298
Chapter 7
fundamental research.
Example 7.9 A simple morphologically structured model for oscillating yeast cultures
Cazzador
et al.
(1990) presented a computer model that predicts spontaneous oscillations of a budding yeast
culture. A previously published computer model (Cazzador and Mariani, 1988) for cell-size distribution at
transient growth is combined with a model for characterization of cells by their mass
m
and genealogical
age
ag.
Unlike chronological age, genealogical age
ag
is a discrete variable and
ag
= 0, 1, . .
. represents the
number of bud scars on the cell. The mass is discretized into a number of groups (or morphological forms),
and in each group cells have an average mass characteristic for that particular group. The critical mass for
budding is assumed to be constant, whereas the critical mass for cell division (m^) is taken to be a
monotonically increasing function of the glucose concentration
s
as shown in the empirical expression Eq.
(1) which also includes an effect of ag. £, and
k2
are positive constants, smaller than 1. In the model the
specific growth rate was assumed to follow the same Monod expression for both unbudded and budded
cells. The yield coefficient of biomass on glucose is, however, different for the two morphological forms,
and it is thereby possible to describe oscillations in the glucose concentration when the budding index, the
fraction of budded cells, varies. On the other hand, the glucose concentration affects the critical mass for
cell division, and oscillations in the glucose concentrations result in oscillations in the budding index. The
computer model has been used to simulate spontaneous oscillations at different dilution rates, and the
simulations give a qualitative fit to experimental data. Despite the incorporation of many of the mechanisms
described by the verbal model, the key formula Eq. (1) is, as mentioned above, completely empirical, but
nothing better can at the present be suggested.
min
^d iv .m in
)
v
s 3 +
K 1
l + £,
1
- k
2 7
(
1
)
From the more detailed computer model Cazzador (1991) derived a simple morphologically structured
model [see also Cazzador and Mariam (1990)]. Despite variations of the intracellular composition within
the groups of budded and unbudded cells considered in the detailed computer model, one can collect all the
cells in each of these two groups of cells into two morphological forms in order to obtain a simple
morphologically structured model that may describe spontaneous oscillations. The two forms are: unbudded
cells (Zu) and budded cells (Zb). Thus there is no discrimination between daughter and unbudded mother
cells, but this kind of simplification is acceptable in a simple model. The metamorphosis reactions are
- z ^ + z . ^ o
;
ub = kbZb
(2)
- 2 „ + z 4 = 0
;
uu = k,Zu
(3)
The reaction shown in Eq. (2) describes cell division, whereas the reaction shown in Eq. (3) describes
budding. Thus the rate of the first metamorphosis reaction (ub) may be interpreted as the rate of cell division
of budded cells, and the rate of the second metamorphosis reaction (uu) may be interpreted as the rate of
budding of unbudded cells. Each of the two morphological forms is also synthesized from the substrate, at a
rate
-
a .s + Z i
= 0
;
fii
where
i =
u, b
(4)
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