The study developed a mathematical model to describe the interactions between bacteriophage T4 and its host Escherichia coli B in continuous culture systems. Several experiments were performed to characterize host and virus properties as well as system parameters. To date, population data has only been gathered for resistant bacteria. Future work will involve collecting data on all three populations to parameterize the model and examine effects of varying nutrient concentrations and flow rates on population dynamics and stability over time.

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 proven
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 its host Escherichia
coli B, in continuous culture chemostat vessels. In this study, we have used
nonstandard numerical methods to discretize a series of coupled differential
equations originally proposed by Bohannan and Lenski1. Various
mathematical parameters were measured experimentally, while others are to
be determined using nonlinear parameter estimation methods 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 every 30 minutes. However, we now know the precise
dilution factors that will enable us to monitor all three populations.
This data will then enable us to find our missing parameters via
nonlinear regression analysis.
After we achieve success, we can extend the study to address
effects of host population size and growth rate by altering nutrient
concentrations and flow rates. These variables will allow us to alter
the density of the bacterial populations and the growth rates. These
studies will broaden our understanding of virus-host and predator-
prey dynamics.
Lastly, stability analysis will be performed in order to elucidate
the global behaviors of the populations over extended periods of
time.
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.
Microbiology, 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 lytic
bacteriophage and its 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
the cell, releasing progeny phage particles (adapted from ref. 1).
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* The following can be determined experimentally: R, NA, NC, P, ω, ε,
αA, β, τ, NA’, P’; the others, stemming from the Monod equation, will be
determined by parameter estimation: ΨA,ΨC, ΚA, ΚC .
MATHEMATICAL MODEL
Figure 2. Discrete equations for population dynamics in a chemostat.
The model assumes that the bacteriophage host-range is fixed, and that
reversion of bacteria from phage-resistant to phage-sensitive is
negligible. Nonstandard numerical methods were employed in order to
generate these equations from differential equations reported in ref. 1.,
such that the dynamics remained similar, and also to account for the fact
that samples could only be measured at discrete time intervals.
GROWTH EFFICIENCY (ε)
Figure 3.Growth efficiency of host strain E. coli B. Overnight cultures
of E. coli B were grown with various concentrations of glucose and cell
mass was measured by vacuum filtration using a Millipore filter. The
bacterial yield is equal to the slope of equation y = 0.0056x + 0.185; R2
= 0.98981. Since growth efficiency is defined as the reciprocal of the
bacterial yield, ε = 178.57 ± 0.01 mg. Error bars, ±1 standard deviation
from the mean.
LATENT PERIOD (τ)
Figure 4. 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 5. Latent period for sensitive bacteria in different media. 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, and which exhibited no
decrease in optical density. (A) Lysogeny Broth (LB) medium, where τ =
20 ± 2 minutes. (B) M9 minimal medium, consisting of inorganic salts
and 20 mM glucose, where τ = 28 ± 2 minutes. Error bars, ±1 standard
deviation from the mean.
GROWTH YIELDS UNDER VARIOUS GLUCOSE CONCENTRATIONS
Figure 6. Kinetic growth yields under various glucose concentrations.
Optical density was measured via a spectrophotometer at 600 nm every
20 minutes until stationary phase was reached. Growth yield is positively
correlated with increasing glucose concentrations ≤ 5%; above 5%
glucose, growth is inhibited. 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 Fisher Scientific mini pump
was calculated by measuring the amount of time
required to reach a volume of 1 mL in a 10 mL
graduated cylinder at the lowest setting. Pump flow
rate was determined to be 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; the first
experiment yielded a burst size of 17 ± 1; the
second experiment resulted in β = 14 ± 3.
These results are consistent with the
literature3.
Right: Flask containing
phage and lysed
bacteria, evidenced by
reduced turbidity; Left:
Control flask containing
uninfected bacteria
FUTURE WORK
REFERENCES
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