Deterministic and Stochastic Models of Biological Systems
57
for very complex biological systems a new kind of biological experimentation, which the author has referred to as "in numero" studies [7, 9, 11], in
contrast to "in vitro" and "in vivo" studies. This methodology of course
applies to other classes of models as well; most notably to stochastic models
wherein numerical methods combined with Monte Carlo procedures allow
the generation of artificial curves referred to also as "simulations". More
details on this aspect will be given in the next section. A number of such
studies have already been undertaken in our laboratory and are referred to
in the list of references; e.g. a theoretical study of glycogen metabolism [18],
a theoretical study of hepatocyte proliferation processes [21, 22], studies in
the simulation of first order kinetic systems, [7, 11] and the computer
simulation of electrocardiographic analysis [10, 17]. Time does not permit
us to discuss all of these studies here.
Mathematical models used in this way and such combinations of experimental and theoretical methodologies, then, allow us to experiment on the
system, numerically or theoretically, without disrupting the actual system,
in order to discover the conditions required within the total complex of the
hypothetical biosystem to account for what is observed in vivo. Such a
technique is essential, for example, in studying reversible reactions involved
in a whole chain of reactions, each single step of which could not be varied
directly, or separated from the rest of the chain. These same in numero
techniques become a means also of simulating abnormalities of medical
significance, providing some extra insight in situations that are not amenable
to more direct experimental study. The previously quoted references
discuss some theoretical studies of diabetes and toxic cirrhosis and their
correlations with in VillO data, all from this point of view.
V. Stochastic Models
An empirical first test for the necessity of a stochastic formulation
would involve repeating a set of experimental, timed determinations on the
system is as a means of detecting the presence of significant, unpredictable
random variations. The superposition of all curves onto a common pair
of axes in such a case might reval a strong central tendency, but with a
detectable spread, or range, of curves to either side, say. It is in fact the
purpose of a stochastic model to provide a mathematical rationale for such
random ensembles.
The most primitive underlying principle of construction in the case of
deterministic models is stated in terms of large collections of the basic units
of composition or structure. But in the stochastic approach, one begins
with the pertinent individual discrete unit of composition C or structure S,
say, introducing, in effect, a probability space for representing the effects of
the associated transformation T of each such unit resulting from possible
57
for very complex biological systems a new kind of biological experimentation, which the author has referred to as "in numero" studies [7, 9, 11], in
contrast to "in vitro" and "in vivo" studies. This methodology of course
applies to other classes of models as well; most notably to stochastic models
wherein numerical methods combined with Monte Carlo procedures allow
the generation of artificial curves referred to also as "simulations". More
details on this aspect will be given in the next section. A number of such
studies have already been undertaken in our laboratory and are referred to
in the list of references; e.g. a theoretical study of glycogen metabolism [18],
a theoretical study of hepatocyte proliferation processes [21, 22], studies in
the simulation of first order kinetic systems, [7, 11] and the computer
simulation of electrocardiographic analysis [10, 17]. Time does not permit
us to discuss all of these studies here.
Mathematical models used in this way and such combinations of experimental and theoretical methodologies, then, allow us to experiment on the
system, numerically or theoretically, without disrupting the actual system,
in order to discover the conditions required within the total complex of the
hypothetical biosystem to account for what is observed in vivo. Such a
technique is essential, for example, in studying reversible reactions involved
in a whole chain of reactions, each single step of which could not be varied
directly, or separated from the rest of the chain. These same in numero
techniques become a means also of simulating abnormalities of medical
significance, providing some extra insight in situations that are not amenable
to more direct experimental study. The previously quoted references
discuss some theoretical studies of diabetes and toxic cirrhosis and their
correlations with in VillO data, all from this point of view.
V. Stochastic Models
An empirical first test for the necessity of a stochastic formulation
would involve repeating a set of experimental, timed determinations on the
system is as a means of detecting the presence of significant, unpredictable
random variations. The superposition of all curves onto a common pair
of axes in such a case might reval a strong central tendency, but with a
detectable spread, or range, of curves to either side, say. It is in fact the
purpose of a stochastic model to provide a mathematical rationale for such
random ensembles.
The most primitive underlying principle of construction in the case of
deterministic models is stated in terms of large collections of the basic units
of composition or structure. But in the stochastic approach, one begins
with the pertinent individual discrete unit of composition C or structure S,
say, introducing, in effect, a probability space for representing the effects of
the associated transformation T of each such unit resulting from possible