Biochemical reactions matrix equations

Matrix operations make it possible to carry out calculations on systems of biochemical reactions with equations like 3.1-14. When the transformed Gibbs energy is used the corresponding equation is... 

In treating systems of biochemical reactions it is convenient to use the fundamental equation for G in matrix form (Alberty, 2000b). The extent of reaction t for a chemical reaction was discussed earlier in Section 2.1. For a system of chemical reactions, the extent of reaction vector , is defined by... 

Matrix Forms of the Fundamental Equations for Biochemical Reaction Systems... 

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... 

This shows that when the pH is specified, the conservation equations can be written in terms of reactants (sums of species), rather than species. We can use linear algebra to show that this is the conservation matrix for a biochemical reaction system with the following single reaction. 

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... 

It is of interest to note that this row reduced conservation matrix is characteristic of all biochemical reactions of the form A + B + C = D + E + F. The last column indicates that adp = glutamate + atp - pi + ammonia - glutamine, which is equation 7.4-4. 

If enough fluxes are measured at a metabolic steady state, MFA [11,12] can be used to estimate the fluxes through the remainder of the metabolic reaction network (Figure 15.1). This analysis is powerful because only the stoichiometry of the biochemical reaction network is required, and no knowledge of the chemical reaction kinetics is needed. MFA is usually formulated as a matrix equation ... 

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. 

The obvious advantage is that the steady-state solution of an S-system model is accessible analytically. However, while the drastic reduction of complexity can be formally justified by a (logarithmic) expansion of the rate equation, it forsakes the interpretability of the involved parameters. The utilization of basic biochemical interrelations, such as an interpretation of fluxes in terms of a nullspace matrix is no longer possible. Rather, an incorporation of flux-balance constraints would result in complicated and unintuitive dependencies among the kinetic parameters. Furthermore, it must be emphasized that an S-system model does not necessarily result in a reduced number of reactions. Quite on the contrary, the number of reactions r = 2m usually exceeds the value found in typical metabolic networks. 

The limiting cases are limvo 0 a = 1 and limy. x a = 0. To evaluate the saturation matrix we restrict each element to a well-defined interval, specified in the following way As for most biochemical rate laws na nt 1, the saturation parameter of substrates usually takes a value between zero and unity that determines the degree of saturation of the respective reaction. In the case of cooperative behavior with a Hill coefficient = = ,> 1, the saturation parameter is restricted to the interval [0, n] and, analogously, to the interval [0, n] for inhibitory interaction with na = 0 and n = , > 1. Note that the sigmoidality of the rate equation is not specifically taken into account, rather the intervals for hyperbolic and sigmoidal functions overlap. 

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