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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas
  
  
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DETERMINISTIC AND STOCHASTIC MODELS

Descartes' machine has been the meta-model on
which most biological models have been patterned. It
is a clockwork machine designed so that a fixed input
will result in a fixed output. The input-output relation
is one-one or many-one, but never one-many. A per-
turbation at any point in the structure results in an
exactly predictable response (including no response) at
every other part of the structure. This meta-model is
widely accepted in biology and molecular biology. The
program of molecular biology is identical with the
program described by Descartes in Part V of the Dis-
course on Method.
The present description of the action
of genes in controlling protein synthesis, which is the
core of molecular biology, is isomorphic with the de-
scription of an automobile factory including quality
control, inventory control, assembly lines, and the like.
There is even a conscious exclusion of ambiguities
when they appear in experimental data, because of the
a priori certainty that the correct model is a Cartesian
one. For example, the present picture of the action
of genes is that each triplet of bases in the DNA mole-
cule specifies a particular amino acid to be added to
the protein. The experimental results that established
the correspondence between triplet and amino acid


246

showed that a particular triplet could cause the incor-
poration of several amino acids in vitro, but one more
than others. It was assumed that this ambiguity was
an experimental artifact and that genes are more exact
than chemists. In general, molecular biologists deal
with all-or-none concepts, with switchings-on and
switchings-off of genes, with repression and de-repres-
sion. While the methods and data of this kind of biology
are quantitative and continuous, the interpretations are
qualitative and discrete. It is not accidental that statis-
tics and probability, the fitting of curves, and the
estimation of parameters, is not part of the apparatus
of modern molecular biology.

A quite different world view is reflected in the
models constructed for the analysis of population phe-
nomena like evolution and ecology. The models are
Laplacean rather than Cartesian. Chance elements are
built into the models, although this does not imply that
the phenomena being modelled are themselves really
uncertain. That is, most biologists adhere to a deter-
ministic view of phenomena but assume that a large
number of small deterministic effects that cannot and
need not be analyzed, give an apparent indeterminacy
to phenomena at a higher level.

In stochastic or probabilistic models the correlative
statements connecting variables are of the form “If X
takes the value x then Y takes the value y with proba-
bility f(y|x).” That is, each rule becomes a table of
probabilities. of y given x, or a function for generating
those probabilities. Realizations of such models then
require generations of “random” sequences, or, more
usually, pseudo-random sequences, which are indistin-
guishable from randomness by nearly any criterion, yet
are produced by an analytic rule.

Such stochastic models can then be set in motion
over and over again to produce empirically an array
of outcomes for any given input, since the random
sequence generator never repeats itself. For example,
in primitive populations, marriages are contracted
according to certain age, clan, and relationship prefer-
ences. These rules are not rigid, however, but can be
expressed as probabilities. If a village has a small pop-
ulation size, the subsequent history of marriages, births,
and deaths cannot be exactly given but various histories
may occur with different probabilities. It is relatively
simple to simulate the history of such a village in a
computer program and then to run hypothetical village
histories over and over again from which an array of
growth rates, pedigrees, total birth and death rates will
be produced. This array is a picture of the probability
distribution of outcomes predicted by the theory.

While this kind of stochastic modelling is very com-
mon in population biology and ecology, it raises a
serious problem. Since a range of outputs will occur
for a given input, only the weakest kind of comparison
with nature is possible. There is only one primitive
tribe with a unique history. Is this history “explained”
by the theory that has been modelled, when that model
produces an array of results among which the observed
history may very well lie? What kind of observed
history would be at variance with a model that pro-
duces an array of results? About the best that can be
done in those areas of biology is to say that a given
observation in nature is reasonable under the theory,
that it is not surprising given the hypothetical struc-
ture. The method of making such judgments is the
method of statistical testing. The model is used to
construct a calculating engine which produces a prob-
ability distribution of outcomes, given the hypothesis.
The actual case is compared with this probability
distribution, and if it is very improbable, the hypothesis
is rejected as an explanation of the natural event. The
difficulty is that many theories, especially in evolu-
tionary genetics, when modelled, turn out to give such
a broad array of results that almost any observation
is compatible with them.

Modelling serves a function that becomes apparent
in biology, but not in the physical sciences. If theories
contain a stochastic element and if only unique real-
izations of particular cases occur, model building may
be used to show that no choice among theories can be
made at some levels of theory making.
These conditions,
which apply in much of population and evolutionary
biology, may also turn out to have the same effect in
social science.