Discrete and Continuous Representations of Metabolic Models
31
Our correspondence principle suggests, therefore, a constructive
approach to such problems as the elucidation of biochemical control pathways in differentiating systems. Using digital descriptions, it is possible to
characterize the class of (digital) systems which will exhibit any pattern of
differentiation which we may specify. In the present state of knowledge,
however, it is impossible to use these digital descriptions to make detailed
predictions concerning actual differentiating systems, because such predictions (and their verification on actual experimental systems) involve
primarily dynamical aspects of system activity. However, by the judicious
applications of a correspondence principle relating digital and discrete
descriptions, we can translate any particular digital description into dynamical terms; this will permit us to make detailed predictions concerning real
systems exhibiting the activities in terms of which the original digital
systems were specified, and to design experiments to test these predictions
on the real systems. This type of systematic investigation offers, to me,
an attractive alternative to the trial-and-error simulations which presently
comprise the bulk of our theoretical efforts directed to the understanding
of differentiation and multi-cellularity.
In another direction, the principle we have sketched may allow us to
directly attack some of the problems raised by STAHL [16] concerning the
role of computability in development and evolution. The translation of
computability into a purely dynamical concept, as we have suggested above,
should enable us to clarify the role of computability in the activity of
dynamical systems. It may well turn out [9, 11, 14] that the class of (realizable) dynamical systems is strictly larger than the class of systems which
can be replaced by TURING machines, and hence that computability per se
does not place any restriction on the developmental potentialities of organisms. In any case, however, the dynamical properties of a dynamical system
which is capable of computing a nonrecursive function must be very complex
topologically; the catastrophic set must be such as to preclude the effective
construction of equivalence classes of forcings upon which our transition
from continuous to digital descriptions was based; otherwise a fortiori the
original continuous system would necessarily be computable.
References
1. ARBIB, M.: J. SIAM Control Ser. A., 3, 206 (1965).
2. - Automatica 3, 161 (1966).
3. - and M. BLUM: Proc. Amer. Math. Soc. 16, 442 (1965).
4. BLUM, M.: Quart. Prog. Rep. RLE, MIT 72, 237 (1964).
5. GOODWIN, B. C. : Temporal Organization in Cells. New York: Academic Press
1963.
6. HEINMETS, F.: Analysis of Normal and Abnormal Cell Growth. New York:
Plenum Press 1967.
31
Our correspondence principle suggests, therefore, a constructive
approach to such problems as the elucidation of biochemical control pathways in differentiating systems. Using digital descriptions, it is possible to
characterize the class of (digital) systems which will exhibit any pattern of
differentiation which we may specify. In the present state of knowledge,
however, it is impossible to use these digital descriptions to make detailed
predictions concerning actual differentiating systems, because such predictions (and their verification on actual experimental systems) involve
primarily dynamical aspects of system activity. However, by the judicious
applications of a correspondence principle relating digital and discrete
descriptions, we can translate any particular digital description into dynamical terms; this will permit us to make detailed predictions concerning real
systems exhibiting the activities in terms of which the original digital
systems were specified, and to design experiments to test these predictions
on the real systems. This type of systematic investigation offers, to me,
an attractive alternative to the trial-and-error simulations which presently
comprise the bulk of our theoretical efforts directed to the understanding
of differentiation and multi-cellularity.
In another direction, the principle we have sketched may allow us to
directly attack some of the problems raised by STAHL [16] concerning the
role of computability in development and evolution. The translation of
computability into a purely dynamical concept, as we have suggested above,
should enable us to clarify the role of computability in the activity of
dynamical systems. It may well turn out [9, 11, 14] that the class of (realizable) dynamical systems is strictly larger than the class of systems which
can be replaced by TURING machines, and hence that computability per se
does not place any restriction on the developmental potentialities of organisms. In any case, however, the dynamical properties of a dynamical system
which is capable of computing a nonrecursive function must be very complex
topologically; the catastrophic set must be such as to preclude the effective
construction of equivalence classes of forcings upon which our transition
from continuous to digital descriptions was based; otherwise a fortiori the
original continuous system would necessarily be computable.
References
1. ARBIB, M.: J. SIAM Control Ser. A., 3, 206 (1965).
2. - Automatica 3, 161 (1966).
3. - and M. BLUM: Proc. Amer. Math. Soc. 16, 442 (1965).
4. BLUM, M.: Quart. Prog. Rep. RLE, MIT 72, 237 (1964).
5. GOODWIN, B. C. : Temporal Organization in Cells. New York: Academic Press
1963.
6. HEINMETS, F.: Analysis of Normal and Abnormal Cell Growth. New York:
Plenum Press 1967.