Modeling of Growth Kinetics
299
Cazzador (1991) performed a stability analysis of this simple morphologically structured model, and he
found that a necessary condition for instability is either Eq. (5) or Eq. (6) (see also Problem 7.4), However,
a stable limit cycle, i.e., presence of sustained oscillations, is obtained only with Eq. (4). Equation (4)
specifies that if the specific growth rate is the same for the two morphological forms, the yield of cell mass
from the substrate has to be larger for the budded cells than for the unbudded cells [notice that this
constraint is used in the computer model of Cazzador
et al.
(1990) described above]. Equation (6) specifies
that if the yields are the same for the two morphological forms, the unbudded cells must have a higher
specific growth rate than the budded cells if instability is to be maintained. However, this constraint does
not ensure sustained oscillations, which is probably a consequence of the simple model structure.
if
Mu = Mb
(5)
Mu
>
Mb
if
a u = a t
(6)
Based on the computer model of Cazzador
et al.
(1990), Cazzador (1991) could specify rates for the two
metamorphosis reactions (2) and (3) such that the morphologically structured model could describe
spontaneous oscillations. The rates of these reactions are functions of the specific growth rate and the
critical mass for cell division [which again is a function of the glucose concentration, according to Eq. (1)].
With the condition of Eq. (5), Cazzador finds the rates of the metamorphosis reactions to be given (see
Problem 7.4) by
, _
h - \
b~ M h\n(h)
(7)
(
8
)
where
h
is defined by Eq. (9):
h =
m
div
m
bud
(9)
is the critical mass of budding, which is taken to be constant, and
mdlv
is calculated by Eq. (1), where
the genealogical age
as
is taken to be zero. The model predicts spontaneous oscillations, but the predicted
oscillation period is too long compared with experimental data, probably because of the extremely
simplified expressions for the rates of the metamorphosis reactions. In the Cazzador model these rates are
functions only of the glucose concentration in the medium, and due to the empiricism of Eq. (1), which is
used in Eqs. (7)-(9), the model does not reveal much of the physiology behind oscillations. The simple
model is, however, valuable for analytical studies of the oscillations._________________________________