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

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Since both

mm and Y are considered to be constants for a speciﬁc system, we also can deﬁne

a new coefﬁcient, the maximum speciﬁc substrate utilization rate:

max

The value of Y can vary tremendously, depending on both the substrate and the micro-

organism. For many carbohydrates, common heterotrophic bacteria have yields of about 0.5

to 0.6 g of dry biomass produced per gram of carbohydrate utilized. A yield value can be

greater than 1.0, and in fact typically is so for hydrocarbons as well as for the oxygen used

as an electron acceptor. On the other hand, the yield for an autolithotrophic nitrifying bac-

teria growing on nitrite may be 0.05 or less. Various units may also be used; for example,

the biomas s and organic substrate could be expressed on a basis of dry weight, carbon, or

chemical oxygen demand (see Section 13.1.3).

The yield described here is the true yield. The observed yield or actual amount of

biomass production will be lower. This is because some of the biomass produced will

be lost through decay. On the other hand, some of the substrate removed may simply

be adsorbed, or even precipitated, rather than utilized. For this reason, some prefer the

term speciﬁc substrate removal rate,orspeciﬁc substrate uptake rate, rather than

utilization.

Equation (11.18) can be rearranged as m ¼ Y

. Subtracting b from both sides and

remembering that m  b ¼ m

gives

¼ YU  b ð11:19Þ

Multiple Substrates It is common to focus on one substrate when examining microbial

growth. In reality, of course, organisms need many substrates: an energy source; an

electron acceptor; sources of carbon, nitrogen, phosphorus, and all the other essential

elements; and perhaps organic growth factors. Most of these will normally be present

in great excess of microbial needs, and hence can be ignored. Baron Justus von Liebig

noted this for plants in 1840, leading to Liebig’s law of the minimum: that the nutrient

in shortest supply will limit growth. For microorganisms, Monod kinetics (discussed

in Section 11.7.2) are generally used to describe this common situation, in which one

substrate is or becomes limiting. However, what if the concentrations of two (or more)

substrates are sufﬁciently low that they both will limit the amount and rate of growth?

One empirical approach used to describe this problem is an interactive, multiplicative

Monod model. For the case of two potentially limit ing substrates, A and B:

m ¼

þ K

ð11:20Þ

Additional substrate terms can be added in the same way, as needed. Note that if

 K

, the second term approaches 1, and the equation reduces to the basic Monod

expression [equation (11.6)]. This can be used to give a quantitative deﬁnition to the

term limiting substrate, as one that is not present in sufﬁcient concentr ation to allow

growth at more than some percentage (perhaps 90 or 95%) of the maximum rate.

328 QUANTIFYING MICROORGANISMS AND THEIR ACTIVITY

Example 11.9 Suppose that a particular microorganism’s growth can be described using

the multiplicative Monod model. Given the substrate concentrations and kinetic coefﬁ-

cients below, what would its growth rate be? Would either of the substrates be considered

limiting (using <90% of the maximum rate as the criterion)?

Answer

¼ 25 mg=L glucose

¼ 10 mg=L ammonium-N

¼ 5:0mg=L

¼ 0:50 mg=L

mm ¼ 6 :0 day

1

m ¼

þ S

¼ 6:0

5:0 þ 25



0:50 þ 10



¼ð6:0Þð0:833Þð0:952Þ¼4:8 day

1

The glucose concentration allows growth at only 83% of the maximum (limiting by the

stated criterion), whereas the ammonium-N concentration allows growth at 95% of the

maximum (not limiting, even though it is at a lower concentration than the gluc ose).

Equation (11.20) for microbial growth has the same form as equation (5.38), which

described enzyme kinetics for two substrates. However, whereas the enzymatic equation

is mechanistic (derived from enzyme kinetics), the growth equation is again empirical.

A somewhat different problem is the presence of multiple substrates fulﬁlling the same

nutritional need. In laboratory systems, microbiologists are able to study the uptake of a

single substrate by a pure culture of bacteria. However, in most environmental systems,

there are likely to be several utilizable organic substances present, for example, and

perhaps several electron acceptors and nitrogen sources as well. Will an organism use

only one at a time, or two or more simultaneously?

This turns out to be a very complex question, and the answer depends on the speciﬁc

substrates and organism involved. On a practical level, this is often addressed by using some

single general measure of organic matter, such as BOD or COD (see Section 13.1.3),

rather than trying to account for the actual mechanisms involved. Electron acceptors

are more likely to be used sequentially, with oxygen preferred when it is available. Howe v er ,

the ability to use a speciﬁc electron acceptor depends on the particular organism.

Of course, there are usually many different microorganisms present as well, not a pure

culture of a single species. This, similarly, is commonly handled by using a general mea-

sure of biomass that lumps all or many different species together. Although these simpli-

ﬁcations severely compromise any mechanistic basis for the models used, they have still

been found in practice to provide useful information.

11.7.5 Continuous Culture and the Chemostat

Although some microbial systems can be represented as batches (Section 11.7.2), many

others have ﬂows of material entering and leaving, includin g streams, lakes, groundwater

GROWTH 329

aquifers, and most biological wastewater treatment plants. Such continuous culture sys-

tems receive inputs of new substrates, but also lose substrates and even organisms in the

outﬂow. Thus, to model a continuous culture, in addition to knowing the reactions that are

occurring, it is necessary to keep track of the inputs to and outputs from the system. This

is usually done by means of a mass balance.

A general mass balance equation for a particular component, A, in a continuous ﬂow

system (Figure 11.25) can be written as

rate of change of mass

in system ¼ðrate of mass

inputÞðrate of mass

outputÞ

þðrate of mass

production through reactionsÞ

or if M

is used for the mass of A and R

for the reaction term, then

¼ðrate of M

inputÞðrate of M

outputÞþR

For a continuous-ﬂow ðQÞ system of volume V, in which concentration ðC ¼ M=VÞ of

component A is measured in the inﬂuent and efﬂuent (subscripts i and e, respectively) this

can be expressed as

dðVCÞ

¼ðQ

ÞðQ

ÞþðVR

Þð11:21Þ

Note that in a batch system, Q

¼ Q

¼ 0 and V ¼ constant, so that

¼ R

Thus, from equations (11.11) and (11.17), for biomass and substrate, respectively, we

see that

¼ m

X ð11:22Þ

¼

X ð11:23Þ

Chemostat One particular type of laboratory continuous culture system in which certain

constraints are imposed is known as a chemostat (Figure 11.26). It is of great utility as a

research tool but is also of interest because many observations about its functioning can be

generalized to other continuous culture systems, such as activated sludge wastewater

treatment.

, C

V, C

Figure 11.25 Schematic of a continuous culture system.

330

QUANTIFYING MICROORGANISMS AND THEIR ACTIVITY

One requirement for a chemostat is that Q

¼ Q

¼ Q ¼ constant. If the inﬂuent and

efﬂuent ﬂows are equal and constant, the chemostat volume will also be constant. It is also

assumed that the substrate concentration in the inﬂuent ðS

Þ will be constant and that this

ﬂow will contain no biomass ðX

¼ 0Þ.

Another require ment for a chemostat is that it must be completely mixed (indicated by

the ‘‘propeller’’ in Figure 11.26), so that the concentration of each constituent in every

drop of water in the reactor is equivalent to its concentration in every other drop. A con-

sequence of complete mixing is that the concentration in the efﬂuent will be equal to the

concentration in the reactor ðX

¼ X; S

¼ SÞ, since a drop leaving the reactor is equiva-

lent to a drop inside the reactor. This means that if the inﬂuent concentration is 1000 mg/L

of substrate, for example, and the efﬂuent concentration is 5 mg/L, the concentration

everywhere within the reactor is 5 mg/L. Thus, a microorganism in this reactor ‘‘sees’’

only 5 mg/L of substrate, never the 1000 mg/L being fed to it. This concept may seem

difﬁ cult to grasp at ﬁrst, but keep in mind that every drop entering the reactor (at 1000 mg/L)

is instantaneously mixed and diluted throughout the reactor volume, and that further, the

substrate in it is being consumed by the microorganisms present.

The general mass balance equation [equation (11.21)] can now be rewritten for

biomass in the chemostat as

dðVXÞ

¼ Q

 Q

þ VR

¼ 0  QX þ Vm

¼

X þ m

¼ m





¼ðm

 DÞX ð11:24Þ

where D ¼ Q=V: D has units of (tim e)

1

and is referred to as the dilution rate. It repre-

sents the number of times the contents of the reactor is replaced per unit of time. The

inverse of D, referred to as the hydraulic residence time (HRT), indicates the amount

of time the water (and hence any soluble materials) spends in the reactor. It is widely

used in environmental engineering and science for a variety of reactor types and systems

and is often represented by the symbol y, so that y ¼ V= Q ¼ 1=D.

= 0

= Q

V, X, S

= X

= S

= Q

Figure 11.26 Schematic of a chemostat.

GROWTH 331

For substrate, the mass balance equation is

dðVSÞ

¼ Q

 Q

þ VR

¼ QS

 QS

 V



¼ DðS

 S

Þ



ð11:25Þ

One characteristic that makes the chemostat so interesting is that once it is inoculated

with microorganisms that can grow in the system and utilize the substrate, it will naturally

move toward a stable condition referr ed to as steady state. Steady state occurs when the

concentrations in the reactor no longer change, that is, when dX=dt ¼ 0 and dS=dt ¼ 0. If

the microorganisms grow faster ðdX=dt > 0Þ, the substrate concentration drops

ðdS=dt < 0Þ, and this slows the growth back down. If the growth is too slow

ðdX=dt < 0Þ, organisms wash out of the system faster than they are growing; this drop

in biomass allows the substrate to increase ðdS=dt > 0Þ, and this leads to faster growth.

Thus, the biomass concentration grows to the point at which it uses up substrate at the

same rate that it is added.

At steady state,

¼ðm

 DÞX ¼ 0 ð11:26Þ

and X does not change. This can occur under two conditions: when m

 D ¼ 0 ðm

¼ DÞ

or when X ¼ 0 (no biomass in the system). If X 6¼ 0, substituting for m

and using

SS for

the steady state value of S, gives

SS þK



 b ¼ D ð11:27Þ

Thus, for the steady-state chemostat reactor, the net growth rate is equal to the dilution

rate. Since D ¼ QN; the dilution rate can be controlled merely by controlling Q for a

given reactor volume. Thus, the biological growth rate can be controlled directly by the

chemostat operator. A similar relationship holds for some wastewater treatment processes,

such as activated sludge (Section 16.1.3), in which the net growth rate can be control led

by the amount of sludge wasting. This gives the process operator a means of direct control

over the biology of the process.

Equation (11.27) can be solved for the efﬂuent substrate concentration:

SS ¼

ðD þ bÞ

mm ðD þ bÞ

ð11:28Þ

Note that since

SS is a function of only constants, it too must be a constant, conﬁrming

that dS=dt ¼ 0:

¼ DðS



SSÞ



¼ 0

332 QUANTIFYING MICROORGANISMS AND THEIR ACTIVITY

Solving for

XX gives

XX ¼

DðS



SSÞY

However, at steady state, m

¼ m b ¼ D (if

XX 6¼ 0), so m ¼ D þ b. Substitution

then gives

XX ¼

DðS



SSÞY

D þ b

ð11:29Þ

Considering equation (11.26) and its meaning shows one value of the chemostat as a

research tool. Once set up and inoculated, a chemostat will automatically run to a steady

state in which X and S will be constant. If the setup conditions are reasonable, so that

XX 6¼ 0, the net growth rate will be equal to the dilution rate ðm

¼ DÞ. Since D ¼ Q=V,

this means that any desired growth rate up to almost

mm can be chosen simply by adjusting

the ﬂow rate ðQÞ in a reactor of constant volume V.

Equation (11.28) shows further that the steady-state value of substrate will depend on

the dilution rate but will be independent of the inﬂuent substrate concentration. A higher

leads to a higher steady-state biomass concentration

XX [equation (11.29)], not a higher

SS. Thus, by selecting Q, chemostat operators can select the efﬂuent substrate concentra-

tion; then choosing the inﬂuent substrate conce ntration allows them to pick the biomass

concentration. This type of relationship holds true similarly for the activated sludge

process.

It is also possible to calcul ate a minimum steady-state substrate concentration ð

min

that can be approached in a chemostat. This will occur when m

¼ D ¼ 0 (Fig ure 11.27).

Substituting into equation (11.28) yields

min

mm  b

min

D = µn = µ − b

Figure 11.27 Steady-state substrate minimum,

min

, in a chemostat.

GROWTH 333

This value can only be approached, since if D ¼ 0, then Q ¼ 0, and the system becomes a

batch, not a chemostat.

On the other hand, if D is too high, the organisms cannot grow fast enough to maintain

themselves in the reactor (Figure 11.28). At this critical dilution rate ðD

Þ, or washout

rate,

XX ¼ 0 and thus

SS ¼ S

. In other words, as D (and hence, growth rate) increases,

also increases until it reaches the inﬂuent concentration. Since it cannot go any higher,

neither can growth rate, and the culture washes out. The highest net growth rate that

can be approached (at steady state) is thus

¼ m

þ K

 b

Determining Microbial Kinetic Coefﬁcients in Chemostats One use of chemostats is

for determining the microbial kinetic coefﬁcients

mm and K

. For this purpose, chemostats

are run several times at different dilution rates and the steady-state substrate concentration

is determined for each. A plot of

SS vs. D is then made (Figure 11.28). Note in the ﬁgure

that at values of D > D

XX ¼ 0 and

SS ¼ S

. Also, at low values of D, the importance of

decay ðbÞ leads to a decrease in

XX.

From the development of equation (11.28), we can see that

D þ b ¼

SS þK

ð11:30Þ

Such nonlinear equations are becoming readily usable on computers, but a traditional

linearization method, the Lineweaver–Bur ke plot, may still be useful to know. (It can

100

150

200

02468

10 12

D (da

−1

)

mg/L

Figure 11.28 Theoretical plots of

SS and

XX vs. D in a chemostat. Values used in equations (11.28)

and (11.29) were

mm ¼ 10 day

1

, K

¼ 5 mg/L, b ¼ 0:05 day

1

, Y ¼ 0:5, and S

¼ 200 mg=L.

334

QUANTIFYING MICROORGANISMS AND THEIR ACTIVITY

also be applied to data collected from batch cultures.) This involves inverting equation

(11.30):

D þ b

SS þK

ð11:31Þ

For values of D  b, a plot of 1=D vs. 1=

SS now gives a straight line with an intercept

of 1=

mm and a slope of K

mm (Figure 11.29). The value of

mm is then obtained by taking the

inverse of the y-intercept value. K

can also be calculated from the slope once

mm is known;

however, for imperfect data with their variability, it is usually better (particularly if b is

not very small; see the ‘‘apparent slope’’ in Figure 11.29) to obtain it from the plot of

SS vs.

D (Figure 11.28). This is done by noting the D value that corresponds to

mm, then ﬁnding

the corresponding S value that gives this growth rate (by deﬁnition, K

). (To improve the

quality of the estimates of the coefﬁcients obtained, the slope K

mm can then be calculated

and used to better draw the Lineweaver–Burke plot, with the new intercept giving a better

value of

mm, which can then be used in turn to better estimate K

from the plot of

SS vs. D.

After two or three iterations, the values will stabilize on the best estimate for the data.)

02468

(L/m

)

1/D

1/(D + b)

apparent

slope

(a)

(b)

1/D (day)

Figure 11.29 Lineweaver–Burke plot. In part ðaÞ the same coefﬁcients are used as in Figure 11.28

to generate the theoretical curves. In part ðbÞ the value of b ¼ 0:15, showing its effect on the

nonlinearity of the 1=D plot.

GROWTH 335

One criticism of the use of Lineweaver–Burke plots is that with some data they do not

give very good estimates of the coefﬁcients (particular ly for K

values obtained from the

slope only rather than the

SS vs. D plot). Nonlinear regression is now a preferred approach

if sufﬁcient data points have been collected.

The coefﬁcients Y and b can also be found from chemostat data. Since m

¼ D in a

chemostat, we can rewrite equation (11.19) as

D ¼ YU  b ð11:32Þ

Thus, by plotting D vs. U, Y (the slope) and b (the negative of the y -axis intercept) can be

determined (Figure 11.30). For this purpose, U can be determined as

U ¼

QðS



SSÞ

DðS



SSÞ

11.7.6 Environmental Factors

In addition to the concentrations of various substrates, other environmental factors, such

as temperature, pH, pressure, moisture, and salinity, can clearly affect microbial growth.

These can be viewed as inﬂuencing the kinetic coefﬁcients, such as

mm, K

, Y, and b, that

are used to describe growth and substrate utilization. (In fact, the observation that they

vary is the reason that they have been referred to in this chapter as coefﬁcients, rather

than constants.) The effects of temperature and pH, which are usually the most important

such factors in systems of interest to environmental engineers and scientists, are discussed

brieﬂy below.

Temperature As discussed in Chapter 10, microorganisms are capable of growth at a

wide variety of temperatures. However, each species has a particular minimum and max-

imum temperature at which it can grow, and an optimum temperature within that range.

−1

0510

U (m

substrate/m

biomass · da

)

D (day

−1

)

−b

Figure 11.30 Determining Y and b. The same coefﬁcients were used to generate the plot as in

Figure 11.28, except that b ¼ 0:15.

336

QUANTIFYING MICROORGANISMS AND THEIR ACTIVITY

These three values are referred to as the organism’s cardinal temperatures (Figure 11.31).

The minimum temperature for growth probably represents the point at which the cell’s

lipid-rich membr anes become too rigid to be functional (somewhat like the increase in

viscosity of an automobile’s engine oil in winter), and/or reactions become too slow to

maintain esse ntial cell processes. The maximum temperat ure is reached when critical

cell components are destroye d; in most cases this results from the denaturation (loss

of three-dimensional structure) of proteins. The optimum usually is taken to be the tem-

perature giving the highest

mm value, but it can also refer to a slightly different temperature

(due to changes in yield or decay, for example) at which doubling time is shortest or

biomass accumulation is most rapid.

The minimum and maximum temperatures for growth may form a narrow band or be

widely separated. Similarly, growth rates may drop off rapidly as temperature departs

from the optimum, or there may be a broad plateau of nearly optimal temperatures. How-

ever, the optimum temperature is always closer to the maximum than it is to the mini-

mum. This is because increases in temperature will lead to increases in reaction rates,

speeding growth, up to the point at which the enzymes or other cell constituents become

unstable.

A common way to incorporate temperature effects into models is to use the following

approximation, which is derived from work by von Arrhenius in the nineteenth century.

Based on equation (5.48), the maximum speciﬁc growth rate

at the temperature of

interest (T,in



C) is estimated from the rate

at 20



Cas



T20

ð11:33Þ

where  is a dimensionless (no units) empirical coefﬁcient, often taken to be 1.05 in the

absence of an experimentally derived value. A related rule of thumb that is sometimes

used is that reaction rates (and hence growth) will double for every 10



C increase in tem-

perature. This is called the Q

rule and corresponds to an  value in equation (11.33) of

about 1.07. However, both of these approximations are suitable for mesophilic organisms

only at temperatures between their minimum and optimum; outside this range (such as

can occur in composting) they can give grossly misleading results.

01020304050

T (°C)

µ (

day

−1

)

minimum

maximum

optimum

Figure 11.31 Dependence of microbial growth on temperature for a hypothetical mesophile.

GROWTH 337