Systems of Biochemical Reactions

4 Calculations of Equilibrium Compositions for Systems of Biochemical Reactions 

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. 

Since concentrations of ATP, ADP, NADox, and NADred may be in steady states, it is of interest to calculate equilibrium compositions that correspond with these steady state concentrations. These calculations are referred to as level 3 equilibrium calculations because they are based on the introduction of [ATP], [ADP], and the like, as natural variables by use of a Legendre transform. 

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 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 

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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... 

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... 

CALCULATIONS OF EQUILIBRIUM COMPOSITIONS FOR SYSTEMS OF BIOCHEMICAL REACTIONS... 

The semigrand partition function can be written for a system of biochemical reactions at specified concentrations of coenzymes, and G" = —kT n T". The fundamental equation for G" can be used to calculate the other thermodynamic... 

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. 

Calculation of equilibrium compositions of a single biochemical reaction or a system of biochemical reactions at specified pH... 

Equilibrium compositions of systems of biochemical reactions can be calculated using the following two programs. The first was written by Fred Krambeck (Mobil Research and Development) and the second was written by Krambeck and Alberty. The Newton-Raphson method is used to iterate to the composition with the lowest possible Gibbs energy or transformed Gibbs energy. 

Quantitative calculations on systems of biochemical reactions are sufficiently complicated that it is necessary to use a personal computer with a mathematical application. Mathematical (Wolfram Research, Inc. 100 World Trade Center, Champaign, IL, 61820-7237) is well suited for these purposes and is used in this book to make calculations, construct tables and figures, and solve problems. The last third of the book provides a computer-readable database, programs, and worked-out solutions to computer problems. The database BasicBiochemData2 is available on the Web at http //www.mathsource.com/cgi-bin/msitem70211-662. 

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. 

BRN Biochemical reaction network. A system of biochemical reactions that interact with each other. 

R. A. Alberty, Equilibrium calculations on systems of biochemical reactions, Biophys. Chem. 42, 117-131 (1992). 

The amounts of apparent components in a system of biochemical reactions can be calculated using... 

In studying the thermodynamics of systems of biochemical reactions it is desirable to obtain a global view. One way to do that is to assume that coenzymes are in steady states because they are involved in so many different reactions. When coenzyme concentrations are specified, a further transformed Gibbs energy G can be defined by (7,8,9)... 

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... 

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