Turnover Compartmentalization
67
Experimentally one tries to realize the steady-state distribution by
keeping the daily uptake of stable calcium equal to the output. Ps(t) is
estimated by injecting a small amount of tracer (e.g. 45Ca2+) and observing
the retention of radioactivity as a function of time. But the observations
must be stopped at some time t = T <{ 00, and the previously mentioned
integral cannot be estimated. Problem: Can any quantities be constructed
that characterize the steady-state distribution of mother substance (here,
stable calcium) and that can be estimated from this kind of tracer data?
I have recently presented a set of such quantities [4], but to determine their
practical usefulness they must be applied to actual experiments, and must
therefore be introduced to the consumer of theory-the experimenter.
Usually, in this field, the experimenters have minimal mathematical training,
and the following is an attempt to give a non-mathematical presentation of
the new concepts; the approach is inspired by the success statistics has
had in presenting itself in terms of white and black balls in an urn.
The upper part of Figure 1 is supposed to represent the cow, with its
input and output of calcium. Steady-state is assumed. Obviously the different calcium ions stay varying lenghts of time in the system, i.e., the ions
have different time of sojourn and in these terms we classify the steady-state
distribution: The black Ca2+ with a time of sojourn between 0 and t 1 , the
white ions with time of sojourn between t1 and t 2 ; the square ions, with time
of sojourn longer than t 2 , are not considered. Steady-state implies that the
numbers of black and white Ca 2 + (m1 and m 2 respectively) are time independent; and so also are the turnovers, r 1 and r 2 , i.e. the number of black
and white particles, respectively, that leave the system per unit of time.
The procedure implies a subdivision of the steady-state population of
particles into two mutually exclusive turnover compartments, K1 and K 2 • The
compartments have well-defined masses or sizes (m1' m 2 ) and turnovers
(r1' r 2 ), but are in general undefined geometrically and physicochemically.
K2 can be subdivided into two parts: an initial part K 21 , and a final part K 22 ,
with the masses m 21 and m 22 (m2 = m 21 + m 22 ). K21 consists of those white
particles that, at the moment of observation, have stayed in the system less
than t 1 , whereas the particles in K22 have stayed longer than t1 and are
therefore just about to leave the system. Only K22 contributes to r 2 ; i.e.,
the particles in K21 have zero exit probability. Note that K1 consists only of
a final part (instead of K1 it would therefore be more logical to write K 12 ).
As shown in Figure 1, the different steady-state quantities (m1' m 21 , m 22 ,
r 1 , r 2 ) can be represented by a turnover diagram: The heights of the columns
give the masses, and the lenghts of the arrows give the turnovers. In
addition we have other quantities, the turnover factors (rate constants) AEI
and AE2 (m 1 )% = r 1 ; m22 ;'E2 = r 2 ); they can be interpreted as conditional
mean exit-probabilities with respect to the steady-state distribution of
particles in K1 und K22 respectively.
5'
67
Experimentally one tries to realize the steady-state distribution by
keeping the daily uptake of stable calcium equal to the output. Ps(t) is
estimated by injecting a small amount of tracer (e.g. 45Ca2+) and observing
the retention of radioactivity as a function of time. But the observations
must be stopped at some time t = T <{ 00, and the previously mentioned
integral cannot be estimated. Problem: Can any quantities be constructed
that characterize the steady-state distribution of mother substance (here,
stable calcium) and that can be estimated from this kind of tracer data?
I have recently presented a set of such quantities [4], but to determine their
practical usefulness they must be applied to actual experiments, and must
therefore be introduced to the consumer of theory-the experimenter.
Usually, in this field, the experimenters have minimal mathematical training,
and the following is an attempt to give a non-mathematical presentation of
the new concepts; the approach is inspired by the success statistics has
had in presenting itself in terms of white and black balls in an urn.
The upper part of Figure 1 is supposed to represent the cow, with its
input and output of calcium. Steady-state is assumed. Obviously the different calcium ions stay varying lenghts of time in the system, i.e., the ions
have different time of sojourn and in these terms we classify the steady-state
distribution: The black Ca2+ with a time of sojourn between 0 and t 1 , the
white ions with time of sojourn between t1 and t 2 ; the square ions, with time
of sojourn longer than t 2 , are not considered. Steady-state implies that the
numbers of black and white Ca 2 + (m1 and m 2 respectively) are time independent; and so also are the turnovers, r 1 and r 2 , i.e. the number of black
and white particles, respectively, that leave the system per unit of time.
The procedure implies a subdivision of the steady-state population of
particles into two mutually exclusive turnover compartments, K1 and K 2 • The
compartments have well-defined masses or sizes (m1' m 2 ) and turnovers
(r1' r 2 ), but are in general undefined geometrically and physicochemically.
K2 can be subdivided into two parts: an initial part K 21 , and a final part K 22 ,
with the masses m 21 and m 22 (m2 = m 21 + m 22 ). K21 consists of those white
particles that, at the moment of observation, have stayed in the system less
than t 1 , whereas the particles in K22 have stayed longer than t1 and are
therefore just about to leave the system. Only K22 contributes to r 2 ; i.e.,
the particles in K21 have zero exit probability. Note that K1 consists only of
a final part (instead of K1 it would therefore be more logical to write K 12 ).
As shown in Figure 1, the different steady-state quantities (m1' m 21 , m 22 ,
r 1 , r 2 ) can be represented by a turnover diagram: The heights of the columns
give the masses, and the lenghts of the arrows give the turnovers. In
addition we have other quantities, the turnover factors (rate constants) AEI
and AE2 (m 1 )% = r 1 ; m22 ;'E2 = r 2 ); they can be interpreted as conditional
mean exit-probabilities with respect to the steady-state distribution of
particles in K1 und K22 respectively.
5'