Apparent conservation 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... 

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... 

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 ... 

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

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... 

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. 

This makes it possible to obtain a basis for the apparent conservation matrix A from the stoichiometric number matrix. 

This is the same row reduced result obtained from the apparent conservation matrix conmatS. This completes the demonstration that we can go from a conservation matrix for a biochemical reaction to a stoichiometric number matrix or from a stoichiometric number matrix to a conservation matrix. 

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 second problem with this program has been discussed in Section 7.3 it is the fact that when h2o is a reactant the stoichiometric number matrix v is inconsistent with the conservation matrix A because [h2o] is not included in the expression for the apparent equilibrium constant. A second program equcalcrx was developed (4) to solve these problems. This program takes advantage of the fact that a basis for the conservation matrix can be calculated from the stoichiometric number matrix for the system. 

The transformation T we adopt is induced by the wave function normalization condition which, in terms of the weights, reads w + W3 = 1. From (3.5), it is apparent that if T sends the vvm set into a new set wm with ivi = vvi + iv3 = 1 as one of its elements, then both the first row and the first column of the transformed polarization component of the solvent force constant matrix K, "/ = T. Kp°r. T (T = T) are zero, since the derivatives of wi are zero. Given the normalization condition and the orthogonality requirement — with the latter conserving the original gauge of the solvent coordinates framework — one can calculate T for any number of diabatic states [42], The transformation for the two state case is... 

The total mass of the solid matrix can be determined quite accurately by weighing on a standard analytical balance. The surface area of a waste granule exposed to the leachant is more difficult to determine. A conservative estimate (lower limit) can be made by assuming smooth surfaces and measuring the apparent geometric area assume an impervious sphere configuration). 

In fact, the photon creation and annihilation events at each molecule appear simultaneous, as far as real experimental measurements with finite time resolution are concerned. However, the time-energy uncertainty relation does permit short-lived states that are not properly energy-conserving. This helps explain why it is necessary to include diagrams corresponding to time sequences in which a virtual photon is created before either real photon arrives. It nonetheless transpires that such apparently unphysical cases produce the smallest contributions to the matrix element. 

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