Deterministic and Stochastic Models of Biological Systems
63
ARLEY:
Of course, I do not know what answer Dr. BARTHOLOMAY himself would give
to your comment. However, personally I would like to stress most emphatically here-as I also do in my textbook (ARLEY, N., and K. R. BUCH: "Introduction to the theory of probability and statistics", Science Edition, Wiley, New
York, 1966)-that the experimental errors-Dr. BARTHOLOMAY'S term Cl (t)
here-can never be got rid of in any actual experimental observation. Personally, I therefore wish to protest very much against the expression "experimentally controllable fluctuations" in Dr. BARTHOLOMAY'S paper at this
point.
BERGNER:
Dr. BARTHOLOMAY has earlier-in his analysis of the invertase reaction-tried
to determine separately the random fluctuations implied by the stochastic
character of the system itself. If I recall it correctly he calculated, on this basis,
the confidence interval (95 %?) for the concentration curve and obtained an
interval that satisfactorily covered the experimental data. Hence, his analysis
implies that the pure experimental error can be neglected. But we know, I
believe, that this is not correct and, consequently, we here have an inconsistency.-One may also raise the question whether BARTHOLOMAY'S result
suggests that the molecular events are not independent of each other. From a
physical viewpoint, however, this is not a very attractive conclusion.
ARLEY:
As a physicist I would also say on this point: certainly it is not since it is one
of the most well established facts of atomic physics that the behaviour of the
single atoms and molecules are independent stochastic events, in any case
when we are neglecting the coupling between them which occurs when they
are bound together in crystal lattices. We therefore need more knowledge
about the state of matter considered in Dr. BARTHOLOMAY'S paper Dr. BERGNER
quotes.
KRUGER:
Does the stochastic approach also include growth phenomena? Is it possible
to relate a mathematical description of macroscopic phenomena-which is
inevitably the starting point for biomathematics-to microscopic events, as
has been done in the transition from classical to statistical mechanics and as
could be exemplified bv the progress of genetics from classical terms to molecular genetics?
ARLEY:
Although I do not know the details of Dr. BARTHOLOMAY'S papers, I would
guess that the answer is yes, if x, as here, denotes a concentration or a number
of molecules in a certain volume, then the equations are valid for any value of
the variables, large or small, i.e. they also cover the macroscopic behaviour of
the phenomenon considered. It is well known even for biologists that any
actual measurements can only give us a finite number of points and that, when
these n points are plotted in a coordinate system (x, y), we always can, in an
infinite number of ways, construct a continuous curve passing exactly through
each of these points (e.g. by a polynomial in the independent variable x of
degree at least equal to n-l). However, anyone of such interpolation curvesthe growth functions being a special class of them-does not give us any new
63
ARLEY:
Of course, I do not know what answer Dr. BARTHOLOMAY himself would give
to your comment. However, personally I would like to stress most emphatically here-as I also do in my textbook (ARLEY, N., and K. R. BUCH: "Introduction to the theory of probability and statistics", Science Edition, Wiley, New
York, 1966)-that the experimental errors-Dr. BARTHOLOMAY'S term Cl (t)
here-can never be got rid of in any actual experimental observation. Personally, I therefore wish to protest very much against the expression "experimentally controllable fluctuations" in Dr. BARTHOLOMAY'S paper at this
point.
BERGNER:
Dr. BARTHOLOMAY has earlier-in his analysis of the invertase reaction-tried
to determine separately the random fluctuations implied by the stochastic
character of the system itself. If I recall it correctly he calculated, on this basis,
the confidence interval (95 %?) for the concentration curve and obtained an
interval that satisfactorily covered the experimental data. Hence, his analysis
implies that the pure experimental error can be neglected. But we know, I
believe, that this is not correct and, consequently, we here have an inconsistency.-One may also raise the question whether BARTHOLOMAY'S result
suggests that the molecular events are not independent of each other. From a
physical viewpoint, however, this is not a very attractive conclusion.
ARLEY:
As a physicist I would also say on this point: certainly it is not since it is one
of the most well established facts of atomic physics that the behaviour of the
single atoms and molecules are independent stochastic events, in any case
when we are neglecting the coupling between them which occurs when they
are bound together in crystal lattices. We therefore need more knowledge
about the state of matter considered in Dr. BARTHOLOMAY'S paper Dr. BERGNER
quotes.
KRUGER:
Does the stochastic approach also include growth phenomena? Is it possible
to relate a mathematical description of macroscopic phenomena-which is
inevitably the starting point for biomathematics-to microscopic events, as
has been done in the transition from classical to statistical mechanics and as
could be exemplified bv the progress of genetics from classical terms to molecular genetics?
ARLEY:
Although I do not know the details of Dr. BARTHOLOMAY'S papers, I would
guess that the answer is yes, if x, as here, denotes a concentration or a number
of molecules in a certain volume, then the equations are valid for any value of
the variables, large or small, i.e. they also cover the macroscopic behaviour of
the phenomenon considered. It is well known even for biologists that any
actual measurements can only give us a finite number of points and that, when
these n points are plotted in a coordinate system (x, y), we always can, in an
infinite number of ways, construct a continuous curve passing exactly through
each of these points (e.g. by a polynomial in the independent variable x of
degree at least equal to n-l). However, anyone of such interpolation curvesthe growth functions being a special class of them-does not give us any new