Discrete and Continuous Representations of Metabolic Models
25
the other hand the basic concepts of the discrete description, such as
computability, have no meaning in the continuous description.
It is most important to find a way of reconciling and comparing the
discrete and continuous descriptions of biological activity and indeed of
systemic activity in general. A variety of attempts at such a reconciliation
have already been proposed of which we may briefly mention several here.
1. Sampled Data Systems
In this case we really do not have two independent descriptions. We
begin with a continuous description and simply discretize it by standard
relaxation methods. In particular, every state of a sampled data system is
already a state of original system. The time scale and the structure of the
state space of the sampled data system, together with all observable properties of that system, are inherited from the continuous description
directly. We shall not consider sampled data systems further in this paper,
although a particular class of such systems will emerge from our analysis.
2. Hybrid Systems
SUGrTA [20] has proposed the consideration of hybrid systems, part of
which can be described only discretely, and part only continuously. Here
again, we do not have independent descriptions of overall system activity;
although hybrid systems may prove convenient for important applications,
it is to be expected that a hybrid description will be superseded by purely
digital and continuous descriptions which may then be directly related.
3. Homologies between System Descriptions
A number of authors [1, 2, 22] have noted that the continuous and
discrete descriptions are related by formal homologies, and have on this
basis attempted to construct a single formalism which could encompass
both kinds of description. These attempts have a number of fruitful
consequences, but from our point of view suffer from one basic difficulty:
they are of an external and formal character, and do not relate the independent digital and continuous descriptions of any particular D'stem. We have
shown [13] that the formal homologies on which these studies are based
arise out of theoretical necessities governing all types of systems descriptions
and are thus incapable of providing direct insight in the question we have
raised.
What is really needed, then, is a kind a correspondence principle which will
allow us to relate these two kinds of descriptions in terms of the intrinsic
properties of any particular system being described, rather than to simply
exhibit formal homologies between the descriptions. We would expect
such a principle to exist from the simple fact, that independent alternate
25
the other hand the basic concepts of the discrete description, such as
computability, have no meaning in the continuous description.
It is most important to find a way of reconciling and comparing the
discrete and continuous descriptions of biological activity and indeed of
systemic activity in general. A variety of attempts at such a reconciliation
have already been proposed of which we may briefly mention several here.
1. Sampled Data Systems
In this case we really do not have two independent descriptions. We
begin with a continuous description and simply discretize it by standard
relaxation methods. In particular, every state of a sampled data system is
already a state of original system. The time scale and the structure of the
state space of the sampled data system, together with all observable properties of that system, are inherited from the continuous description
directly. We shall not consider sampled data systems further in this paper,
although a particular class of such systems will emerge from our analysis.
2. Hybrid Systems
SUGrTA [20] has proposed the consideration of hybrid systems, part of
which can be described only discretely, and part only continuously. Here
again, we do not have independent descriptions of overall system activity;
although hybrid systems may prove convenient for important applications,
it is to be expected that a hybrid description will be superseded by purely
digital and continuous descriptions which may then be directly related.
3. Homologies between System Descriptions
A number of authors [1, 2, 22] have noted that the continuous and
discrete descriptions are related by formal homologies, and have on this
basis attempted to construct a single formalism which could encompass
both kinds of description. These attempts have a number of fruitful
consequences, but from our point of view suffer from one basic difficulty:
they are of an external and formal character, and do not relate the independent digital and continuous descriptions of any particular D'stem. We have
shown [13] that the formal homologies on which these studies are based
arise out of theoretical necessities governing all types of systems descriptions
and are thus incapable of providing direct insight in the question we have
raised.
What is really needed, then, is a kind a correspondence principle which will
allow us to relate these two kinds of descriptions in terms of the intrinsic
properties of any particular system being described, rather than to simply
exhibit formal homologies between the descriptions. We would expect
such a principle to exist from the simple fact, that independent alternate