Here QGR is in units of kcal/h, while QO2 is in millimoles of O2/h.

Metabolic heat released during fermentation can be removed by circulating cooling

water through a cooling coil or cooling jacket in the fermenter. Often, temperature control
(adequate heat removal) is an important limitation on reactor design (see Chapter 10). The
ability to estimate heat-removal requirements is essential to proper reactor design.

6.3. QUANTIFYING GROWTH KINETICS

6.3.1. Introduction

In the previous section we described some key concepts in the growth of cultures. Clearly,
we can think of the growth dynamics in terms of kinetic descriptions. It is essential to re-
call that cellular composition and biosynthetic capabilities change in response to new
growth conditions (unbalanced growth), although a constant cellular composition and bal-
anced growth can predominate in the exponential growth phase. If the decelerating growth
phase is due to substrate depletion rather than inhibition by toxins, the growth rate de-
creases in relation to decreasing substrate concentrations. In the stationary and death
phases, the distribution of properties among individuals is important (e.g., cryptic death).
Although these kinetic ideas are evident in batch culture, they are equally evident and im-
portant in other modes of culture (e.g., continuous culture).
Clearly, the complete description of the growth kinetics of a culture would involve
recognition of the structured nature of each cell and the segregation of the culture into in-
dividual units (cells) that may differ from each other. Models can have these same attrib-
utes. A chemically structured model divides the cell mass into components. If the ratio of
these components can change in response to perturbations in the extracellular environ-
ment, then the model is behaving analogously to a cell changing its composition in re-
sponse to environmental changes. Consider in Chapter 4 our discussion of cellular
regulation, particularly the induction of whole pathways. Any of these metabolic re-
sponses results in changes in intracellular structure. Furthermore, if a model of a culture is
constructed from discrete units, it begins to mimic the segregation observed in real
cultures. Models may be structured and segregated, structured and nonsegregated, un-
structured and segregated, and unstructured and nonsegregated. Models containing both
structure and segregation are the most realistic, but they are also computationally
complex.
The degree of realism and complexity required in a model depends on what is being
described; the modeler should always choose the simplest model that can adequately de-
scribe the desired system. An unstructured model assumes fixed cell composition, which
is equivalent to assuming balanced growth. The balanced-growth assumption is valid pri-
marily in single-stage, steady-state continuous culture and the exponential phase of batch
culture; it fails during any transient condition. How fast the cell responds to perturbations
in its environment and how fast these perturbations occur determine whether pseudobal-
anced growth can be assumed. If cell response is fast compared to external changes and if
the magnitude of these changes is not too large (e.g., a 10% or 20% variation from initial
conditions), then the use of unstructured models can be justified, since the deviation from

Sec. 6.2 Batch Growth 175

balanced growth may be small. Culture response to large or rapid perturbations cannot be
described satisfactorily by unstructured models.
For many systems, segregation is not a critical component of culture response, so
nonsegregated models will be satisfactory under many circumstances. An important ex-
ception is the prediction of the growth responses of plasmid-containing cultures (see
Chapter 14).
Because of the introductory nature of this book, we will concentrate our discussion
on unstructured and nonsegregated models. The reader must be aware of the limitations
on these models. Nonetheless, such models are simple and applicable to some situations
of practical interest.

6.3.2. Using Unstructured Nonsegregated Models to Predict

Specific Growth Rate

6.3.2.1. Substrate-limited growth. As shown in Fig. 6.11, the relationship

of specific growth rate to substrate concentration often assumes the form of saturation ki-
netics. Here we assume that a single chemical species, S, is growth-rate limiting (i.e., an
increase in S influences growth rate, while changes in other nutrient concentrations have
no effect). These kinetics are similar to the Langmuir–Hinshelwood (or Hougen–Watson)
kinetics in traditional chemical kinetics or Michaelis–Menten kinetics for enzyme reac-
tions. When applied to cellular systems, these kinetics can be described by the Monod
equation:
m S
mg = m (6.30)
Ks + S

where mm is the maximum specific growth rate when S >> Ks. If endogeneous metabolism
is unimportant, then mnet = mg. The constant Ks is known as the saturation constant or half-
velocity constant and is equal to the concentration of the rate-limiting substrate when the
specific rate of growth is equal to one-half of the maximum. That is, Ks = S when mg =
 mmax. In general, mg = mm for S >> Ks and mg = (mm/Ks)S for S < < Ks. The Monod equation
is semiempirical; it derives from the premise that a single enzyme system with
Michaelis–Menten kinetics is responsible for uptake of S, and the amount of that enzyme
or its catalytic activity is sufficiently low to be growth-rate limiting.

Figure 6.11. Effect of nutrient concentra-

tion on the specific growth rate of E. coli.
(With permission, from R. Y. Stanier,
M. Doudoroff, and E. A. Adelberg, The
Microbial World, 5th ed., Pearson Education,
Upper Saddle River, NJ, 1986, p. 192.)

176 How Cells Grow Chap. 6

 This simple premise is rarely, if ever, true; however, the Monod equation empiri-
cally fits a wide range of data satisfactorily and is the most commonly applied unstruc-
tured, nonsegregated model of microbial growth.
The Monod equation describes substrate-limited growth only when growth is slow
and population density is low. Under these circumstances, environmental conditions can
be related simply to S. If the consumption of a carbon–energy substrate is rapid, then the
release of toxic waste products is more likely (due to energy-spilling reactions). At high
population levels, the buildup of toxic metabolic by-products becomes more important.
The following rate expressions have been proposed for rapidly growing dense cultures:
mmS
mg = (6.31)
K s 0 S0 + S

or
mmS
mg = (6.32)
K s1 + K s 0 S0 + S

where S0 is the initial concentration of the substrate and Ks0 is dimensionless.

Other equations have been proposed to describe the substrate-limited growth phase.
Depending on the shape of m–S curve, one of these equations may be more plausible than
the others. The following equations are alternatives to the Monod equation:
Blackman equation: mg = mm , iff S ≥ 2 K s
m (6.33)
m g = m S, iff S < 2 K s
2 Ks

Tessier equation: m g = m m (1 - e - KS ) (6.34)

mmSn - n -1
Moser equation: mg = = m m (l + Ks S ) (6.35)
Ks + S n

mmS
Contois equation: mg = (6.36)
K sx X + S

Although the Blackman equation often fits the data better than the Monod equation, the
discontinuity in the Blackman equation is troublesome in many applications. The Tessier
equation has two constants (mm, K), and the Moser equation has three constants (mm, Ks,
n). The Moser equation is the most general form of these equations, and it is equivalent to
the Monod equation when n = 1. The Contois equation has a saturation constant propor-
tional to cell concentration that describes substrate-limited growth at high cell densities.
According to this equation, the specific growth rate decreases with decreasing substrate
concentrations and eventually becomes inversely proportional to the cell concentration in
the medium (i.e., mg μ X-1).
These equations can be described by a single differential equation as
du b
= Ku a (1 - u) (6.37)
dS

Sec. 6.3 Quantifying Growth Kinetics 177

 TABLE 6.2 Constants of the Generalized
Differential Specific Growth Rate Equation
6.34 for Different Models

a b K

Monod 0 2 1/Ks
Tessier 0 1 1/K
Moser 1 – 1/n 1 + 1/n n/Ksl/n
Contois 0 2 1/Ksx

where u = mg/mm, S is the rate-limiting substrate concentration, and K, a, and b are

constants. The values of these constants are different for each equation and are listed in
Table 6.2.
The correct rate form to use in the case where more than one substrate is potentially
growth-rate limiting is an unresolved question. However, under most circumstances the
noninteractive approach works best:

m g = m g ( S1 ) or m g ( S2 ) or . . . m g ( Sn ) (6.38)

where the lowest value of mg(Si) is used.

6.3.2.2. Models with growth inhibitors. At high concentrations of sub-

strate or product and in the presence of inhibitory substances in the medium, growth be-
comes inhibited, and growth rate depends on inhibitor concentration. The inhibition
pattern of microbial growth is analogous to enzyme inhibition. If a single-substrate
enzyme-catalyzed reaction is the rate-limiting step in microbial growth, then kinetic
constants in the rate expression are biologically meaningful. Often, the underlying mecha-
nism is complicated, and kinetic constants do not have biological meanings and are ob-
tained from experimental data by curve fitting.
1. Substrate inhibition: At high substrate concentrations, microbial growth rate is
inhibited by the substrate. As in enzyme kinetics, substrate inhibition of growth may be
competitive or noncompetitive. If a single-substrate enzyme-catalyzed reaction is the rate-
limiting step in microbial growth, then inhibition of enzyme activity results in inhibition
of microbial growth by the same pattern.
The major substrate-inhibition patterns and expressions are as follows:

mm
Noncompetitive substrate inhibition: mg =
Ê1 + K s ˆ Ê1 + S ˆ (6.39)
Ë Á ˜
S ¯Ë KI ¯

mmS
Or if K I  K s , then: mg = (6.40)
Ks + S + S 2 /KI

178 How Cells Grow Chap. 6

 mmS
For competitive substrate inhibition: mg =
Ê Sˆ (6.41)
K s Á1 + ˜ + S
Ë KI ¯

Note that eq. 6.41 differs from 6.39 and 6.40, and KI in 6.40 and 6.41 differ. Substrate inhibi-
tion may be alleviated by slow, intermittent addition of the substrate to the growth medium.
2. Product inhibition: High concentrations of product can be inhibitory for micro-
bial growth. Product inhibition may be competitive or noncompetitive, and in some cases
when the underlying mechanism is not known, the inhibited growth rate is approximated
to exponential or linear decay expressions.
Important examples of the product inhibition rate expression are as follows:
mmS
Competitive product inhibition: mg =
Ê Pˆ
K s Á1 + +S (6.42)
Ë K p ˜¯

mm
Noncompetitive product inhibition: mg =
Ê1 + K s ˆ Ê1 + P ˆ (6.43)
Ë S ¯ ÁË K p ˜¯

Ethanol fermentation from glucose by yeasts is a good example of noncompetitive prod-

uct inhibition, and ethanol is the inhibitor at concentrations above about 5%. Other rate
expressions used for ethanol inhibition are
n
mm Ê Pˆ
mg = Á1 - ˜
Ê1 + s ˆ Ë
K Pm ¯ (6.44)
Ë S¯

where Pm is the product concentration at which growth stops, or

mm - P/ K p
mg = e
Ê1 + K s ˆ (6.45)
Ë S¯
where Kp is the product inhibition constant.
3. Inhibition by toxic compounds: The following rate expressions are used for compet-
itive, noncompetitive, and uncompetitive inhibition of growth in analogy to enzyme inhibition.
mmS
Competitive inhibition: m g =
Ê I ˆ
K s Á1 + ˜ + S (6.46)
Ë KI ¯
mm
Noncompetitive inhibition: mg =
Ê1 + s ˆ Ê1 + I ˆ
K (6.47)
Ë Á ˜
S ¯Ë KI ¯

Sec. 6.3 Quantifying Growth Kinetics 179

 mmS
Uncompetitive inhibition: mg =
Ê Ks Ê I ˆ
+ Sˆ Á1 + ˜ (6.48)
ÁÊ I ˆ ˜ Ë KI ¯
Á Á1 + ˜ ˜
ËË KI ¯ ¯

In some cases, the presence of toxic compounds in the medium results in the inactivation
of cells or death. The net specific rate expression in the presence of death has the follow-
ing form:
mmS
mg = - kd¢ (6.49)
Ks + S

where k¢d is the death-rate constant (h-1).

6.3.2.3. The logistic equation. When plotted on arithmetic paper, the batch
growth curve assumes a sigmoidal shape (see Fig. 6.3). This shape can be predicted by
combining the Monod equation (6.30) with the growth equation (6.2) and an equation for
the yield of cell mass based on substrate consumption. Combining eqs. 6.30 and 6.2a and
assuming no endogenous metabolism yields
dX m S
= m X (6.50)
dt K s + S
The relationship between microbial growth yield and substrate consumption is
X - X0 = YX/S ( S0 - S ) (6.51)
where X0 and S0 are initial values and YX/S is the cell mass yield based on the limiting nu-
trient. Substituting for S in eq. 6.50 yields the following rate expression:

dX m m (YX/S S0 + X0 - X )
= X (6.52)
dt ( K S YX/S + YX/S S0 + X0 - X )

The integrated form of the rate expression in this phase is

( K S YX/S + S0 YX/S + X0 ) Ê Xˆ K S YX/S

ln Á ˜ - ln{(YX/S S0 + X0 - X )/YX/S S0} = m m t (6.53)
(YX/S S0 + X0 ) Ë X0 ¯ (YX/S S0 + X0 )

This equation describes the sigmoidal-shaped batch growth curve, and the value of X
asymptotically reaches to the value of YX/SS0 + X0.
Equation 6.52 requires a predetermined knowledge of the maximum cell mass in a
particular environment. This maximum cell mass we will denote as X•; it is identical to
the ecological concept of carrying capacity. Equation 6.53 is implicit in its dependence
on S.
Logistic equations are a set of equations that characterize growth in terms of carry-
ing capacity. The usual approach is based on a formulation in which the specific growth
rate is related to the amount of unused carrying capacity:

180 How Cells Grow Chap. 6

 Ê X ˆ
m g = k Á1 - ˜ (6.54)
Ë X •¯

Thus,
dX Ê X ˆ
= kX Á1 - ˜ (6.55)
dt Ë X• ¯

The integration of eq. 6.55 with the boundary condition X(0) = X0 yields the logistic
curve.
X0 e kt
X=
X (6.56)
1 - 0 (1 - e kt )
X•

Equation 6.56 is represented by the growth curve in Fig. 6.12.

Equations of the form of 6.56 can also be generated by assuming that a toxin gener-
ated as a by-product of growth limits X• (the carrying capacity). Example 6.2 illustrates
the use of the logistic approach.
Example 6.2. Logistic Equation
Ethanol formation from glucose is accomplished in a batch culture of Saccharomyces cere-
visiae, and the following data were obtained.
Time (h) Glucose (S), g/L Biomass (X), g/L Ethanol (P), g/L

0 100 0.5 0.0

2 95 1.0 2.5
5 85 2.1 7.5
10 58 4.8 20.0
15 30 7.7 34.0
20 12 9.6 43.0
25 5 10.4 47.5
30 2 10.7 49.0

a. By fitting the biomass data to the logistic equation, determine the carrying-capacity
coefficient k.
b. Determine yield coefficients YP/S and YX/S.
Solution
a) Equation 6.55 can be rewritten as:
1 dX Ê X ˆ
= k Á1 - ˜
X dt Ë X• ¯

or
1 DX Ê X ˆ
k= ∏ Á1 -
X Dt Ë X• ˜¯

–
where X is average biomass concentration during Dt, and X• is about 10.8 g/L, since
growth is almost complete at 30 h. Thus:

Sec. 6.3 Quantifying Growth Kinetics 181

 Figure 6.12. Logistic growth curve.

1 Ê X ˆ
Á1 - X ˜
– DX / Dt ( h -1 )
Dt (h) X(g/L) Ë k (h-1)
X •¯

2 0.75 0.333 0.931 0.36

3 1.55 0.236 0.856 0.28
5 3.45 0.156 0.681 0.23
5 6.25 0.093 0.416 0.22
5 8.65 0.044 0.200 0.22
5 10.00 0.016 0.074 0.22
5 10.55 0.0057 0.023 0.25

A value of k = 0.24 h-1 would describe most of the data, although it would slightly under-
estimate the initial growth rate. Another approach would be to take the log of the above
equation to give:
1 dX Ê X ˆ
log = log k + log Á1 - 0 ˜
X dt Ë X• ¯

and to fit the data to this equation and estimate k from the intercept. In this case k would be
about 0.25 h-1.
b) The yields are estimated directly from the data as:
- D P -( 49 - 0) gP
YP/S = = = 0.5
DS (2 - 100) gS
- D X -(10.7 - 0.5) gX
YX/S = = = 0.104
DS (2 - 100) gS
The above estimate of YX/S is only approximate, as maintenance effects and endogenous
metabolism have been neglected.

182 How Cells Grow Chap. 6