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ESM 219

Lecture 5: Growth and Kinetics

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Microbial Growth

Region 1:Lag phase

microbes are adjustingto the new substrate(food source)

Region 2Exponential growth

phase, microbes have

acclimated to theconditions

Region 3Stationary phase,

limiting substrate orelectron acceptor limitsthe growth rate

Region 4Decay phase,

substrate supply hasbeen exhausted Time

log [ X]32 41

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During

exponential

phase growth, a

log-linear plot

produces astraight line.

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Generation time, a.k.a. doubling time, is the time

required for the population to double.

The calculation is: td = ln(2)/Q

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Exponential Phase Growth

XdtdX Q!

Log phase growth is first order, ie

Growth rate w to population sizeSo lnX vs. t is linear, slope = Q Qunits are 1/t (i.e. hr-1)

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Monod Growth Kinetics

SK

S

s

max

Q

!Q

Relates specific growth rateQ, to substrate concentrationEmpirical---no theoretical basisit just fits!

Have to determine Qmax and Ks in the labEach Q is determined for a different starting S

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Monod Growth Kinetics First-order region,

SKS, the equationcan be approximatedby Q = Qmax

S, mg/L

Q, 1/hr

Qmax

SKS

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Determining Monod parameters

Double reciprocal plot (Lineweaver Burke)

Commonly used

Caution that data spread are often insufficient Other linearization (Eadie Hofstee)

Less used, better data spread

Non-linear curve fitting

More computationally intensive Progress-curve analysis (for substrate depletion)

Less lab work (1 curve), more uncertainty

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Michaelis Menten Kinetics Used when microbe population is constant

= non-growing (or short time spans)

Derivable from first principles (enzyme-

substrate binding rates and equilibria

expressions)

Parameter determination methods used for

Monod calculations (i.e. Lineweaver Burke)

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Km/Vmax

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Monod vs. Michaelis-Menten:

recap of differences Monod

Growth

Empirical

Ks

Q, 1/t

Michaelis Menten

No growth; constant E

Derived from theory

Km

v, mg/L-t

Simlarities are shape of curves, form of function, parameter

estimation techniques.

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SubstrateD

epletion Kinetics The rate of biodegradation or biotransformation is

a focus of environmental studies

Substrate consumption rates have often beendescribed using Monod kinetics

Sis the substrate concentration [mg/L]

Xis the biomass concentration [mg/ L]

kis the maximum substrate utilization rate [sec-1]

KSis the half-saturation coefficient [mg/L]

SK

kSX

dt

dS

s !

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SubstrateD

epletion Kinetics Since

And

Then

And

SK

kS

dt

dS

s

!

SKS

s

max

Q!Q

dt

dSY

dt

d!Q!

Y)SK(

S

Ydt

dS

s

max

Q!

Q!

Where k =

Y

maxQ

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Modeling SubstrateD

epletion Three main methods for modeling

Monod kinetics (mid range concentrations)

First-order decay (low concentration of S,

applicable to many natural systems)

Zero-order decay (substrate saturated)

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Modeling First-OrderD

ecay dS/dt= kS where k is a pseudo firstorder

constantGenerally assumes nothing about

limiting substrates or electron acceptors Degradation rate is proportional to the

concentration

Generally used as a fitting parameter,encompassing a number of uncertainparameters

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Monod Kinetics First-order region,

SKS, the equationcan be approximated bylinear decay(C= C0kt)

dS

dt

S

First-orderregion

Zero-orderregion

dS

dt!

kSX

Ks

dS

dt! kX

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Microbial Kinetics in Modeling

Fate of a Substrate Use mass balance framework for modeling

fate of substance, S

Choose appropriate ideal reactor analogy

(usually batch or complete mix)

Substitute appropriate reaction expression

into the framework

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Mass Balance: Batch example

CV

Verbal: In Out + Reaction = Accumulation

Math: 0 0 rV (t (S V

Units: m/l3-t l3 t m/l3 l3

Rearrange: r V = (S/(t V

pClosedpWell-mixed

pConstant volume

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Mass Balance: Batch exampleTake limits as (S and (t p 0 r =

dt

d

Substitute a rate equation for r

e.g. 1st order decay of S: -kS

So, -kS =

dt

dS!t

0

t

0

tkS

dSS

S

Rearrange, integrate:

ktSlnS

S

t

0!

kt

0

t eS

S ! kt

0t eSs

!pp

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Mass Balance: Batch

example of exponential decayS0 = 100 mg/L, k =-0.2/hrConcentration versus time

0

20

40

60

80

100

120

0 1 2 3 4 5 6

time, hours

Concentration(mg/L)

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Mass Balance: CFSTRCV


Math: QS0 (t - QS (t + r V (t = (S V

Units: l3/t m/l3 t m/l3t t l3 m/l3 l3

Rearrange: Q/V (S0 S) + r =(S/(t

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Mass Balance: CFSTRTake limits as (S and (t p 0

dt

dS

pSubstitute a rate equation for re.g. 1st order decay of S: -kS

pMakesteady state (SS) assumption

(no net accumulation or depletion:pRearrange

0dt

dS!

Q/V (S0 S) + r =

? A)Q/V(k10SS

!

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Chemostat: CFSTR for

Microbial GrowthCV


Math: QX0 (t - QX (t + r V (t = (X V

Units: l3/t m/l3 t m/l3t t l3 m/l3 l3

Rearrange: Q/V (X0 X) + r =(X/(t

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Chemostat: CFSTR for

Microbial GrowthTake limits as (X and (t p 0

dt

dX

pSubstitute exponential growth equation for rpSet X0 = 0 (no influent cells)pMakesteady state (SS) assumption

(no net accumulation or depletion):p Let Q/V = D = dilution ratepRearrange:

0dt

dX!

Q/V (X0 X) + r =

XXV

QQ! Q!

V

Q D = Qp p

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Chemostat:CFSTR for

Microbial

Growth

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Environmental Factors Temperature

pH

Salinity

Oxygen Concentration

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Environmental Factors

Extremophiles can tolerate or perhaps require

extreme conditions in any of the above.

Cellular compensation outside of their optimacan reduce growth rate and yield.

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Temperature

effects

on growth

rate.

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Classifications of microbes according to temperature optima.

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Classification

of microbes

according

to toleranceofpH

extremes

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Classification

of microbes

according

to salinitytolerances.

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To equilibrate

their internal

solute concentration

with the external,

microbes make

compatiblesolutes.

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Classification

of microbes

according totheir oxygen

responses.

a. Aerobic

b. Anaerobic

c. Facultative

d. Microaerobic

e. aerotolerant

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Oxygen tolerance is conferred by enzymes that scavenge and scrub

toxic free radicals. Enzymes include superoxide dismutase, catalase

and peroxidase.

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