Computation of Equilibria and Kinetics of
Chemical Systems with Many Species
H. J. BREMER MANN
Abstract
An example is given to show that the solutions of a system of differential
equations, describing chemical reactions of the second order, are generally of a
non-elementary nature. Thus, the solutions cannot merely be analytically "written
down"; at best approximate solutions of limited accuracy can be obtained. The
number of variables required for metabolic studies might turn out to be quite
considerable. The numerical solution is a non-trivial problem. It should be advantageous to start by calculating the equilibrium concentrations which satisfy
a system of non-linear equations. Theory and practice of the numerical solutions
of such equation systems are underdeveloped~especially if the number of
variables is considerable. The author's experimental calculatory methods, still in
the process of development, are outlined briefly.
Consider a second order chemical system: Let Xl> ... Xn denote the
concentrations of the interacting chemical species, lXi' {3ij, Yijk constants, Xi
the time derivatives. The equations for the reaction rates have the form:
.
n
n
Xi =Xi + L Pu Xj + LYijk J0 Xk, i = 1, ... n
(1)
j I
j,k~l
In addition the Xi have to satisfy mass balance equations of the form
m
LCijXj + CiQ = 0,
j~1
i = 1, ... m,
(2)
where the cij and CiQ are constants. Also, the conditions Xi ~ ° must be
satisfied.
For some metabolic systems a set of equations (1) and (2) gives an adequate description while others must take into account a distribution of Xi
between several compartments and transfer mechanisms between compartments [7a; 16a, bJ. Equations (1) also arise in ecology (VOLTERA
theory) (comp. GARFINKEL [16]).
If the equations (1) reduce to linear equations, then their general
solution can be written as the sum of at most n different exponential func6 3. Symp. Quant. BioI.
Chemical Systems with Many Species
H. J. BREMER MANN
Abstract
An example is given to show that the solutions of a system of differential
equations, describing chemical reactions of the second order, are generally of a
non-elementary nature. Thus, the solutions cannot merely be analytically "written
down"; at best approximate solutions of limited accuracy can be obtained. The
number of variables required for metabolic studies might turn out to be quite
considerable. The numerical solution is a non-trivial problem. It should be advantageous to start by calculating the equilibrium concentrations which satisfy
a system of non-linear equations. Theory and practice of the numerical solutions
of such equation systems are underdeveloped~especially if the number of
variables is considerable. The author's experimental calculatory methods, still in
the process of development, are outlined briefly.
Consider a second order chemical system: Let Xl> ... Xn denote the
concentrations of the interacting chemical species, lXi' {3ij, Yijk constants, Xi
the time derivatives. The equations for the reaction rates have the form:
.
n
n
Xi =Xi + L Pu Xj + LYijk J0 Xk, i = 1, ... n
(1)
j I
j,k~l
In addition the Xi have to satisfy mass balance equations of the form
m
LCijXj + CiQ = 0,
j~1
i = 1, ... m,
(2)
where the cij and CiQ are constants. Also, the conditions Xi ~ ° must be
satisfied.
For some metabolic systems a set of equations (1) and (2) gives an adequate description while others must take into account a distribution of Xi
between several compartments and transfer mechanisms between compartments [7a; 16a, bJ. Equations (1) also arise in ecology (VOLTERA
theory) (comp. GARFINKEL [16]).
If the equations (1) reduce to linear equations, then their general
solution can be written as the sum of at most n different exponential func6 3. Symp. Quant. BioI.