An Empirical Investigation of Delayed Growth Response in

Escherichia coli

Nariman GhoochanJerald D. HendrixSean EllermeyerDepartment of Biological and Physical SciencesDepartment of MathematicsKennesaw State University

Overview Continuous culture of bacteria can

be achieved in a chemostat Chemostat: A broth culture system

in which fresh nutrient is continuously added at a constant rate (and used broth is removed at the same rate)

Basic Chemostat System

Our Chemostat System

Overall Objectives of Our Work To develop and refine mathematical

models that predict the growth of bacteria in continuous culture

To test the predictions of the models under a variety of experimental conditions

Mathematical Models of Continuous Bacterial Culture

Factors that affect bacterial population growth in continuous culture: Relationship between the organism’s growth

rate and the limiting nutrient concentration The amount of bacteria produced per unit

mass of nutrient (yield) Concentration of limiting nutrient in the feed Flow rate and vessel volume

Mathematical Models of Continuous Bacterial Culture

The classic model (Monod model) of continuous culture: Is a set of differential equations That predict changes in bacterial

concentration and limiting nutrient concentration over time.

The Monod model assumes that the bacterial growth rate responds instantaneously to a change in nutrient concentration

Mathematical Models of Continuous Bacterial Culture

The Monod model is given in the equations:

)()(

)(1))(( tx

tsK

tsμ

YtssD

dt

ds

h

mf

)()()(

)(tDxtx

tsK

tsμ

dt

dx

h

m

Mathematical Models of Continuous Bacterial Culture

We have modified the Monod model to account for a delayed response of growth rate to a change in nutrient concentration

We have determined a preliminary fit of this model to continuous culture of E. coli 23716, under conditions of limiting glucose concentration

Mathematical Models of Continuous Bacterial Culture

Our delayed response model:

)()(

)()( tx

tsC

tsνtssD

dt

ds

h

mf

)()()(

)(exp tDxτtx

τtsC

τtsYνDτ

dt

dx

h

m

τμY

μν

m

mm

exp

hhh KτμC 1exp2

Experimental Details E. coli 23716

Grown in Davis minimal broth with glucose as the limiting nutrient and sole carbon source

Starter (batch) culture in chemostat vessel grown to early stationary phase

Continuous culture in Virtis chemostat (1500 ml) at 37°C, with stirring and aeration

Flow rate of 3 ml/min, with varying glucose concentrations in the feed

Bacterial concentration determined by measuring A425. The absorbance measurements were calibrated and converted to dry mass equivalents (g/L)

Results: Estimation of m & Kh

m: Estimated by determining the growth

rate of 23716 in excess glucose (0.2%) in batch culture in the reaction vessel

Kh: Estimated by determining the growth

rate in a series of glucose concentrations (0.005 – 0.1% glucose)

Results: Estimation of m & Kh

Growth of E. coli 23716, 9-20-01 batch culture y = 0.0187e

0.0069x

R2 = 0.9928

0.01

0.1

1

10

0 200 400 600 800 1000 1200 1400 1600

time, min

A42

5

Results: Estimation of m & Kh

Growth of E. coli 23716on varying [glucose]

0.00000

0.00050

0.00100

0.00150

0.00200

0.00250

0.00300

0.00350

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

[glucose], %

gro

wth

rat

e, 1

/min

Results: Estimation of m & Kh

Double-reciprocal plot of glucose vs. growth rate

y = 1.1631x + 285.66

R2 = 0.9871

-100

0

100

200

300

400

500

600

-400 -300 -200 -100 0 100 200 300

1/[glucose], 1/%

1/(g

row

th r

ate)

, min

0.1% glucose run using Monod Model (tau=0) and 28% yield

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 500 1000 1500 2000 2500

Time (min)

bac

teri

a (g

/l)

observed

initial data

predicted

0.1% glucose run using Monod Model (tau=0) and 28% yield

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 500 1000 1500 2000 2500

0.1% glucose run assuming 20 min delay and 28% yield

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 500 1000 1500 2000 2500

Time (min)

bac

teri

a (g

/l)

observed

initial data

predicted

0.1% glucose run assuming 20 min delay and 28% yield

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 500 1000 1500 2000 2500

0.000

0.050

0.100

0.150

0.200

0.250

0 500 1000 1500 2000 2500

time (min)

bac

teri

a (g

/l)

observed

predicted

0.05% glucose run using Monod model (tau=0) and 28% yield

0.000

0.050

0.100

0.150

0.200

0.250

0 500 1000 1500 2000 2500

0.000

0.050

0.100

0.150

0.200

0.250

0 500 1000 1500 2000 2500

time (min)

bac

teri

a (g

/l)

observed

initial data

predicted

0.05% glucose run assuming delay of 20 min and 28% yield

0.000

0.050

0.100

0.150

0.200

0.250

0 500 1000 1500 2000 2500

0.000

0.050

0.100

0.150

0.200

0.250

0 1000 2000 3000 4000 5000 6000

time (min)

bac

teri

a (g

/l)

observed

predicted

0.025% glucose run using Monod model (tau=0) and 28% yield

0.000

0.050

0.100

0.150

0.200

0.250

0 1000 2000 3000 4000 5000 6000

0.000

0.050

0.100

0.150

0.200

0.250

0 1000 2000 3000 4000 5000 6000

time (min)

bac

teri

a (g

/l)

observed

initial data

predicted

0.025% glucose run assuming delay of 20 min and 28% yield

0.000

0.050

0.100

0.150

0.200

0.250

0 1000 2000 3000 4000 5000 6000

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0 200 400 600 800 1000 1200 1400 1600 1800 2000

time (min)

bac

teri

a (g

/l)

observed

predicted

0.01% glucose run using Monod model (tau=0) and 28% yield

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0 200 400 600 800 1000 1200 1400 1600 1800 2000

time (min)

bac

teri

a (g

/l)

observed

initial data

predicted

0.01% glucose run assuming delay of 20 min and 28% yield

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.1% glucose run using Monod Model (tau=0) and 49% yield

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 500 1000 1500 2000 2500

Time (min)

bac

teri

a (g

/l)

observed

predicted

0.1% glucose run using Monod Model (tau=0) and 49% yield

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0 500 1000 1500 2000 2500

0.1% glucose run assuming 360 min delay and 49% yield

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 500 1000 1500 2000 2500

Time (min)

bac

teri

a (g

/l)

observed

initial data

predicted

0.1% glucose run assuming 360 min delay and 49% yield

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 500 1000 1500 2000 2500

0.000

0.050

0.100

0.150

0.200

0.250

0 500 1000 1500 2000 2500

time (min)

bac

teri

a (g

/l)

observed

predicted

0.05% glucose run using Monod model (tau=0) and 49% yield

0.000

0.050

0.100

0.150

0.200

0.250

0 500 1000 1500 2000 2500

0.000

0.050

0.100

0.150

0.200

0.250

0 500 1000 1500 2000 2500

time (min)

bac

teri

a (g

/l)

observed

predicted

initial data

0.05% glucose run assuming delay of 360 min and 49% yield

0.000

0.050

0.100

0.150

0.200

0.250

0 500 1000 1500 2000 2500

0.000

0.050

0.100

0.150

0.200

0.250

0 1000 2000 3000 4000 5000 6000

time (min)

bac

teri

a (g

/l)

observed

predicted

0.025% glucose run using Monod model (tau=0) and 49% yield

0.000

0.050

0.100

0.150

0.200

0.250

0 1000 2000 3000 4000 5000 6000

0.000

0.050

0.100

0.150

0.200

0.250

0 1000 2000 3000 4000 5000 6000

time (min)

bac

teri

a (g

/l)

observed

initial data

predicted

0.025% glucose run assuming delay 360 min and 49% yield

0.000

0.050

0.100

0.150

0.200

0.250

0 1000 2000 3000 4000 5000 6000

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0 200 400 600 800 1000 1200 1400 1600 1800 2000

time (min)

bac

teri

a (g

/l)

observed

predicted

0.01% glucose run using Monod model (tau=0) and 49% yield

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0 200 400 600 800 1000 1200 1400 1600 1800 2000

time (min)

bac

teri

a (g

/l)

observed

initial data

predicted

0.01% glucose run assuming delay 360 min and 49% yield

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Conclusions Using the empirically-estimated

value of = 20 min: Good fit of the predictions of the time-

delay model with experimental data (using Y = 0.28)

Very little difference between predictions of the time delay and Monod models

Conclusions Using the value of = 360 min:

Good fit of the predictions of the time-delay model with experimental data (using Y = 0.49)

Large differences between predictions of the time delay and Monod models

Continuing Research Is there a short or a long time delay

during chemostat runs? Analysis of glucose concentration over

time Analysis of model with a different limiting

nutrient (e.g., with a tryptophan auxotroph)

Isolation of “time delay” mutants with varying values