Deterministic 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 DELAY DIFFERENTIAL EQUATIONS (DDE) based on the system proposed by Beretta. 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

S Susceptible bacterial population Bacteria/volume

I Infected population Bacteria/volume

P Free phages Phage/volume

alpha Growth rate constant of bacteria population 1 hr^-1

C Carrying capacity 1.25e+09 Bacteria/volume

k Adsorption rate 2.1000e-007 ml/hr

b Burst size 200 Number of phages

mp Phage decay rate 2.1000e-007 2.1000e-007 hr^-1

mi Infected bacteria death rate Parameter to modified in our project hr^-1

tau Latency period 0.2 hr

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

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.