Discrete and Continuous Representations of Metabolic Models
27
of e, then any property of ~ is necessarily specified in terms of the values
assumed on S by a (finite) set of real-valued functions on S. Hence, in
particular, it follows that the d-states themselves must necessarily be so
specified. Let us denote the c-observables in terms of which the d-states are
defined by VI' V2, .•. , l'r'
Obviously, any two c-states 0'1,0'2 E S such that Vj (0'1) = Vi (0'2)' i =
1, ... , r will not be distinguished as separate states by the d-description.
Hence the d-states can be regarded as inducing a decomposing of S into
equivalence classes, in such a way that the set Sd of d-states is isomorphic
to S / R, where R is the equivalence relation on 5, defined by writing
0'1 Ra 2 iff I Vj(a 1 ) = Vi (a 2 ) , i = 1, ... , r.
We can get further information regarding the equivalence classes in
5/ R by looking at the kinetic properties of the d-description. These kinetic
properties are embodied in statements of the form
;) (s,a) = s'
where we know that s, s' E Sd correspond to equivalence classes of c-states
in S. We do not yet know to extend this correspondence to the d-inputs a.
But we can observe that, in the d-description, the system e does not leave
a d-state until it is presented with an input symbol at one of the abstract
time instants of the system. Retreating to the c-description, this means that
atry c-state transitions occurring in real time (i.e. in c-time) after the system has
assumed the d-state s E Sd but before it has received a new input symbol a
must be restricted to the equivalence class corresponding to s.
Moreover, the application of the input ~ymbol a to the {lstem must correspond
in the c-description to some activity which will be capable of causing a c-state transition from one equivalence class to another.
If we think about these two statements for a moment, we will realize
that they must be interpreted in the following manner: The d-states of e,
corresponding to individual equivalence classes of c-states of e, are simply
what THOM [21] has called the basins of stable orbits of the c-equations of
motion of e, and hence are completely determined by the c-kinetic properties
of e. The d-inputs a must correspond to transient forcings or controls.
Let us make this last remark more precise. It is intuitively to be expected
that, since the d-states of e; correspond to equivalence classes of c-states, the
d-inputs will correspond likewise to equivalence classes of forcings of e,
rather than to individual such forcings. This turns out in fact to be the
case. In greater detail, let the basins of the c-description be enumerated as
B 1 , B 2 , ••. , Br (we may assume that the c-description has a finite set of
basins, since otherwise an independent finite d-description could not exist,
1 "iff" means: "if and only if", a notation originally introduced by P. R.
HALMOS (Measure Theory, Princeton, N. J. 1964).
27
of e, then any property of ~ is necessarily specified in terms of the values
assumed on S by a (finite) set of real-valued functions on S. Hence, in
particular, it follows that the d-states themselves must necessarily be so
specified. Let us denote the c-observables in terms of which the d-states are
defined by VI' V2, .•. , l'r'
Obviously, any two c-states 0'1,0'2 E S such that Vj (0'1) = Vi (0'2)' i =
1, ... , r will not be distinguished as separate states by the d-description.
Hence the d-states can be regarded as inducing a decomposing of S into
equivalence classes, in such a way that the set Sd of d-states is isomorphic
to S / R, where R is the equivalence relation on 5, defined by writing
0'1 Ra 2 iff I Vj(a 1 ) = Vi (a 2 ) , i = 1, ... , r.
We can get further information regarding the equivalence classes in
5/ R by looking at the kinetic properties of the d-description. These kinetic
properties are embodied in statements of the form
;) (s,a) = s'
where we know that s, s' E Sd correspond to equivalence classes of c-states
in S. We do not yet know to extend this correspondence to the d-inputs a.
But we can observe that, in the d-description, the system e does not leave
a d-state until it is presented with an input symbol at one of the abstract
time instants of the system. Retreating to the c-description, this means that
atry c-state transitions occurring in real time (i.e. in c-time) after the system has
assumed the d-state s E Sd but before it has received a new input symbol a
must be restricted to the equivalence class corresponding to s.
Moreover, the application of the input ~ymbol a to the {lstem must correspond
in the c-description to some activity which will be capable of causing a c-state transition from one equivalence class to another.
If we think about these two statements for a moment, we will realize
that they must be interpreted in the following manner: The d-states of e,
corresponding to individual equivalence classes of c-states of e, are simply
what THOM [21] has called the basins of stable orbits of the c-equations of
motion of e, and hence are completely determined by the c-kinetic properties
of e. The d-inputs a must correspond to transient forcings or controls.
Let us make this last remark more precise. It is intuitively to be expected
that, since the d-states of e; correspond to equivalence classes of c-states, the
d-inputs will correspond likewise to equivalence classes of forcings of e,
rather than to individual such forcings. This turns out in fact to be the
case. In greater detail, let the basins of the c-description be enumerated as
B 1 , B 2 , ••. , Br (we may assume that the c-description has a finite set of
basins, since otherwise an independent finite d-description could not exist,
1 "iff" means: "if and only if", a notation originally introduced by P. R.
HALMOS (Measure Theory, Princeton, N. J. 1964).