1. Experimental and Mathematical Analysis of Bacteria and Bacteriophage Dynamics in a Chemostat
John Jeffrey Jones, Victor Rodriguez, Frank Healy1 and Saber Elaydi2
1Department of Biology, Trinity University, San Antonio, TX
2Department of Mathematics, Trinity University, San Antonio, TX
The ecological dynamics between viruses and their hosts have proved
important to our understanding of evolutionary processes. In order to explore
viral-host ecological dynamics, we have developed a mathematical model to
describe the interactions between bacteriophage T4 and Escherichia coli strain
B in continuous culture chemostat vessels. A system of difference equations
derived using nonstandard numerical methods from the differential equations
proposed by Bohannan and Lenski1. Various mathematical parameters were
measured experimentally, while others were determined by nonlinear regression
analysis using math software, R. Several experiments were performed in order
to characterize host and virus properties as well as chemostat parameters. This
work describes the results of these studies and sets the stage for pending work
involving comparative studies between experimental and simulated datasets.
PHAGE-HOST INTERACTION
To date, we have only managed to gather population data for the
resistant bacteria. Without prior knowledge of appropriate dilution
factors for plating, we were not able to detect the sensitive
population over the course of a seven hour experiment with
sampling occurring every 30 minutes. However, we now know the
precise dilution factors that will enable us to monitor all three
populations. This population data will then enable us to find our
missing parameters via nonlinear regression analysis.
After we achieve success, we will hopefully introduce 3 more
chemostats, with which we will manipulate the glucose
concentration as well as the flow rate. These variables will allow us
to alter the density of the bacterial populations and the dilution rate.
In turn, these will affect the parameters accordingly.
PARAMETERS
1. Bohannan, B. & Lenski, R. (2000) Linking genetic change to
community evolution: insights from studies of bacteria and
bacteriophage. Ecology Letters, 3, 362-377.
2. Chao, L., Levin, B.R. & Stewart, F.M. (1977). A complex community in
a simple habitat: an experimental study with bacteria and phage.
Ecology, 58, 369-378.
3. Hadas, H., Einav, M., Fishov, I. & Zaritsky, A. (1997). Bacteriophage
T4 development depends on the physiology of its host Escherichia coli.
Mircobiology, 143, 179-185.
4. Lenski, R.E. (1984). Two-Step Resistance by Escherichia coli B to
Bacteriophage T2. Genetics, 107, 1-7.
Figure 1. Flow chart for a typical interaction between a bacteriophage
and a bacterial host. This is known as the lytic viral replication cycle,
during which the virion first attaches to a host’s receptor via its tail
fibers, injects its genome through the bacterial cell wall, replicates by
arresting the metabolism of the host, and finally lyses through the cell
membrane.
7.06
7.08
7.1
7.12
7.14
7.16
7.18
7.2
0 1 2 3 4 5 6 7 8 9
log[pfu/mL]
Time (minutes)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
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0.17
0.18
0.19
0.2
0.21
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0.24
0.25
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
OpticalDensity(AU)
Time (minutes)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
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0.28
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
OpticalDesity(AU)
Time (minutes)
0.17
0.175
0.18
0.185
0.19
0.195
0.2
0.205
0.21
0.215
0.22
0 1 2 3 4 5 6
CellMass
[Glucose]
Table 1. Symbols with corresponding definitions used in the
mathematical model. The following can be determined experimentally:
R, NA, NC, P, ω, ε, αA, β, τ, NA’, P’; the others, stemming from the
Monod equation, via parameter estimation: ΨA,ΨC, ΚA, ΚC .
MATHEMATICAL MODEL
Table 2. Difference equations for population dynamics in a chemostat.
This model assumes that the bacteriophage exhibits no host-range, i.e., it
does not mutate in response to bacteria that become resistant to wild-
type phage and also that the mutation rate is negligible since resistant
phenyotypes of bacteria are initially present in the immense chemostat
population. Nonstandard numerical methods were employed in order to
transform the aforementioned differential equations, such that the
dynamics remained similar, and also to account for the fact that samples
could only be measured at discrete time intervals. We have yet to verify
this model.
GROWTH EFFICIENCY (ε)
Figure 2. Overnight batch culture growth efficiency for E. coli strain B.
Cell mass was measured at various glucose concentrations by vacuum
filtration using a Millipore filter holder and a Millipore filter with a pore
size of 0.45 μ. The bacterial yield is equal to the slope of equation y =
0.0056x + 0.185; R2 = 0.98981. Since growth efficiency is defined at the
reciprocal of the bacterial yield, ε = 178.57 ± 0.01 mg. Error bars, ±1
standard deviation from the mean.
LATENT PERIOD (τ)
Figure 3. Adsorption rate of phage on sensitive bacteria. At two minute
intervals, two 100-fold dilutions were performed which effectively stops
the density-dependent process of phage adsorption, and then three drops
of CHCl3 were added, since chloroform kills the bacteria and the phage
that have adsorbed to them but leaves free (unadsorbed) phage
unaffected. The adsorption coefficient was estimated from the slope of
the exponential decay in concentration of free phage estimated by the
regression of the log of free phage against time, corrected for the density
of bacterial cells on which adsorption occurs4; thus, αA = 7.67 X 10-7
mL/hr. Error bars, ±1 standard deviation from the mean.
ADSORPTION RATE (αA)
Figure 4. Latent period for sensitive bacteria. The latent period is
defined as the time elapsed between infection and burst during which
phage particles are assembled. Controls were used in each experiment,
in which no phage was added, which exhibited no decrease in optical
density. (A) Growth in Lysogeny Broth (LB) medium, where τ = 20 ± 2
minutes. (B) Growth in M9 minimal medium, consisting of inorganic
salts and 20 mM glucose, where τ = 28 ± 2 minutes. Error bars, ±1
standard deviation from the mean.
0
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1
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1.6
1.8
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 520 540 560
OpticalDesity(AU)
Time (minute)
GROWTH RATES UNDER VARIOUS GLUCOSE CONCENTRATIONS
Figure 5. Kinetic growth rates under various glucose concentrations.
Optical density was measured via a spectrophotometer at 600 nm every
20 minutes until stationary phase was reached. The glucose
concentrations, in increasing order:
[0.125]<[0.25]<[0.5]<[1]<[2.5]<[15]<[10]<[5]; the data suggests that the
more rapid the substrate consumption, the more likely inhibitory
metabolites are released. The slopes of each exponential phase yielded an
average doubling time of 54 ± 3 minutes for E. coli strain B.
A
B
ABSTRACT
FLOW RATE (ω)
The flow rate for this Fischer Scientific mini pump
was calculated by measuring the amount of time
reach a volume of 1 mL in a 10 mL graduated
cylinder at the lowest possible speed in order to
maximize the reagents used to create the reservoir.
We obtained a value of 0.333 mL/min.
Knowing that the specific growth rate, defined as
the increase in cell mass per unit time (speed of
cell division), for our strain was equal to 0.0068
min-1, it was also important to find the minimum
culture volume such that the bacteria could
maintain a stable population (steady-state) and not
wash out. Thus, the dilution rate, defined as the
medium flow rate divided by the culture volume,
must be ≤ the specific growth rate and the culture
volume be no less than 49 mL.
Mini pump
Chemostat
apparatus
BURST SIZE (β)
Burst size is defined as the total number of phage
progeny released per bacterial cell. We performed
two independent experiments in order to
accurately determine the burst size and
corroborate the results; one followed the protocol
according to the one-step growth experiment []
and yielded a burst size of 17 ± 1; the other
followed our own protocol using a multiplicity of
infection (MOI) of 100, resulting in β = 14 ± 3.
These results happen to be consistent with
literature3.
Right: Flask
containing phage
that exhibits lysis
(reduced turbidity)
Left: Control
FUTURE WORK
REFERENCES