6 3 quantifying growth kinetics6.3. quantifying growth...
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6 3 Quantifying Growth Kinetics6.3. Quantifying Growth Kinetics
ModelModelUnstructured Structured .N
Most idealized caseCell population treated as Multicomponent
Nonsegr
BalancedGrowth
one component solute average cell description
regated
(approximation)Average cell Average cell
single component Multicomponent
Seg Balanced
approximation approximation
g p pheterogeneous description of cell-to-cellindividual cells heterogeneity
gregated
Growth
(approximation)Actual case
d .
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6.3.2. Using Unstructured Nonsegregated Modelsto Predict Specific Growth Rate
6 3 2 1 Substrate-limited growth6.3.2.1. Substrate-limited growth
Monod equationMonod equation
µ =µm S_____ µ Ks + S
µm : maximum specific growth rateKs : saturation constant
Other equations
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Other equationsfor substrate-limited growth
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When more than one substrate is growth rate limiting
Interactive or multiplicativeµ/µ = [µ(s )] [µ(s )] [µ(s )]µ/µm = [µ(s1)] [µ(s2)] … [µ(sn)]
AdditiveAdditiveµ/µm = w1 [µ(s1)] + w2 [µ(s2)] + … + wN [µ(sn)]
Noninteractive( ) ( ) ( )µ = µ(s1) or µ(s2) or … µ(sn)where the lowest value of µ(si) is used.µ( i)
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6 3 2 2 Models with growth inhibitors6.3.2.2. Models with growth inhibitors
The inhibition pattern of microbial growth is analogous to enzyme inhibition.a a ogous to e y e b t o
Substrate inhibition
Product inhibition
Inhibition by toxic compound
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Substrate Inhibitionof Cell Growth
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Enzyme InhibitionEnzyme Inhibition
N titi i hibitiNoncompetitive inhibition
Competitive inhibition
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Product Inhibition of Cell GrowthProduct Inhibition of Cell Growth
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Cell Growth in Ethanol FermentationCell Growth in Ethanol Fermentation
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Inhibition by Toxic CompoundsInhibition by Toxic Compounds
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Enzyme InhibitionEnzyme Inhibition
Competitive inhibitionCompetitive inhibition
Noncompetitive inhibitionNoncompetitive inhibition
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Enzyme InhibitionEnzyme Inhibition
Uncompetitive inhibition
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Inhibition by Toxic CompoundsInhibition by Toxic Compounds
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6 3 2 3 The logistic equation6.3.2.3. The logistic equation
dX/dt XdX/dt = µ X
dX/dt k (1 X/X ) X X(0) XdX/dt = k (1-X/X∞) X, X(0)=X0
X0 ektX =
X0 e1- (X0 /X∞) (1 - ekt)
______________
Sigmoidal(stationary phase involved)
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6.3.2.4. Growth modelfor filamentous organisms
In suspension culture : formation of microbial pelletOn solid medium : colony (long and highly branched cellsy ( g g y
In the absence of mass transfer limitations
R∝ t R: radius of a microbial pelletR t R: radius of a microbial pelletradius of a mold colony
dR/dt = kp = constant Eq(1)
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Growth model for filamentous organismsGrowth model for filamentous organisms
M : mass of pellet
dM/dt (4 R2) (dR/dt)dM/dt = ρ (4πR2) (dR/dt)
= kp ρ (4πR2) (by Eq(1))p ρ ( ) ( y q( ))
= kp (36πρ)1/3 (4/3 πR3ρ)2/3
= γ M2/3 Eq(2)
where γ = kp (36πρ)1/3
M = 4/3 πR3ρ
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Growth model for filamentous organismsGrowth model for filamentous organisms
E (2)Eq(2)
dM/dt = γ M2/3 Eq(2)γ q( )
M-2/3 dM = γ dt
3 [M1/3] = γ t
M = (M 1/3 + (γ t)/3)3
M
M0
M = (M01/3 + (γ t)/3)3
≈ (γ t /3)3
M0 (the initial biomass) is usually very smallcompared to Mcompared to M.
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6 3 3 Chemically Structured Model6.3.3. Chemically Structured Model
Since it is not practical to write material balances on every cell component, we must select skillfully the key variables and processes of major interest in a particular application when formulating aof major interest in a particular application when formulating a structured kinetic model.
Mass balance inside the cell
Batch VR : total volume of liquid in the reactorC : extrinsic concentration of component iCi : extrinsic concentration of component iX : extrinsic concentration of biomass
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Intrinsic and Extrinsic ConcentrationsIntrinsic and Extrinsic Concentrations
Intrinsic concentrationThe amount of a compound per unit cell mass (or cellThe amount of a compound per unit cell mass (or cell volume)
Extrinsic concentrationThe amount of a compound per unit reactor volumeThe amount of a compound per unit reactor volume
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Intrinsic and Extrinsic ConcentrationsIntrinsic and Extrinsic Concentrations
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Intrinsic and Extrinsic ConcentrationsIntrinsic and Extrinsic Concentrations
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Single-cell model for E. coli by Shuler and colleagues
Fig. 6.14
The model contains the order of 100 stoichiometric and kinetic parameters almost allstoichiometric and kinetic parameters, almost all of which can be determined from previous literature an the biochemistry of E coli growthliterature an the biochemistry of E. coli growth.
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Single-cell model for E. coli by Shuler and colleagues