46
A. F. BARTHOLOMAY
system as a means of promoting the construction process of mathematical
models and emphasizing their underlying biological bases. Its overall aims
are: 1. the suggestion of a general formulation of the notion of mathematical model of a biological system, with a view toward further biomathematical development along axiomatic lines; and 2. an elucidation of the
common features and associated scientific methodologies of mathematical
models which, in the absence of specific operational algorithms for such
constructions, may serve as a useful guide. Within this general framework
a number of different approaches to the construction, and a variety of
different classes of such models are discernible. The latter portion of this
paper will confine itself to an elaboration of the two contrasting classes of
deterministic and stochastic models.
The concepts from molecular-set theory transfer in obvious fashion to a
large number of biological transformations by expanding the context of
molecular sets A to include appropriate sets of biological interacting units
and material transformations T governing the interconversions. In the next
section, in fact, some of these extensions of molecular-set theory to obtain
a biomathematical description of the notion of biological system are
subsumed.
II. Biological System or "Biosystem" @5 (5, C, E, F, t)
Informally speaking, by "biological system" will be understood
organized, dynamic collections of sets of material objects-either molecules,
submolecular particles, cells, tissues, organs, organisms; or populations or
mixtures of these units-which stand in certain physicochemical, mechanical, morphological or topological, and temporal relations to each other and
to their environment, and with which are associated certain specific functions or sets of biological activities, evolving and ordered in some "natural"
biological sense according to their relative times of occurence on interrelated time scales. The biological activities are associated with material
transformations of the biological objects or with information processed by
the system, which is assumed to be imbedded in an environment containing
other systems with which it may communicate in this functional sense. The
term biological system is thus meant to include any of the so-called systems
studied in biology such as the circulatory system, the nervous system, the
endocrine system, the cardiovascular system, the protein synthetic system,
systems of growing and competing organisms, etc.
At the outset, it should be stressed that certain practical considerations,
limitations and consequences of the present interpretation, and indeed of
any notion of biolugical system itself, are well in mind. For example, any
description or specification of a "biological system" is a relative or arbitrary
matter depending, for example, on the current state of biological and per-
A. F. BARTHOLOMAY
system as a means of promoting the construction process of mathematical
models and emphasizing their underlying biological bases. Its overall aims
are: 1. the suggestion of a general formulation of the notion of mathematical model of a biological system, with a view toward further biomathematical development along axiomatic lines; and 2. an elucidation of the
common features and associated scientific methodologies of mathematical
models which, in the absence of specific operational algorithms for such
constructions, may serve as a useful guide. Within this general framework
a number of different approaches to the construction, and a variety of
different classes of such models are discernible. The latter portion of this
paper will confine itself to an elaboration of the two contrasting classes of
deterministic and stochastic models.
The concepts from molecular-set theory transfer in obvious fashion to a
large number of biological transformations by expanding the context of
molecular sets A to include appropriate sets of biological interacting units
and material transformations T governing the interconversions. In the next
section, in fact, some of these extensions of molecular-set theory to obtain
a biomathematical description of the notion of biological system are
subsumed.
II. Biological System or "Biosystem" @5 (5, C, E, F, t)
Informally speaking, by "biological system" will be understood
organized, dynamic collections of sets of material objects-either molecules,
submolecular particles, cells, tissues, organs, organisms; or populations or
mixtures of these units-which stand in certain physicochemical, mechanical, morphological or topological, and temporal relations to each other and
to their environment, and with which are associated certain specific functions or sets of biological activities, evolving and ordered in some "natural"
biological sense according to their relative times of occurence on interrelated time scales. The biological activities are associated with material
transformations of the biological objects or with information processed by
the system, which is assumed to be imbedded in an environment containing
other systems with which it may communicate in this functional sense. The
term biological system is thus meant to include any of the so-called systems
studied in biology such as the circulatory system, the nervous system, the
endocrine system, the cardiovascular system, the protein synthetic system,
systems of growing and competing organisms, etc.
At the outset, it should be stressed that certain practical considerations,
limitations and consequences of the present interpretation, and indeed of
any notion of biolugical system itself, are well in mind. For example, any
description or specification of a "biological system" is a relative or arbitrary
matter depending, for example, on the current state of biological and per-