Experimental and Mathematical Analysis of Bacteria and Bacteriophage Dynamics in a ChemostatJohn Jeffrey Jones, Victor Rodriguez, Frank Healy1 and Saber Elaydi2

1Department of Biology, Trinity University, San Antonio, TX2Department 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.

1 2 3 4 5 6 7 8 97.04

7.06

7.08

7.1

7.12

7.14

7.16

7.18

Time (minutes)

log[

pfu/

mL]

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 1800

0.010.020.030.040.050.060.070.080.090.1

0.110.120.130.140.150.160.170.180.190.2

0.210.220.230.24

Time (minutes)

Opt

ical

Den

sity

(AU

)

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 1700

0.020.040.060.080.10.120.140.160.180.20.220.240.260.280.30.320.340.360.380.40.420.440.46

Time (minutes)

Opt

ical

Des

ity (

AU)

0 1 2 3 4 5 60.17

0.175

0.18

0.185

0.19

0.195

0.2

0.205

0.21

0.215

0.22

[Glucose]

Cell

Mas

s

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 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 5400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (minute)

Opt

ical

Des

ity (

AU)

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