Deterministic and Stochastic Models of Biological Systems
47
sonal knowledge; on what is regarded as the major or "target" level of
biological abstraction being studied; and, where mathematical formulation
is an aim, on the convenient selection of a minimum number of variables
consistent with the biological integrity of the system and ultimately with
mathematical tractability. These matters will be elaborated further in the
sequel.
Along slightly more formal lines, partly in anticipation of the formation, ultimately, of an appropriate and comprehensive axiomatic
framework, in this paper a biosystem ~ (C, S, E, P, t) will be taken as
a kind of primitive concept entailing five major components, which are
describable, using the biomathematical-set theoretic language and point
of view, as follows:
a. Composition C. At a given instant the composition C of the biological
system is expressible as the biomathematical-set theoretic sum of subsets
C I , C 2 , ••• , en:
(1 )
where each C i is made up of a collection of well-defined chemical or biological elements or units, each possessing identical, or at least similar,
biological and physicochemical characteristics. This partitioning is arbitrary
in the sense that for purposes of a particular kind of study of system ~ the
component sets may include either macroscopic, microscopic, or submicroscopic elements or all of these. In other cases limited knowledge of ~
may be the determining factor for the numbers and kinds of such compositional components. Thus, in population studies the C's might be different
interacting populations of whole organisms. In pharmacological studies
using in vivo tracer kinetic techniques, the components might be "biological
black boxes" containing whole "subsystems" of individual organisms. In
Mendelian genetical studies the system might be specified down to the level
of chromosomes or genes. In protein synthetic systems the C's might be
biological macromolecules, metabolites, ribosomal particles, etc.
It is, of course, understood that some or all of these components may
be interconverting so that some may vanish or new ones arise, depending
on the controlling conditions and "states" of the system at different times.
For these reasons, it is more accurate to enlarge the set C given in eq. (1) in
the biomathematical mixed-Cartesian Product sense [4, 8] to include both a
temporal set I of biologically meaningful time instants t, as well as a collection pa of parametric sets PI' P 2 , ••• , Pm each consisting of different values
of pressure, temperature, pH, etc., all of which would be included in the
more extensive biomathematical-set theoretic statement format:
C = (CI U ('2 U ... u en) x (PI X P 2 X ... x Pm) X I
(2)
3 See also section 2 e.
47
sonal knowledge; on what is regarded as the major or "target" level of
biological abstraction being studied; and, where mathematical formulation
is an aim, on the convenient selection of a minimum number of variables
consistent with the biological integrity of the system and ultimately with
mathematical tractability. These matters will be elaborated further in the
sequel.
Along slightly more formal lines, partly in anticipation of the formation, ultimately, of an appropriate and comprehensive axiomatic
framework, in this paper a biosystem ~ (C, S, E, P, t) will be taken as
a kind of primitive concept entailing five major components, which are
describable, using the biomathematical-set theoretic language and point
of view, as follows:
a. Composition C. At a given instant the composition C of the biological
system is expressible as the biomathematical-set theoretic sum of subsets
C I , C 2 , ••• , en:
(1 )
where each C i is made up of a collection of well-defined chemical or biological elements or units, each possessing identical, or at least similar,
biological and physicochemical characteristics. This partitioning is arbitrary
in the sense that for purposes of a particular kind of study of system ~ the
component sets may include either macroscopic, microscopic, or submicroscopic elements or all of these. In other cases limited knowledge of ~
may be the determining factor for the numbers and kinds of such compositional components. Thus, in population studies the C's might be different
interacting populations of whole organisms. In pharmacological studies
using in vivo tracer kinetic techniques, the components might be "biological
black boxes" containing whole "subsystems" of individual organisms. In
Mendelian genetical studies the system might be specified down to the level
of chromosomes or genes. In protein synthetic systems the C's might be
biological macromolecules, metabolites, ribosomal particles, etc.
It is, of course, understood that some or all of these components may
be interconverting so that some may vanish or new ones arise, depending
on the controlling conditions and "states" of the system at different times.
For these reasons, it is more accurate to enlarge the set C given in eq. (1) in
the biomathematical mixed-Cartesian Product sense [4, 8] to include both a
temporal set I of biologically meaningful time instants t, as well as a collection pa of parametric sets PI' P 2 , ••• , Pm each consisting of different values
of pressure, temperature, pH, etc., all of which would be included in the
more extensive biomathematical-set theoretic statement format:
C = (CI U ('2 U ... u en) x (PI X P 2 X ... x Pm) X I
(2)
3 See also section 2 e.