332
Chapter 8
Within the population of hyphal elements there will be more or less fragmentation at different positions in
the individual hyphal elements. This fragmentation occurs when the local shearing forces become larger
than the tensile strength of the hyphal wall (van Suijdam and Metz, 1981). The tensile strength of the hyphal
wall depends on the morphological state of the individual cell in the hyphal element, e.g., the tensile
strength of the hyphal cells could be smaller than that of apical cells, but here it is assumed that it is constant
and independent of the state of the hyphae. The breakage function and the partitioning function are therefore
both taken to be independent of the morphological state of the individual cells.
In a culture with a very large number of hyphal elements, the average ratio of hyphal length to the number
of actively growing tips in the two ’’daughter” fragments is identical, i.e.,
(l) i(n)
=
U’ ~ ^ { n*
~ n),
if
fragmentation occurs with equal probability at any position on the hyphal elements. It is therefore assumed
that
11 n —(l
—/)/(«*
—ri)
(or
lin = V i n ' \
whereby the partitioning function can be stated as a
function of
f
and / only. Furthermore, since the tensile strength of the hyphal wall is assumed to be
constant, there is an equal probability of fragmentation at any position in the hyphal element. The
partitioning function is therefore given by
I//*
0
if
l '> l
and
l i n —V ln
in all other cases
(3)
The breakage function specifies the rate of fragmentation, and this is taken to be a linear function of the
total length of the hyphal element, i.e.,
b(l,n,t)=y/(t)l
(4)
yf(t)
is the specific rate of fragmentation, which is here taken to be a function of the environmental
conditions only.
w(t)
is determined by the number of times the hyphal elements enter the zones where the
local shearing forces are larger than the tensile strength of the hyphal element, e.g., the impeller zone, and it
is therefore determined both by the circulation pattern and the shear force distribution in the bioreactor.
However, in a simple model describing growth in a stirred tank reactor, one can assume that
W{t)
is a
function only of the energy input (which can be calculated from the stirring speed), i.e., it is constant for
constant stirring speed (van Suijdam and Metz, 1981).
By inserting the partitioning function of Eq. (3) and the breakage function of Eq. (4) in Eq. (2) we get
h(l,n,t) = 2tf/{t) \$ f{t',n * yt)dl*
)
(5)
/
In order to solve the dynamic balance of Eq. (1) for the distribution function, it is necessary to specify
proper boundary conditions, and these are given in Eqs. (6)-(8). Fragmentation never results in the
formation of hyphal elements with zero length or no actively growing tips, and the boundary conditions in
Eq. (6) therefore hold. Since the breakage function is linearly dependent on the length of the hyphal
element, no hyphal elements have a infinite length. Furthermore, due to the assumption
l* in —lin
there
are also no hyphal elements with an infinite number of tips. Finally,
fo(m,n)
specifies the distribution