Conversion of Chemical Equations to Biochemical Equations Robert A. Alberty Massachusetts Institute of Technology, Cambridge, MA02139 When chemical change takes place in a system, the thing we know for sure is that the numben of various types of atoms is not changed. The conservation of atoms is expressed by a conservation equation for each element, and the coefficients in these conservation equations form the conservation matrix A, with a row for each element and a column for each snecies. A set of independent chemical reactions for the system also expresses the conservation of the various tmes of atoms. The stoichiometric number matrix v is madeup of the stoichiometric numbers of a set of independent chemical equations with a row for each species and a column for each reaction. These two ways of expressing the conservation of atoms are related by the equation ( 1 ) which provides the means for calculating v if A is known, and vice versa. The v matrix corresponding with an A matrix is called the null space of the A matrix. It can be calculated by hand for a small A matrix, but a computer is needed for larger A matrices. The use of eq 1 can be illustrated with the hydrolysis of adenosine triphosphate to adenosine diphosphate and inorganic phosphate. As a simplification, we will consider the system under wnditions where only eight species have to be considered. The chemical equations for a reference reaction and three acid dissociations are
This same row reduced vT can be obtained from reactions 2-5. This is the proof that eqs 2-5 form a n independent set of reactions that can provide for all possible changes in composition that can occur in the system. In order to calculate the equilibrium composition of this system, given the initial wmposition and the four equilibrium constants, the four equilibrium equations and the four conservation eauations have to be solved simultaneously. IIowtver, in iiochemistry the situation is different because the DII iii measured at chemical eauilihrium, rather than being calculated from the eight si&ultaneous eauations. Since that is the case, we can ima&ne that the bibchemical experiment is carried out in a celiwith a semipermeable membrane (permeable only to H+ and a nonEeacting anion to maintain electrical neutrality) separating the reaction system from a reservoir a t the specified pH. Now when the reaction oecurs, H is not conserved in the reaction system. This takes the hydrogen row and the Hi column out of the conservation matrix and leaves
where redundant columns have been eliminated in the second form. Now there are four "species": ATP, ADP, Pi, and water. Calculation of the null space shows that there is only one reaction, which can be written and ADP3- reprewhere A T P represents Cl&I1201sN~P~P sents Cl&IlzOl&P23-. If the species are arranged in the order A T P , HATP", ADP3-, HADP", HP04'; HzPOQ,HzO, and HI, and the elements are arranged in the order C, H, 0 , and P, the A matrix is 10 10 12 13 13 13 ~ = [ 33
10 12 10 2
10 0 0 0 0 13 1 2 2 1 10 4 4 1 0 2 loo]
'
(61
The nitrogen row has been omitted because it is redundant; C and N always occur in a 2:l ratio. When the null space ofthis A matrix is calculated using Mathematica and row reduced, the following transposed u matrix is obtained.
ATP + H,O = ADP + Pi
(9)
The apparent equilibrium constant for reaction 9 is IC = [ADPIIPil/[ATPl,which is written in terms of sums of species. The equilibrium composition at a specified pH in terms of ATP, ADP, and Pi can be calculated by use of the. initial composition in terms of ATP, ADP, and Pi and the value of K'. without the necessitv of solvina eiaht simultaneous equ&ons! In discussing tke chemis& in a biochemical reaction it is more useful to write it in terms of snecies. like reaction 2, with at least part of the structure ihown; hut with a clear understanding that such a reference reaction does not tell the whole story about chemical equilibrium. Literature Cited 1. Alberty, a.A.J Ckem Educ. ISSl,6B, 984
Volume 69 Number 6 June 1992
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