PROBLEMS

SUGGESTIONS FOR FURTHER READING

155

(6.2a)1 dX

!-Lne' =Xdt

How Cells Grow

6

For rnicrobes, growth is their most essential response to their physiochernical environ­

ment. Growth is a result of bothreplication and change in cen size. Microorganisms can

grow under a variety of physical, chernical, and nutritional conditions. In a suitable nutri­

ent medium, organisms extract nutrients from the medium and convert them into biologi­

cal compounds. Part of these nutrients are used for energy production and part are used

for biosynthesis and product formation. As a result of nutrient utilization, rnicrobial mass

increases with time and can be described simply by

substrates + cells ~ extracellular products + more cens (6.1)LS+X~LP+nX

Microbial growth is a good example of an autocatalytic reaction. The rate ot growth is di­

rectly related to cen concentration, and cellular reproduction is the normal outcome of thisreaction.

The rate of rnicrobial growth is characterized by the net specific growth rate, de­fined as

6.1. INTRODUCTION

Major Metabolic Pathways154

B. Specific information on interaction ofmetabolism and productformation

CRUEGER,w., ANDA. CRUEGER,Biotechnology. A Textbook o/lndustrial Microbiology (T. D. Brock,

00., English edition), 2d 00., Sinauer Associates, Inc., Sunderland, MA, 1990.

NEWAY,J. O., Fermentation Process Development o/lndustrial Organisms, Marcel-Dekker, Inc.,NewYork,1989.

5.1. Cite the ATP-consurning and ATP-generating steps in glycolysis.

5.2. Briefly specify major functions of the TCA cycle.

5.3. What are the major control sites in glycolysis?

SA. What is the Pasteur effect? Explain in terms of regulation of metabolic flow into a pathway.

5.5. How is glucose synthesized from pyruvate?

5.6. Explain the major functions of the dark and light reaction phases in photosynthesis.

5.7. What are the major differences in photosynthesis between microbes and plants?

5.8. What is transarnination? Provide an example.

5.9. Briefly explain the Crabtree Effect.

5.10. What are the major steps in aerobic metabolism of hydrocarbons? What are the end prodUc1

5.11. What is nitrogen fixation? Compare the aerobic and anaerobic nitrogen fixation mechanis

A. General information on metabolic pathways

ALBERTS,B., D. BRAY,A. JOHNSON,1. LEWlS,M. RAFF,K. ROBERTS,ANDP. WALTER,Essential Cell

Biology: An lntroduction to the Molecular Biology o/ the Cell, Garland Publ., Inc., New York,1998.

BLACK,1. G., Microbiology: Principles and Applications, 3d 00., Prentice Hall, Upper Saddle River,

NJ,1996.

MADIGAN,M. T., 1. M. MARTINKO,ANDJ. PARKER,Brock Biology o/ Microorganisms, 8th 00., Pren-tice Hall, Upper Saddle River, NJ, 1991:

MORAN,L. A., K. G. SCRlMGEOUR,H. R. HORTON,R. S. QcHS, AND1. D. RAWN,Biochemistry, 2d00., Prentice Hall, Upper Saddle River, NJ, 1994.

6.2. BATCH GROWTH

157

Figure 6.1. Diagram of a particle counter

using the electrica! resistance method for

measuring cell number and cell size distribu­

lion. The ratio of volumes of a nonconduCl­

ing particle to the orifice volurne (altered bychanging orifice diameter) detenTUnes the

size of the voltage pulse. (Adapted with per­mission. from D. I. C. Wang and others, Fer­

mentation and Enzyme Technology, John

Wiley & Sons, New York, 1979, p. 64.)

o

E~~u~

Batch Growth

ApertureHole

coIony may not be formed from a single cell. This method (plate counts) is more suitable forbacteria and yeasts and much less suitable for molds. A large number of colonies must be

counted to yield a statistically reliable number. Growth media have to be selected carefully,since some media support growth better than others. The viable count may vary, dependingon the composition of the growth medium. From a single cell, it may require 25 generationsto form an easily observable colony. Unless the correct medium and culture conditions arechosen, some cells that are metabolically active may not form colonies.

In an altemative method, an agar-gel medium is placed in a small ring mounted ona rnicroscope slide, and cells are spread on this rniniature culture dish. After an incubationperiod of a few doubling times, the slide is exarnined with a microscope to count cells.

This method has many of the same limitations as plate counts, but it is more rapid, andcells capable of only lirnited reproduction will be counted.

Another method is based on the relatively high electrical resistance of cells

(Fig. 6.1). Commercial particle counters employ two electrodes and an electrolyte solu­tion. One electrode is placed in a tube containing an orifice. A vacuum is applied to the

inner tube, which causes an electrolyte solution containing the cells to be sucked throughthe orifice. An electrical potential is applied across the electrodes. As cells pass throughthe orifice, the electrical resistance increases and causes pulses in electrical voltage. Thenumber of pulses is a measure of the number of particles; particle concentration is known,since the counter is activated for a predetermined sample volume. The height of the pulseis a measure of cell size. Probes with various orifice sizes are used for different cell sizes.This method is suitable for discrete cells in a particulate-free medium and cannot be usedfor mycelial organisms.

The number of particles in solution can be determined from the measurement of

scattered light intensity with the aid of a phototube (nephelometry). Ught passes through

Sec. 6.2

(6.3)

(6.2b)

Chap.6'How Cells Grow

I dN

IlR= N dt

~net = ~g -kd

156

where IlR is the net specific replicatiolÍ rate (h-I). If we ignore cell death, kd, then we usethe symbol ~'R; and in cases where cell death is unimportant, IlR wil1 equalll' R'

In this chapter we will discuss how the specific growth rate changes with its envi­ronment. First, we wil1 consider growth in batch culture, where growth conditions areconstantly changing.

Batch growth refers to culturing cells in a vessel with an initial charge of medium that isnot altered by further nutrient addition or removal. This form of cultivation is simple andwidely used both in the laboratory and industrially.

6.2.1. Quantifying Cell Concentration

6.2.1.1. Detennining cell number density. A Petroff-Hausser slide or ahemocytometer is often used for direct cell counting. In this method, a calibrated grid isplaced over the culture chamber, and the number of cells per grid square is counted usinga microscope. To be statistically reliable, at least 20 grid squares must be counted and av­eraged. The culture medium should be clear and free of particles that could hide cells orbe confused with cells. Stains can be used to distinguish between dead and live cells. Thismethod is suitable for nonaggregated cultures. It is difficult to count molds under the mi­croscope because of their mycelial nature.

Plates containing appropriate growth medium gelled with agar (Petri dishes) are usedfor counting viable cells. (The word viable used in this context means capable of reproduc­tion.) Culture samples are diluted and spread on the agar surface and the plates are incu- .,.bated. Colonies are counted on the agar surface following the incubation period. The resultsare expres sed in terms of colony-forming units (CFU).1f cells form aggregates, then a single

where X is cell mass concentration (gll), t is time (h), and Iln •• is net specific growth rate

(h-I). The net specific growth is the difference between a gross specific growth rate, ~g

(h-l), and the rate of loss of cell mass due to cell death or endogenous metabolism, kd (h-1).

Microbial growth can also be described in terms of cell number concentration, N, aswell as X. In that case

The quantification of cell concentration in a culture medium is essential for the determina­tion of the kinetics and stoichiometry of rnicrobial growth. The methods used in the quan­tification of cell concentration can be classified in two categories: direct and indirect. Inmany cases, the direct methods are not feasible due to the presence of suspended solids orother interfering compounds in the medium. Either cell number or cell mass can be quan­tified depending on the type of information needed and the properties of the system. Cellmass concentration is often preferred to the measurement of cell number density whenonly one is measured, but the combination of the two measurements is often desirable.

159

l.D =in

J

1

3.0

2.0

3.0

J211'J6 t1

50

40

40

'----------------

20 30Time. Hours

10

10

A o

1..š'21•2

j..V

J..

Batch Growth

116

12••

tu

J..

20 30T_. Hours

Figure 6.2. The time-dependent changes in cell composition and cell size for Azotobac­

ter vinelandii in batch culture are shown. (With pennission, from M. L. Shu1er and H. M.

Tsuchiya, "Cell Size as an Indicator of Changes in Intracellular Composition of Azotobac­

ter vinelandii," Can. J. Microbiol. 31:927, 1975 and National Research Counci1 ofCanada, Ottawa.)

Sec. 6.2How Cells Grow158

6.2.1.2. Determining cell mass concentration. Direet methods. Detenni­

nation of cellular dry weight is the most commonly used direct method for determiningcen mass concentration and is applicable only for cells grown in solids-free medium. ffnoncellular solids, such as molasses solids, cellulose, or com steep liquor, are present, thedry weight measurement will be inaccurate. Typically, samples of culture broth are cen­trifuged or filtered and washed with á"buffer solution or water. The washed wet cell massis then dried at 80°C for 24 hours; then dry cell weight is measured.

Paeked eell volume is used to rapidly but roughly estimate the cell concentration ina fermentation broth (e.g., industrial antibiotic fermentations). Fermentation broth is cen.trifuged in a tapered graduated tube under standard conditions (rpm and time), and thevolume of cells is measured.

Another rapid method is based on the absorption of light by suspended cells in sam~ple culture media. The intensity of the transmitted light is measured using a spectrometer.Turbidity or optieal density measurement of the culture medium provides a fast, inexpen­sive, and simple method of estimating cell density in the absence of other solids or light·absorbing compounds. The extent of light transmission in a sample chamber is a functionof cell density and the thickness of the chamber. Light transmission is modulated by boabsorption and scattering. Pigmented cells give different results than unpigmented ones,Background absorption by components in the medium must be considered, particularlyabsorbing dissolved species are taken into cells. The medium should be essentially p,cle free. Proper procedure entails using a wavelength that minimizes absorptionmedium components (600- to 700-nm wavelengths are often used), "blanking" agaimedium, and the use of a ca1ibration curve. The ca1ibration curve relates optical densil(OD) to dry-weight measurements. Such ca1ibration curves can become nonlinear at hiOD values (> 0.3) and depend to some extent on the physiological state of the cells.

lndireet methods. In many fermentation processes, such as mold fermentations,rect methods cannot be used. In such cases indirect methods are used, which aremainly on the measurement of substrate consumption and/or product formation duringcourse of growth.

Intracellular components of cells such as RNA, DNA, and protein can be measas indirect measures of cell growth. During a batch growth cycle, the concentrationsthese intracellular components change with time. Figure 6.2 depicts the variation oftain intracellular components with time during a batch growth cycle. Concentratio~RNA (RNA/cell weight) varies significantly during a batch growth cycle; however,and protein concentrations remain fairly constant. Therefore, in a complex medium.concentration can be used as a measure of microbial growth. Cellular protein measments can be achieved using different methods. Total amino acids, Biuret, Lowry (flreagent), and Kjeldah1 nitrogen measurements can be used for this purpose. Tota!acids and the Lowry method are the most reliable. Recently, protein determinationfrom several vendors have been developed for simple and rapid protein measurem'

the culture sample, and a phototube measures the light scattered by cells in the sample.The intensity of the scattered light is proportional to cell concentration. This method givesbest results for dilute cell and particle suspensions.

161

CIo

15

cO.I-l" O

0.01

D'QthOr

Deeline

l

10

Time (h)

5

Batch Growth

Growth Phase

..J 10'E

....•.

~CD

.Q

~ 101Z

D'CII'rotlon

tlI EXPGj'ntlol ,1, Stotlonory

10'O

"ii010"1:1'

,g

Figure 6.3. Typical growth curve for a bacterial population, Note lhat the pbase of

growth (sbown bere for cell number) depends On the parameter used to monitor growth.

The lag phase occurs immediately after inoculation and is a period of adaptation ofcells to a new environment. Microorganisms reorganize their molecular constituents whenthey are transferred to a new medium. Depending on the composition of nutrients, newenzymes are synthesized, the synthesis of some other enzymes is repressed, and the inter­

nal machinery of cells is adapted to the new environmenta] conditions. These changes re­flect the intracellular mechanisms for the regulation of the metabolic processes discussedin Chapter 4. During this phase, celI mass may increase a little, without an increase in cellnumber density. When the inoculum is small and has a low fraction of cells that are vi­able, there may be a pseudolag phase, which is a result, not of adaptation, but of small in­oculum size or poor condition of the inoculum.

Low concentration of some nutrients and growth factors may also cause a long lagphase. For example, the lag phase of Enterobacter aerogenes (formerly Aerobacter aero­genes) grown in glucose and phosphate buffer medium increases as the concentration of

Mg2+,which is an activator of the enzyme phosphatase, is decreased. As another example,even heterotrophic cells require CO2 fixation (to supplement intermediates removed from

keyenergy-producing metaboIic cycles during rapid biosynthesis), and excessive spargingcan remove metabolically generated CO2 too rapidly for cellular restructuring to be ac­compIished efficiently, particularly with a small inoculum.

Sec. 6.2

(6.4)

How Cells Grow

1 'f' O ATP lucilerase l' hUCl enn + 2 + ----> 19 t

160

However, many media contain proteins as substrates, which limits the usefulness of thisapproach.

The intracellular ATP concentration (mg ATP/mg cells) is approximately constant fora given organism. Thus, the ATP concentration in a fermentation broth can be used as a mea­sure ofbiomass concentration. The method is based on luciferase activity, which catalyzesoxidation ofluciferin at the expense of oxygen andATP with the emission oflight.

6.2.2. Growth Pattems and Kinetics in Bateh Culture

When a liquid nutrient medium is inoculated with a seed culture, the organisms setively take up dissolved nutrients from the medium and convert them into biomass. Acal batch growth curve includes the following phases: (l) lag phase, (2) logariexponential growth phase, (3) deceleration phase, (4) stationary phase, and (5)phase. Figure 6.3 describes a batch growth cycle.

When oxygen and luciferin are in excess, totallight emission is proportional to total ATPpresent in the sample. Photometers can be used to detect emitted light. Small concentra­tions of biomass can be measured by'this method, since very low concentrations of ATP(10-12g ATPIl) can be measured by photometers or scintillation counters. The ATP con­tent of a typical bacterial cell is 1mg ATP/g dry-weight cell, approximately.

Sometimes, nutrients used for cellular mass production can be measured to followmicrobial growth. Nutrients used for product formation are not suitable for this purposeiNitrate, phosphate, or sulfate measurements can be used. The utilization of a carbonsource or oxygenuptake rate can be measured to monitor cellular growth when cell massis the major product.

The products of cell metabolism can be used to monitor and quantify cellular.growth. Certain products produced under anaerobic conditions, such as ethanol and lacticacid, can be related nearly stoichiometrically to microbial growth. Products must be eithetgrowth associated (ethanol) or mixed growth associated (lactic acid) to be correlated withmicrobial growth. For aerobic fermentations, CO2 is a common product and can be relatedto rnicrobial growth. In some cases, changes in the pH or acid-base addition to control pHcan be used to monitor nutrient uptak.e and microbial growth. For example, the utilizationof ammonium results in the release of hydrogen ions (H+) and therefore a drop in pH. Theamount of base added to neutralize the H+ released is proportional to ammonium uptakeand growth. Similarly, when nitrate is used as the nitrogen source, hydrogen ions are re-..

moved from the medium, resulting in an increase in pH. In this case, the amount of acidadded is proportional to nitrate uptake and therefore to microbial growth.

In some fermentation processes, as a result of mycelial growth or extracellular posaccharide formation, the viscosity of the fermentation broth increases during the coof fermentation. If the substrate is a biodegradable polymer, such as starch or cellulothen the viscosity of the broth decreases with time as biohydrolysis continues. Changesthe viscosity of the fermentation broth can be correlated with the extent of micro' .growth. Although polymeric broths are usually non-Newtonian, the apparent viscomeasured at a fixed rate can be used to estimate cell or product concentration.

/'

163

(6.6)

(6.7)

(6.8)'t' _lnZd--/lR

ln Z 0.693'td=--=--/lne' /lne'

8atch GrowthSec. 6.2

determined from either celI number or cell mass would be the same. Since the nutrient

concentrations are large in this phase, the growth rate is independent of nutrient concen­tration. The exponential growth rate is first order:

dX-=/lne,X, X=Xo at t=O (6.5)dt

lntegration of eq. 6.5 yields

where 'td is the doubling tirne of celI mass.Similarly, we can ca!culate a doubling tirne based on celI numbers and the net spe­

cific rate of replication. Thus

Xln-=" t or X= X eU''''

X rne" oo

where X and Xo are ceH concentrations at time t and t = O.The time required to double the microbia! mass is given by eq. 6.6. The exponential

growth is characterized by a straight line on a semilogarithrn plot of ln X versus time:

where t'd is tbe doubling time based on the replication rate. During ba!anced growth 'tdwill equa! t'd, since the average cell composition and size will not change with tirne.

The deceleration growth phase foHows the exponentia! phase. ln this phase, growthdecelerates due to either depletion of one or more essential nutrients ar the accumulationof toxic by-products of growth. For a typica! bacteria! culture, these changes occur over avery short period of time. The rapidly changing environment results in unbalancedgrowth. During unba!anced growth, celI composition and size will change and 'td and t'dwill not be equa!. ln the exponentla! phase, the celIular metabolic control system is set toachieve maximum rates of reproduction. In the deceleration phase, the stresses induced bynutrient depletion or waste accumulation cause a restructuring of the ceH to increase theprospects of cellular surviva! in a hostile environment. These observable changes are theresult of the molecular mechanisms of repression and induction that we discussed inChapter 4. Because of the rapidity of these changes, celI physiology under conditions ofnutrient limitation is more easily studied in continuous culture, as discussed later in thischapter.

The stationary phase starts at the end of the deceleration phase, when the netgrowth rate is zero (no celI division) or when the growth rate is equa! to the death rate.Even though the net growth rate is zero during the stationary phase, celIs are stilI metabol­ica!ly active and produce secondary metabolites. Primary metabolites are growth-relatedproducts and secondary metabolites are nongrowtb-related. In fact, the production ofcertain metabolites is enhanced during tbe stationary phase (e.g., antibiotics, some hor-

o tnones) due to metabolite deregulation. During tbe course of the stationary phase, one or'i .. inore of the folIowing phenomena may tak:eplace:

How Cells Grow

Figure 6.4. lnfluence of Mg2+ co:lion on tbe lag phase in E. aerogenes

(Witb permission, from A. C. R.C. Hinshelwood, Growth. FunctiOn.

Regulation in Bacterial Cells. OxfordLondon, 1966, p. 55.)

o ~,---'---'"O S 10 IS 20 2S

Concentration of magnesium sulphate. mg!I

...

j"óI

162

14

12

The age of the inoculum culture has a strong effect on the length of lag phase. The

age refers to how long a culture has been maintained in a batch culture. Usually, the lagperiod increases witb the age of tbe inoculum. In some cases, there is an optimal inoculumage resulting in minimum lag period. To minimize the duration of tbe lag phase, cellsshould be adapted to tbe growth medium and conditions before inoculation, and celIsshould be young (or exponential phase cells) and active, and the inoculum size should be

large (5% to 10% by volume). The nutrient medium may need to be optimized and certaingrowth factors included to minimize the lag phase. Figure 6.4 shows an example of varia­tion of the lag phase witb MgS04 concentration. Many commercial fermentation plants

rely on batch culture; to obtain high .productivity from a fixed plant size, tbe lag phasemust be as short as possible.

Multiple lag phases may be observed when the medium contains more tban one car-bon source. This phenomenon, known as diauxic growth, is caused by a shift in metabolic

patbways in the middle of a growth cycle (see Example 4.1). After one carbon sourceexhausted, the cells adapt their metabolic activities to utilize tbe second carbon source.The first carbon source is more readily utilizable than the second, and tbe presence ofmore readily available carbon source represses the syntbesis of tbe enzymes required fortbe metabolism of the second substrate.

The exponential growth phase is a!so known as the logarithmic growth phase.

this phase, the cells have adjusted to their new environment. After this adaptation pen,cells can multiply rapidly, and cell mass and cell number density increase exponentiwith ume. This is a period of balanced growth, in which a!l components of a celI growthe same rate. That is, tbe average composition of a single cell remains approximalconstant during this phase of growth. During ba!anced growth, the net specific growth r:

.'

-kdtdX == -kdK or K == Ksoedt

(6.13)

(6.14)

lÍPYp/s ==- M

Y _ LU'x/oz ----

d02

LU'

YX1S =-- (6.11)M

At the end of the batch growth period, we have an apparent growth yield (or ob­served growth yield). Because culture conditions can a!ter patterns of substrate utilization,the apparent growth yield is not a true constant. For exarnple, with a compound (such asglucose) that is both a carbon and energy souree, substrate may be consumed as:

M == MassimiJation + Massirnilated +Mgrowth energy + Mmaintenance

mtobiomass i,to an e,OIgy (6.12)extJ'acellu)arproduct

In the section on continuous culture, we wilI differentiate between the true growthyield (which is constant) and the apparent yield. Yield coefficients based on other sub­strates or product formation may be defined; for exarnple,

properties arnong individua!s in a population. With a narrow distribution, cell death wil1

occur nearly simultaneously; with a broad distribution, a subfraction of the populationmay survive for an extended period. It is this subfraction that would dominate the reestab­lishment of a culture from inoculum derived from stationary or death-phase cultures.Thus, using an old inoculum may select for variants of the origina! strain having alteredmetabolic capabilities.

To better describe growth kinetics, we define some stoichiometrica!ly related pa­rarneters. Yield coefficients are defined based on the amount of consumption of anothermateria!. For exarnple, the growth yield in a fermentation is

165

For organisms growing aerobica!ly on glucose, YXIS is typical1y 0.4 to 0.6 g/g for mostyeast and bacteria, while YXI02 is 0.9 to 1.4 g/g. Anaerobic growth is less efficient, and theyield coefficient is reduced substantially (see Fig. 6.5). With substrates that are more orless reduced than glucose, the value of the apparent yield coefficient will change. Formethane, YXIS would assume values ofO.6 to 1.0 g/g, with the cOrresponding YX!O? decreas­ing to about 0.2 g/g. In most cases the yield of biomass on a carbon-energy source is 1.0 ±0.4 g biomass per g of carbon consumed. Table 6.1 lists some exarnples of YXIS and YXQ2for a variety of substrates and organisms.

A maintenance coefficient is used to describe the specific rate of substrate uptakefor cellular maintenance, or

m == - [dS~t]m (6.15)

However, during the stationary phase where little external substrate is available, endoge­neous metabolism of biomass components is used for maintenance energy.

Sec.6.2 Batch GrowthHow Cells Grow

N==N e-k;tsordN ==-k'Ndt d

where Ns is the concentration of cells at the end of the stationary phase and k!d is Uleorder death-rate constant. A plot of ln N versus t yields a line of slope -k!d' Duril1,death phase, cells may or may not lyse, and the reestablishment of the culture may besible in the early death phase if cells are transferred into a nutrient-rich medium. 111the death and stationary phases, it is important to recognize thal there is a

164

During the stationary phase, the cel! catabolizes cel!ular reserves for new building blocksand for energy-producing monomers. This is called endogenous metabolism. The celJmust a!ways expend energy to maintain an energized membrane (i.e., proton-motiveforce) and transport of nutrients and for essentia! metabolic functions such as motility aru:!repair of darnage to cellular structures. This energy expenditure is ca!led maintenance en·

ergy. The appropriate equation to describe the conversion of cell mass into maintenance,energy or the loss of cell mass due to celllysis during the stationary phase is

1. Total cel! mass concentration may stay constant, but the nurnber of viable cel!s maydecrease,

2. Celllysis may occur and viable cell mass may drop, A second growth phase mayoccur and cel!s may grow on Iysis products of lysed cells (cryptic growth),

3. Cells may not be growing but may have active metabolism to produce secondarymetabolites, Cellular regulation changes when concentrations of certain metabolites(carbon, nitrogen, phosphate) are low. Secondary metabolites are produced as are­sult of metabolite deregulation.

where kd is a first-order rate constant for endogeneous metabolism, and Kso is the ce

mass concentration at the beginning of the stationary phase. Because S is zero, Ilgis ze.in the stationary phase.

The reason for termination of growth may be either exhaustion of an essentia! nUlent or accurnulation of toxic products. ff an inhibitory product is produced and acculates in the medium, the growth rate wm slow down, depending on inhibitor producti,and at a certain level of inhibitor concentration, growth wi11stop. Ethanol productionyeast is an exarnple of a fermentation in which the product is inhibitory to growth. Diltion of toxified medium, addition of an unmetabolizable chernica! compound complexiwith the toxin, or simultaneous removal of the toxin would alleviate the adverse effectsthe toxin and yield further growth.

The death phase (or decline phase) follows the stationary phase. However, SOI

cell death may start during the stationary phase, and a clear demarcation between thtwo phases is not always possible. Often, dead cells Iyse, and intracellular nutrientsleased into the medium are used by the living organisms during stationary phase. Atend of the stationary phase, because of either nutrient depletion or toxic product acclati on, the death phase begins.

The rate of death usua11yfollows first-order kinetics:

How Cells GroW 167

gjg

1.50

1.18

1.46

1.11

1.32

1.35

0.85

1.46

0.97

0.98

0.62

0.97

0.37

0.48

0.31

0.70

0.46

0.61

0.42

0.56

0.53

0.44

0.29

0.20

0.19

0.17

YX1O:la

gjg-C

1.03

1.32

1.05

1.01

1.28

1.08

0.95

1.12

1.25

0.88

0.62

1.16

0.46

0.49

0.43

0.90

0.70

1.30

0.93

1.01

1.28

1.09

1.34

1.06

0.80

0.75

YXIS

gjmol

149.2

95.2

76.1

72.7

91.8

77.4

68.4

81.0

90.0

53.2

29.7

41.8

16.6

17.9

10.5

21.0

16.8

31.2

22.5

12.2

15.4

13.1

16.2

12.8

9.6

gjgSubstrate

Batch Growth

Organism

Maltose 0.46

Mannitol 0.52

Fructose 0.42

Glucose 0.40

Glucose 0.51

Glucose 0.43

Glucose 0.38

Glucose 0.45

Glucose 0.50

Ribose 0.35

Succinate 0.25

Glycerol 0.45Lactate 0.18

Pyruvate 0.20Acetate 0.18

Acetate 0.36

Acetate 0.28

Ethanol 0.68

Ethanol 0.49

Methanol 0.38

Methanol 0.48

Methanol 0.41

Methane 1.01

Methane 0.80

Methane 0.60

Methane 0.56 9.0

• yXA:>2 is the yield factor relating grams of cells forrned per gram of O2 consumed.With permission, from S. Nagai in Advances in Biochemical Engineering. vol. 11, T. K. Ghose, A. Fiechter,

and N. Blakebrough, eds., SpriJiger-Verlag, NewYork, p. 53,1979.

Enterobacter aerogenes

qp=a~g+~ (6.18)

Lactic acid fermentation, xanthan gum, and some secondary metabolites from cellculture are exarnples of mixed-growth-associated products. Equation 6.18 is a

Candida ulilis

Pseudomonas fluorescensCandida ulilis

Pseudomonas fluorescens

Klebsiella sp.

Methylomonas sp.

Pseudomonas sp.

Methylococcus sp.

Pseudomonas sp.

Pseudomonas sp.Pseudomonas methanica

qp = ~ = constant (6.17)

Many secondary metabolites, such as antibiotics (for example, penicillin), are non­growth-associated products.

3. Mixed-growth-associated product formation takes place during the slow growth andstationary phases. In this case, the specific rate of product formation is given by thefollowing equation:

TABLE 6.1 Summary ofYield Factors for Aerobic Growth of Different Microorganismson Various Carbon Sources

Candida ulilis

Penicillium chrysogenum

Pseudomonas fluorescens

Rhodopseudomonas spheroides

Saccharomyces cerevisiae

Enterobacter aerogenes

AlIlIeI'Obic

Aerobic

YpUCOSC- 58.2Imol-'

10

Substrate c:oncentration (mM)o

800

:::-I1.;i

1 dP =YpIX~gqp = X dt

Figure 6.5. Aerobic and anaerobic growth yields of srreptocOCcus faecalis with glucosess substrate. (With permission, from B. Atldnson and F. Mavituna. Biochemical Engineer­

ing and Biotechnology Handbook, Macmillan, lne., New York, 1983.)

The production of a constitutive enzyme is

product.2. Nongrowth-associated product formation takes place during the stationarY

when the growth rate is zero. The specific rate of product formation is

1. Growth-associated products are produced simultaneously with microbial gr'The specific rate of product formation is proportional to the specific rate of

~g' Note that ~g differs from ~, the net specific growth rate, when endoge'metabolism is nonzero.

166

Cellular maintenance represents energy expenditures to repair darnaged Cen

components, to transfer some nutrients and products in and out of cells, for motilily,to adjust the osmolarity of the cells' interior volume. Microbial growth, product.Jo:tion, and substrate utilization rates are usually expressed in the form of specific rates (e.normalized with respect to X), since bioreactions are autocatalytic. The specific ratesused to compare the effectiveness of various fermentation schemes and biocatalysts.

Microbial products can be classified in three major categories (see Fig. 6.6):

How Cells GroW169

(6.19)

(6.20)

dN =(Il~-k;)Ndt

II' =Ae-E,IKT k' -A' -EdiKTr-R , d - e

Batch GrowthSec. 6.2

6.2.3. How Environmental CondltionsAffect Growth Kinetics

where Ea and Ed are activation energies for growth and thermal death. The activation en­ergy for growth is typically 10 to 20 kcal/mol, and for thermal death 60 to 80 kcal/mol.That is, thermal death is more sensitive to temperature changes than microbial growth.

Temperature also affects product formation. However, the temperature optimum forgrowth and product formation may be different. The yield coefficient is also affected bytemperature. In some cases, such as single-cell protein production, temperature optimiza­tion to maximize the yield coefficient (Yx;s) is critical. When temperature is increasedabove the optimum temperature, the maintenance requirements of cells increase. That is,the maintenance coefficient (see eq. 6.15) increases with increasing temperature with anactivation energy of 15 to 20 kcal/mol, resulting in a decrease in the yield coefficient.

Temperature also may affect the rate-limiting step in a fermentation process. Athigh temperatures, the rate of bioreaction might become higher than the diffusion rate,and diffusion would then become the rate-limiting step (for example, in an immobilizedcell system). The activation energy of molecular diffusion is about 6 kcal/mol. The activa­tion energy for most bioreactions is more than 10 kcal/mol, so diffusionallimitations mustbe carefully considered at high temperatures. Figure 6.7 depicts a typical variation ofgrowth rate with temperature.

Hydrogen-ion concentration (pH) affects the activity of enzymes and therefore themicrobial growth rate. The optimal pH for growth may be different from that for productformation. Generally, the acceptable pH range varies about the optimum by ±1 to 2 pHunits. Different organisms have different pH optima: the pH optimum for many bacteriaranges from pH = 3 to 8; for yeast, pH = 3 to 6; for molds, pH = 3 to 7; for plant cells, pH= 5 to 6; and for animal cells, pH = 6.5 to 7.5. Many organisms have mechanisms to

At high temperatures, the thermal death rate exceeds the growth rate, which causes a netdecrease in the concentration of viable cells.

Both Il'R and k'd vary with temperature according to the Arrhenius equation:

The patterns of microbial growth and product formation we have just discussed are in­fluenced by environmental conditions such as temperature, pH, and dissolved-oxygenconcentration.

Temperature is an important factor affecting the performance of cells. According totheir temperature optima, organisms can be classified in three groups: (1) psychrophiles

(Top! < 20°C), (2) mesophiles (Top! = from 20° to 50°C), and (3) thermophiles (Top! > 50°C). As the temperature is increased toward optimal growth temperature, the growth rateapproximately doubles for every 10°C increase in temperature. Above the optimal temper­ature range, the growth rate decreases and thermal death may occur. The net specific repli­cation rate can be expressed by the following equation for temperature above optimallevel:

Time

(c)

ltz'"/p,Time

(b)

eellGlueose

Time

eoncentrationeoneentration

(h)

(gIl)(gIl)

O

1.25100

9

2.45 97

16

5.1 90.4

23

10.5 76.9

30

22 48.1

34

33 20.6

36

37.5 9.38

40

41 0.63

Tíme(a)

X

orp

a. Calculate the maximumnet specificgrowthrate.b. Calculate the apparent growth yield.c. What maximum cell concentrationcould one expect if 150 g of glucose were

with the same size inoculum?Solution A plot of ln Xversus t yields a slope of 0.1 h.

a) =lnX2-1nX1 = ln 37.5-ln5.1 '= 0.lh-1linet t -t 36-162 1

!:J.X 41-1.25b) Y = __ = ------ '= 0.4 g cellslg substrate

!'>S 0.625-100

c) Xmu = Xo+ YSo = 1.25+ 0.4(150)= 60.25 g cellsll

Figure 6.6. Kínetiepattemsof'1l!0wthandproductforrnationin batehfermentations:(a)growth-associatedproduetforrnation,(h)mixed-growth-associatedproduetfonnation,and(e)nongrowth-associatedproductfonnation.

Luedeking-Piret equation. If a = O, the product is only non-growth associated, and

if ~ = O, the product would be only growth associated, and consequently lX wouldthen be equal to YpJX•

168

Some of the concepts concerning growth rate and yield are illustrated in Ex­

ample 6.l.

Example 6.1.

A strain of mold was grown in a batch culture on glucose.and the following data weretained.

8

.--....•. ,'\

\\\

IIIIIIIf

IJ9

",,­.;,,­

,,-"III/IIIII Wlth

/ AdoptotionIIII

1.0

0.9Oimensionless

SpecificGrowth

Rote,p.lp.rn 0.5

0.4

0.3

0.20.1

O

171

6pH

Figure 6.8. Typica! variation of specific growth rate with pH. The units are aIbitrary.With some microbia! cultures, it is possible to adapt cuItures to a wider range of pH va!­ues if pH changes are made in smal1 increments from culture transfer to transfer.

DO level is below the critical DO concentration. In this case, another medium component(e.g., glucose, ammonium) becomes growth-extent lirniting. For example, with Azotobac­ter vinelandii at a DO = 0.05 mgll, the growth rate is about 50% of maximum even if alarge amount of glucose is present. However, the maximum amount of cells formed is notdeterrnined by the DO, as oxygen is continual1y resupplied. If glucose were total1y con­sumed, growth would cease even if DO = 0.05 mg/l. Thus, the extent of growth (mass of

cel!s formed) would depend on glucose, while the growth rate for most of the culture pe_riod would depend on the value of DO.

The critical oxygen concentration is about 5% to 10% of the saturated DO concen­tration for bacteria and yeast and about 10% to 50% of the saturated DO concentration formold cultures, depending on the pellet size of molds. Saturated DO concentration in waterat 25°C and 1 atm pressure is about 7 ppm. The presence of dissolved salts and organicscan alter the saturation value, while increasingly high temperatures decrease the saturationvalue.

Oxygen is usualIy introduced to fermentation broth by sparging air through thebroth. Oxygen transfer from gas bubbles to cells is usually limited by oxygen transferthrough the 1iquid film surrounding the gas bubbles. The rate of oxygen transfer from thegas to liquid phase is given by

N02 =kLa(C*-CL)=OlR (6.21)

where kL is the oxygen transfer coefficient (cmlh), a is the gas-liquid interfacial area(cm2/cm3), kLa is the volumetric oxygen transfer coefficient (h-1), c* is saturated DO

Sec, 6.2 Batch GrowthCHow Cells Grow

Figure 6.7. Arrhenius plot of growth rale

of E. coli B/r. Individua! data points are

marked with corresponding degrees Ce1sius.

E. coU B/r was grown in a rich cornp1ex

medium (e) and a glucose-mineral salts

medium (O). (With pennission, after S. L.

Herendeen, R. A. VanBoge1en, and F. C.

Neidhardt, "Leve1s of Major Protein of Es­

chericma coli durlng Growth at Different

Temperatures," 1. Bacteriol. 139: 195, 1979,as drawn in R. Y. Stanier and others, The

3.5 crobial World, 5th ed., Pearson Education,Upper Saddle River, NI, 1986,207.)

3.3 3.41,OOOIT (OK)

39°

3.2

170

3.0

4.0

maintain intracellular pH at a relatively constant level in the presence of f1uctuations inenvironmental pH. When pH differs from the optimal value, the maintenance-energy 1'(\­

quirements increase. One consequence of different pH optima is that the pH of th,medium can be used to select one organism over another.

In most fermentations, pH can vary substantially. Often the nature of the nitroglsource can be important. If ammonium is the sole nitrogen source, hydrogen ions are re-.leased into the medium as a result of the rnicrobial utilization of ammonia, resulting in adecrease in pH. If nitrate is the sole nitrogen source, hydrogen ions are removed from ttIé.medium to reduce nitrate to ammonia, resulting in an increase in pH. Also, pH can chan,because of the production of organic acids, the utilization of acids (particularly aminlacids), or the production of bases. The evolution or supply of CO2 can alter pH greatlysome systems (e.g., seawater or animal cell culture). Thus, pH control by means ()fbuffer or an active pH control system is important. Variation of specific growth rate wipH is depicted in Fig. 6.8, indicating a pH optimum.

Dissolved oxygen (DO) is an important substrate in aerobic fermentations andbe a lirniting substrate, since oxygen gas is sparingly soluble in water. At high cel! ccentrations, the rate of oxygen consumption may exceed the rate of oxygen supply, le,ing to oxygen lirnitations. When oxygen is the rate-limiting factor, specific growthvaries with dissolved-oxygen concentration according to saturation kinetics; below acal concentration, growth or respiration approaches a first-order rate dependence ondissolved-oxygen concentration.

Above a critical oxygen concentration, the growth rate becomes independent ofdissolved-oxygen concentration. Figure 6.9 depicts the variation of specific growthwith dissolved-oxygen concentration. Oxygen is a growth-rate-lirniting factor when

0.2

.,1:?,tIo~~<.!l

(a) ~. rin"ondil

173

(6.25)

Batch Growth

Growth rate varies nearly Iinearly with the oxygen transfer rate under oxygen-transferlimitations. Among the various methods used to overcome DO Iimitations are the use ofoxygen-enriched air or pure oxygen and operation under high atmospheric pressure (2 to3 atm). Oxygen transfer has a big impact on reactor design (see Chapter 10).

The redox potential is an important parameter that affects the rate and extent ofmany oxidative-reductive reactions. In a fermentation medium, the redox potential is acomplex function of DO, pH, and other ion concentrations, such as reducing and oxidiz­ing agents. The electrochemical potential of a fermentation medium can be expressed bythe following equation:

E E' 2.3RTI D 2 RTI +h = 0+-- og"-02+ .3- og (H )4F F

Sec. 6.2

where the electrochemical potential is measured in miIIivolts by a pHlvoltmeter and P02 isin atmospheres.

The redox potential of a fermentation media can be reduced by passing nitrogen gas orbythe addition ofreducingagents such as cysteineHCI or Na2S, Oxygen gas can be passed orsome oxidizing agents can be added to the fermentation media to increase the redox potential.

Dissolved carbon dioxide (DC02) concentration may have a profound effect on per­formance of organisms. Very high DC02 concentrations may be toxic to some cells. Onthe other hand, cells require a certain DC02 level for proper metaboIic functions. The dis­solved carbon dioxide concentration can be controlled by changing the CO2 content of theair supply and the agitation speed.

The ionic strength of the fermentation media affects the transport of certain nutri­ents in and out of cells, the metaboIic functions of cells, and the solubiIity of certain nutri­ents, such as dissolved oxygen. The ionic strength is given by the following equation:

1 2

I="2I.CjZ; (6.26)

where C is the concentration of an ion, Z; is its charge, and I is the ionic strength of themedium.

High substrate concentrations that are significantly above stoichiometric require­ments are inhibitory to cellular functions. Inhibitory levels of substrates vary dependingon the type of cells and substrate. Glucose may be inhibitory at concentrations above 200gIl (e.g., ethanol fermentation by yeast), probably due to a reduction in water activity.Certain salts such as NaCI may be inhibitory at concentrations above 40 gIl due to highosmotic pressure. Some refractory compounds, such as phenol, toluene, and methanol, areinhibitory at much lower concentrations (e.g., I gIl). Typical maximum noninhibitory con­centrations of some nutrients are glucose, 100 gIl; ethanol, 50 gIl for yeast, much less formost organisms; arnmonium, 5 gll; phosphate, 10 gIl; and nitrate, 5 gIl. Substrate inhibi­tion can be overcome by intermittent addition of the substrate to the medium.

6.2.4. Heat Generation by Microbial Growth

About 40% to 50% of the energy stored in a carbon and energy source is converted to bio­logical energy (ATP) during aerobic metaboIism, and the rest of the energy is released asheat. For actively growing cells, the maintenance requirement is low, and heat evolution isdirectly related to growth.

(6.

Figure 6.9. Growth-rate dependence on00 for (a) Azotobacter vinewndii, a strietly

aerobic organisrtl, and (b) E. coli, which is

faculative. E. coli"grows anaerobical1y at a

rate of about 70% of ils aerobic growth in

nunimal medium. .-J.1* J.1-J.1m

J.1:~-J.1:--

(With permission, fromJ. Chen,A. L. Tan­nabili, and M. L. Shu1er, Biotechnol. Bioerig,27: 151, 1985, and John Wuey & Sons, Inc••

NewYork.)

1.0 1.2

IlgX = kLa(C* -CL)YX102

00 _~_o.o

g~o

o

0.1 0.2 0.3 0.40.0. (mg/l)

0.2 0.4 0.6 0.80.0. (mg/U

oo

0.95'- o

(b) €. f!l!i.1.00

0.10

0.90p.*

0.85

172

dX

di' = YXI02kLa(C*-CL)

or

How Cells Grow

concentration (mg/l), CL is the actual DO concentration in the broth (mgll), and the Nf>].

the rate of oxygen transfer (mg Oi1·h). A1so, the term oxygen transfer rate (OTR) is us'The rate of oxygen uptake is denoted as OUR (oxygen uptake rate) and

IlXOUR =Q02X=-g­

YXI02

where Q is the specific rate of oxygen consumption (mg Oig dw cells·h), YXI02is

yie1d coe~cient on oxygen (g dw cells/g O2), and X is cell concentration (g dw cellsll).When oxygen transfer is the rate-1imiting step, the rate of oxygen consumption

equal to the rate of oxygen transfer. If the maintenance requirement of O2 is negligicompared to growth, then

175Batch GrowthSec. 6.2

6.3.1. Introduction

Here QGR is in units of kcal/h, while Q02 is in millimoles of Oih.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). Theability to estimate heat-removal requirements is essential to proper reactor design.

In the previous section we described some key concepts in the growth of cultures. Clearly,we can think of the growth dynarnics in terms of kinetic descriptions. It is essential to re-

. call that cellular composition and biosynthetic capabilities change in response to newgrowth conditions (unbalanced growth), a!though a constant cellular composition and ba!­anced growth can predorninate in the exponentia! growth phase. If the decelerating growthphase is due to substrate depletion rather than inhibition by toxins, the growth rate de­creases in relation to decreasing substrate concentrations. In the stationary and deathphases, the distribution of properties among individuals is important (e.g., cryptic death).Although these kinetic ideas are evident in batch culture, they are equa!ly evident and im­portant in other modes of culture (e.g., continuous culture).

Clearly, the complete description of the growth kinetics of a culture would involverecognition of the structured nature of each ceH and the segregation of the culture into in­dividua! units (cells) that may differ from each other. Models can have these same attrib­utes. A chernicaHy structured model divides the cell mass into components. If the ratio ofthese components can change in response to perturbations in the extracellular environ­ment, then the model is behaving analogously to a cell changing its composition in re­sponse to environmenta! changes. Consider in Chapter 4 our discussion of cellularregulation, particularly the induction of whole pathways. Any of these metabolic re­sponses results in changes in intracellular structure. Furthermore, if a model of a culture isconstructed from discrete units, it begins to rnirnic the segregation observed in realcultures. Models may be structured and segregated, structured and nonsegregated, un­structured and segregated, and unstructured and nonsegregated. Models containing bothstructure and segregation are the most realistic, but they are a!so computationallycomplex.

The degree of realism and complexity required in a model depends on what is beingdescribed; the modeler should always choose the simplest model that can adequately de­scribe the desired system. An unstructured model assumes fixed cell composition, whichis equivalent to assurning balanced growth. The balanced-growth assumption is valid pri­marily in single-stage, steady-state continuous culture and the exponentia! phase of batchculture; it fails during any transient condition. How fast the cell responds to perturbationsin its environment and how fast these perturbations occur deterrnine whether pseudoba!­anced growth can be assumed. If cell response is fast compared to external changes and ifthe magnitude ofthese changes is not too large (e.g., a 10% or 20% variation from initialconditions), then the use of unstructured model s can be justified, since the deviation from

6.3. QUANTIFYING GROWTH KINETICS

(6.27a)

Combustionof mlcrobiolcells

6Hc(kJ/g cel!)

Microbiol cells+

Mlcroblol

growth ondresplrotion

I/YH(kJ/g cel!)

CO2 + H20

where VL is the liquid volume (1)and X is the cell concentrauon (gll).In aerobic fermentations, the rate of metabolic heat evolution can roughly be corre"

lated to the rate of oxygen uptake, since oxygen is the final electron acceptor.

QGR == 0.12Q02

Totol cambustion _ C02 + H~Substrote + ~ 6H_(kJ/g substrate)

174

Figure 6.10. Hea! balance on nucrobial utilization of substrate.

Ml, =Ml +..!-YX1S e YH

where Ml, is the heat of combustion of the substrate (kJ/g substrate), YXlS is the substrate

yield coefficient (g cell/g substrate), Mle is the heat of combustion of cells (kJ/g cells),and l/YH is the metabolic heat evolved per gram of cell mass produced (kJ/g cells).

Equation 6.27a can be rearranged to yield

YX1SYH=----Ml, - YX1SMle

Ml, and Mli:can be determined from the combustion of substrate and cells. Typical

Me values for bacteria! cells are 20 to 25 kJ/g cells. Typical values of YH are glucose, 0.42glkca!; malate, 0.30 glkca!; acetate, 0.21 g/kcal; ethanol, 0.18 glkca!; methanol, 0.12glkca!; and methane, 0.061 g/kcal. Clearly, the degree of oxidation of the substrate hasstrong effect on the amount of heat released.

The total rate ofheat evolution in a batch fermentauon is

1

QGR = VLlloot Xr-H

How Cells Grow

The heat generated during microbia! growth can be calculated using the heat of

combustion of the substrate and of cellular materia!. A schematic of an enthalpy balancefor microbia! utilization of substrate is presented in Fig. 6.10. The heat of combusuon ofthe substrate is equal to the sum of the metabolic heat and the heat of combustion of thecellular material.

6.3.2. Using Unstroctured Nonsegregated Models to PredictSpecific Growth Rate

177

(6.33)

(6.34)

(6.35)

(6.32)

iff S ~ 2Ks

iff S < 2Ks

Ilg=Ilm(l-e-KS)

Ilg= Ilm'

Ilm SIl =-,g 2Ks

Ilg= IlmSn -Ks + sn -llm(1 + KsS-n)-1

Moser equation:

Tessier equation:

Quantifying Growth KineticsSec. 6.3

IlmSIl =-~-

g KsJ + KsoSo +S

where So is the initial concentration of the substrate and Ksa is dimensionless.Other equations have been proposed to describe the substrate-limited growth phase.

Depending on the shape of Il-S curve, one of these equations may be more plausible thanthe others. The following equations are alternatives to the Monod equation:

Blackman equation:

Contois equation: Ilg= IlmS (6.36)KsxX+S

Although the Blackman equation often fits the data better than the Monod equation, thediscontinuity in the Blackman equation is troublesome in many applications. The Tessierequation has two constants (Ilm, K), and the Moser equation has three constants (Ilm, Ks,

n). The Moser equation is the most general form of these equations, and it is equivalent tothe Monod equation when n = 1. The Contois equation has a saturation constant propor­tional to cell concentration that describes substrate-limited growth at high cell densities .According to this equation, the specific growth rate decreases with decreasing substrateconcentrations and eventually becomes inversely proportional to the cell concentration in

the medium (Le., Ilgoe AJ).These equations can be described by a single differential equation as

du b-=Ku·(l-u) (6.37)dS

or

This simple premise is rarely, if ever, true; however, the Monod equation empiri­cally fits a wide range of data satisfactorily and is the most commonly applied unstruc­tured, nonsegregated model of microbial growth.

The Monod equation describes substrate-limited growth only when growth is slowand population density is low. Dnder these circumstances, environmental conditions canbe related simply to S. ff the consumption of a carbon-energy substrate is rapid, then therelease of toxic waste products is more likely (due to energy-spilling reactions). At highpopulation levels, the buildup of toxic metabolic by-products becomes more important.The following rate expressions have been proposed for rapidly growing dense cultures:

IlmS

Ilg = K S S (6.31)so 0+

How Cells Grow

Figure 6.11. Effectof nutrient con,tion on the specific growth rate of E. c,

(With petnussion, from R. Y. Stanier.M. Doudoroff. and E. A. Adelberg,

Microbial World, 5th ed., pearson Ed'

Upper Saddle River, NJ, 1986, p. 192.)

2520

IlmS

Ilg=[(+šs

Glucose coneentralion (!.lM)

o.aL----- --------------------

í<Š

0.6.! l!!~0.4

'i e(.!)0.2

0.0' •.

0151015

176

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 (Le., anincrease in S influences growth rate, while changes in other nutrient concentrations ha'no effect). These kinetics are similar to the Langmuir-Hinshelwood (or Hougen-Watsokinetics in traditional chemical kinetics or Michaelis-Menten kinetics for enzyme retions. When applied to cellular systems, these kinetics can be described by the Mon,

equation:

where !lm is the maximum specific growth rate when S » Ks· ff endogeneous metabolisis unimportant, then Jlnet = Ilg. The constant Ks is known as the saturation constant or ha,velocity constant and is equal to the concentration of the rate-limiting substrate whenspecific rate of growth is equal to one-half of the maximum. That is, Ks = S when 11&

illmax' In general, Ilg = !lm for S »Ks and Ilg = (JlnIKs)S for S < < Ks· The Monod equati:is semiempirical; it derives from the premise that a single enzyme system wilMichaelis-Menten kinetics is responsible for uptake of S, and the amount of that enor its catalytic activity is sufficiently low to be growth-rate limiting.

balanced growth may be small. Culture response to large or rapid perturbations cannot bedescribed 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 (seeChapter 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 limitationson these models. Nonetheless, such models are simple and applicable to some situations

of practical interest.

TABLE 6.2 Constants of the GeneralizedDifferential Specific Growth Rate Equation6.34 for Different Models

(6.42)

(6.43)

179

(6.44)

(6.47)

I!mS11 =----

g K{I+ ~J+s

I!m ...

I1g = ( K)( P J1+-;- 1+ Kp

11m

I1g (1+ ~s )(1+ ;J

Competitive product inhibition:

Noncompetitive iD.hibition:

Noncompetitive product inhibition:

Quantifying Growth Kinetics

Ethanol fermentation from glucose by yeasts is a good exarnple of noncompetitive prod­uct inhibition, and ethanol is the inhibitor at concentrations above about 5%. Other rateexpressions used for ethanol inhibition are

Fo.-<ompeti." oub""" mhibitioo, " o (".s )_K 1+~ +S (6.41)

, K1

Note that eq. 6.41differs from 6.39and 6.40,and K1 in 6.40and 6.41differ. Substrate inhibi­tion rnay be alleviated by slow, intermittent addition of the substrate to the growth medium.

2. Product inhibition: High concentrations of product can be inhibitory for micro­bial growth. Product inhibition may be competitive or noncompetitive, and in some caseswhen the underlying mechanism is not known, the inhibited growth rate is approximatedto exponential or linear decay expressions.

Important examples of the product inhibition rate expression are as follows:

". o(I:~)(1- :. Jwhere Pm is the product concentration at which growth stops, or

_ 11m -PIKp

I1g - ( 1e1+ Ks I (6.45)S

where Kp is the product inhibition constant.

3. lnhibition by toxic compounds: The following rate expressions are used for compet­itive, noncompetitive, and uncompetitive inhibition of growth in analogy to enzyme inhibition.

Competiti" mhitioo, '. o ( •• :]K 1+- +S (6.46)

s K1

Sec. 6.3

. .. I!g(S.)

How Cells Grow

I!mS

I!g = K +S+S2IK1s

I!m

I!g = ( K)( S)1+-;- 1+ K1

ar if K1 ~ K" then:

I!g = I!g (Sl) or I!g (S2) or

abK

Monod

O2lIK,Tessier

O1lIKMoser

1 -lIn1 + lInn/K'{"

Contois

O2l/K",

Noncompetitive substrate inhibition:

where the lowest value of I!g(Sj) is used.

178

"

where U = I!/I!m' S is the rate-limiting substrate concentration, and K, a, and b areconstants. 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 potentialljigrowth-rate limiting is an unresolved question. However, under most circumstancesnoninteractive approach works best:

6.3.2.2. Models with growth inhibitors. At high concentrationsstrate or product and in the presence of inhibitory substances in the medium, growthcomes inhibited, and growth rate depends on inhibitor concentration. The iD.hipattem of microbial growth is analogous to enzyme inhibition. If a single-subslenzyme-catalyzed reaction is the rate-limiting step in microbial growth, then kiconstants in the rate expression are biologically meaningful. Often, the underlying mnism is complicated, and kinetic constants do not have biological meanings and aretained from experimental data by curve fitting.

1. Substrate inhibition: At high substrate concentrations, microbial growthinhibited by the substrate. As in enzyme kinetics, substrate inhibition of growth ma'competitive or noncompetitive. If a single-substrate enzyme-catalyzed reaction is thelimiting step in microbial growth, then inhibition of enzyme activity results in iD.hi".of microbial growth by the same pattem.

The major substrate-inhibition pattems and expressions are as follows:

where kd is the death-rate constant (h-1).

ln 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:

(6.54)!lg =+- {]

T11Ile (h) Glucose (S), gILBiomass (X), gIL&hanol (P), gILO

100 0.50.02 95 1.02.55 85 2.17.510 58 4.820.015 30 7.734.020 12 9.643.025 510.447.530 210.749.0

181Quantifying Growth Kinetics

where Jl is average biomass concentration during t1t, and X~ is about 10.8 gIL, sincegrowth is aImost complete at 30 h. Thus:

Thus,

dX = ~JI_~) (6.55)dt ~'l X~

The integration of eq. 6.55 with the boundary condition X(O) = Xo yieIds the logisticcurve.

X _ Xoekt

I-..&(I-ekt) (6.56)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 gener­ated as a by-product of growth limits X~ (the carrying capacity). Example 6.2 illustratesthe use of the logistic approach.

Example 6.2. Logistic EqUatiOD

Ethanol formation from g1ucose is accomplished in a batch culture of Saccharomyces cere­visiae, and the following data were obtained.

k=..!.. t1X +(1- X)X t1t X~

orJ...dX=k(I_X]Xdt X~

a. By fitting the biomass data to the logistic equation, determine the carrying-capacitycoefficient k.

b. Determine yield coefficients YPIS and Y.\'l.l'.SoIUtiOD

a) Equation 6.55 can be rewritten as:

Sec. 6.3

(6.48)

How Cells Grow

!lmS _--

Uncompetitive inhibition: !lg - -[-K--'-~](--- I )(1+ ~J+5 1+ K1

180

This equation describes the sigmoidal-shaped batch growth curve, and. the value of:_asymptotically reaches to the value of Y~o + Xo. ' .~

Equation 6.52 requires a predetermined knowledge of the maximum cell masS il!',!

particular environment. This maximum cell mass we will denote as X_; it is identical,{

the ecoIogical concept of carrying capacity. Equation 6.53 is implicit in its depende '.onS.

Logistic equations are a set of equations that characterize growth in terms of c

ing capacity. The usual approach is based on a formulation in which the specific

rate is related to the amount of unused carrying capacity:

(KsYx/s + SoYx/s + Xo) ln (.!...) KsYx/s ln{ (Yx/sSo + Xo - X)/Yx,sSol= I!mt(y S +Xo) Xo (Yx/sSo +Xo)x/s o

I!m5 -k~!lg = K +S," s

The integrated form of the rate expression in this phase is

where Xo and 50 are initial values and Yx/s is the cell mass yield based on the limiting ni

trient. Substituting for S in eq. 6.50 yields the following rate expression:

_dX_= __ I!_m_(Y_X_/s_S_o _+_X_o_-_X_)_Xdt (KsYxIS + Yx/s50 + Xo -X)

6.3.2.3. The logistic equation. When pIotted on arithmetic paper, the batch

growth curve assumes a sigmoidal shape (see Fig. 6.3). This shape can be predicted by

combining the Monad 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 = !lm5 Xdt Ks +S

The relationship between microbial growth yieId and substrate consumption is

X-Xo = Yx/s(50 -5)

183

(6.57)

(6.59)

(6058b)

dR = kp = constantdt

Quantifying Growth Kinetics

In terms of growth rate of a mold colony, eq. 6.57 can be expressed as

Sec. 6.3

dM = p47tR2 dR = kp47tR2p (6058a)dt dt

In most practical applications of rnicrobial cultures, the environmental or culture condi­

tions can shift, dramatically leading to changes in cellular composition and biosynthetic

capabilities. These cellular changes are not instantaneous but occur over an observable pe­

riod of time. In this section we examine models that can describe or predict such tirne­

dependent (or transient) changes.

6.3.2.4. Growth models for filamentous organisms. Filamentous organ­

isms such as molds often form microbial pellets at high cell densities in suspension cul­

ture. Cells growing inside pellets may be subject to diffusion limitations. The growth

model s of molds should include the simultaneous diffusion and consumption of nutrients

within the pellet at large pellet sizes. This problem is the same one we face in modeling

the behavior of bacteria or yeasts entrapped in spherical gel particles (see Chapter 9).

Alternatively, filamentous cells can grow on the surface of a moist solid. Such

growth is usually a compIicated process, involving not only growth kinetics but the diffu­

sion of nutrients and toxic metabolic by-products. However, for an isolated colony grow­

ing on a rich medium, we can ignore some of these complications.ln the absence of mass-transfer lirnitations, it has been observed that the radius of a

rnicrobial pellet in a submerged culture or of a mold colony growing on an agar surface

increases linearly with time.

6.3.3. Models forTransient Behavior

6.3.3.1. Models with time delays. The unstructured growth models we have

described so far are lirnited to balanced or pseudobalanced growth conditions. These un­

structured models can be improved for use in dynamic situations through the addition of

time delays. The use of time delays incorporates structure implicitly. It is built on the

prernise that the dynamic response of a cell is dorninated by an internal process with a

time delay on the order of the response time under observation. Other internal processes

dM =yM2/3dt

or

where y= kp(361tp)1I3. Integration of eq. 6058b yields

M=( M6/3 +~J..(~JThe initial biomass, Mo, is usually very small compared to M, and therefore M varies with

~. This behavior has been supported by experimental data.

Ch

15

036028023022022022025

k (h-l)

How Cells Grow

v,Y310,8560-6810-4160,2000,0740,023

10

l1- :Jne

Tlme (h)

!.MIM (h-J)X

xClD-------

./~./" //

Figure 6.12. Logistic growth curve,

f

X(gIL)M(h)

x(g It)

2 0,75 03333 155 02365 3-45 0,1565 625 0,0935 8-65 0,0445 10,00 0,0165 10,55 0,0057

A value of k = 0.24 h-1 wou1d describe most of the data, although it wou1d slightly under.estimate the initial growth rate. Another approach would be to take the log of the abov1

equation to give:

1dX (x)10g-- = log k + log 1--!LXdt X_

and to fit the data to this equation and estimate k from the intercept. In this' case k wouldabout 0.25h-1•

b) The yie1ds are estimated directly from the data as:

-U -(49-0) gPYp/s =-- =- =0.5-6.S (2-100) gS

y = -ÁX -(10.7-0.5) =O.l04gXx/s ÁS (2-100) gS

The above estimate of YXlS is only approximate, as maintenance effects and endogettl

metabolism have been neg1ected.

182

are assumed to be too fast (essentially always at a pseudoequilibrium) or too slow to influ­ence greatly the observed response. By using black-box techniques equivalent to the tradi­tional approach to the control of chemical processes, it is possible to generate transferfunctions that can represent the dynarnic response of a culture. An example of the resultsfrom such an approach are given in Fig. 6.13. However, it should be recognized that suchapproaches are lirnited to cultures with sirnilar growth histories and subjected to qualita­tively sirnilar perturbations.

185

(6.60)

rtixVRX

tOlalbiomass

in reactor

Quantifying Growth Kinetics

Figure 6.14. An idea1ized slretch of the model E coli B/rA growing in a glucose-ammonium saIts

medium with g1ucose or ammonia as the Iimiting nument. At the time shown the cell has just com­

pleted a round of DNA replication and initiated cross-wall fonnation and a new round of DNA repli­cation. Solid lines indicate the flow of materia1, while dashed lines indicate flow of infonnation. The

symbols are: AI' arnmonium ion; Az, glucose (and associated compounds in the cell); W, waste prod­

ucts (COz, HzO, and acetate) formed from energy metaboIism during aerobic growth; P1, amino

acids; Pz, ribonucleotides; P3, deoxyribonucleotides; P4, cell envelope precursors; Ml' protein (bothcytoplastnic and envelope); M2RIl, imrnature "stable" RNA; M2RTM, mature "stable" RNA (r-RNA

and t-RNA-assume 85% r-RNA throughout); MZM, messenger RNA; M3, DNA; M4, nonprotein

part of cell envelope (assume 16.7% peptidoglycan, 47.6% lipid, and 35.7% pOlysaccharide); Ms,g1ycogen; PG, ppGpp; E1, enzymes in the conversion of Pz to P3; Ez, ~, molecules involved in di­

recting cross-wall fonnation and cell envelope synthesis; GLN, glutamine; E4, glutamine synthetase;* indicates that the materia! is present in the externa! environment. (With permission, from M. L.

Shuler and M. M. Domach, in Foundations oj Biochemical Engineering, H. W. Blanch, E. T. Papout­salds, and G. Stephanopoulos, ed., ACS Symposium Senes 207, American Chetnica1 Society,Wash_ington, DC, 1983, p. 93.)

concentrations. A second consideration, closely related to the first, is that the dilution ofinfrinsic concentration by growth must be considered.

The appropriate equation to use in a nonflow reactor is

d[VRCiJ

df

rate of changein amount of iin the reactor

rate offonnation

ofi perunit biomass

based on intrinsicconcentrations

where VR is the tot81 volume in the reactor, X is the extrinsic biomass concentration, andCi is the extrinsic concentration of component i.

Sec. 6,3

If

,.

1210• •Time lh I

-o

i

.S o.. c;

., .1I~' _ EllPItl'i_teIl

e. ,.! .... •••O_II IIlOlMII~-I_o , ..••",_ -- MonocI --I-.. -- - _ s

!.. I·t - --..---~ 1 I , , I 1

iI

How Cells Grow

Figure 6.13. Comparison of predictions from a model derived from a system-ana1ysis per­

spective, predictions from a Monod model, and experiment. The experimenta1 system was a

chemostat for a glucose-limited culture of Saccharomyces cerevisiae operating at a dilution

rate of 0.20 h-1. ln this particular experiment, the system was perturbed with a stepwise in­

crease in feed glucose concentration from 1.0 and 2.0 g 1-1. X is biomass concentration, II is

growth rate, and S is substrate concentration. (With permission, from T. B. Young lIJ and

H. R. Bungay, "Dynamic Ana1ysis of a Microbia1 Process: A Systems Engineering Ap­proach," BiotechnoL Bioeng. 15:377, 1973, and John Wiley & Sons, Inc., NewYork.)

1

6.3.3.2. Chemical/v stNctured models. A much more general approachwith much greater a priori predictive-'power is a model capturing the important kinetic in­teractions among cellular subcomponents. Initially, chernically structured models werebased on two components, but at least three components appear necessary to give good re­sults. More sophisticated models with 20 to 40 components are being used in many labu­ratories. A schematic of one such model is given in Fig. 6.14.

Writing such models requires that the modeler understand the physical system at lilevel of greater detail than that at which the model is written, so that the appropriate as-•.•surnptions can be made. A detailed discussion of such models is appropriate for more ad­vanced texts. However, two important guidelines in writing such models should beunderstood by even the beginning student. The first is that all reactions should be ex­pressed in terms of intrinsic concentrations. An intrinsic concentration is the amount of a:compound per unit cell mass or cell volurne. Extrinsic concentrations, the amount ofcompound per unit reactor volurne, cannot be used in kinetic expressions. Althoughmay seem self-evident, all the early structured models were flawed by the use of extri

184

187

Figure 6.15. Prediction of transient re­

sponse in a continuous-flow system to per­twbations in Iimiting substrate concentration

Orflow rate. Note that the predictions are not

fitted to the data; predictions are completelya priori. (a) Transient glucose concentration

in response to step change in glucose feed

Concentration from 1.0 to 1.88 gII in anaero­

bic Continuous culture of E. coli B/r at dilu­

tion rate of 0.38 h- \ • and ••. resuJts of

two duplicate experimental runs; solid line,

cornputer prediction. (h) Transient glucose

concentration in response to step change in

dilution rate from D == 0.16 to D == 0.55 h-1;

• and ., results oftwo duplicate expeO­

mental runs; solid line, cornputer prediction.(With pennission, from M. M. Ataai and

M. L. ShuJer, Biotechnol. Bioeng. 27: 1051,1985, and John Wi1ey & Sons, Inc., NewYork)

~3 ••(h)

•• 6Tlrne (h)

oO

~'20!:8::s

(5

oO 2

(b)

I(5

Quantifying Growth Kinetics

(a)

Solution When measuring the concentration of A, P, and S*, the natural Ul1Ítsare grams perliter of reactor volurne. These are extrinsic concentrations. The total biomass concentrationwill be

X=A+P+S=A+P (~

since the interna! concentration of S will be low [< O.OOI(A + P)J. By considering S* and S

separately, transients in S can be captured. However, these transients will be fast in compari­son to changes in A and P. The Monod and other unstructured models assume an instanta­neous equilibrium between S* and S.

A number of kinetic descriptions would be defensible. However, let us assume the fol­lowing pseudochemical equations:

A + (XjS + ... ~ 2A +. . . (b)

where (Xl g of S is consumed to malce I g of A plus by-products; the reaction requires both Aas a "catalyst" and S for buiJding blocks. SimiJarly, for p,

A+lXzS+···~P+A+... (c)

where P is formed reversibly in this Parallel reaction requiring A as the catalyst, and U2 is a

stoichiometric coefficient between the monorner S and the polymer P (g . S consumed/g . pformed).

(6.2a)

How Cells GroW

I dX

J.1== X dr

d(C/X) =(...!.. dCi)_J.1C;1Xdt Xdt

Recalling

we have

186

In eq. 6.63, the rft terrn must be in terrns of intrinsic concentrations and the terrn llnetC/Xrepresents dilution by growth. These concepts are illustrated in Example 6.3.

The model indicated in Fig. 6.14 is that for a single cello The single-ceIl model re"

sponse can be directly related to culture response if aIl cells are assumed to behave identf­

cally. In this case each ceIl has the same division cyc1e. A population will have, at st,

state, twice as many cells at "birth" as at division. The average concentrations in the

ture will be at the geometric mean (a time equa1 to .Ji mu1tiplied by the division time) ft

each ceIl component. Used in this way, the model is a structured, nonsegregated model.

however, a cellular population is divided into subpopulations, with each subpopulatI

represented by a separate single-cell model, then a population model containing

high level of structure, as well as aspects of segregation, can be built. Such a fini

representation technique has been used and is capable of making gooda priori predicti

of dynamic response in cu1tures (see Fig. 6.15). When used in this context, at least

random element in the cell cycle must be included to lead to realistic prediction of dis

utions. Some examples of such randomness include placement of the ceIl cross wal1,

ing of the initiation of chromosome synthesis or cross-wa11 synthesis, and distributio

plasmids at division.

With this section, the reader should have a good overview of the basic conc

modeling and some simple tools to describe microbial growth. We need these tools to

quate1y discuss the cultivation of cells in continuous culture. Example 6.3 illustrates

a structured model might be developed.

Example 6.3.

Write the equations describing the following system. An organism consists of active bÍ,and a storage component. The storage component is made when the internal concen'the carbon-energy sources is high and is degraded when it is low. Use the symbolsAbiomass, P = polymeric storage compound, S* = external concentration of S, and S =concentration of S. Assume that S is the growth-rate-limiting nutrient.

d(C/X) IlnetCi

-d-t-==rft - -X

and, substituting eq. 6.60 for (l/X)(dC/dt) after dividing eq. 6.60 by VRX and assuming VR

is a constant,

Equation 6.60 can a1so be rewritten in terrns of intrinsic concentrations. For simplic­

ity we wi1l use mass fractions (e.g., C/X). Note that

d(C;lX) ==(...!.. dCi)_(C/XdX/dt)dt X dt I X (6.61)

The actual reaction mechanisms implied in these reactions are Iikely very complex.Most reactions in a cell are saturable and subject to feedback contro!.For these reactions, theintracellular level of S is the main control variable.Thus, a reasonable set of equations basedon eq. 6.60 for constant VR is

189Quantifying Growth Kinetics

6.3.4. Cybemetic Models

the reciprocal of the yield coefficientsand can be readily estimated frorn growth experirnentsand cornpositionaldata on P.

The rate forms used here are by no rneans the only correct solution.A more extensivebase of experirnentalobservationswould be needed to eliminate other possible formulations.

Sec. 6.3

6.4.1. Introduction

Another modeling approach has been developed primarily to predict growth under condi­tions when several substrates are available. These substrates may be complementary (e.g.,carbon ar nitro gen) or substitutable. For example, glucose and lactose would be substi­tutable, as these compounds both supply carbon and energy. As the reader will recall, wediscussed the diauxic phenomenon for sequential use of glucose and lactose in Chapter 4.That experimental observation led us to an understanding of regulation of the lac operonand catabolite repression. This metabolic regulation was necessary for the transition fromone primary pathway to another. The reader might infer that the culture had as its objec­tive function the maximization of its growth rate.

Dne approach to modeling growth on multiple substrates is a cybemetic approach.Cybemetic means that a process is goal seeking (e.g., maximization of growth rate).While this approach was initially motivated by a desire to predict the response of a micro­bial culture to growth on a set of substitutable carbon sources, it has been expanded toprovide an altemative method of identifying the regulatory stmcture of a complex bio­chemical reaction network (such as cellular metabolism) in a simple manner. Typically asingle objective, such as maximum growth rate, is chosen and an objective-oriented math­ematical analysis is employed. This analysis is similiar to many economic analyses for re­source distribution. For many practical situations this approach describes satisfactorilygrowth of a culture on a complex medium. However, the potential power of this approachis now being realized in efforts in metabolic engineering and in relating information onDNA sequences in an organism to physiologic function (see Chapter 8).

This approach has limitations, as the objective function for any organism is maxi­mizing its long-term survival as a species. Maximization of growth rate or of growth yieldare really subobjectives which can dominate under some environmental conditions; theseconditions are often of great interest to the bioprocess engineer. Consequently, the cyber­netic approach is often a valuable tool. lt is too complex for us to describe in detail in thisbook; the interested reader may consult the references at the end of this chapter.

The culture environment changes continually in a batch culture. Growth, product forma­tion, and substrate utilization terminate after a certain time interval, whereas, in continu­

ous culture, fresh nutrient medium is continually supplied to a well-stirred culture, andproducts and cells are simultaneously withdrawn. Growth and product formation can bemaintained for prolonged periods in continuous culture. After a certain period of time, the

6.4. HOW CELLS GROW IN CONTINUOUS CULTURE

(e)

(d)

k4KpSd (P/X)]KpSd+S/X

dS =XJ~ _ Ctj I0.S/X (A/X)dr lKxs' + S* KAS + S/Xrate of rate of S useduptake to form A

CtJ "-3S/X (A/X) _ k4KpSd (p,x)]lLKpsf + S/X KpSd+S/X IInet rate of Sconsumptionto form P

dP = X í ~S/X (A/X)dr LKpSf+S/X

Equations f and g describe the rate of formation of A and P.The value of k1 could be determinedby the uptake of a nonmetabolizableanalogof the'·

substrate, use of vesicles, or material balance information on exponentiilllygrowing cells•Values of 10.and k3 can be estimated from measurementson total biomass and polymerunderconditions where S/X should be large (e.g., exponential growth), while k4 and ks can be esti.mated from time-dependentmeasurementsof cell compositionin the stationaryphase, wSlX shouldbe low.If experimentalvalues of S/X are availableat various times, then more~fined estimates of kz, k3, k4, and kscan be made, as well as estimatesof the saturationparamcters. The rate form chosen in eq. (g) requires that KpSd « KpSf' so that P is made oulyS/X is in excess and degraded ouly when SlX is low.The values for Ctj and Ctz are essential1:

How Cells Grow

where ks is a first-ordercoefficient(h") and

where kz is g A formedlg . A present-h, k3 is g P formedlg . A present-h, and k4 is a first-orderdegradation rate for P (h"). The parameters Ct, and Ctz are defined in eqs. (b) and (c), whileKAS>KpSf, and KpSd are saturationparameters in units of mass fraction. Similarly,

dA = xl I0.S/X (A/X) - ks(A/X)]dr LKAS+S/Xrate of decomposition

formation of A to provideof A maintenanceenergy

dS*=X~dr Kxso +S*

where k, is rate of uptake per unit cell mass (g S transportedlgX-h) and Kxs' is the saturationparameter for uptake. Since S* is not a cellular component, extrinsic concentrations can beused in the rate expression. For S we write

188

191

(6.64)dX

VR'dfFXo - FX + VRllgX - VRkdX

r - -- -~QNTRQIJ&t!D- - - - - - -,i I

II FEEDBACKICONTROLISIGNAL FOR

IFEED PUMPIII

IlIGHTI=lJo--...JJ1j PHOTOMUL TIPLIER

RECIRCULATIONLOOP

CONSTANTVOL. aTEMP.

How Cells Grow in Continuous Culture

AIR

STERlLEMEDIUM

Sec. 6.4

An ideal chemostat is tbe same as a perfectly rnixed continuous-flow, stirred-tank reactor(CFSTR). Most chemostats require some control elements, such as pH and dissolved­oxygen control units, to be useful. Fresh sterile medium is fed to tbe completely mixedand aerated (if required) reactor, and cen suspension is removed at tbe same rate. Liquidvolume in tbe reactor is kept constant.

Figure 6.18 is a schematic of a simplified chemostat. A matenal balance on the ceHconcentration around tbe chemostat yields

6.4.3. The Ideal Chemostat

Figure 6.17. Typieallaboratory setup for a turbidostat. (With pernrission, from D. 1. C.

Wang and others, Fennentation and Enzyme Technology, John Wiley & Sons, lne., New

York, 1979, p. HJO.)

pump is activated and fresh medium is added. Tbe culture volume is kept constant by re­moving an equal amount of culture fluid. Tbe turbidostat is less used than tbe chemostat,since it is more elaborate tban a chemostat and because tbe environment is dynamic. Tur­bidostats can be very useful in selecting subpopu1ations able to witbstand a desiredenvironmental stress (for example, high ethanol concentrations), because tbe ceH concen­tration is maintained constant. Tbe selection of variants or mutants with desirable proper­ties is very important (see Chapter 8).

A plug flow reactor (PFR) can also be used for continuous cultivation purposes.Since there is no backmixing in an ideal PFR, fluid elements containing active ceHs can­not inoculate otber fluid elements at different axial positions. Liquid recycle is requiredfor continuous inoculation of nutrient media. In a PFR, substrate and ceH concentrationsvary with axial position in the vessel. An ideal PFR resembles a batch reactor in whichdistance along tbe fermenter replaces incubation time in a batch reactor. In waste treat­ment, some units approach PFR behavior, and multistage chemostats tend to approachPFR dynarnics if tbe number of stages is large (five or more).

c

llllt r-;;-\AI, -~•

1..-.Id I

•..,.,01' I._J

____J~:~t.,

How Cells GroW

Figure 6.16. A eontinuous-culture laboratory setup (ehemostat). (With pernrission, fromD. 1. C. Wang and others, Fennentation and EnzYme Technology, John Wiley & Sons, lne.,

New York, 1979, p. 99.)

190

St.,I"•• dlulll, ••••. "01'

system usually reaches a steady state where cell, product, and substrate concentrationsremain constant. Continuous culture provides constant environmental conditions for

growth and product formation and supplies uniform-quality product. Continuous cultureis an important tool to deterrnine the response of microorganisms to their environment and

to produce the desired products under optimal environmenta1 conditions. In Chapter 9we wm compare batch and continuouS culture in terms of their suitability for large-scale

operation.

6.4.2. Some Specific Devices for Continuous Culture"

Tbe primary types of continuous cultivation devices are the chemostat and turbidostat, a1-though plug flow reactors (PFR) are used. In some cases these units are modified by recy-cle of ceHs.

Figure 6.16 is a schematic of a continuous culture device (chemostat). CeHular growthis usual1y Hrnited by one essential nutrient, and other nutrients are in excess. As we willshow, when a chemostat is at steady state, the nutrient, product, and ceH concentrations areconstant. For this reason, the name chemostat refers to constant chernical environment.

Figure 6.17 is a schematic of a turbidostat in which the ceH concentration in the cul­ture vesse1 is maintained constant by monitoring the optica1 density of the cu1ture andcontrol1ing the feed flow rate. When the turbidity of the medium exceeds the set point, a

How Cells Grow

(6.67)

(6.71)

(6.70)

(6.72)

(6.68)

(6.69)

193

(6.73a)

(6.73b)

D(So - S) = ~gXY/1s

~g =D= ~mSK,+S

D=~g-kd=llnel

Ilg=D+kd

1 1 dSFSo - FS - VR~gX -;;r - VRqpX- = VR-

Yx/s Yp/s dt

How Cells Grow in Continuous CultureSec. 6.4

or

Using eq. 6.68, the steady-state ceU concentration can be expressed as

X = Yfrs(So _ K,D )11m-D

ln eqs. 6.71 and 6.72, the yield coefficient (Yx/s) is assumed to be constant, which isan approximation, since endogenous metabolism has been neglected. Usually, Yx/s varieswith the limiting nutrient and growth rate. Consider the effect the inclusion of endogenousmetabolism wilI have; eq. 6.66 becomes

Since growth rate is limited by at least one substrate in a chemostat, a simple de­scription of chemostat performance can be made by substituting the Monod equation (eq.

6.30) for ~g in eq. 6.66:

Since ~g = D at steady state if kd = O,

X = Y::s(So - S)

where S is the steady-state limiting substrate concentration (g/l). If D is set at a value greaterthan ~, the culture cannot reproduce quickly enough to maintain itself and is washed out.

Equation 6.67 is identical to that for Michaelis-Menten kinetics and, as we discussed in

Chapter 3, a plot of l/Jlg versus I/S can be used to estimate values for ~m and Ks.Using eq. 6.67, we can relate effiuent substrate concentration to dilution rate for D <~.

S= KsD~m-D

A material balance on the limiting substrate in the absence of endogenous metabolismyields

where So and S are feed and effiuent substrate concentrations (g/l), qp is the specific rate

of extracellular product formation (g P/~g cells h), and Y~ and YP/s are yield coefficients(g celllg S and g Plg S). The use ofthe superscript M on YXlS denotes a maximum value ofthe yield coefficient; such a superscript will be important in discussing the effects ofmaintenance energy.

When extracellular product formation is negligible and the system is at steady state(dSldt= O),

Liquid Volume :Working Volume:

VR

FSXP

00 O

S,X, p.'

Figure 6.18. Simplified schematic of a chemostat,

GosSp0l'Qe

FSoXo=OPo=O

where F is the flow rate of nutrient solution (lfh), VR is the culture volume (1) (assumedconstant), X is the cell concentration (gIl), and ~g and kd are growth and endogenous (ardeath) rate constants, respectively (ni). The reader should note that if cell mass is the pri­mary parameter, it is difficult to differentiate cell death from endogenouS metaboUsm.When we use kd, we imply that endogenouS metaboUsm is the primary mechanism for cemass decrease, With k'd' we imply that cell death and lysis are the primary mechanisms adecrease in mass. The reader should also note that, if eq. 6.64 had been written in termScell number, kd could only be a cell death rate. When balances are written in terms of c'number, the influence of endogenouS metaboUsm can appear only in the substrate bal

equation, Since most experiments are done by measuring total cell mass rather than nber, we write our examples based on X, However, the reader should be aware of the aru'

guity introduced when equations are written in terms of X.Equation 6.64 can be rearranged as

dX_ =DXo +(~g -kd -D)Xdt

where D is dilution rate and D = FNR• D is the reciprocal ofresidence time,UsuaUy, the feed media are sterile, Xo = O, and if the endog~nous metaboUsl

death rate is negligible compared to the growth rate (kd « ~g) and if the system'steady state (dXldt = O), then

~g = D (if kd = O) (6.~.]'1

ln a chemostat, cells are removed at a rate equal to their growth rate, and the growth~.of ceUs is equal to the dnution rate. This property allows the investigator to manipa,

growth rate as an independent parameter and makes the chemostat a powerful expe:' ,tal too1.

192

How Cells GroW195

4 5

•/.9 glucose = m.9 cells-hr

Slope = 0.2

3.0

2.0

2.5

I/. APYX/S

DP =o qpX (6.80)

where qp is described by eqs. 6.16, 6.17, or 6.18. For nongrowth-associated product for­mation, qp is a constant (~), while for growth-associated products it is a function of Ilg.

For substrate balance, eq. 6.69 becomes

Figure 6.19. Graphical approach to estimating y~ and m, for chemostat data for E. coU

growing on glucose as the linuting nument.

model, we must view this conversion as instantaneous. Otherwise, a perceptible period ofsubstrate uptake, prior to conversion to product, would cause a change in the amount of X.This is unlike maintenance, since the period between substrate uptake and use for mainte­nance functions may be long, so we allow S to become incorporated into X and X is thendegraded when needed; the following equations assume instantaneous conversion of sub­strate to extracellular product, with the cell acting as a catalyst. The balance on productformation is

1 1D(So -S)=o~(D+kd)X +-qpX (6.81)

Yx/s Yp/s

The biomass balance is unchanged from the case with endogenous metabolism and yields

. Sec. 6.4 How Cells Grow in Continuous Culture

1 m-V +.J.Yx/s D

S K,(D + kd)~m -D-kd

IYAPx/s

( 1) D kd

D- _---=0Y~ Y~s Y~s

1 kd 1-V + r;r- = ---;;;pYXIS YXIS' D Yx/s

v(So;S) - lfY~s(D+kd)=O

Dx=lf,s[So-S]' ­D + kd

In addition to the effects of maintenance, it is often appropriate to consi,

conversion of extracellular substrate into extracellu1ar product. With an uns'

and

km =_d_

, lf,s

where m, is the maintenance coefficient based on substrate S. Y~ is the apparent yiel,When YX/S is written, it should be interpreted as Y~. Whi1e YM-sis a constant, Y~ vwith growth conditions if kd > O.

Values of YM-s and m, can be obtained from chemostat experiments by p1o'

lIY~ against lID. The slope is m, and the intercept is l/YM-s (see Fig. 6.19).In the presence of endogenous metabolism (Ilnet = ~/(K, + S) - kd), we caI1

show that

where

or

By substituting eq. 6.73b into the steady-state substrate balance, assuming no extracellu­lar product fonnation (eq. 6:70), we find

D(So _S)-lfY~s(D+kd)X = O (6.73c)

where YM-sdenotes the maximum yield coefficient (no endogenouS metabolism or mainte­nance energy). YM-shas a single constant value independent of growth rate. This approachdoes not allow the direct conversion of S into maintenance energy. Rather, S must be fust

incorporated into the cell mass, where it is degraded by endogenous metabolism. Thisviewpoint is the proper one when working with an unstructured model. With a structuredmodel, which explicitly recognizes intracellular S as a subcomponent of the biomass, thedirect consumption of S for maintenance functions could be modeled. However, with un­structured models, substrate that is no longer extracellular becomes part of the biomass.

Equation 6.73c Can be rearranged to

(a)

(a)

Yx"Js=O.l~g subs

1=0.05 gIl

1(1+ 0.05)D.-Ml0.5-D

D = /.l - k = /.lmS _ kg d Ks +S d

Ks=lgll

Ks(l+.2....)DS- KlI1m-D

S=~O.5-D

For the seeond step, reeall that

So= 10 gIJ

Xo=O

Kl"" 0.01 gIl Ilm = 0.5 h-1 kd = O

a. Determine X and S as a funetion of D when 1 = O.

b. With inhibitor added to a chemostat, determine the effluent substrate eoneentrationandX as a funetion of D.

c. Determine the cell productivity, DK, as a function of dilution rate.

Then

The specific growth rate for inhibited growth in a chemostat is given by the following equation:/.lm S

/.l =-- _8 Ks +S+/Ks/K[

where

a)S= Kp ""~

11m-D 0.5-D

X=Yx~s(SO-S)=0.1(10 - -2-)0.5-Db) In tbc presence of inhibitor

- IlmS D

118 - ( I)Ks l+KI +S

How Cells Grow in COntinuous Culture197

x=yx~s(So- S) = 0.{1O - o.:~D)

1 1 Ks 1-=-+--D+ kd /.lm !lm S (b)

We now plot 1/(D + kd) versus 1/S. The intereept is 1.25 h or 11m = 0.8 h-1. The slope is 100h/(mgIJ). Thus, Ks/Ilm= 100 or Ks = 80 mgll.

ExampIe 6.5.

SoJution

(6.78)

(6.82)

366

407

408

404

371

299

59

Cell

concentration (mg/!)

16.7

33.5

59.4

101

169

298

702

s= K,(D+kd)Jlm-D-kd

Carbon substrate

concentration (mgII)

0.1

0.2

0.30.4

0.5

0.6

0.7

Dilution cate (h-')

How Cells Grow

Solution The first step is to plot l/yt; versus liD. ytsis ealeulated from X/(So- SJ. ,;;:.tereept is l/Y~ = 1.58 or Y~ = 0.633 g X/g S. The slope is 0.06 g S/g X-h, whieh is ~~.

for ms. Recallms=kdIY~; thenkd=msY~s=O.06 g S/gX-h· 0.633 gX/gS=0.038h-l.:~

196

Dopt = Jlm(l- ~ K,~ sJ

Example6A.

A new strain of yeast is being eonsidered for biomass produetion. The following dataobtained using a ehemostat An influent substrate eoneentration of 800 mg/l and an ex,oxygen were used at a pH of 5.5 and T = 35°C. Using the following data, ealeulate 11m,

Yts, kd, and ml' assuming I!net = !!mS/(Ks+ S) - kdo

Equation 6.81 can be so1ved for X:

X= Y~s(So -s)[ DM]

YXlS

D+kd+qp-Yp1S

The productivity of a chemostat for product and biomass (PrJ can be found from DP and

DX, respectively. The dilution rate that maximizes productivity is found by differentiating

DP or DX with respect to D and setting the derivative equa! to zero. The optimal value of ...•

D (Dopt) wiII depend on whether endogenous metabolism andlor product formation are

considered. When kd = O and qp = O, eqs. 6.7I and 6.68 apply. Then Dopt for biomass pro­duction (DX) becomes

Since So is usuaIly much greater than Ks' Dopt wilI approach D = !lm or the washout poillStable chemostat operation with D "" Jlm is very difficult, uruess the flow rate and liqui,

volume can be maintained exactly constant Consequently, a value of D slightly less th:

Dopt may be a good compromise between stability and biomass productivity. It should al•.

be apparent that Dopt for biomass formation wiII not necessarily be optimal for prodformation.

Examples 6.4 and 6.5 iIIustrate the use of these equations to characterize the pemance of chemostats.

6.4.4. The Chemostat as aTool

c) Prx=DX=DYx/s(So-S)=O.ID(lO - ~)O.5-D

199

Xl +(I-U)D'Il ••-Lx, =X,K,+Sz

Figure 6.20. Simple two­

companmental model for an incom­pletely mixed "chemostat."

Biomass balance

X, +aD'/l ••~X, =(1+ DD')X,K,+5í

F,X1,S,

Xl = Y(So - SI)

Substrate balance

X, -X, = Y(SI-S,)

aV, X/,S,

Region 1

Region 2

{I-QJ V, X2, S2

F', X"S, I I F', X2,S2

':;So

ln an imperfectly mixed chemostat with a stagnant biotic phase, the dilution rate can

exceed 11m without washout. The washout dilution rate is obtained by setting So = SI :: S2andis

D = /!mSo [ (1- a)2llmSo + 1]wo Km +SO Ks +So -(l-a)D'llmSo

where Dwo Ís the washout dilution rate.

During batch cultivation, a population of cells typically exhibits severa] different growthphases. During the lag phase, the celI builds the biosynthetic pathways necessary for max­ima] growth rates in the fresh medium. During exponential growth, cel! replication ismaxima], and the chemica] composition ofthe cel! population is nearly constant (e.g., bal­anced growth). When substrate is nearly exhausted or when toxic metabolic by-productshave built to a critical level, the growth rate begins to drop rapidly, causing significantchanges in biosynthetic pathways. In the stationary phase, there is no net growth; cel!snow reorient their metaboJic machinery to increase the probabiJity of long-term survivaJ.At some point, some cel!s can no longer obtain enough energy from their reserves orenough of another critical resource, and the culture enters the death phase. Dead cells do

not have an energized membrane and often lyse (or break apan). Nutrients released bylysed cel!s can be utilized by survivors, aJJowing cryptic growth.

Products formed by cells can be related to this batch-culture growth cycle. Primaryproducts are growth associated. Secondary products are nongrowth associated and aremade in the stationary phase. Some products have both growth-associated and nongrowth_associated components.

CelI-growth kinetics can be modeled. Models that are structured break the popula­tion into distinct subcomponents. Unstructured models quanti/y celI mass as a single

Sec. 6.5 SummaryHow Cells Grow198

In fermentations such as the utilization or production of polysaccharides or mycelial ti

mentations, the fermentation broth may be highly viscous and it may be difficult totain complete mixing. Also, certain cells tend to grow in the form of a film on s'surfaces in fermenters, such as on fermenter walls or probe surfaces. Incomplete mixithe role in industrial-sca1e fermenters. The presence of incompletely mixed regions"chemostat" culture is common. The term "chemostat" is actually applicable onIperfectly mixed vessels. Here we denote nonidea1ity by putting chemostat in qtion marks. A segregated reactor model can be used to analyze an incompletelycontinuous- flow fermenter.

A simple, two-compartmental model of a "chemostat" culture is depicFig. 6.20, where the reactor is divided into two regions: a well-mixed region and anant region. Feed and effluent streams pass through the well-mixed region 1, and m:

change takes place between the two regions. The biomass and substrate balances forregions at steady state are as follows:

6.4.5. Deviations from Ideality

The chemostat can be used as a tool to study the mutation and selection of cultures andalso to study the effect of changes in the environment on cell physiology. The molecular

aspects ofmutation and selection w!Jl be discussed in Chapter 8.Natural or induced mutations can take place in a chemostat culture. Errors in DNA

replication take place with an average frequency of about 10-6to Ia-s gene per generation.With a cell concentration of 109 cells/ml in culture, the probability is high in a chemostatthat a wide variety of mutant cells will be formed. The vast majority of natural mutationsin a chemostat are of little significance, unless the mutation alters the function of a proteininvolved in growth in the chemostat environment. ff the specific growth rate of the mutantis larger than that of the wiJd type, then the mutant outgrows the wild type in a chemostat.This selection for a variant cell type can be accomplished by creating a more favorable en­vironment for growth of the mutant organism.

A chemostat culture can be used for the selection of special organisms. Selection orenrichment nutrient media need to be used for this purpose. For example, if it is desired toselect an organism growing on ethanol, a nutrient medium containing ethanol and mineraJSalts is used as a feed to a chemostat culture. An organism capable of oxidizing sometoxic refractory compounds can be selected from a mixed culture by slowly feeding" .compound to a chemostat. A thermophiJic organism can be selected from a natural POP'

lation by operating a chemostat at an elevated temperature (e.g., 50° to 60°C). Selectionchemostats also presents significant problems in the culture of cells containing recomnant DNA. The most productive cells often grow more slowly and are displaced by J,productive cells. We will discuss this problem in more detail in Chapter 14.