Kinetics of Growth in Batch and Continuous Culture

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CHAPTER 3:

KINETICS OF GROWTH

IN BATCH AND

CONTINOUS CULTURE

ERT 317 BIOCHEMICAL ENGINEERINGEM 1 2012/13



Lecture Outline

Introduction

Batch Growth Characteristics

Growth Stages, Effects of Environmental Conditions,

Product Formation, Mathematical Models

Continuous Growth Characteristics

Dilution Rate, Optimum Operation



Cell growth

Microbial growth is an autocatalytic reaction

The rate of growth is directly related to cell concentration

Characterized by the net specific growth rate:

nXPXS

CellsMoreProductslarExtracelluCellsSubstrate

dt dX

X net

1



Cell growth

The net specific growth rate is the difference between

a gross specific growth rate (μg, h-1) and the rate of

loss of cell mass due to cell death (kd, h-1):

Microbial growth can also be described in terms of

cell number, N:

where μR is the net specific replication rate (h-1

)

d g net k

dt

dN

N R

1



Batch Growth – Determining Cell

Number Density

Hemocytometer

Direct microscopic count

Counts all cells present (viable and non-viable)

Immediate result

Agar plates Counts only living cells

Delayed result

Assumption: each viable cell will yield 1 colony

Results expressed in CFUs(colony-forming units) Particle counters

Counts all cells present (viable and non-viable)

Suitable for discrete cells in a particulate-free medium

Can distinguish between cells of different sizes



Hemocytometer



Viable Cell Count





Coulter Particle Counter



Determining Cell Mass Concentration – Direct

Methods

Dry cell weight (DCW)

A sample of fermentation broth is centrifuged, washed,

and dried at 80°C for 24hrs

Off-line measurement; wet cell weights (WCW) canperformed in-process

Packed cell volume

Like wet cell weight, but measures cell pellet volume

Optical density (OD)

Turbidity – based on the absorption of light by suspended

cells in culture media



Determining Cell Mass Concentration – Indirect

Methods

In many fermentation processes, particularly with moulds,

direct methods cannot be used

Indirect methods are therefore employed, based on the

measurement of substrate consumption and/or productformation

Intracellular components of cells such as RNA, DNA and protein

can be measured as indirect indicators of cell growth

Concentration of RNA/cell weight varies significantly during abatch growth cycle, while DNA and protein concentrations per

cell weight remain fairly constant, and can therefore be used

as reasonable measures of cell growth



Time-Dependent Changes in Cell

Composition and Cell Size

Azotobacter vinelandii Growth in Batch Culture



Batch Growth



Batch Growth Curve

Growth Phases

1. Lag

2. Exponential

3. Deceleration

4. Stationary

5. Death/Decline



Lag Phase

Occurs immediately after inoculation and is aperiod of adaptation for the cells to their newenvironment

New enzymes are synthesized, synthesis of otherenzymes is repressed

Intracellular machinery adapts to the new conditions

May be a slight increase in cell mass and volume, butno increase in cell number

The lag phase can be shortened by high inoculumvolume, good inoculum condition (high % of living cells),age of inoculum, nutrient-rich medium



Influence of [Mg2+] on Lag Phase Duration in E.

aerogenes Culture

E. aerogenes requires Mg2+ to

activate the enzyme

phosphatase, which is

required for energy

generation by the organism

The concentration of Mg2+ in

the medium is indirectly

proportional to the durationof the lag phase



Exponential Growth Phase

In this phase, the cells have adjusted to their newenvironment

At this point the cells multiply rapidly (exponentially)

Balanced growth – all components of a cell grow at the same

rate Growth rate is independent of nutrient concentration, as

nutrients are in excess

The first order exponential growth rate expression is:

t

net

net

net e X X t X

X

t X X X dt dX

0

0

0

orln

0atwhere



Exponential Growth Phase (cont’d)

An important parameter in the exponential phase is

the doubling time (time required to double the

microbial mass)

A graph of ln X versus t produces a straight line on asemi-logarithmic plot:

The doubling time based on cell number is expressed

as:

maxmax

693.02ln

d

R

d

2ln'



Exponential Growth Phase (cont…)

t



Deceleration Phase

Very short phase, during which growth decelerates

due to either:

Depletion of one or more essential nutrients, or,

The accumulation of toxic by-products of growth (e.g.Ethanol in yeast fermentations)

Period of unbalanced growth: td=td’

Cells undergo internal restructuring to increase theirchances of survival

Followed quickly by the Stationary Phase



Stationary Phase

Starts at the end of the Deceleration Phase, when thenet growth rate is zero (no cell division, or growth rateis equal to death rate)

Cells are still metabolically active, and can producesecondary metabolites

Primary metabolites are growth-related products, whilesecondary metabolites are non-growth-related

Many antibiotics and some hormones are produced as

secondary metabolites Secondary metabolites are produced as a result of

metabolite deregulation



Stationary Phase (cont’d)

During this phase, one or more of the following

phenomena may occur:

Total cell mass concentration may stay constant, but the

number of viable cells may decrease Cell lysis may occur, and viable cell mass may drop. A

second growth phase may occur as cells grow on lysis

products from the dead cells (cryptic growth)

Cells may not be growing, but may have activemetabolism to produce secondary metabolites



Stationary Phase (cont’d)

During the stationary phase, the cell catabolizes cellularreserves for new building blocks and for energy-producing monomers

This is called endogenous metabolism

The cell must expend maintenance energy in order to stayalive

The equation that describes the conversion of cellular mass intoenergy, or the loss of cell mass due to lysis during the

stationary phase is:

t k

SOd d e X X t k

dt

dX or



Death Phase

The death or decline phase is characterized by theexpression:

Where Ns is the concentration of cells at the end ofthe stationary phase, and is the first-order death-rateconstant

A plot of ln N versus t yields a line of slope – kd’

t k

S d d e N N t k

dt

dN '

or'



Death Phase

1. Cell lysis (spillage) may occur

2. Rate of cell decline is first-order

where: – kd = 1st order death rate constant,

Xs = conc. of cell at end of stationary phase

3. Growth can be re-established by transferring to fresh media



Yield Coefficients

Growth kinetics are generally further described bydefining stoichiometrically related parameters

Yield coefficients are defined based on the amount ofconsumption of a given material

For example, the growth yield coefficient is:

For organisms growing aerobically on glucose, Yx/s istypically 0.4 to 0.6 g/g, for most yeast and bacteria;anaerobic growth is much less efficient

S

X Y S X

/

A bi d A bi G th Yi ld f



Aerobic and Anerobic Growth Yields of

S. faecalis on Glucose



Yield Coefficients

At the end of a batch growth period, there is an

apparent or observed growth yield:

The apparent yield is not a true constant for

compounds that can be used as both a carbon andenergy source, but the true growth yield (YX/S) is

constant ΔS

energy

maintence

energy

growth

productlar extracellu

aninto

onassimilati

biomassinto

onassimilati S S S S S



Yield Coefficients

Yield coefficients can also be defined for othersubstrates or for product formation:

YX/O2 is typically 0.9 to 1.4 g/g for most yeast andbacteria, but is much lower for highly reducedsubstrates (e.g. methane, CH4)

S

P Y

O

X Y

S P

O X

/

2/ 2



Summary of Yield Factors for Aerobic

Growth



The Maintenance Coefficient

The maintenance coefficient is used to describe the specificrate of substrate uptake for cellular maintenance:

However, during the Stationary Phase, where little external

substrate is available, endogenous metabolism of biomass

components is used for maintenance energy

Maintenance energy is the energy required to repairdamaged cellular components, to transfer nutrients and

products in and out of cells, for motility, and to adjust the

osmolarity of the cells’ interior volume

X

dt dS m m/



Microbial Products

Microbial products can be classified into three majorcategories

Growth-associated products

Non-growth-associated products

Mixed-growth-associated products

Growth-associated products

These products are produced simultaneously with microbialgrowth

Specific rate of product formation is proportional to thespecific growth rate, μg

Note that μg is not equal to μnet, the net specific growth rate,when endogenous metabolism is occurring



Growth-Associated Products

The rate expression for product formation in

growth-associated production is:

Where qp is the rate of product formation (h-1)

The production of a constitutive (continuously

produced, as opposed to inducible) enzyme is anexample of a growth-associated product

g X P p Y

dt

dP

X

q /

1



Non-Growth-Associated Products

Non-growth-associated product formation takes

place during the Stationary Phase, when the growth

rate is zero

Specific rate of product formation is constant:

Many secondary metabolites, such as mostantibiotics (e.g. penicillin), are non-growth-

associated products

constant pq



Mixed-Growth-Associated Products

Mixed-growth-associated product formation takes place duringthe Deceleration (slow growth) and Stationary Phases

The specific rate of product formation is given by the

Luedeking-Piret equation:

If α= 0, the product is completely non-growth associated; If β=

0, the product is completely growth-associated Examples: lactic acid fermentation, production of xanthan gum,

some secondary metabolites

g pq



Product Yield Coefficients (cont…)

a) Growth-associated product formation

b) Non-growth-associated product formation

c) Mixed-growth-associated product formation



Environmental Factors

Patterns of microbial growth and product formationare influenced by environmental factors such astemperature, pH and dissolved oxygen concentration(D.O.)

Microorganisms can be classified by their optimumgrowth temperatures, Topt Psychrophiles: (Topt< 20°C)

Mesophiles: (20°C < Topt

< 50°C)

Thermophiles: (Topt> 50o°C)

As the temperature increases towards Topt, the growthrate doubles every ~10°C



Optimum Growth Temperature



Optimum Growth Temperature



Effect of Temperature on Cell Growth

Above Topt the growth rate decreases and thermal deathmay occur

The net specific replication rate for temperatures above Topt isexpressed by:

Both and vary with temperature according tothe Arrhenius equation:

Where:

Ea =activation energy for growth ≈ 10-20 kcal/mol

Ed =activation energy for death ≈ 60-80 kcal/mol

N k dt

dN

d R

''

RT E

d

RT E

R

aa

Aek Ae/'/'

'

R '

d k



Arrhenius Plot of Growth Rate of E. Coli

Legend:

(●) Growth onrich, complex

medium(○) Growth onglucose-mineral

salts medium



Effect of pH on Cell Growth

pH affects the activity of enzymes, and therefore

the microbial growth rate

Acceptable pH’s for growth are typically within 1 or

2 pH units of the optimum pH pH range varies by organism:

bacteria (most) pH = 3 to 8

yeast pH = 3 to 6 plants pH = 5 to 6

animals pH = 6.5 to 7.5



Effect of pH on Cell Growth

The optimal pH for growth may be different from

the optimal pH for product formation (e.g. Pichia

pastoris)

Microorganism have the ability to control pH insidethe cell, but this requires maintenance energy

pH can change due to:

Utilization of substrates; NH4+ releases H+, NO3-consumes H+

Production of organic acids, amino acids, CO2, bases



Effect of pH on Cell Growth (cont…)



Effect of Dissolved O2 on Cell Growth

At high cell concentrations, the rate of oxygen

consumption may exceed the rate of O2 supply

When oxygen is the rate-limiting factor, specific growth rate

varies with [DO] according to saturation (Michaelis-Menten)

kinetics

Below a critical concentration, growth approaches a

first-order rate dependence on DO (oxygen is a limiting

substrate) Above a critical concentration, the growth rate becomes

independent of DO (oxygen is non-limiting))



Effect of Dissolved O2 on Cell Growth (cont…)

Obligate aerobic cells

Saturation kinetics

Facultative aerobic cells

Saturation kinetics



Effect of Dissolved O2 on Cell Growth

The saturated DO concentration for water at 25°Cand 1 atm is ~7 ppm

The presence of dissolved salts and organics can alterthe saturation value

Increasing temperatures decrease the saturation value

The critical oxygen concentration is about 5%-10%of the saturated DO concentration for bacteria and

yeast, and about 10%-50% of [DO]sat

for moulds,since they grow as large spheres in suspendedculture (diffusion issues)



Other Effects on Cell Growth

Dissolved CO2 can have a profound effect on theperformance of microorganisms Very high DCO2 concentrations can be toxic to some cells

On the other hand, cells require a certain minimum DCO2 levelfor proper metabolic function

Ionic strength (I); too high dissolved salts is inhibitory tomembrane function (membrane transport of nutrients, osmoticpressure):

where : Ci = molar concentration of ion i

Zi = ion charge



Other Effects on Cell Growth

The redox potential is an important parameter that affects the rate andextent of many oxidative-reductive reactions

In fermentation media, the redox potential is a complexfunction of DO, pH, and other ion concentrations, such asreducing and oxidizing agents

Substrate concentrations significantly above stoichiometric requirementsare inhibitory to cellular functions

Inhibitory levels of substrates vary depending on cell type andsubstrate

Typical maximum non-inhibitory concentrations of somenutrients are – glucose, 100 g/l; ethanol, 50 g/l for yeast, muchless for other organisms; ammonium, 5 g/l; phosphate, 10 g/l;nitrate, 5 g/l



Heat Generation by Growth

About 40% to 50% of the energy stored in a carbon and energysource is converted to biological energy (ATP) during aerobic

metabolism, and the rest of the energy is released as heat

For actively growing cells, the maintenance requirement is low, and heat

evolution is directly related to growth The heat of combustion of the substrate is equal to the sum of the metabolic

heat and the heat of combustion of the cellular material:

Where ΔHS is the heat of combustion of the substrate (kJ/g substrate), ΔHC is the heat of combustion of cells, and 1/YH is the metabolic heat evolved

per gram of cell mass produced (kJ/g cells)



Energy Balance on Microbial Utilization

of Substrate



Heat Generation by Growth

The above equation in heat generation can berearranged to become:

ΔHS and ΔHC can be determined from the combustion ofsubstrate and cells

Typical ΔHC values for bacterial cells are 20-25 kJ/g cells

Typical values of YH are: glucose, 0.4 g/kcal; malate, 0.3

g/kcal; acetate, 0.21 g/kcal; ethanol, 0.18 g/kcal; methanol,0.12 g/kcal; and methane, 0.061 g/kcal

Clearly, the degree of oxidation of the substrate has a strongeffect on the amount of heat released



Heat Generation by Growth (cont…)

Substrate, S ∆Hs (kJ/g S) YH (g dcw/kJ)

Glucose 15.64 0.072

Methanol 22.68 0.029

Ethanol 29.67 0.043

n-Decane 47.64 0.038

Methane 55.51 0.015

For substrates:

The oxidation state of S has a large effect on 1/ Y H

Rate of Heat Generation by Growth



Rate of Heat Generation by Growth,

QGr

The total rate of heat evolution in a batchfermentation is:

where: VL = liquid volume

In aerobic fermentations, the rate of metabolic heatevolution can roughly be correlated to the rate of

oxygen uptake:

where: QGR is in kcal/h, and QO2 is in mM of O2/h



Heat Generation by Microbial Growth

Metabolic heat released during a fermentation can

be removed by circulating cooling water through a

cooling coil within the fermenter, or a cooling jacket

surrounding the fermenter Temperature control is a critical limitation on reactor

design

The ability to estimate heat removal is essential to

proper reactor design

C li il d W t J k t d F t



Cooling coils and Water Jacketed Fermenter

Documents

Kinetics of Growth in Batch and Continuous Culture