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... 

Further transformed Gibbs energies of formation are especially useful in calculating equilibrium compositions by computer programs that accept conservation matrices and vectors of initial amounts, as discussed in the next section. 

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... 

This looks different from the 3x4 matrix on the left side of equation 7.1-4. However, the fact that these two conservation matrices have the same information content can be demonstrated by looking at the row-reduced forms of the two conservation matrices ... 

This shows that for the system of two reactions, the components can be taken to be atoms of the elements C, H, and O or molecules of CH4, O2, and CH2O. The last two columns indicate that CO2 = - CH4 + 2CH2 O and H2 O = CH4 + O2 -CH2 O. The second reaction is the same as reaction 7.1-15, but the first reaction is not the same as reaction 7.1-14. However, it does balance atoms. The use of NuIlSpace and RowReduce above provides a more organized way to compare conservation matrices with stoichiometric matrices for larger systems. 

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). 

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... 

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 set of chemical reactions is not unique for example, the reference reaction can be written with H2P04. Additional reactions are involved if Mg2 + or other cations are bound reversibly by these species. The conservation matrix for this... 

The rows for nitrogen and electric charges are redundant, and therefore are omitted. The row-reduced conservation matrix is... 

As discussed in Chapter 4, biochemists are generally more interested in reactions at specified pH. At specified pH, hydrogen is not conserved, and so this row and column of matrix 5.1-32 are omitted to obtain the following conservation matrix... 

This is referred to as an apparent conservation matrix to distinguish it from the conservation matrix in equation 5.1-32. Thus specifying the pH has the effect of simplifying the conservation matrix of the system by reducing the number of rows by one and the number of columns by four. The matrix in equation 5.2-2 is not unique. An equivalent apparent conservation matrix can be obtained more simply by conserving adenosine groups this leads to... 

A basis for the null space v of conservation matrix 5.2-5 at specified pH obtained with equation 5.1-19 or with a computer is... 

The product of the apparent conservation matrix A and the column vector of amounts of reactants (pseudoisomer groups) gives the column vector ric of the amounts of the apparent components ... 

This is like the product of the conservation matrix A and the amounts n of species, which gives the amounts of components nc (equation 5.1-12). The apparent components in equation 5.2-4 are ATP, H20, and ADP. 

The use of NullSpace yields the following row reduced conservation matrix ... 

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 . 

Transformation matrix. When the conservation matrix a for a system is written in terms of elemental compositions, the elements are used as components. But we can change the choice of components (change the basis) by making a matrix multiplication that does not change the row-reduced form of the a matrix or its null space. Since components are really coordinates, we can shift to a new coordinate system by multiplying by the inverse of the transformation matrix between the two coordinate systems. A new choice of components can be made by use of a component transformation matrix m, which gives the composition of the new components (columns) in terms of the old components (rows). The following matrix multiplication yields a new a matrix in terms of the new components. 

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. 

These three biochemical reactions are catalyzed by hexokinase (EC 2.7.1.1), glucose-6-phosphate isomerase (EC 5.3.1.9), and 6-phosphofructokinase (EC 2.7.1.11), respectively. The EC numbers are from Enzyme Nomenclature (Webb, 1992). The first step is to write the conservation matrix for this reaction system at specified pH because that will show how to calculate the further transformed Gibbs energies of formation at specified [ATP] and [ADP]. 

At specified pH the apparent conservation matrix for this system is... 

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... 

Conservation matrix A that corresponds to this stoichiometric matrix is obtained by calculating the null space of (v )T, as indicated by equation 6.3-4. In order to obtain a conservation matrix with identifiable rows, RowReduce is used again and the result is shown in Fig. 6.2. The figure shows that Glu, ATP, ADP, NAD0X, NADred, and Pj can be taken as the six components for glycolysis. This... 

Figure 6.2 Transposed apparent conservation matrix (A ) for glycolysis at specified pH in dilute aqueous solution, calculated from the apparent stoichiometric number matrix in the previous figure. This conservation matrix shows the composition of the noncomponents (the last 10 rows) in terms of components (see Problem 6.3). [With permission from R. A. Alberty, J. Phys. Chem. B 104, 4807-4814 (2000). Copyright 2000 American Chemical Society.]...

When the concentration of a component is held constant in an equilibrium calculation, its row and column in the conservation matrix A are deleted. When the rows and columns for ATP, ADP, NADox, NADred, and P are deleted, the remaining apparent conservation matrix is dramatically reduced, in fact it is reduced to a vector, namely 1,1,1, which applies to the... 

The equilibrium composition for an enzyme-catalyzed reaction or a series of enzyme-catalyzed reactions can be calculated by using equcalcc or equcalccrx. The first of these programs requires a conservation matrix. The second requires a stoichiometric matrix. The second program is recommended, especially when water is involved as a reactant, because the convention that when dilute aqueous solutions are considered, the activity of water is taken to be unity, means that a second Legendre transform is necessary. 

Since the activity of water is taken as unity in dilute aqueous solutions independent of the extent of reaction, the H2 O column and the oxygen row must be deleted from the conservation matrix ... 

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