296
Chapter 7
number of subpopulations (e.g mother cells
m
and daughter cells
d),
and if the generation time of
one subpopulation is an integer multiple o f the generation time of the other then synchrony can
be obtained. Thus, if the creation of a mother cell from a new bom daughter cell takes exactly 2 (
tb + tm)
then the ratio of mother cells to daughter cells in the population will be constant and
equal to (1 + V5)/2 as shown by Hjortso and Nielsen (1994).
Note 7.7 Relation between
Tosc
and the dilution rate in continuous culture
The result of Hjortso and Nielsen (1994) is a special case of the following:
Let
nt
be the population of mother cells,
d
the population of daughter cells. New mother- and daughter
cells are bom in each generation. Thus at
t = (n+
1) T«*:
+
(or m,+2 = /V i + dn+i)
(1)
dn+t = p m 0
(2)
if the mother cell has the probability
p
to develop a new daughter cell exactly at
t =
(«+1)
Tosc.
Eliminating rfn+1 between (1) and (2) yields a second order difference equation:
tf*n+2 -
-
p m» =
0
(3)
But the whole culture grows exponentially with time, and consequently after one cycle time
Tosc
in a
continuous culture:
m
D+1
=
exp(p
T ^)
(4)
When (4) is inserted in the difference equation one obtains the following algebraic equation for p = exp
{D Toscy.
p
:-p-p =
0
o
r
p=!4 (l+(l+4 p)l/2)
(5)
from which a relation between
D
and
is derived :
To«.
= In p / £>
(6)
For
p
= 1 (i.e all mother cells start to make daughter cells immediately after they have lost a bud) the
result is identical to what is found in the Hjortso and Nielsen (1994) model. With the extra parameter
p
Duboc and von Stockar (2000) could fit experimental data better. For
p = Vi
the value of p decreases
from 1.618 to 1.366 and at a given
D
a smaller oscillation time
is predicted In both cases the
oscillation time is, however predicted to decrease inversely proportional to
D.
This last feature is
confirmed by experiments, but the experiments also seem to show that different values of
p
between 2
and
'/2
should be used at different
D
values (see figure 7 of Duboc and von Stockar (2000)). The value
p
= 2 is of course not allowed in the Duboc and von Stockar hypothesis, but the resulting relation
Tosc =
In
2
ID
corresponds to the cell cycle of prokaryotes where daughter cells start to develop into mother cells
immediately after cell division has taken place. For the reasons outlined below stable oscillations are not
likely to develop for these organisms.'
Finally it should be mentioned that Beuse
et at.
(1999) explained synchrony in the culture by a