a conservation matrix, and that is by use of equation 7.2-6. The stoichiometric number matrix for reaction 7.4-4 is... 

The associated free energy produced or consumed in each reaction is captured in units of adenosine triphosphate (ATP). The ATP stoichiometry is usually obtained from biochemical tables since the energy has to be also balanced for the cell. Thus for Eq. (7-148) the stoichiometric ATP requirement to convert one C-mole of glucose to one C-mole of glycerol is 5. In calculations of the carbon flux distribution in different pathways this ATP requirement has to be added on the left-hand side of the equation. Again the other form of the cofactor ATP is usually left out to simplify the reaction equation. 

Microbial growth is the core of the biochemical reactions in the TBC model. This growth is linked to substrate and electron acceptor concentrations via Monod-terms. The back-coupling between microbial growth and reactive species consumption is performed via turnover coefficients and stoichiometric relationships. The basic equations are exemplified for a single microbial group X, one substrate S and one electron acceptor E ... 

At this stage, we must make the following important observation regarding the Z-scheme for photosynthesis (summarizing all the primary biochemical reactions of the metabolism, ie, the light reactions of photosynthesis). If we consider only the stoichiometric photons involved in this scheme, then we can simply define the mean quantum yield using the data on the stoichiometric coefficient v o-x, direcdy Unked by the structured equations (eg, Eq. (121)) to the stoichiometric coefficient Vmadph,h -x and to the value of the P/2e ratio from... 

Stoichiometric models describe the metabolic network as a set of stoichiometric equations representing the biochemical reactions in the system. The model is often represented as a stoichiometric matrix with the elements representing stoichiometric coefficients of the different metabolites in the metabolic network. 

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