82
H. J. BREMERMANN
tions. The world, however, does not seem to be linear, while mathematics
ceases to be elementary when we have to deal with non-linearities in many
variables. M. ROSENLICHT (Berkeley) has communicated to the author the
following example which demonstrates this fact: Consider:
x = x, y = xy, Z = )'.
(3)
As a general solution we obtain successively through integration:
x(t) = e t + a , y(t) = Cee t + a , and Z(t) = C !ee t + a dt, where a and Care
constants. Substituting u = e t + a the integral becomes !(eu/u) duo ROSENLICHT [17] has given a simple proof that !(e U ju) du is not an elementary function: it cannot be obtained through finitely many algebraic operations on
rational functions, exponentials, logarithms, the trigonometric functions
and their inverses. This example shows that the equations (1) in general
(except for n = 1 and possibly n = 2) do not have elementary solutions.
No amount of ingenuity will produce formulas for the Xi (t) that are algebraic
expressions in t and the elementary transcendentals (as is the case when the
equations are linear). In order to obtain a numerical approximation of a
solution to (1) one must use computers.
The metabolism of E. coli involves, according to WATSON [18], some
2000 - 3000 different species of molecules. Even if all of the species and
their reactions were known, the numerical solution would transcend the
present state of the art of computation. Analog computers are limited to a
small number of variables by hardware problems. Analog work is being
done by HEINMETS [12] and by J. J. HIGGINS of the Johnson Foundation,
Univ. Pennsylvania, Philadelphia, Penna. At the same institution GARFINKEL is doing digital work (Comp. [15] and the bibliography there); he
uses a straightforward stepwise integration method (EULER'S method).
D. F. DeTAR [8], working independently on similar problems with similar
integration methods, reports: "It turns out in practice that these quite
primitive methods of numerical integration give very good results and
are actually more efficient for kinetics problems than many of the more
elaborate methods".
Difficulties arise when steady state concentrations of some species are
very small while their reaction rates are large. In this case it can be of
advantage to compute steady state concentrations by different numerical
methods. (Comp. [10].) For a bibliography of computer methods see BARD
[2], p. 178. In addition there are methods reported in [6, 7, and 9]; the methods of [7] and [6] are based on minimizing the free energy of the system
and the application of non-linear programming techniques.
Knowledge of the steady state concentrations (which we will denote by
X/OJ) permits transformation of the equations (1) into a more convenient
form. We substitute Xi = Y i + X/OJ. The new variables Y i are the deviations from equilibrium. For sufficiently small Yi the linear terms dominate
H. J. BREMERMANN
tions. The world, however, does not seem to be linear, while mathematics
ceases to be elementary when we have to deal with non-linearities in many
variables. M. ROSENLICHT (Berkeley) has communicated to the author the
following example which demonstrates this fact: Consider:
x = x, y = xy, Z = )'.
(3)
As a general solution we obtain successively through integration:
x(t) = e t + a , y(t) = Cee t + a , and Z(t) = C !ee t + a dt, where a and Care
constants. Substituting u = e t + a the integral becomes !(eu/u) duo ROSENLICHT [17] has given a simple proof that !(e U ju) du is not an elementary function: it cannot be obtained through finitely many algebraic operations on
rational functions, exponentials, logarithms, the trigonometric functions
and their inverses. This example shows that the equations (1) in general
(except for n = 1 and possibly n = 2) do not have elementary solutions.
No amount of ingenuity will produce formulas for the Xi (t) that are algebraic
expressions in t and the elementary transcendentals (as is the case when the
equations are linear). In order to obtain a numerical approximation of a
solution to (1) one must use computers.
The metabolism of E. coli involves, according to WATSON [18], some
2000 - 3000 different species of molecules. Even if all of the species and
their reactions were known, the numerical solution would transcend the
present state of the art of computation. Analog computers are limited to a
small number of variables by hardware problems. Analog work is being
done by HEINMETS [12] and by J. J. HIGGINS of the Johnson Foundation,
Univ. Pennsylvania, Philadelphia, Penna. At the same institution GARFINKEL is doing digital work (Comp. [15] and the bibliography there); he
uses a straightforward stepwise integration method (EULER'S method).
D. F. DeTAR [8], working independently on similar problems with similar
integration methods, reports: "It turns out in practice that these quite
primitive methods of numerical integration give very good results and
are actually more efficient for kinetics problems than many of the more
elaborate methods".
Difficulties arise when steady state concentrations of some species are
very small while their reaction rates are large. In this case it can be of
advantage to compute steady state concentrations by different numerical
methods. (Comp. [10].) For a bibliography of computer methods see BARD
[2], p. 178. In addition there are methods reported in [6, 7, and 9]; the methods of [7] and [6] are based on minimizing the free energy of the system
and the application of non-linear programming techniques.
Knowledge of the steady state concentrations (which we will denote by
X/OJ) permits transformation of the equations (1) into a more convenient
form. We substitute Xi = Y i + X/OJ. The new variables Y i are the deviations from equilibrium. For sufficiently small Yi the linear terms dominate