56
A. F. BARTHOLOMAY
becomes available to the problem of relating the mathematical interpretation
implied to the biology itself through the medium of experiment and observation. The total ,a-procedure may be diagrammed as follows:
~~ml
l "e I P
Pe .L
~x@:----+m=mx~
(10)
The left-hand portion of this diagram refers to the imbedding fle of the
original system ~ in an experimental context resulting in the augmentation
of ~ by the Experimental system @:, represented by the "Cartesian product"
notation ~ X @:. And the lower line refers to the correspondence fie set up
between the experimentally studied system ~ X @: and an appropriately,
statistically enlarged mathematical model m = m X ~.
A further use and modification of such models (and indeed of other
classes of models as well), is required for rather complex situations such as
the mathematical representation of the kinetics of a metabolic process, in
which the mathematical model may consist of a system of non-linear
differential equations. Because it is not possible usually to obtain the solution
of such systems, numerical methods must be tried. But numerical approximations yield only special solutions and are tedious to apply. For such purposes one turns to an electronic digital or analog computer. And so, the
mathematical model m (or ml now may be understood) must undergo
further transformations fl. The first of these may consist of introducing
scaling factors into the model m, transforming the variables and using the
methods of numerical analysis to put the mathematical expressions into a
form that may be implemented on a computer-call this entire procedure
fll' and the result mI' Then for later convenience the entire numerical
procedure may be flow-diagrammed (procedure fl.;), resulting in a third
form m 2 ; viz., the computer-orientated form of the model m 2 • Then m 2 is
programmed (e.g., using FORTRAN) into a symbolic language (procedure
fl3)' The resulting program in effect is a fourth form, call it m a , of the model.
Implementation of the program on a computer results in the translation
of the symbolism (procedure flJ into machine language resulting in an
electronic realization of the model, m 4 • And finally the computer generates
(procedure fl5) outputs such as sets of curves m5 which allows us to visualize,
say, graphically the predictions of the representation. These steps are
summarized in the following diagram:
"
"1
"2
"3
"4
"5
)
~ ---+ m ---+ m1 ---+ m 2 ----+ ma ----+ m4 ----+ mls
(11
In this sense, then, such a mathematical model is seen to be a prerequisite
to the programming of such biological problems on a computer. This
combination of mathematical and digital computer methodologies creates
A. F. BARTHOLOMAY
becomes available to the problem of relating the mathematical interpretation
implied to the biology itself through the medium of experiment and observation. The total ,a-procedure may be diagrammed as follows:
~~ml
l "e I P
Pe .L
~x@:----+m=mx~
(10)
The left-hand portion of this diagram refers to the imbedding fle of the
original system ~ in an experimental context resulting in the augmentation
of ~ by the Experimental system @:, represented by the "Cartesian product"
notation ~ X @:. And the lower line refers to the correspondence fie set up
between the experimentally studied system ~ X @: and an appropriately,
statistically enlarged mathematical model m = m X ~.
A further use and modification of such models (and indeed of other
classes of models as well), is required for rather complex situations such as
the mathematical representation of the kinetics of a metabolic process, in
which the mathematical model may consist of a system of non-linear
differential equations. Because it is not possible usually to obtain the solution
of such systems, numerical methods must be tried. But numerical approximations yield only special solutions and are tedious to apply. For such purposes one turns to an electronic digital or analog computer. And so, the
mathematical model m (or ml now may be understood) must undergo
further transformations fl. The first of these may consist of introducing
scaling factors into the model m, transforming the variables and using the
methods of numerical analysis to put the mathematical expressions into a
form that may be implemented on a computer-call this entire procedure
fll' and the result mI' Then for later convenience the entire numerical
procedure may be flow-diagrammed (procedure fl.;), resulting in a third
form m 2 ; viz., the computer-orientated form of the model m 2 • Then m 2 is
programmed (e.g., using FORTRAN) into a symbolic language (procedure
fl3)' The resulting program in effect is a fourth form, call it m a , of the model.
Implementation of the program on a computer results in the translation
of the symbolism (procedure flJ into machine language resulting in an
electronic realization of the model, m 4 • And finally the computer generates
(procedure fl5) outputs such as sets of curves m5 which allows us to visualize,
say, graphically the predictions of the representation. These steps are
summarized in the following diagram:
"
"1
"2
"3
"4
"5
)
~ ---+ m ---+ m1 ---+ m 2 ----+ ma ----+ m4 ----+ mls
(11
In this sense, then, such a mathematical model is seen to be a prerequisite
to the programming of such biological problems on a computer. This
combination of mathematical and digital computer methodologies creates