PART 1 Introduction
6.3. QUANTIFYING GROWTH KINETICS 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 structurednature of each cell and the segregationof 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
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 Sinfluences 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:
(6.30) where mmis the maximum specific growth rate when S>>Ks. If endogeneous metabolism is unimportant, then mnet= mg. The constant Ksis known as the saturation constantor half- velocity constantand 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=Swhen mg= mmax. In general, mg= mmfor S >> Ksand mg=(mm/Ks)Sfor 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.
m m
g= +
m s
S
K S
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:
(6.31) or
(6.32) where S0is the initial concentration of the substrate and Ks0is dimensionless.
Other equations have been proposed to describe the substrate-limited growth phase.
Depending on the shape of m–Scurve, one of these equations may be more plausible than the others. The following equations are alternatives to the Monod equation:
(6.33)
(6.34) (6.35)
(6.36) 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
(6.37) d
dSu K a b
u u
= (1- ) Contois equation: mg m
= +
m sx
S K X S Moser equation: mg m l
m
= m+ = + - -
n
s
n m s
S n
K S ( K S ) 1
Tessier equation: mg=mm(1-e-KS) Blackman equation: iff
iff < 2
g g
m m
m m
= ≥
=
m s
m s
s
S K
K S S K
, ,
2 2
m m
g= m
s s
S K1+K S0 0+S
m m
g=
+
m s
S K S0 0 S
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:
(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:
(6.39)
(6.40) Or if I then: g
I
K K S
K S S K
s
m s
,
m m /
= + + 2 Noncompetitive substrate inhibition: g
I
m m
= Ê + Ë
ˆ
¯Ê + ËÁ ˆ
¯˜
m
Ks
S
S
1 1 K
mg=mg( ) S1 or mg(S2) or . . . mg(Sn)
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/Kl/ns
Contois 0 2 1/Ksx
178 How Cells Grow Chap. 6