Stoichiometric number matrix

Biochemical reactions balance the atoms of all elements except for hydrogen, or of metals when they are bound reversibly and their ionic concentrations are held constant. Thus a system of biochemical reactions can be represented by an apparent conservation matrix or an apparent stoichiometric number matrix. The adjective apparent is used because hydrogen ions are omitted in the apparent conservation matrix since they are not conserved. Hydrogen ions are also omitted in the apparent stoichiometric number matrix since they do not appear in biochemical reactions. The conservation and stoichiometric number matrices for a system of biochemical reactions can be derived from the conservation matrix... 

Apparent conservation matrices A and apparent stoichiometric number matrices v at specified pH have the properties indicated by equations 5.1-10 and 5.1-22 so that... 

The primes on the amounts are needed to indicate that they are amounts of reactants, which are sums of species that are pseudoisomers at specified pH. The primes on the stoichiometric number matrices and extents of reaction column matrices are needed to indicate that these matrices are for biochemical reactions written in terms reactants (sums of species). The primes are needed on the transformed chemical potentials to distinguish them from chemical potentials of species. 

Krambeck (1978, 1991) in APL and in Mathematica and by Smith and Missen (1982) in Fortran. Krambeck wrote the program equcalc for use on gaseous mixtures involved in petroleum processing and also adapted it to solution reactions as equcalcc and equcalcrx. These latter programs, which are given in BasicBiochemData2, have the advantage that they operate with conservation matrices and stoichiometric number matrices, respectively, so that they can be used with systems of any size. 

Systems of biochemical reactions can be represented by stoichiometric number matrices and conservation matrices, which contain the same information and can be interconverted by use of linear algebra. Both are needed. The advantage of writing computer programs in terms of matrices is that they can then be used with larger systems without change. 

A Problem with Conservation Matrices and Stoichiometric Number Matrices When a Reaction or a System of Reactions Involves HjO... 

Printing Enzyme-eatalyzed Reactions from Stoichiometric Number Matrices References... 

Equilibrium compositions of systems of chemical reactions or systems of enzyme-catalyzed reactions can only be calculated by iterative methods, like the Newton-Raphson method, and so computer programs are required. These computer programs involve matrix operations for going back and forth between conservation matrices and stoichiometric number matrices. A more global view of biochemical equilibria can be obtained by specifying steady-state concentrations of coenzymes. These are referred to as calculations at the third level to distinguish them from the first level (chemical thermodynamic calculations in terms of species) and the second level (biochemical thermodynamic calculations at specified pH in terms of reactants). 

In the A matrix there is a column for each species and a row for each component. Note that the components are taken to be atoms of C, H, and O. (In equation 5.1-15 we will see that other choices of components can be made.) The stoichiometric number matrix corresponding with equation 5.1-5 is... 

Note that the matrix product of the CxiVs conservation matrix and the N x R stoichiometric number matrix is a C x R zero matrix. 

We can add a column to the conservation matrix for C02 and a column to the stoichiometric number matrix for this reaction to obtain... 

In matrix 5.1-15 the three components are CO, H2, and CH4. However, if the order of the columns were changed, other components would be chosen. Thus the conservation matrix is not unique. A set of components must contain all the elements that are not redundant. The rank of the stoichiometric number matrix is equal to the number of independent reactions. 

Next we want to consider the fact that a stoichiometric number matrix can be calculated from the conservation matrix, and vice versa. Since Av = 0, the A matrix can be used to calculate a basis for the stoichiometric number matrix v. The stoichiometric number matrix v is referred to as the null space of the A matrix. When the conservation matrix has been row reduced it is in the form A = [/C,Z], where /c is an identity matrix with rank C. A basis for the null space is given by... 

Equation 5.1-19 shows that the stoichiometric number matrix corresponding with conservation matrix 5.1-15 is... 

Equation 5.1-10 provides the means for calculating a basis for the stoichiometric number matrix that corresponds with the conservation matrix. Similarly the transposed stoichiometric number matrix provides the means for calculating a basis for the transposed conservation matrix. This is done by using the following equation, which is equivalent to equation 5.1-10 ... 

This can be illustrated by starting with the transposed stoichiometric number matrix 5.1-14, which is... 

A basis for the stoichiometric number matrix can be calculated using... 

This does not correspond with reactions 5.1-28 to 5.1-32, but it is equivalent because the row-reduced form of equation 5.1-34 is identical with the row-reduced form of the stoichiometric number matrix for reactions 5.1-28 to 5.1-32 (see Problem 5.2). The application of matrix algebra to electrochemical reactions is described by Alberty (1993d). 

The rank of the A matrix is the number C of apparent components, and the rank of the apparent stoichiometric number matrix is the number R of independent biochemical reactions. 

Glutamate + ATP + ammonia = glutamine + ADP + Pj The transposed stoichiometric number matrix for this reaction is... 

Where coni is the component represented by equation 5.3-6. Row reduction yields equation 5.3-5, which shows that the stoichiometric number matrix and conservation matrix are equivalent. The last column of equation 5.3-5 shows that there is a single reaction and that it agrees with equation 5.3-3. When coupling introduces additional conservation equations, components can be chosen in such a way that the conservation relations are all expressed in terms of conservations of reactants that are chosen as components. Thus equation 5.3-5 utilizes the five components glutamate, ATP, ammonia, ADP, and P . 

This equation is useful for setting up the fundamental equation for consideration of a chemical reaction system described by a particular stoichiometric number matrix. 

It is hard to divide the six chemical work terms into three terms for three chemical reactions, but this can be done using equation 5.4-4 with the following stoichiometric number matrix ... 

Systems of biochemical reactions like glycolysis, the citric acid cycle, and larger and smaller sequential and cyclic sets of enzyme-catalyzed reactions present challenges to make calculations and to obtain an overview. The calculations of equilibrium compositions for these systems of reactions are different from equilibrium calculations on chemical reactions because additional constraints, which arise from the enzyme mechanisms, must be taken into account. These additional constraints are taken into account when the stoichiometric number matrix is used in the equilibrium calculation via the program equcalcrx, but they must be explicitly written out when the conservation matrix is used with the program equcalcc. The stoichiometric number matrix for a system of reactions can also be used to calculate net reactions and pathways. 

The relation between a stoichiometric number matrix v for a set of R reactions involving N reactants and the stoichiometric number matrix v et for a net reaction is a system of linear equations that is represented by the following matrix multiplication (Alberty, 1996) ... 

This net reaction is obtained by multiplying the first five reactions of glycolysis by 1, the second five reactions by 2, and adding. This causes the intermediates to cancel. Alternatively, this net reaction can be calculated by multiplying the stoichiometric number matrix v for the 10 reactions of glycolysis by the pathway matrix s, where (,v )r = 1,1,1,1,1,2,2,2,2,2, according to equation 6.1-3. 

Since equation 6.1-2 represents a set of linear equations, the path can be calculated from the stoichiometric number matrix and a particular net reaction by solving the set of linear equations (Alberty,1996a). In Mathematica this can be done with LinearSolve ... 

Problem 6.2 illustrates the use of equation 6.2-1 by applying it to four net reactions that represent the oxidation of glucose to carbon dioxide and water (1) the net reaction for glycolysis, (2) the net reaction catalyzed by the pyruvate dehydrogenase complex, (3) the net reaction for the citric acid cycle, and (4) the net reaction for oxidative phosphorylation. The v in equation 6.2-1 is the apparent stoichiometric number matrix for these four reactions. The net reaction is... 

The apparent stoichiometric number matrix v" can be obtained from the row-reduced form of A" by use of the analogue of equation 5.1-19 or by calculating a basis for the null space using a computer program. 

When equcalcrx[nt,lnkr,no] is applied to a system of R independent chemical reactions, it requires a R x N transposed stoichiometric number matrix nt, a vector of natural logarithms of the equilibrium constants of independent reactions, and a vector no of the initial concentrations. It can be used at a specified pH by using a R x N transposed stoichiometric number matrix nt, a vector lnkr of natural logarithms of the apparent equilibrium constants of independent biochemical reactions, and a vector no of the initial concentrations. 

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... 

Figure 6.1 Apparent stoichiometric number matrix v for the 10 reactions of glycolysis at specified pH in dilute aqueous solutions, (see Problem 6.3) [With permission from R. A. Alberty, J. Phys. Chem. B 104, 4807-4814 (2000). Copyright 2000 American Chemical Society.]...

When using a computer, a net reaction is obtained more conveniently by use of a matrix multiplication (see Section 6.1). H20 is put in parentheses because its stoichiometric number is not used in the stoichiometric number matrix, but it is involved in the calculation of K for this net reaction using ArG ° = —RT nK. ... 

In writing the stoichiometric number matrix for glycolysis, there is a choice as to the order of the reactants. To make Glc, ATP, ADP, NAD0X, NADred, and P components, they are put first in the rows for reactants in the apparent stoichiometric number matrix, followed by the rest of the reactants ending with Pyr. The stoichiometric number matrix for glycolysis is shown in Fig. 6.1. To check that these 10 reactions are indeed independent, a row reduction of the transposed stoichiometric number matrix can be used. Another way to test the correctness of this matrix is to calculate the net reaction using equation 6.1-3. 

It is convenient to use the fundamental equation in matrix form (see Chapter 5), The stoichiometric number matrix v for reactions 7.1-3 to 7.1-6 is... 

A biochemical reaction can be represented by a vector of its stoichiometric numbers. A system of biochemical reactions is represented by a stoichiometric number matrix. This stoichiometric number matrix can be used to print out the reactions. The programs that can be used to print out the biochemical reactions are mkeqm and nameMatrix. 

This system of reactions is represented by the following stoichiometric number matrix. 

It is more convenient to use equcalcrx because it takes a stoichiometric number matrix and a vector of the apparent equilibrium constants for a set of independent reactions in the system. Note that this program calculates a consevation matrix that is consistent with the stoichiometric number matrix, and uses it in equcalc. The transposed stoichiometric number matrix nt for the reactin without H2 O is given by... 

It can be considered to be the sum of two reactions, (a) Write the stoichiometric number matrix for this enzyme-catalyzed reaction and use NullSpace to obtain a basis for the conservation matrix., (b) Write a conservation matrix that includes a constraint to couple the two subreactions, and row reduce it to show that it is equivalent to the stoichiometric number matrix obtained in (a). 

Now we start with the transpose of the stoichiometric number matrix, nutr Transpose[nu]... 

Now calculate a basis for the transpose of the conservation matrix from the transposed stoichiometric number matrix. 

The rank of the stoichiometric number matrix is equal to the number of independent reactions, which is 2. 

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