Population Balance Equations
319
Kothari
et al
(1972) applied a population model originally derived by Eakman
et al
(1966) for description
of the size distribution of the yeast
Schizosaccharomyces pombe
at steady state in a chemostat. Thus they
use a one-dimensional distribution function
fm , t\
which at steady state is given by
J)] _
2
f
b(m* ,s)p(m* ,m )f{m *) dm* ~ b (m ,s)f (m) - D f (m)
(1)
dm
J
df{m ) _
2
dm
r(m,.s)
jb(m *,s)p(m *,m )f(m *) dm* -
r(m ,s)
V
dm
/ ( « )
(2)
where
r(m ,s)
is the growth rate for cells with mass
m
(grams per cell per hour). Both the growth rate and
the breakage function are taken to be functions of the substrate concentration
s
in the surrounding medium.
In the model it is assumed that the distribution of division masses around the mean division mass
md is
of a
Gaussian type, and the breakage function is therefore given by
b(m,s)
2e-[im-m*v *f r{m ,s)
£yfnetfc[(m
-
md
) / £■]
(3)
For the partitioning function, it is furthermore assumed that the distribution of daughter cell mass
m
is also
of the Gaussian type, with a median of half the mass of the parent cell at division,
m*.
Thus
p{m* ,m)
Çyfnerf (m*
/ 2£)
(4)
These definitions of the breakage and partitioning functions give the right trends. It is, of course, not
biologically reasonable that 6(0, s ) ^ 0 and
p(m*,m)
^ 0, but this does not influence the conclusions
drawn by Kothari
et al.
(see below).
By comparing model calculations with experimental data for the mass distribution (obtained using a Coulter
counter), Kothari
et al
estimated the parameters in the model, i.e., the average mass at division md and the
standard deviations (e and %) for the functions in Eqs. (3) and (4). Furthermore, they examined different
models for
r(m)
(s is constant at a certain dilution rate in the chemostat), and found that a model where
r(m)
is constant and independent of
m,
i.e.,
r(m)=k
corresponding to zeroth-order growth kinetics for the cell
mass, gave the best fit to the experimental data obtained at different dilution rates (see Fig. 8.1). With a
model where
r(m)
is first-order in
m,
i.e.,
r(m)=km
corresponding to exponential growth of the single-cell
mass, the calculated distribution function could not be fitted to the measured profile. Thus the application of
the population model based on number revealed that
the growth rate o f individual cells is not proportional
to their mass.
This does not contradict the dictum that the growth rate of a microbial culture is proportional
to the
total
biomass concentration, since