Deterministic and Stochastic Models of Biological Systems
59
gives the probability that a single transformation will take place in (I, 1 +
Ll t) among the n-molecules of A assumed to be present at time t.
The relation of this result to the deterministic Mass Action Principle has
been conceived [3] as follows. Substituting n for x in equation (5), and
approximating d n by the infinitesimal .1 n:
I J n I = k 11 Ll t + ° (Ll!)
(14)
where ° (Ll t) is a higher order infinitesimal, vanishing in the limit as Ll t ~ 0.
This may be interpreted as the approximate number of A-molecules transformed in the interval (I, t + Ll t), so that, where n is the cardinality of A at
time I, the ratio
~ = k Ll t + ° (Ll!)
(15)
n
may be interpreted in the context of the frequency theory of probabilities
as the probability of a simple transformation in this time interval. Then by
identifying the deterministic rate constant k with the stochastic parameter fl,
equation 15 is seen to correspond exactly to the basic probability measure
(eq. 12), of the current representation.
The discrete, integral-valued time developing random variable x is then
defined over this space, where x = 0, 1, 2, ... , Xo gives the number of
molecules of A per fixed arbitrary volume of the homogeneous reaction
space. Previous studies of stochastic models [2] suggested the reasonableness of identifying the random variable x with a continuous-time Markov
process with finitely many states i = 0,1,2, ... , Xo and stationary transition
probabilities (see references [1, 12, 13, 15, 16]). Using this as the basic
mathematical model ';JJ(, the so-called "transition probabilities" {Pik (r)}
(where Pik (r) is the probability of a transition in time r from any state i
occurring at arbitrary time t to a state k :s:: i), which contain the information
for defining the whole process, were deduced to be:
(16)
And from this we obtained the prior probability:
(17)
that, given an initial cardinality xu' at subsequent time t > ° the cardinality
will have been reduced to the value x. This distribution considered as a
function of t was shown to have a mean value
(18)
and variance function
(19)
59
gives the probability that a single transformation will take place in (I, 1 +
Ll t) among the n-molecules of A assumed to be present at time t.
The relation of this result to the deterministic Mass Action Principle has
been conceived [3] as follows. Substituting n for x in equation (5), and
approximating d n by the infinitesimal .1 n:
I J n I = k 11 Ll t + ° (Ll!)
(14)
where ° (Ll t) is a higher order infinitesimal, vanishing in the limit as Ll t ~ 0.
This may be interpreted as the approximate number of A-molecules transformed in the interval (I, t + Ll t), so that, where n is the cardinality of A at
time I, the ratio
~ = k Ll t + ° (Ll!)
(15)
n
may be interpreted in the context of the frequency theory of probabilities
as the probability of a simple transformation in this time interval. Then by
identifying the deterministic rate constant k with the stochastic parameter fl,
equation 15 is seen to correspond exactly to the basic probability measure
(eq. 12), of the current representation.
The discrete, integral-valued time developing random variable x is then
defined over this space, where x = 0, 1, 2, ... , Xo gives the number of
molecules of A per fixed arbitrary volume of the homogeneous reaction
space. Previous studies of stochastic models [2] suggested the reasonableness of identifying the random variable x with a continuous-time Markov
process with finitely many states i = 0,1,2, ... , Xo and stationary transition
probabilities (see references [1, 12, 13, 15, 16]). Using this as the basic
mathematical model ';JJ(, the so-called "transition probabilities" {Pik (r)}
(where Pik (r) is the probability of a transition in time r from any state i
occurring at arbitrary time t to a state k :s:: i), which contain the information
for defining the whole process, were deduced to be:
(16)
And from this we obtained the prior probability:
(17)
that, given an initial cardinality xu' at subsequent time t > ° the cardinality
will have been reduced to the value x. This distribution considered as a
function of t was shown to have a mean value
(18)
and variance function
(19)