1. Mathematics of Microbial Growth and Chemostat Operation
Exponential Growth
Although many analyses of microbial growth can be done
graphically or algebraically, as discussed in Chapter 9, for some
purposes it is convenient to use a differential equation that
expresses the quantitative relationships of growth :
dX / dt =μX (1)
where X may be cell number or some specific cellular component
such as protein and μ is the instantaneous growth-rate constant.
If this equation is integrated, we obtain a form which reflects the
activities of a typical batch culture population :
InX = InXo + μ(t) (2)

2. where In refers to the natural logarithm (logarithm base e) , X0
is cell number at time 0. X is cell number at time t, and is elapsed
time during which growth is measured. This equation fits experi-
mental data from the exponential phase of bacteria cultures very
well, such as those in Figure 9.2. Taking the antilogarithm of each
side gives
X = Xo е u t (3)
This equation is useful because it allows prediction of population
density at a future time by knowing the present value and μ, the
growth-rate constant. As discussed in Section 9.2, an important
constant parameter for an exponentially growing populations the
doubling(or generation) time. Doubling of the population has
occurred when X / X0 = 2. Rearranging and substituting this value
into equation 3 gives
2 = e u (t gen ) (4)

3. Taking the natural logarithm of each side and rearranging gives
μ = In2 / t gen = / t gen (5)
The generation time t gen may be used to define another growth
parameter, k, as follows :
k = 1 / t gen (6)
where k is the growth-rate constant for a batch culture. Combining
Equations 5 and 6 shows that the two growth-rate constants, μ and
k, are related :
μ = k (7)
It is important to understand that μ and k are both reflections of
the same growth process of an exponentially increasing population.
The difference between them may be seen in their derivation above
: μ is an average value for the population over a finite period of
time.

4. This distinction is more than a mathmatical point. As was
emphasized in Chapter 9, microbial growth studies must deal
with population phenomena, not the activities of individual cells.
The constant k reflects this averaging assumption. However, the k
reflects this averaging assumption. However, the constant μ, being
instantaneous, is a closer approximaiton of the rate at which
individual activities are occurring. Further, the instantaneous
constant μ allows us to consider bacteria growth dynamics in a
theoretical framework separate from the traditional batch culture.
Mathematical Relationships of Chemostats
An especially important application of the instantaneous growth-
rate constant μ is, of course, the chemostat, a culture device
(Section 9.6) in which population size and growth rate may be
maintained at constant values of the experimenter’s choosing
over a wide range of values, As we saw in Section 9.5, the rate of
bacterial growth is a function of nutrient concentration. This
function represents a saturation process which may be described
by the equation

5. μ = μ max [ S / ( Ks + S ) ] (8)
where μ max is the growth rate at nutrient saturation, S is the
concentration of nutrient, and Ks is a saturation constant which
is numerically equal to the nutrient concentration at which μ =
½ u max. Equation 8 is formally equivalent to the Michaelis-
Menten equation used in enzyme kinetic analyses. Estimates of
the unknown parameters μ max and Ks can be made by plotting
1 / μ on the y axis versus 1 / s on the x axis. A straight line will be
obtained which intercepts the y axis, and the value at the intercept
is 1 / μ. The slope of the line is Ks / μ max, and since μ max is
known, Ks can then be calculated. Values for Ks as a rule are
very low. A few examples are 2.1*10-5M for glucose (Escherichia
coli), 1.34*10-6M for oxygen (Candida utilis), 3.5*10-8M for
phosphate (Aspergillum spp.), and 4.7*10-13M for thiamine (Cryp-
tococcus albidus). The key to the operation of the chemostat is
that, in a steady-state population, the nutrient concentration is
very low, usually very near 0. Therefore, each drop of fresh
medium entering the chemostat supplies nutrient which is
consumed almost instantaneously by the population and the

6. experimenter has direct control over the growth rate : if nutrient
is added more rapidly, μ decreases ; if it is added more slowly, μ
decreases. In a chemostat the rate at which medium enters is equal
to the rate at which spent medium and cell exit through the
overflow. This rate is called the dilution rate, D, and is defined :
D = F / V ( 9 )
where F is the flow rate ( in units of volume per time, usually ml/
hour) and V is chemostat volume (in ml). Therefore, D has units
of time-1 (usually hour-1) as does μ. In fact, at steady state, μ = D.
That μ must equal D may be seen by consideration of Equation 8
and 9 and the discussion above. When S is low, μ is a direct
function of S. But since nutrient is instantly consumed upon
addition, the value of S at any moment is controlled by the rate of
medium addition, that is, the dilution rate D. A further crucial
consequence of the relation μ = D is that a chemostat is a self-
regulating system within broad limits of D. Consider a steady-state
chemostat functioning with μ = D.

7. If μ were to increase momentarily due to some variation in the
culture, S would necessarily decline, as the increased growth
caused increased nutrient consumption. The lower value of S would
in turn cause u to decrease back to its stable level. Conversely, if μ
were to decrease momentarily, then S would increase as “extra”
nutrient went unconsumed. This rise in S would lead to higher
values of μ until steady state is again reached. This biological
feedback system functions well in practice, as it does in theory.
Chemostat populations may be maintained at constant growth
rates (μ) for long periods of time : months or even years.
There is another important aspect to chemostat studies and that
is population density. The relation between cell growth and
nutrient consumption may be defined :
Y = dX / dS (10)

8. where Y is the yield constant for the particular organism on the
particular medium and dX / dS is the amount of cell increase per
unit or substrate consumed. The yield constant is in units of
Y = weight of bacteria formed / weight of substrate consumed
and is a measure of the efficiency with which cells convert nutrient
to more cell material. The composition of the medium in the
reservoir is control to the successful steady-state operation if the
chemostat. All components in this medium are present in excess,
except one, the growth-limiting substrate. The symbol for the
concentration of that substrate in the reservoir is SR, whereas that
for the same substrate in the culture vessel is S. Taking into
account Equations 1, 8, and 10, the following considerations can
be made.
When a chemostat is inoculated with a small number of bacteria,
and the dilution rate does not exceed a certain critical value(Dc),
the number of organisms will start to increase. The increase is
given by

9. dX / dt = growth – output or dX / dt = μX –DX (11)
since initially μ >D, dX / st is positive. However, as the population
density increases, the concentration of the growth-limiting
substrate decreases, causing a decrease of μ (Equation 8). As
discussed above, if D is kept constant, a steady state will be reached
in which μ = D and dX / dt = 0
Steady states are possible only when the dilution rate does not
exceed a critical value Dc. This value depends on the concentration
of the growth-limiting substrate in the reservoir(SR) :
Dc = μ max [ SR / ( Ks + SR) ] (12)
Thus, when SR >> Ks (which is usually the case), steady states can
Be obtained at growth rates close to μ max . The change in S is given By
dS / dt = SRD - sD - μo X / Y ( 13 )

10. When a steady state is reached, dS / dt = 0.
from Equation 11 and 13, the steady-state values of organism
concentration ( X ) and growth-limiting substrate ( S ) can be
calculated :
X = Y ( SR - S ) (14)
S = Ks [ D / (u max – D)] (15)
These equations have the constants, SR, Y, Ks and μ max . Once
their values are known, the steady-state values of and can be
predicted for any dilution rate. As can be seen from Equation 15,
the steady-state substrate concentration (S ) depends on the
dilution rate (D) applied, and is independent of the substrate
concentration in the reservoir(SR) Thus K determines S, and
determines.
A specific example of the use of the equations to predict chemostat
behavior can be obtained by study of Figure 9.10 In that figure, the
constants are u max = 10 hour – 1 ; y = 0.5 ; Ks = 0.2 g / liter ; SR =
10g / liter. With these constants, the curves in Figure 9.10 can be
generated.

11. Experimental Uses of the Chemostat
One of the major theoretical advantages of a chemostat is that this
device allows the experimenter to control growth rate (u) and
population density ( ) independently of each other. Over rather
wide ranges (see above and Section 9.6) any desired value of u can
be obtained by alteration of D. In practice this means changing
the flow rate, since volume is usually fixed (Equation 9). Similarly,
the population density may be determined by varying nutrient
concentration in the reservoir (Equation 14). This independent
control of these two crucial growth parameters not possible with
conventional batch cultures.
A practical advantage to the chemostat is that a population may
be maintained in a desire growth condition ( u and ) for long
periods of time. There fore, experiments can be planned in detail
and performed whenever most convenient. Comparable studies
with batch cultures are essentially impossible, since specific growth
conditions are constantly changing.

12. Mathematics of the Stationary Phase
The differential equation (Equation 1) expressing the exponential
growth phase can be modified to include the trend toward the
stationary phase, which occurs at high population density.
A second term is added so the equation that expresses the maximum
population attainable for that organism under the environmental
conditions specified, here called Xm. The equation then assumes a
form often called the logistics equation :
dX / dt = uX – (u/Xm) X2 (16)
The second term in this equation essentially expresses self-crowding
Effects, such as nutrient depletion or ingibitor buildup. In integrated
Form, this equation will graph as typical microbial growth curve,
Showing exponential and stationary phases. The length of the
exponential phase will be determined by the size of Xm. When X is
small, the second term will have little (dX/dt) will approach zero.

13. The logistics equation is widely used by ecologists studying higher
organisms to express the growth rate and carrying capacity of
the habitat for animals and plants. The equation is equally
applicable to microbial situations.
Mathematics of Disinfection
An equation can also be developed expressing the death rate of
unicellular organisms upon treatment with a disinfectant or other
lethal agent. If Xd represents the number of dead cells at time t,
then X0-Xd will represent the survivors. The death rate is
proportional to the survivors :
dX / dt = k (X0 – Xd) (17)
Upon integration, this equation gives a typical first-order
relationship ;
kt = In [X0 / (X0 – Xd) ] ( 18 )

14. and when the logarithm of X0-Xd (that is, the number of survivors)
is plotted against time, a straight line will be obtained. This
equation is useful when determining the length of time necessary
to allow a disinfectant to completely sterilize something ( for
example, the contact time for chlorine gas in a water purification
plant)