Bacterial Growth Kinetics

8/13/2019 Bacterial Growth Kinetics

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2-Bacterial Growth Kinetics_F12

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BACTERIAL GROWTH KINETICS

Growth Kinetics of Pure Bacterial Culturein a Batch System (VH, p.444)

- Growth Phases of a Simple Batch

Culture

1) Lag phase2) Accelerating growth phase3) Log (exponential growth) phase4) Declining growth phase5) Stationary phase6) Death or declining death phase7) Log death phase

1. L ag Phase

a. Adaptation (acclimation) period - the lag phase is the initial phase which represents the period (time) required for bacteria to

adapt to their new environment. b. Constant number of cells

- during this phase, the individual bacterial cells increase in size, but the number of cellsremains unchanged.

c. Physiologically active - they are very active physiologically and are synthesizing new enzymes and activating

factors.

2. Accelerating Growth Phase

a. Transition period from the lag phase to the log phase. b. Cell is beginning to grow (increase in numbers) noticeably as enzyme systems are gearing up.

3. Log (Exponenti al Gr owth) Phase

a. Exponential growth- during this phase, the bacterial cells divide regularly at a constant rate.

b. Straight line on semilog scale- The logarithms of the number of cells plotted against time results in a straight line.

c. Maximum Rate of Substrate utilization- A maximum growth rate occurs under optimal conditions, and substrate is removed from

the medium at the maximum rate.- The growth rate is limited only by the bacteria's ability to process the substrate.- food is in excess (not limiting) so that the rate of growth is only limited by the ability to

process the food- Sometimes called "0-order growth" and growth rate is constant and maximum.

Endogenous Phase

5 64

37

1

2 S

Time

S u

b s t r a

t e - - - -

____ L

o g

( N o .

o f V i a b l e C e l

l s )


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4. Declini ng Growth Phase

a. Transition period from the log phase to the stationary phase. b. Decreasing growth rate

c. Exhaustion of essential nutrientsd. Accumulation of toxic metabolic products

- the growth rate can be limited either by the exhaustion of essential nutrients or by theaccumulation of toxic metabolic products.

- food becomes limiting factor and therefore growth rate and mass of bacteria are dependenton the amount of food present.

5. Stationar y Phase

a. The number of cells remains constant perhaps as a results of complete cessation of division or

the balancing of reproduction rate by an equivalent death rate.

- growth of new cells is balanced by the death of old cells.- no increase in cell mass- population is "stable"- net growth rate, r g = 0

6. Death or declini ng Phase

a. The number of viable cells decreases slowly due to the fact that the death rate exceeds the production rate of new cells (while the total mass may remain constant).

b. Depletion of essential nutrientsc. Accumulation of inhibitory products.

- Death occurs primarily as a result of depletion of essential nutrients and/or the accumulationof inhibitory products.

7. Log Death Phase

a. Exponential death - "wholesale die-off"- system is dead- even if you add food, you will get no growth.


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En dogenous Phase

a. Near-starvation condition- That portion of the bacterial growth curve encompassing parts of the stationary and

declining phases in which microorganisms are in near-starvation is more frequently calledthe endogenous phase. b. Most activated sludge treatment systems are operated in this phase since the microorganisms

flocculate rapidly and settle out of solution by gravity.- food is scarce and the bacteria are forced to metabolize storage products and dead, lysed cells- growth does not cease, but death exceeds it, net growth rate is negative.

REMARKS:

a. Growth curve is for a batch system (fed once) and pure culture.

- real biological treatment systems are typically continuously fed (and wasted).

b. Growth of bacteria is dependent upon many other factors which will be discussed in moredetail later.

c. Activated sludge and other biological treatment systems contain many types of organisms.- i.e., biological treatment systems are ecosystems and as such the growth characteristics of

one organism may be affected by other organisms.

d. In general, cells settle much better when they are in endogenous phases.

e. In real biological treatment systems, you must operate at a positive growth rate or you willeventually lose your bacteria.


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Relationship between: (A) biological growth phase, (B) biological flocculation. Also shown arethe accumulation of the exocellular polymer (ECP) to microorganisms (Tenny and Verhoff)

Biomass vs. time

Turbidity vs. time

Exocellular polymer vs.time

ECP/Biomass vs. time


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Filamentous organisms observed in activated sludge

Common characteristics of filamentous organisms observed in activated sludge; 1000 x phasecontrast(Water Pollution Control Federation)


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Common characteristics of filamentous organisms observed in activated sludge; 1000 x phasecontrast(Water Pollution Control Federation)


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Bacterial Growth Kinetics

a. For generalization of results, quantitative descriptions of bacterial growth

- To generalize results, quantitative descriptions of bacterial growth in a culture

are essential.

b. For evaluation and interpretation of experimental results

- Mathematical models have been successfully used in evaluation and interpretation ofexperimental results.

c. The experimental data may be rendered more meaningful and concise if they are analyzed interms of the various growth kinetic parameters.

d. The basic kinetic constants include:

1) specific growth rate, 2) maximum specific growth rate, m3) maximum rate of substrate utilization per unit mass of organisms, k4) half-saturation constant or Monod constant, Ks5) yield coefficient, Y6) endogenous decay coefficient (death rate constant), k d

Batch Experiment

It has been observed that if one of the essential requirements for growth is present in only limitedamounts, the limitimg factor affects on the rate of growth. For example, if substrate is the growthlimiting factor, the varing growth of the bacteria may be observed by running a batch experimentusing different concentrations of the substrate.

S1 < S2 < S3 < S4 < S5


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2)

S5 S5 S4S4

No. of Log 5 S3

cells S 3 No. of Cells

S2 S2 S1 S1

1

Time Time

The Monod Model

a. The effect of substrate concentration on the specific growth rate is shown in Figure 1.

(a) The Monod Plot

3

S1 S2 S3 S4 S5S

1

2


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S

K S

m

m/2

Fig. 1. Effect of substrate concentration on the specific growth rate.

(3rd

: DC 350; 4th

DC 459 Fig. 6-18) Monod growth rate constant as a function of limiting foodconcentration

b. It has been observed that if one of the essential requirements (e.g, substrate) for growth is present in only limited amounts, the effect of a limitimg factor on the rate of growth can bedefined by the equation proposed by Monod (1942):

m S = ----------- (1)

Ks + S

where = specific growth rate, T -1 (day -1)m = maximum specific growth rate, T -1(day -1)S = growth rate limiting substrate concentration, ML -3 (mg/L)Ks = half-saturation coefficient; substrate concentration at one-half the maximum growth

rate, ML -3 (mg/L)= half-velocity constant (by Metcalf & Eddy)= Monod constant

c. The Monod expression provides a continuous transition between first and zero order kinetics based on growth limitimg substrate concentration.

d. If Ks >> S, 'm s

S k S

K

where k =m / Ks

the model would be reduced to a first order expression


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e. If Ks


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The Eadie PlotEadie plot

m S = ------------

Ks + S

(Ks + S) = m S Ks + S = m S Ks = ( m - ) S

m - m -- = ------------ = ------- -----S K s K s K m

1 m -- = - ---- + ------

S K s K s

Rate of Bacterial Growth, , g g

dX r

dt

a. In the batch culture, the specific growth rate ( ) is defined in the exponentical growth phase. b. Since bacteria increase their number at a constant rate in this phase, the rate of growth is

defined by the relationship:

m g

g

S dX r X X

dt Ks S

where r g = rate of bacterial growth, M L -3 T -1 (mg/Ld)X = concentration of viable bacteria (in VSS) at time t, ML -3 (mg/L) = specific growth rate, T -1 (d -1)

m

s

S K S

The parameter represent the rate of growth per unit concentration of bacteria and has thedimension of reciprocal time:

1

g

dX X dt

m/Ks

/S

Slope = 1/K s


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Rate of Substrate Utilization, , suu

dS r

dt

, suu

dS r

dt

= rate of substrate utilization, ML -3 T-1 (mg/L-d)

Cell Yield, Y

a. Through bacterial growth, a portion of substrate is converted to new bacterial cells and a portion is oxidized to end product releasing usable energy and heat energy.

g u

X S t t

b. The number of bacterial cells or the amountof biomas formed per unit amount ofsubstrate utilized has been reported to benearly constant for a given organism andsubstrate.

X2 X1 -------- = Y = yield coefficientS1 S2

X2 X1 ---------- = Y

S2 S1

where Y = bacterial cell yield = yield coefficient = maximum yield coefficient, MM -1 (mgVSS/mg BOD)

= the ratio of the mass of cells formed to the mass of substrate consumedX2 = concentration of viable bacteria at time t = 2, M L -3 X1 = concentration of viable bacteria at time t = 1, M L

-3 S2 = concentration of growth limiting substrate at time t = 2, M L -3 S1 = concentration of growth limiting substrate at time t = 1, M L -3

X2 X1 S2 S1 ---------- = -Y -----------

t2 - t 1 t2 t1

dX dS Y

dt dt

So X

Xo S

1Time, t

X , S

2


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g u

dX dS Y

dt dt

where suu

dS r

dt

= rate of substrate utilization, M L -3 T -1

Note: since utilization has meaning of negative, the sign is changed.

g

u

dX dt

Y dS

dt

Cell yield depends on (ME, p 371)

1) the oxidation state of the carbon source and nutrient elements2) the degree of polymerization of the substrate3) pathways of metabolism4) growth rate5) various physical parameters of cultivation

The Relationship between the Rate of Bacterial Growth and the Rate of SubstrateUtilization

c. The relationship between the rate of growth and rate of substrate utilization can be expressed

as:

g u

dX dS Y

dt dt

or r g = Y r su

1 1 m su

u g s s

dS dX SX kSX r

dt Y dt Y K S K S

where k = maximum rate of substrate utilization per unit mass of organisms, T -1 (d -1)

mk Y or m kY

m g

g

dX S kYS r X X X

dt Ks S Ks S


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Note: In a continuous flow reactor, CSTR,

suu

dS So S r

dt

u

dS So S dt

U X X

Other Rate Expressions

The substrate utilization rate, r su r kSX

Ks S su

1) When S >> Ks, r kSX

Ks S kX su .. 1

st order expression

- If the system is in endogenous phase and excess microbes,r su = - k .. zero order expression (8 -15), ME, p 373

2) When S


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m

s

S K S

(2)

dX dt

Y dS dt g u

(3)

mo

o o

o

t P X X

Q YS X X YS

( ) ln ( ) ln (4)

where

P K Y S Y X

YS X s o o

o o

Q K Y

YS X s

o o

Equations (1), (2), and (3) are the basic expression of bacterial growth kinetics. Using these basic kinetic expressions, develop the bacterialgrowth model [ Equation (4)] that describes the temporal bacterial growth.

Show clearly your work step by step. Define all the terms used in the equation, and give all the assumptions used.

Bacterial Death Rate, Endogenous Death (Decay) Rate, ,d d

dX r

dt

d d d

dX r k X

dt

wherer d = endogenous decay rate, , ML -3 T -1(mg L -1 d -1)

k d = death rate constant = engogenous decay coefficient, T-1

(d -1

)

Endogenous Phase - near starvation phase

dX Net rate of bacterial growth, r g , (-----) g net

dt

Net Growth Rate = Growth Rate - Death Rate

'net

g g g d

dX dX dX r dt dt dt

Where , d g d

dX dX X k X

dt dt


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' ( )net

g d net g

dX r k X X

dt

in which

mnet d d

s s

S Y k S k k

K S K S

where k = m / Y= maximum rate of substrate utilization per unit mass of organisms, T -1 (d -1)

'net

g net d

g s

dX Y k S r X k X

dt K S

dXEndogenous Phase Net rate of bacterial growth, r g , (-----) g net

dt

dX dX dXr g= ( ----) g net = (----) g (-----) d = X k d X = ( k d) X = net X = X

dt dt dt

d u

dS Y k X dt

dXr g= ( -----) g net = net rate of bacterial growth, ML -3 T -1

dt

where = net = net specific growth rate, T -1

dX m S X dS

since (----) g = ----------- = Y (----) u dt Ks + S dt

dX m S X dSr g = (----) g

net = ----------- - k d X = Y (----) u - k d X = Y r su - k d Xdt Ks + S dt


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dX m S kY Sr g = (----) g net = (---------- k d ) X = net X = (------------ k d) X

dt Ks + S Ks + S

where m = k Y

Note:m S Y ( m /Y) S

= net = ----------- k d = ---------------- k d Ks + S Ks + S

Y k S

= ----------- k d since m /Y = k m = Y kKs + S

The above expression will be used in developing mass balance on X (see CSTR model).

Observed Growth Yield, Y obs

(dX/dt) g net

Yobs = --------------(dS/dt) u

Substrate Utilization Rate, r su

1 1 m su

u g s

s

S dS dX r X

dt Y dt Y K S

kSX K S

Specific Substrate Utilization Rate, U

u

dS dt

U X

(1)


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where (dS/dt) u has positive value

u

dS kSX dt Ks S

(2)

Dividing (2) by X yields

u

dS kS dt

U X Ks S

kS U

Ks S (3)

Rearranging (3) for S as a function of U

U(Ks +S) = kSUKs + US = kSkS US = U KsS(k-U) = UKs

U KsS

k U

This is an expression for S as a function of U.- as U increases, S decreases

In a CSTR,

( )u

dS So S So So S dt

U X X X So

QSo So S F E

VX So M

where E = (So - S)/ So = substrate removal efficiency = V/Q = hydraulic retention time

= aeration time, hrF/M = Food to microorganisms ratio

----------------------------------------Find S as a function of U:

SincekS

U Ks S


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U (Ks+S) = kS

UKs US = kS

kS US = U Ks

S (k U) = U Ks

UKsS

k U

As U increases, S decreases

Evaluation of k and Ks

Linearlize Equation (3), Specific Substrate Utilization Rate eqn.

kS U

Ks S (1)

1 Ks S Ks S U kS kS kS

1 1 1 KsU k S k

Or, dividing Eq. (1) by S gives

U k S Ks S

Inverse the eqn

1S Ks S KsS

U k k k

1/U Slope = Ks/k

Intercept = =1/k

1/S

S/U Slope = 1/k

Intercept = =Ks/k

S

Bacterial Growth Kinetics

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