318
Chapter 8
where
h+
(y>/) represents the rate of formation of cells with property y,
and similarly
h
(y ,0 represents the rate of disappearance o f these ceils as they divide. Cell division is
normally a singular event, which occurs quite independently of what happens to the other cells in
the population. Let
b(y,t)
represent the division frequency (or
breakage frequency
), i.e.,
b(y,t) dt
is the probability that a cell with property y at time
t
divides in the interval
t
to
t + dt.
Thus
h~(y,t) = b (y ,t)f(y ,t)
(8.4)
In order to identify
h+{y ,t)
, we must consider the (average) number of cells arising from division
of a cell with property y. This is normally 2, independent of the cellular properties and the
environmental conditions, i.e., two new cells are formed upon cell division.* Next we define the
function
p (
y ,y V ) to represent the probability of the formation of cells with properties y and
y* ~ y , respectively, upon division of a cell with property y
}
The rate of formation of cells with
property y is then given by
M (y,0 = 2 f
b(y\t)p(y,y\t)f(y\t) dy'
(8.5)
**
V1
The function
p (y,y* ,0
is called the
partitioning Junction.
It satisfies the constraints/K y.yV ) = 0
whenever one of the elements
y t
in the property vector y is larger than
y*,
and if is scaled by
\ v P(y.y\t)dy = l
(8.6)
Combining Eqs. (8.3)-(8.5) gives
M y>0 = 2 f
b(y\t)p(y,y\t)f(y\t) dy* -b(y,t)f(y,t)
(8.7)
No direct influence of the environmental conditions is included in Eq. (8.7). However, both
r
and
h
are normally functions of the concentrations of substrate and metabolic products in the surrounding
medium. Application of the population balance and an example of the breakage frequency and the
partitioning function are illustrated in Example 8.1.
Example 8.1 Specification of the partitioning function and the breakage frequency
1
For the meiosis of eucaryotes, four cells are formed in a cell cycle, but this special situation will not be considered here.
2 This holds only when the cell properties are conserved upon cell division. There are many cell properties for which this
is not the case, e.g., cell age and surface area. However, here the
h
function can often be described explicitly (e.g., as a
Dirac delta function) as illustrated in Example 6.2.