Cell Growth Models

Lecture 6

Fall 2007

Quantifying Growth Kinetics

• Structured vs Unstructured - – S - model divides the cell mass into components– U - assumes fixed cell composition - exponential

growth phase of batch or in continuous

• Segregated vs Nonsegregated– S - assumes different types of cells exist– N - all cells are the same type

Unstructured, Nonsegregated Models(Monod)

• Assumptions– One limiting substrate

– Semi empirical relationship

– Single enzyme system with Michaelis-Menten kinetics is responsible for the uptake of substrate

– Amount of enzyme is sufficiently low to be growth limiting

– Low population density

• Most commonly used expression for growth

SK

S

S

m

m - maximum growth rate when S >> Ks 

• Ks - saturation constant - concentration of the rate-limiting substrate when the specific rate of growth is equal to one half of the maximum.

• Ks = S when = 1/2m • In general = m for S >> Ks 

and for S << Ks S

m

K

S

Cell Growth - Monod

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25

Time (hr)

Ce

ll C

on

ce

ntr

ati

on

(g

/L)

Does not account for death phase

Does not account for lag phase

Unstructured, Nonsegregated Models

• Other models - Section 6.3.2.1 in text assuming one limiting substrate– Blackman– Tessier– Moser– Contois

• Multiple Substrates are growth limiting– Usually do not use unstructured, nonsegregated

model

Inhibition Models(Very similar to enzyme models)

• Substrate Inhibition– High substrate concentration inhibits growth– If a single-substrate enzyme catalyzed reaction is the

rate-limiting step then inhibition of enzyme activity results in inhibition of microbial growth.

Noncompetitive Substrate Competitive Substrate

I

S

m

KS

SK

11

S

KS

K

S

IS

m

1

Inhibition Models - cont.

• Product Inhibition– High concentrations of product can be inhibitory– Underlying mechanism of product inhibition is unknown– Approximated as exponential or linear decay functions

Noncompetitive Product Competitive Product

P

S

m

KP

SK

11

SKP

K

S

PS

m

1

Inhibition Models - cont.• Inhibition by Toxic Compounds

– inhibition of growth is analogous to enzyme inhibition

Noncompetitive Competitive

Uncompetitive Cell Death

I

S

m

KI

SK

11

S

KI

K

S

IS

m

1

II

S

m

KI

SKI

K

S

1)/1(

'd

S

m kSK

S

Batch Reactors• Cell Growth

• Substrate Utilization

• Product (cometabolic contaminants use negative sign)

XSK

SX

dt

dXr

S

mX

SXS

m

SXS Y

X

SK

S

Y

X

dt

dSr

//

SK

S

dt

dP

Xq

Ydt

dP

Xq

S

mp

ggXPp

1

1/

Logistic Equation

• Batch Growth Equation

• Combines Batch growth, Monod and Yield Coefficients

• No maintenance0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20

Time (hr)

Cel

l Co

nce

ntr

atio

n (

g/L

)

Rate Expression for Growth

(1)

Yield Expression

(2)

Substituting into Eq. 1 for S from Eq 2:

Integrating

Logistic Equation

XSK

SX

dt

dXr

S

mX

)( 0/0 SSYXX SX

XXXSYYK

XXSY

dt

dX

SXSXS

SXm

)(

)(

00//

00/

tSYXXSYXSY

YK

X

X

XSY

XSYYKmSXSX

SX

SXS

SX

SXSXS

0/00/

00/

/

000/

00// /)(ln)(

ln)(

)(

Describes Sigmoidal shape batch growth curve

Unstructured, Nonsegregated Models

• Disadvantage of Unstructured, Nonsegregated Models– No attempt to utilize or recognize knowledge about cellular

metabolism and regulation

– Show no lag phase

– give no insight to the variables that influence growth

– assume a black box

– assume dynamic response of a cell is dominated by an internal process with a time delay on the order of the response time

– most processes are assumed to be too fast (psuedo ss) or too slow to influence the observed response.

Filamentous Organisms• Types of Organisms

– mold

– bacteria or yeast entrapped in a spherical gel particle

– formation of microbial pettlets in suspension

• Model - no mass transfer limitations

– R - radius of the cell floc or pellet or mold colony

Then the growth of the biomass (M)can be written as

constkdt

dR

3/2

22 44

Mdt

dM

or

Rkdt

dRR

dt

dMp

3/1)36( pkWhere:

Filamentous Organisms - cont.

• Integrating the equation:

• M0 is usually very small then

• Model is supported by experimental data.

333/1

0 33

ttMM

3tM

Chemically Structured Models

• Improvement over nonstructured, nonsegregated models• Need less fudge factors, inhibitors, substrate inhibition, high

concentration different rates etc.• Model the kinetic interactions amoung cellular subcomponents• Try to use Intrinsic variables - concentration per unit cell mass-

Not extrinsic variables - concentration per reactor volume• More predictive• Incorporate our knowledge of cell biology

In Class Exercise

• The results shown below correspond to typical batch culture dynamics. Calculate the following:

• A)    max assuming Monod growth

• B)       YX/S

• C)    and assuming mixed growth and

• non-growth associated product formation