Deterministic and Stochastic Models of Biological Systems
61
Time does not permit our going further into the details and results of
this study, some of which are in the references previously cited. Suffice it to
say, that agreements between experiment and the theoretical, in numero procedures have been obtained which satisfy both qualitative and statistical criteria
of significance. This work is still continuing in an effort to point up the necessity of considering that, in equations such as (8), the random component
I': (I) must at least be considered to have two components: 1':1 (I) due to
experimental and hence extraneous, controllable fluctuations; and 1':2 (I),
inherently and uncontrollably related to the mechanism of transformation.
Most recently attention has also been given to the study of the characteristics of the experimental error component 1':1 (I), so that eventually we may
obtain a deeper knowledge of the interrelationships of these sources of
error and their experimental significance.
Those interested in stochastic models feel that the neglect of the
probabilistic aspect of biological systems is not justifiable even in the face
of mathematical difficulties, some of which are newer than those encountered
in ordinary analytical work and hence may appear larger than equivalent
analytical difficulties with deterministic models for which direct or indirect
treatments already exist. Hearsay to the contrary, it is not true that those
interested in the stochastic approach do this to the exclusion of the deterministic approach. In cases where no real discrepancies may result, it
is readily agreed that an easier deterministic treatment commends itself-in fact, is usually the place where one begins. On the other hand, a
doctrine of complete determinism leads one to ignore or minimize the
importance of well-established evidence of inherent biological variability in
biosystems; and at the substratum of molecular biology such neglect can
at times lead to costlv misinterpretations of kinetic data.
VI. Conclusion
It is hoped that these remarks and examples may be of some use in
obtaining an overview of the application of mathematical methodology
and rationale to the study of biological systems generally. It is also hoped
that the biomathematical formulations and conceptualizations may be
productive in suggesting at least a possible approach to the construction
of a whole theory, or axiomatization, of mathematical models. What has
so far been presented is of course merely an indication that such a theory
is conceivable. Whether or not the particular approach suggested turns
out to be fruitful in this direction is really less important at this time
than the principal aim of the present paper which is to provide hopefully a little insight into what constitutes a mathematical model and to
the differences between the important classes of deterministic and stochastic models.
61
Time does not permit our going further into the details and results of
this study, some of which are in the references previously cited. Suffice it to
say, that agreements between experiment and the theoretical, in numero procedures have been obtained which satisfy both qualitative and statistical criteria
of significance. This work is still continuing in an effort to point up the necessity of considering that, in equations such as (8), the random component
I': (I) must at least be considered to have two components: 1':1 (I) due to
experimental and hence extraneous, controllable fluctuations; and 1':2 (I),
inherently and uncontrollably related to the mechanism of transformation.
Most recently attention has also been given to the study of the characteristics of the experimental error component 1':1 (I), so that eventually we may
obtain a deeper knowledge of the interrelationships of these sources of
error and their experimental significance.
Those interested in stochastic models feel that the neglect of the
probabilistic aspect of biological systems is not justifiable even in the face
of mathematical difficulties, some of which are newer than those encountered
in ordinary analytical work and hence may appear larger than equivalent
analytical difficulties with deterministic models for which direct or indirect
treatments already exist. Hearsay to the contrary, it is not true that those
interested in the stochastic approach do this to the exclusion of the deterministic approach. In cases where no real discrepancies may result, it
is readily agreed that an easier deterministic treatment commends itself-in fact, is usually the place where one begins. On the other hand, a
doctrine of complete determinism leads one to ignore or minimize the
importance of well-established evidence of inherent biological variability in
biosystems; and at the substratum of molecular biology such neglect can
at times lead to costlv misinterpretations of kinetic data.
VI. Conclusion
It is hoped that these remarks and examples may be of some use in
obtaining an overview of the application of mathematical methodology
and rationale to the study of biological systems generally. It is also hoped
that the biomathematical formulations and conceptualizations may be
productive in suggesting at least a possible approach to the construction
of a whole theory, or axiomatization, of mathematical models. What has
so far been presented is of course merely an indication that such a theory
is conceivable. Whether or not the particular approach suggested turns
out to be fruitful in this direction is really less important at this time
than the principal aim of the present paper which is to provide hopefully a little insight into what constitutes a mathematical model and to
the differences between the important classes of deterministic and stochastic models.