3b kinetics

8/4/2019 3b Kinetics

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ChEn 5751 -Fall 2008

Prof. Wei-Shou Hu 1

Prof. Wei-Shou Hu

ChEn 5751

Kinetics of Growth and Product

Formation

Outline

Quantitative description of growth and product formation Growth, expansion of population

Distributed properties

Description of batch cultures

Stoichiometric limiting and growth rate limiting substrate

Monod model and its variants Monod model

Batch culture kinetics

Maintenance energy

Kinetics of product formation

Growth kinetics of animal cells Anchorage dependence and contact inhibition

Senescene

Material balance on high density culture

Time

Cell Number (brown) Protein/Cell

(red) DNA/Cell

In a synchronized culture, increases in cell numbers, cell biomass

and cellular components are not in sync

Synchronized Culture

1.0 -

2.0 -

0 -

td (doubling time)

Cell Cycle in Eucaryotes

2 sets of geneticmaterial

DNA synthesis

Increase in size and material,

except DNA

Check point


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Prof. Wei-Shou Hu 2

Distributed properties

vs Average properties

DNA content

Cell size

F

G2

G1

S

1 2

isoclion

(same frequency)

Growth, Increase in Cell Concentrationa) In terms of biomass concentration,x

a) In terms of cell number, n

x and n are Specific Growth Rates

With initial condition,

When

is the doubling time

dx xdt

=

ndt

dnn=

0,

( )

o

o

t x x

x x E x p t

= =

=

then,

0

0

0

2

22

ln

ln 0.691

d

d

x xx

tx

t

=

=

= =

dt

24 60 hoursHuman cell culture

1 hourYeast

20 minute (rich medium) 37c4 hours (min medium) 37c

E. Coli

td

Typically plot in semi log

1 dxx dt

=

Growth Stages and Composition Changes

time

Log x

Log n

Stationary Phase

Exponential (log)

phase

Lag Phase

Decline Phase

nx

RNA per

biomass

DNA per cell or per biomass

x/n

Balanced Growth

At some stages of growth, or in a continuous culture at a steady

state, all the components in cells remain in the same proportion:

This is called balanced growth

i: the concentration of cellular component i, (mass ofi/culture volume)

1 1 1i

i

dn dx d

n dt x dt dt

= = =


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Prof. Wei-Shou Hu 3

Change in a population is caused by growth,

death, and migration

n: specific growth rate based on cell

number and concentration

d: Specific death rate (based on cell

number)

m: specific net migration rate

nVdt

nVd

dt

dNmdn +== )(

)(

Monod Model of Growth Kinetics

If S Ks, then max Two parameters, (max and ks),

define the relationship between and limiting substrate concentrations. is also a function of pH,temperature, nutritional status (i.e.serum), waste products; it may alsodepend upon cell density foranchorage-dependent cells whichare subject to contact inhibition.

max

s

s

K s

=+

Saturation

kinetic

Determination of kinetic parameters

max max

1 1 1

[ ]

sK

s = +

1

1

[ ]s

max

sK

max

1

A. Double reciprocal plot:

Plot vs.

Slope: Intercept on y axis

B. Regression analysis

Michaelis-Menten Eq.

can be rearranged to:

1/[s]

1/

The same enzyme at two different concentrations

Two regions ofreaction can beapproximated

-Zero order region:

-First order region:

Michaelis-Mentenkinetics-a saturationtype kinetics

Two enzymes at the same rmax but differ in Km

maxr r

)([ ] ms K>>

max

[ ]

[ ]m

sr r

K

([ ] )ms k0, the final cell concentration is limited by the

availability of Si

Example: Growth ceases when glucose is exhausted, while all the other nutrients (NH3, PO4-4,SO4

-4, etc) are still available Note: this is true only if there is no other substrate that can substitute glucose as a carbon source. For

example, if there is also glutamate, cells may continue to grow on glutamate after glucose isexhausted

Growth rate limiting: At t ime t, S it decreases so that (Sit)


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Prof. Wei-Shou Hu 5

max

s

dx sx x

dt K s

= =

+

max

/ / x s s x s

sds x xmx mx

dt Y K s Y

= =

+

Must impose s as non negative value

in simulation

x

s0

s0

One problem of

maintenance energy

concept is that it

leads to a negative

substrate

concentration,

because as

approaches 0, the

consumption term is

still positive.

Modifications of the Monod Model

Substrate inhibition:

Product Inhibition:

max

n

s I

s

K s K s

=

+ +

max

( ( ))( ( ) )n

s I

s

K s K P

=

+ +

s

s

Description of Product Formation

p

dpq x

dt=

+=pq

= growth-associated specific productivity

= non growth-associated specific productivity

Growth-associated products: e.g., ethanolNegatively growth associated products: secondary metabolites

qp

Mixed mode

Growth associated

Non growth-associated

Negative-growth associated

Conserved vs. non-conserved substrate

Non-conserved substrate:

Are used as energy sources (oxidizedproducts are excreted), in whole or in part

Conserved substrate:

Are used only for biomass and product

synthesis

i

sxf

Yi

1/ = (on a mass basis)


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Prof. Wei-Shou Hu 6

High Cell Density Culture Must Consider

Water Balance

Some substrate and non-substrate solutes in the mediummay even increase inconcentration in the abioticphase in culture

w: density of abiotic phase

xx m

dt

dm=

w

xw

f

m

dt

dm =

x

sx

si mYdt

dm

i

/

1=

xp

pmq

dt

dm

i

i =

outcoxco

coCQmq

dt

dm

22

2 =

)(,2,2

2

outin OO

oCCQ

dt

dm=

max( / )

( / )

S w w

s S w w

m m

K m m

=

+

Variation in Growth Kinetics

Examples:

1. Linear growth

Limited by other factors (e.g., mass transfer)

Non-single cellular microorganisms, e.g., mold

growing as pellets

2. Diauxic growth

Growth of E. coli in glucose and lactose

Diauxic growth

E coli grow on glucose, not utilizing lactose, until glucose is exhausted

After glucose depletion, there is a short period during which newenzymes are synthesized (i.e. induced) to adjust cells machinery to uselactose

So, lactose utilizing enzyme (beta-galactosidase) along with its transport

protein are repressed in the presence of glucose, and is induced by thepresence of lactose.

time

Concentration

Cell growth

glucose

lactose

What are the

evolutionary

advantage of such

regulation?

Animal Cells in Culture

ChE 5751 F ll 2008


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Prof. Wei-Shou Hu 7

Hayflick Phenomenon

Cell Senescence

Cell Doublings or Passages

Specific

Growth

Rate

Total cell death

crisis

Cell Strain Continuous

Cell Line

Normal diploid aneuploid

Normal diploid cells have limited life span in culture; after a number of

population doubling, they will lose the capability to grow. Even if they

are frozen and kept for years, upon thawing and returning to culture,

they remember how many more doublings they can go through.

Hayflick

Cultivation of Normal Diploid Cells

g r o w u n t il c o n t a c ti n h i b i t e d

T r y p s i n iz e t or e l e a s e c e l ls

p l a t e o n t ola r g e r s u r f a c e

d i s s o c ia t e f r o m t i s s u e

Can repeat the

process of culturing

them onto larger

surface, but only in

limited number of

times. For mouse

cells, about 60

times.

Those cells can befrozen down,

thawed years later

and grow again. But

total number of

replication is limited

and relatively

constant.

What are the elements to be included in a

model for growth of normal animal cells?Kinetic Models

Why models?

Summarizing large amount of data

Simulate system behavior to facilitateunderstanding

Predictive value

At least two classes of models

Emperical models Monod model and its variant

Mechanistic models Based on mechanism, more predictive

ChE 5751 F ll 2008


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Prof. Wei-Shou Hu 8

Analysis of Experimental Data

In-Process data analysis Plot measurement data for data quality evaluation

Calculate derived variables (specific rates, stoichiometric

ratios, yields etc)

Post-Process data analysis Data regression on entire time profiles of data

Calculate derived variables base on regressed data

3b kinetics

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