Discrete and Continuous Representations of
Metabolic Models
R. ROSEN
Abstract
A wide variety of important biological systems admit both a continuous and
a discrete representation. For example, the central nervous system can be described
in terms of both a continuous two-factor theory and a discrete theory of neural
nets; biochemical control systems admit a continuous theory of chemical kinetics
and a discrete theory of "biochemical automata". Each of these representations
possesses an intrinsic character not shared by the other; for example we may talk
about transient behavior and stability in continuous systems, and about computability in discrete systems, but we cannot at present transfer such concepts readily
from one sphere to the other. Nevertheless, from the fact that both kinds of theories
describe the same systems, there must be some kind of "correspondence principle"
by means of which transfer of concepts can be made between discrete and continuous models. We shall be concerned with the development of such a principle,
and the implications of this principle for the description of metabolic control and
regulation.
I. Introduction
The behavior of biological systems simultaneously admits a number of
independent types of description. For instance, it is possible to construct
models for the central nervous system in terms of quantities which are
continuously variable in time. The kinetic properties of such systems are
then described by a system of ordinary differential equations (see e.g., [10]).
This type of description allows us to speak meaningfully of intensity
gradations of stimuli and responses, and of the stability or instability of
system behavior. On the other hand, the same central nervous system
may be independently described in terms of a mathematically discrete theory
[8]; here, ideas of continuity, continuous variability and stability are meaningless, and are replaced by entirely new concepts: computability, regularity [7J and ease of computation [3, 4]. An exactly analogous situation
has arisen in the theory of cellular control and regulation (see [12] for an
elaboration of the analogous features). Here too, we have both continuous
descriptions [5, 6, 15] and independent discrete descriptions [16, 18, 19].
Again, the discrete descriptions are characterized by the fact that continuity
and stability have no intrinsic meaning within these descriptions, while on
Metabolic Models
R. ROSEN
Abstract
A wide variety of important biological systems admit both a continuous and
a discrete representation. For example, the central nervous system can be described
in terms of both a continuous two-factor theory and a discrete theory of neural
nets; biochemical control systems admit a continuous theory of chemical kinetics
and a discrete theory of "biochemical automata". Each of these representations
possesses an intrinsic character not shared by the other; for example we may talk
about transient behavior and stability in continuous systems, and about computability in discrete systems, but we cannot at present transfer such concepts readily
from one sphere to the other. Nevertheless, from the fact that both kinds of theories
describe the same systems, there must be some kind of "correspondence principle"
by means of which transfer of concepts can be made between discrete and continuous models. We shall be concerned with the development of such a principle,
and the implications of this principle for the description of metabolic control and
regulation.
I. Introduction
The behavior of biological systems simultaneously admits a number of
independent types of description. For instance, it is possible to construct
models for the central nervous system in terms of quantities which are
continuously variable in time. The kinetic properties of such systems are
then described by a system of ordinary differential equations (see e.g., [10]).
This type of description allows us to speak meaningfully of intensity
gradations of stimuli and responses, and of the stability or instability of
system behavior. On the other hand, the same central nervous system
may be independently described in terms of a mathematically discrete theory
[8]; here, ideas of continuity, continuous variability and stability are meaningless, and are replaced by entirely new concepts: computability, regularity [7J and ease of computation [3, 4]. An exactly analogous situation
has arisen in the theory of cellular control and regulation (see [12] for an
elaboration of the analogous features). Here too, we have both continuous
descriptions [5, 6, 15] and independent discrete descriptions [16, 18, 19].
Again, the discrete descriptions are characterized by the fact that continuity
and stability have no intrinsic meaning within these descriptions, while on