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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
Recommended