Some General Ideas on Deterministic and Stochastic
Models of Biological Systems
A. F. BARTHOLOMAy l.2
Abstract
The paper discusses the formal expression in biomathematical terms of a
general approach to the construction and analysis of mathematical models in
biological systems. It points in the direction of a possible axomatization of such
constructs and their associated methodologies.-The underlying general notion or
"primitive concept" of biosystem tel (C, S, E, P, t) is assumed to include: composition C, structure S, environment E, biological activities or functions P, and time
relations t. In relation to tel a mathematical model is conceived as an associated
mathematical construct m1 such that there exists a correspondence II: tel -+ 9)1
1. between the biological objects in tel and the mathematical symbols in me and 2. between the biological activities in tel and the allowable, biologically interpretable
mathematical operations in the mathematical domains containing me.-Deterministic models are obtained by associating time-dependent, algebraic variables x
with quantifiable aspects of tel; and stochastic models by associating random
variables x, usually defined with reference to a basic probability space corresponding to a discrete biological event. Stochastic and deterministic models for
the kinetics of unimolecular processes are discussed in some detail as examples,
and compared with each other.
I. Introduction
The purpose of the mathematical biologist or biomathematician in constructing a mathematical model is the provision of a deductive or numerical
over structure defined to the biological process in terms of which the knowledge of the systems may be deepened and extended by taking advantage of
(a) the formal deductive mathematical machinery inherent in the mathematical representation, and (b) the quantitative or statistical frame of
reference which it provides for experiments.
The present paper begins with a general biomathematical-or molecular-set theoretic [4, 8, 9] (v.i.) formulation of the notion of biological
1 The work was supported partially by Biomathematics Research Grant No.
GM-l0002 and Training Grant No. 5-Tl-GM-984 from the National Institutes of
Health, Institute of General Medical Sciences, U.S. Department of Health,
Education and Welfare; and by the Howard Hughes Medical Institute.
2 Paper read at the Symposium by N. ARLEY.
Models of Biological Systems
A. F. BARTHOLOMAy l.2
Abstract
The paper discusses the formal expression in biomathematical terms of a
general approach to the construction and analysis of mathematical models in
biological systems. It points in the direction of a possible axomatization of such
constructs and their associated methodologies.-The underlying general notion or
"primitive concept" of biosystem tel (C, S, E, P, t) is assumed to include: composition C, structure S, environment E, biological activities or functions P, and time
relations t. In relation to tel a mathematical model is conceived as an associated
mathematical construct m1 such that there exists a correspondence II: tel -+ 9)1
1. between the biological objects in tel and the mathematical symbols in me and 2. between the biological activities in tel and the allowable, biologically interpretable
mathematical operations in the mathematical domains containing me.-Deterministic models are obtained by associating time-dependent, algebraic variables x
with quantifiable aspects of tel; and stochastic models by associating random
variables x, usually defined with reference to a basic probability space corresponding to a discrete biological event. Stochastic and deterministic models for
the kinetics of unimolecular processes are discussed in some detail as examples,
and compared with each other.
I. Introduction
The purpose of the mathematical biologist or biomathematician in constructing a mathematical model is the provision of a deductive or numerical
over structure defined to the biological process in terms of which the knowledge of the systems may be deepened and extended by taking advantage of
(a) the formal deductive mathematical machinery inherent in the mathematical representation, and (b) the quantitative or statistical frame of
reference which it provides for experiments.
The present paper begins with a general biomathematical-or molecular-set theoretic [4, 8, 9] (v.i.) formulation of the notion of biological
1 The work was supported partially by Biomathematics Research Grant No.
GM-l0002 and Training Grant No. 5-Tl-GM-984 from the National Institutes of
Health, Institute of General Medical Sciences, U.S. Department of Health,
Education and Welfare; and by the Howard Hughes Medical Institute.
2 Paper read at the Symposium by N. ARLEY.