Vaccari D.A., Strom P.F., Alleman J.E. Environmental Biology for Engineers and Scientists

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If the logarithm of X is plotted vs. time ðtÞ for an exponentially growing culture, a

straight line results (Figure 11.21). Using natural logarithms (log

, or ln), the equation

for a straight line then gives

ln X

¼ mt þ ln X

ð11:1Þ

where X

is the value of X at time t, m is the slope of the line, and X

is the starting value of

X (at time 0). Differentiating equation (11.1) (assuming that m is constant):

¼ m ð11:2Þ

¼ mX ð11:3Þ

Cell Number

Time

Figure 11.20 Theoretical stair-stepped batch bacterial growth curve.

Time

ln X

Figure 11.21 Similar plot of exponential growth.

318

QUANTIFYING MICROORGANISMS AND THEIR ACTIVITY

From equation (11.2) it can be seen that m is equal to the rate at which the biomass

concentration is growing, divided by the biomass conce ntration at that time. Therefore,

m is referred to as the speciﬁc growth rate and has units of time

1

(inverse time, or

‘‘per time’’). Growth rates are typically expressed per minute or per hour for bacteria

in the laboratory, but more often per day in environmental systems and for other micro-

organisms, since growth rates are lower.

Returning to equation (11.1) and rearranging yileds

ln X

 ln X

¼ mt

Exponentiating each side gives

lnðX

¼ e

¼ X

ð11:4Þ

There are thus three forms of the exponential growth equation: the differential form

[equation (11.3)]; the logarithmic form [equation (11.1), which gives a straight line on a

semilog plot]; and the exponential form [equation (11.4), useful for calculating the bio-

mass concentration at a time in the near future].

Equation (11.3) is the most fundamental form of the three. This is because, unlike the

other two, it is not dependent on an assumption that m is constant. It simply states that the

rate of biomass increase is proportional to the speciﬁc growth rate and to the current

amount of biomass.

Assuming again that m is constant, suppose that we look at X

after one doubling time,

when it would have doubled from its starting value to 2X

¼ 2X

¼ X

2 ¼ e

ln 2 ¼ mt

Thus, the relationship between speciﬁc growth rate and doubling time is

m ¼

ln 2

ð11:5Þ

At the minimum doubling time, the maximum speciﬁc growth rate will be achieved.

The symbol

mm (read ‘‘mu-hat’’) or m

max

is commonly used for this term. Using E. coli’s

minimum t

value of 20 min, for example:

mm ¼

ln 2

20 min



60 min

3ln2

¼ 2:08 h

1

¼ 50 day

1

GROWTH 319

Example 11.6 What is the meaning of the speciﬁc growth rate, m?

Answer It is the rate of change of biomass concentration in a batch culture (one with

no inﬂow or outﬂow) at any instant, divided by the concentration at that point in time

[equation (11.2)]. Suppose that m ¼ 0:05 h

1

in a culture that at a certain time has a

concentration of 100 mg/L. From equation (11.3):

¼ mX ¼ 0:05 h

1

 100 mg=L ¼ 5mg=L  h

In other words, at that instant, the concentration would be increasing at the rate of 5 mg/L

per hour. At a later time, the concentration might reach 200 mg/L, and then

¼ mX ¼

0:05

 200

¼ 10

L  h

So, as the concentration increased, the rate of increase also increased, even though the

speciﬁc growth rate itself did not change.

The doubling time is easily computed from m by rearranging equation (11.5):

ln 2

0:693

0:05 h

1

¼ 13:86 h

Using equation (11.4), the biomas s concentration at a future time for an exponentially

growing culture can be calculated based on the present concentration and the speciﬁc

growth rate. As an example, starting with a single cell with a mass of 10

12

gin1L

of medium, and using the maximum speciﬁc growth rate of E. coli, after 10 hours the

biomass concentration would be

¼ X

¼ 10

12

g=L  e

ð2:08=hÞð10 hÞ

 1mg=L

However, this equation predicts that after 2 days, the biomass of the same culture would

be 2 10

g, or 2 10

metric tons. This is about 4000 times the mass of Earth;

obviously, this exponential growth model has its limits! In the next section we show

how an improved model can be developed.

11.7.2 Batch Growth Curve

A batch system is one in which all nutrients are present at the beginning and are not

resupplied—there is no inﬂow or outﬂow, except perhaps for aeration. The ﬂask of med-

ium above in which E. coli was growing is a good example. In nature, a fresh deposit of

cow manure on soil or the death of a ﬁsh in a pond would represent ‘‘batches’’ of nutrients

made available at one time.

As the E. coli example demonstrated, exponential growth cannot continue for very long

in a batch system. Depletion of substrates and/or buildup of inhibitory products will soon

lead to decreasing speciﬁc growth rates. Thus, much of the time, microorganisms are

likely to be growing at rates that are far below their

mm value. Equation (11.3) can still

be used to describe this curve, but since m 6¼ constant, it can no longer be integrated to

give equations (11.1) and (11.4) .

320 QUANTIFYING MICROORGANISMS AND THEIR ACTIVITY

Figure 11.22 shows a typical growth curve for a newly inoculated batch laboratory cul-

ture. For convenience, it can be broken up into the ﬁve phases shown and is discussed

below. A similar response would be expected for other batch cultures.

Lag Phase The transfer of a small inoculum of ‘‘seed’’ cells into a fresh batch of med-

ium generally involves new conditions for the organisms, such as different organic sub-

strates, form of nitrogen, pH, oxygen tension, and/or salt concentration. It will frequently

take some time, referred to as a lag, for the cells to become adapted to this new environ-

ment before they start growing exponentially. This may involve production of enzymes to

metabolize new organic substrates or utilize nutrients in another form, or simply replace-

ment of cell constituents that had been depleted previously. Depending on how big a

change is required and how rapidly the organism can respond, the lag period may vary

from seconds to many hours.

Some scientists differentiate between a lag period, during which there is no growth

ðm ¼ 0Þ, and the lag phase, which lasts until exponential growth begins. For a short

time between these two points, the culture may appear to be growing at a rate that is linear

in a plot of X vs. t. This sometimes is referred to as an arithmetic growth phase.How-

ever, it should be noted that increases in biomass during exponential growth start off very

slowly, and unless very precise and sensitive measurements are made, this may mimic a

lag or arithmetic growth period.

Exponential Growth Phase If an appropriate medium has been provided, once the

organisms adapt to their new environment they will grow exponentially (m ¼ c, where

c is a constant greater than 0). Plenty of substrate and nutrients are available, and no harm-

ful products of metabolism have yet accumulated. Thus, m will approach the maximum

speciﬁc growth rate

mm for the organism in that particular environment (in Section

11.7.6 we discuss some factors that affect

mm). However, this phase can last for only a

relatively short period of time before the ideal conditions deteriorate.

Cell Density

Time

Lag

Declining

Rate of Growth

Exponential

Growth

Decay

Stationary

Figure 11.22 Typical batch bacterial growth curve.

GROWTH 321

Declining Rate of Growth Phase An old adage states that ‘‘all good things must come

to an end,’’ and this certainly applies to unrestricted growth within a batch reactor. Even-

tually, one or more factors (usually, substrate depletion, but perhaps product buildup, pH

shift, nutrient deﬁciency, and/or something else) trigger an inevitable decline in cell

growth rate. Biomass still increases, but at a continually decreasing rate (m > 0, but

decreasing).

Most oft en this decrease in m is the resu lt of substrate depletion. This is commonly

modeled by Monod kinetics, in which

m ¼

S þ K

ð11:6Þ

where S is the substrate concentration (mg/L) and K

is the half-saturation coefﬁcient

(mg/L). The half-saturation coefﬁcient is the substrate concentration that allows for

half of the maximum speciﬁc growth rate, as can be seen in Figure 11.23. At high sub-

strate concentrations,

ðS  K

Þ;

S þ K

 1; and m 

mm ð11: 7Þ

At low substrate concentrations,

ðS  K

Þ;

S þ K



; so m 

S ð11:8Þ

When S ¼ K

m ¼

mm ð11:9Þ

At low S [equation (11.8)], the Monod expression approaches ﬁrst order; that is, the

rate is proportional to S to the ﬁrst power ðS

¼ SÞ. Because S is often low in environmental

Figure 11.23 Monod kinetics.

322

QUANTIFYING MICROORGANISMS AND THEIR ACTIVITY

systems, growth-related kinetic expressions are sometimes approximated by ﬁrst-order

equations of the form kS, where k is a constant for that system. However, at high substrate

concentrations [equation (11.7) ], Monod kinetics approach zero order (since S

¼ 1, the

rate is unaffected by substrate concentration). Table 11.5 provides some half-saturation

and othe r coefﬁcients. (Although the discussion so far has focused on the carbon and

energy source as the limiting nutrient, the same approach can be used for other require-

ments, such as dissolved oxygen, as indicated in the table; see ‘‘Multiple Substrates’’ in

Section 11.7.4.)

Note that the Monod equation has the same form as the Michaelis–Menten equation

used to describe enzyme kinetics (Section 5.3.1). However, unlike Michaelis–Menten

kinetics for enzymes, which can be derived from fundamental chemical kinetic principles,

the Monod expression for growth is empirical— an approximate ﬁt to observed results.

Example 11.7 Assuming that an organism grows according to Monod kinetics with a

maximum growth rate of 10 per day and a half-saturation coefﬁcient of 5 mg/L, how

fast would it be growing if the substrate concentration were 100 mg/L? What if it dropped

to 20 mg/L?

Answer At S ¼ 100 mg/L:

m ¼

þ S

¼ 10 day

1



100

5 þ 100

¼ 10

100

105



¼ 9:52 day

1

TABLE 11.5 Examples of Monod Kinetic Coefﬁcients

mm K

Organism Substrate

(



C) (day

1

) (mg/L) (mg/L) (day

1

)

Mixed heterotrophic Sewage COD 20 6 20 0.2 0.67 0.05

bacteria

Mixed heterotrophic Monoethanol amine 20 8.8 9  0.5 0.03

bacteria

Sphaerotilus natans Glucose 20 6.5 10 0.01 0.53 0.05

Citrobacter sp. Glucose 20 9.2 5 0.15 0.55 0.15

Escherichia coli Glucose 37 35 2  

Lactose 37 20 20

Pseudomonas putida Benzene 20 8.4 0.4  1.2

0.5

Toluene 20 8.9 0.2 1.3

1.4

Ammonium-oxidizing Ammonium-N 20 0.8 1.0 0.5 0.24

n.a.

bacteria

Nitrite-oxidizing Nitrite-N 20 0.8 1.3 0.7

bacteria

mm, maximum growth rate at indicated temperature ðTÞ; K

, half-saturation coefﬁcient for the energy source; K

half-saturation coefﬁcient for dissolved oxygen; Y, yield (COD basis); b, decay coefﬁcient.

Substrate used as energy source. COD, chemical oxygen demand (see Section 13.1.3).

Note that yields can be greater than 1.0, since during growth the organism incorporates oxygen and nutrients as

well as the benzene or toluene carbon and hydrogen.

Based on International Association on Water’s Activated Sludge Model No. 1 (ASM1); ASM1 lumps

ammonium oxidation with nitrite oxidation. A typical value for the decay coefﬁcient for autotrophs was not

available.

GROWTH

323

At S ¼ 20 mg/L:

m ¼

þ S

¼ 10 day

1



5 þ 20

¼ 10



¼ 8:00 day

1

Note that the growth rate is over 95% of the maximum with 100 mg/L of substrate, and is

still 80% of the maximum when the substrate concentration is 20 mg/L.

Stationary Phase Once the substrate is sufﬁciently depleted, growth nearly stops

ðm  0Þ. This plateau in biomass concentration is referred to as the stationary phase.

Actually, the cells may still be growing slowly, but this is counterbalanced by the loss

in mass through decay (the next phase).

Decay Phase If a pers on stops eating, he or she gradually loses weight. This is also true

of microorganisms. Their mass decreases, or decay occurs, if substrate is unavailable. The

cell continues to carry on some metabolic proce sses, and if there are no external sources

of energy, it must utilize internal ones. This is calle d endogenous metabolism. It will at

ﬁrst be based on storage materials but will then progress to include nonessential cell com-

ponents, and eventually, essential ones. (The need to regenerate these components can be

one cause of a lag phase when organisms from an old culture are transferred to fresh med-

ium.) At some point, sufﬁcient damage may be done so that the cell cannot recover and is

thus no longer viable.

Energy is also used for cell maintenance during growth. Thus, the observed or net

speciﬁc growth rate ðm

Þ actually represents a higher true growth rate combined with

some decay. Decay is often modeled as a ﬁrst-order reaction with respect to biomass,

with the decay coefﬁcient constant for a given system. In the past, the symbol k

was

often used for this coefﬁcient, but now b is more common, so that

¼ m  b ð11:10Þ

Like growth rate, the decay coefﬁcient has units of inverse time. Although it can

vary considerably, in many wastewater treatment and other systems the decay coefﬁcient

may have a value of around 0.05 day

1

. This means that about 5% of the biomas s

will decay away each day. [Actually, the amount that decays in 1 day is

ð1  e

ð1 day Þð0:05 day

1

Þ100% ¼ 4:88%. This is less than 5% becau se as the amount

of biomass present decreases, so does the amount that decays.]

Specialized resting stages, such as spores or cysts, can be formed by some microorgan-

isms. By greatly slowing down cell metabo lism, they decr ease decay rates, sometimes

very dramatically.

Overall Equation The growth equation (11.3) can now be rewritten incorporating

Monod kinetics [equation (11.6)] and decay [equation (11.10)]:

¼ m

¼ðm  b ÞX

S þ K

 b



X ð11:11Þ

324 QUANTIFYING MICROORGANISMS AND THEIR ACTIVITY

This expression will approximate the batch growth curve after the lag phase. The expo-

nential phase occurs when S  K

, so that m 

mm. As substrate is depleted, m decreases

and the culture enters the declining growth rate phase. Once m  b, then m

 0 and the

culture enters stationary phase. Finally, as S decreases further, m < b, so that m

< 0 and

the decay phase occurs. An example is shown in Figure 11.24.

Example 11.8 If

mm ¼ 10 day

1

, K

¼ 5 mg/L, and b ¼ 0:05 day

1

, at what substrate

concentration will net growth equal zero?

Answer

S þ K

 b ¼ 0

S ¼ K

mm  b

¼ 5mg=L

0:05 day

1

10 day

1

 0:05 day

1



¼ 0:025 mg=L

If the substrate concentration falls below this value, the culture will be in the endogen-

ous phase.

11.7.3 Death, Viability, and Cryptic Growth

In Section 11.7.2 we described the batch growth curve based on biomass (Figure 11.22). If

cell numbers are used instea d, the curve will have a similar shape. However, cell number

100

0510

Time (hours)

Biomass Concentration (mg/L)

Figure 11.24 Batch growth curve using equation (11.11), with

mm ¼ 10 day

1

, K

¼ 5 mg/L,

Y ¼ 0:6 (explained below), b ¼ 0:1 day

1

, and starting values of S and X ¼ 120 and 5 mg/L,

respectively.

GROWTH 325

does not decrease because of decay. It will, on the other hand, decrease as a result of cell

death. Thus, this ﬁnal phase is also sometimes referred to as the death phase.

Note that as decay does not necessarily lead to death, so death, or loss of viability, does

not necessarily lead to decay. Also like decay, death does not necessarily occur only in the

ﬁnal phase; rather, it becomes more apparent there because it is not masked by high

growth rates. Despite these similarities, the mechanisms of death (such as predation,

lysis, acute toxicity, and lethal mutations) and decay are potentially very different.

When using biomass as a measure of microorganism concentration, viability may be of

importance (although it is often ignored). In this sense, viability refers to the fraction or

percent of the biomass that is still ‘‘alive,’’ or capable of growth. Thus, if we consider X

as the living biomass and X

as the dead biomass, the total biomass ðX

Þ and viability ðvÞ

will be

¼ X

þ X

v ¼

ð11:12Þ

The ‘‘growth’’ equation for each fraction can be considered using the same subscripts

for the coefﬁcients (although only m

is a real growth rate). First, the rate of change of the

concentration of the living organisms is dependent on their rate of growth minus their rate

of death:

¼ m

 m

The rate of change of the concentration of the dead biom ass is dependent on the rate of

formation from the death of living cells:

¼ m

What is typically observed is the rate of change of the total biomass:

¼ m

ð11:13Þ

However, the rate of change of the total biomas s can also be expressed as the sum of the

rates for the two fractions:

¼ m

ð11:14Þ

The right-hand side of equation (11.14) also makes sense, in that only the living organ-

isms can grow to produce more total biomass. However, this means that equations (11.13)

and (11.14) can be set equal:

¼ m

or m

¼ m

ð11:15Þ

326 QUANTIFYING MICROORGANISMS AND THEIR ACTIVITY

Substituting in equation (11.12) yields

¼ vm

Thus, the growth rate observed for the total biomass of a culture with a 10% viability is

only 10% of the actual growth rate of the living organisms. This means that the living

organisms are in fact growing 10 times faster than the apparent overall growth rate,

in order to produce all of both the living and the dead biomass.

If cells die and lyse (or lyse and die), their cell constituents become available as

substrates for other microorganisms. This is sometimes referred to as cryptic growth.

11.7.4 Substrate Utilization

In many cases, to the environmental engineer and scientist it might actually be the sub-

strate that is of greater direct concern than the microorganisms. For example, in biological

wastewater treatment, the ‘‘substrates’’ are the constituents of the waste that are to be

removed. Similarly, in polluted aquifers or streams, the contaminants may be the sub-

strates. In such cases, the growth of the microorganisms themselves may be of only

secondary interest.

Microorganisms may utilize substrates as a source of cellular constituents, to supply

energy, and/or as an electron acceptor. Thus, the rate of substrate utilization often is con-

sidered to be proportional to the rate of growth:

/mX ð11:16Þ

where the negative sign indicates that substrate is decreasing and the Monod expression

[equation (11.6)] is frequently used for m.

Yield Coefﬁcient The proportionality coefﬁcient for equation (11.16) is called the yield,

Y. It is often reasonable to expect that for a given amount of substrate utilized, a given

amount of biomass can be formed. Thus, Y has units of biomass produced per mass of

substrate utilized, and the equation can be rewritten as

¼

X ð11:17Þ

The expression m=Y, referred to as the speciﬁc substrate utilization rate, is given the

symbol U, or sometimes k or q:

U ¼

ð11:18Þ

Using Monod kinetics gives

¼

S þ K

GROWTH 327