58
A. F. BARTHOLOMAY
random aspects of the mechanism in the system, and then translates the
unitary probabilistic assumptions into probabilistic statements about the
number of such units, i.e., into suitably defined "random" variables.
For example, when we consider the kinetic aspects of a biological
system, the usual stochastic ,u-procedure consists of imposing on each of
the discrete events of the pertinent transformation T in F of @), a homogeneous probability measure or distribution relative to which the cardinalities of the sets of C- or S-components of F are conceived as random
variables. In this way the kinetic aspects of the underlying system become
mathematically expressed in terms of a unified collection of random variables {x (t)} (as opposed to the ordinary algebraic variables of the deterministic model), one corresponding to each possible value of time t (either
discrete or continuous); or in a multivariate case, of collections {Xl (t)},
{x 2 (t)}, ... of such families. This formulation makes available to the theoretical study of @), the entire mathematical machinery of the theory of
probabilities and stochastic processes, and also a statistical basis for a complete discussion ultimately of random fluctuations in related experimental
data. In practice one attempts first of all, to match up the properties of the
theorized underlying random biological process with a known stochastic
process containing an established mathematical formalism from which
further properties of the system may be deduced.
Very extensive use of the so-called "Markov Process" has been made
in studies of biological as well as chemical kinetics [1, 12, 15, 16]. A
whole mathematical theory has grown up around this particular kind of
Process, which makes it especially useful. And in some of these cases (see
[2, 3, 5, 6]) very close relations between the alternative deterministic model
and the stochastic model have been demonstrated.
Example: A Stochastic Model of the Kinetics of the Unimolecular Reaction Process.
In this case an immediate relation with the earlier deterministic model is
established by incorporating the Mass Action principle in the probability
space corresponding to the transformation of each molecule. Thus, following the preceding general suggestions, the first step here is the assignment
to each molecular element a E A of a basic probability:
P (Jt) = ,u Jt + 0 (Jt),
(12)
the same for each molecule and for any time t after initiation of the reaction,
that in time (t, t + J t) this molecule becomes transformed into b = T (a),
an element of the new component B of the evolving system @). This very
simple probability space independently associated with each molecule of A
then leads to the construction of a probability space for the whole process
by deducing from a statistical assumption of independence that
,un J t + 0 (M)
(13)
A. F. BARTHOLOMAY
random aspects of the mechanism in the system, and then translates the
unitary probabilistic assumptions into probabilistic statements about the
number of such units, i.e., into suitably defined "random" variables.
For example, when we consider the kinetic aspects of a biological
system, the usual stochastic ,u-procedure consists of imposing on each of
the discrete events of the pertinent transformation T in F of @), a homogeneous probability measure or distribution relative to which the cardinalities of the sets of C- or S-components of F are conceived as random
variables. In this way the kinetic aspects of the underlying system become
mathematically expressed in terms of a unified collection of random variables {x (t)} (as opposed to the ordinary algebraic variables of the deterministic model), one corresponding to each possible value of time t (either
discrete or continuous); or in a multivariate case, of collections {Xl (t)},
{x 2 (t)}, ... of such families. This formulation makes available to the theoretical study of @), the entire mathematical machinery of the theory of
probabilities and stochastic processes, and also a statistical basis for a complete discussion ultimately of random fluctuations in related experimental
data. In practice one attempts first of all, to match up the properties of the
theorized underlying random biological process with a known stochastic
process containing an established mathematical formalism from which
further properties of the system may be deduced.
Very extensive use of the so-called "Markov Process" has been made
in studies of biological as well as chemical kinetics [1, 12, 15, 16]. A
whole mathematical theory has grown up around this particular kind of
Process, which makes it especially useful. And in some of these cases (see
[2, 3, 5, 6]) very close relations between the alternative deterministic model
and the stochastic model have been demonstrated.
Example: A Stochastic Model of the Kinetics of the Unimolecular Reaction Process.
In this case an immediate relation with the earlier deterministic model is
established by incorporating the Mass Action principle in the probability
space corresponding to the transformation of each molecule. Thus, following the preceding general suggestions, the first step here is the assignment
to each molecular element a E A of a basic probability:
P (Jt) = ,u Jt + 0 (Jt),
(12)
the same for each molecule and for any time t after initiation of the reaction,
that in time (t, t + J t) this molecule becomes transformed into b = T (a),
an element of the new component B of the evolving system @). This very
simple probability space independently associated with each molecule of A
then leads to the construction of a probability space for the whole process
by deducing from a statistical assumption of independence that
,un J t + 0 (M)
(13)