60
A. F. BARTHOLOMAY
The fact that the mean value function of the stochastic model, on identification of the parameter f1 with the deterministic rate constant k, is identical
with the negative exponential smooth time course (see equation 6) demonstrates consistency of this particular stochastic model with the deterministic
model. This theoretical property would seem to guarantee that the sample
curves of the stochastic process should appear at least qualitatively very
similar to experimental runs on such reactions which had already been
classified as satisfying the accepted deterministic criterion of unimolecularity.
For further comparison, one could therefore begin with the ordinary
variable x of the earlier deterministic model and transform it into a "stochasticized" variable x' which reflects the hypothesized intrinsic stochastic
component of the present model by writing
(20)
where now 8 2 (t) is the subsumed unimolecular stochastic process "normalized" to the mean value Xo e- kt • Then, a further statistical enlargement of
this expression to accommodate an experimental error process, call it 8 1 (t)
would lead to:
(21)
Comparison of (21) with (20) and (8) shows that it represents a combination
of the two models, in the sense that if 8 1 is negligible then (21) reduces to
(20); and if 82 is negligible, then (21) reduces to (8) (where 8 1 corresponds
to e used in eq. 8).
This raises the practical question of assigning the actual origin to fluctuations that may be observed in experimental determinations of first order
kinetics. Accordingly, if one wishes to verify experimentally and directly
the stochastic hypothesis of the present model, it would be necessary to
perform error-free in vitro experiments. Since this is not possible, in numero
studies have been initiated for studying each random component independently; i.e. for obtaining simulations of both kinds of random processes amenable to statistical analyses. Such studies have been carried out using the IBM
1620 and have recently been reported briefly [7, 11]. In the case of the
stochastic model itself (the 8 2 process), the method whereby the simulations
were effected is a Monte Carlo Procedure based on the transition probability
functions (eqs. 16 and 17) of the stochastic model adjusted to the conditions
of two classical well-established unimolecular reactions: the sucrose inversion
reaction and the decomposition of di-t-butyl peroxide. The data published by
PENNYCUICK [19] on the first of these and the data published by RALEY,
RUST and VAUGHAN [20] on the second had been analyzed quite intensively (see [3]) and established as classical examples of unimolecular
reactions.
A. F. BARTHOLOMAY
The fact that the mean value function of the stochastic model, on identification of the parameter f1 with the deterministic rate constant k, is identical
with the negative exponential smooth time course (see equation 6) demonstrates consistency of this particular stochastic model with the deterministic
model. This theoretical property would seem to guarantee that the sample
curves of the stochastic process should appear at least qualitatively very
similar to experimental runs on such reactions which had already been
classified as satisfying the accepted deterministic criterion of unimolecularity.
For further comparison, one could therefore begin with the ordinary
variable x of the earlier deterministic model and transform it into a "stochasticized" variable x' which reflects the hypothesized intrinsic stochastic
component of the present model by writing
(20)
where now 8 2 (t) is the subsumed unimolecular stochastic process "normalized" to the mean value Xo e- kt • Then, a further statistical enlargement of
this expression to accommodate an experimental error process, call it 8 1 (t)
would lead to:
(21)
Comparison of (21) with (20) and (8) shows that it represents a combination
of the two models, in the sense that if 8 1 is negligible then (21) reduces to
(20); and if 82 is negligible, then (21) reduces to (8) (where 8 1 corresponds
to e used in eq. 8).
This raises the practical question of assigning the actual origin to fluctuations that may be observed in experimental determinations of first order
kinetics. Accordingly, if one wishes to verify experimentally and directly
the stochastic hypothesis of the present model, it would be necessary to
perform error-free in vitro experiments. Since this is not possible, in numero
studies have been initiated for studying each random component independently; i.e. for obtaining simulations of both kinds of random processes amenable to statistical analyses. Such studies have been carried out using the IBM
1620 and have recently been reported briefly [7, 11]. In the case of the
stochastic model itself (the 8 2 process), the method whereby the simulations
were effected is a Monte Carlo Procedure based on the transition probability
functions (eqs. 16 and 17) of the stochastic model adjusted to the conditions
of two classical well-established unimolecular reactions: the sucrose inversion
reaction and the decomposition of di-t-butyl peroxide. The data published by
PENNYCUICK [19] on the first of these and the data published by RALEY,
RUST and VAUGHAN [20] on the second had been analyzed quite intensively (see [3]) and established as classical examples of unimolecular
reactions.