Deterministic population dynamics model

As a first approach for our problem, the infection was mathematically modeled with a system of equations.

It is important to consider that the amount of phages on a point in time depends on the amount of phages on a previous point in time due to the latency period (once the phage has inserted its genome, it requires a period of time to redirect the molecular machinery of the bacteria, reproduce and start assembling). To tackle this problem, we modeled the phage infection using a system of [http://en.wikipedia.org/wiki/Delay_differential_equation DELAY DIFFERENTIAL EQUATIONS (DDE)] based on the system proposed by Beretta[1]. The use of DDE allows us to update the system according to the states of the systems in previous points in time.

It is noteworthy that the success of system, on a population level, depends on the efficiency of our suicide system after a bacteria has been infected by a phage. To include this in our model, our system of equations must consider the mortality rate of bacteria after they have been infected by a phage (it is precisely this parameter which we are trying to modify experimentally).

In an infection we have three distinct populations:

• Not infected bacteria (susceptible to be infected)

• Bacteria that have already been infected

• Bacteriophages

System of nonlinear differential equations

System of delay differential equations

• Bacteria (either susceptible or infected) grows logistically with a carrying capacity C.
• According to the law of mass action, when a P (phage) encounters a S (susceptible bacterium), it attaches itself to the cell wall of the bacterium. The bacterium becomes I (infected) at rate K (Bacteriophage Adsorption Rate).
• Infected bacteria, now under control of phage, reproduce inside them a large number of phages copies (burst size) to release when the Infected population die within a time \tau.
• The term mi take into account the death rate caused by the suicide system. So the term e is the probability that infected bacteria do not die in the course of infection because suicide system.
• If the suicide system do not kill the infected bacteria at previous time tau, it will result in a number b of phages.

Results

The system was solved with matlab.

Normal growth of E. coli without phages in the medium. Bacteria population density grows according to a logistic equation.

Behavior of an infection in a wild type E. coli.

When μi es increased host bacteria dies before the phage can reproduces copies of itself. Death parameter by toxin is five times the inverse of latency period (μi=25).

Assumptions

In this modeling approach we assume that:

• One phage is only able to infect one bacterium.
• Bacteria and phages are well mixed (in equilibrium), neglecting the spatial considerations

&mu &alpha

References

[1] E. Beretta, Y. Kuang (2001): Modeling and Analysis of a Marine Bacteriophage Infection with Latency Period. Nonlinear Analysis : Real World Applications, 2, 35-74
[2] Heineman, R., Springman, R., Bull, J. (2008). Optimal Foraging by Bacteriophages through Host Avoidance.. The American Naturalist, 171(4), E149-E157.
[3] De Paepe M, Taddei F (2006) Viruses' life history: Towards a mechanistic basis of a trade-off between survival and reproduction among phages. PLoS Biol 4(7): e193. DOI: 10.1371/journal.pbio.0040193.