Deterministic population dynamics model

Some bacteriophages are parasites of bacteria, and as such, must prudently exploit their resources (in this case bacteria) to avoid killing bacterium before reproduce enough copies of itself. It has been suggested that parasites have evolved to tune their degree of virulence (amount of damage the parasite causes to the host) to achieve a balance between rapid reproduction and a prudent use of resources [1]. It is this fine balance which we intend to break, increasing the virulence of phage in such a way that kills the bacterium so fast that the phage is unable to assemble their own copies.

As a first approach, the infection process was mathematically modeled with a system of differential equations.

It is important to consider that the amount of phages at a given moment 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 [2]. The use of DDE allows us to update the system depending on the states of the system in previous points in time.

It is noteworthy that the success of the system, on a population level, depends on the efficiency of our suicide system after a bacterium 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 the one we are trying to modify experimentally).

Populations

Population

Population	Descirption	Units
S	Susceptible bacterial population	Bacteria/volume
I	Infected population	Bacteria/volume
P	Free phages	Phage/volume

Description of the system of nonlinear differential equations

System of delay differential equations

Bacteria (either susceptible or infected) grow 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 the control of phages, produce a large number of phages (burst size) that will be released when the infected population dies within a time τ(tau).
The term mi takes into account the death rate caused by the suicide system. The term e is the probability that the infected bacteria do not die in the course of infection due to the suicide system.
If the suicide system doesnt kill the infected bacteria at previous time τ(tau), it will result in a number b of phages.

The system was solved using dde23 solver in Matlab [3].

Parameters

Parameter

Parameter	Description	Value	Units	Reference
α	Growth rate constant of bacteria population	1	hr^-1	Experimentally estimated
C	Carrying capacity	1.25E+09	Bacteria/volume	Experimentally estimated
k	Adsorption rate	2.10E-07	ml/hr	[4]
b	Burst size	176	Number of phages	Molecular simulations
μi	Infected bacteria death rate	[0 25]	hr^-1	Estimated
μp	Phage decay rate	2.10E-07	hr^-1	[5]
τ	Latency period	0.2	hr	[5]

Table 2 contains the parameters used for the equations.

Results

The initial population was coded as a population vector [Susceptible Infected Phages] in Matlab.

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

Initial population vector set to [6e05 0 0].
As there are no phages present, the infected and phage populations does not increase and E. coli grows normally.

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

Initial population vector set to [6e05 0 1].
Susceptibles bacteria becomes infected when phages are added. Bacterial death is only caused by phage lysis product, therefore, the death rate is set to one time the inverse of the latency period (μi=5).

Behavior of an infection in an engineered E. coli.

Initial population vector set to [6e05 0 1].
when T7 infects a bacterium which has the suicide circuit, μi (mortality rate caused by suicide system) is increased, host bacteria die before the phages reproduce itself. The mortality parameter estimated caused by the toxin is five times the inverse of the latency period (μi=25).

Assumptions

In this modelling 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] Frank, S. A. 1996 Models of parasite virulence. Q. Rev. Biol. 71, 37–78.
[2] E. Beretta, Y. Kuang (2001): Modeling and Analysis of a Marine Bacteriophage Infection with Latency Period. Nonlinear Analysis : Real World Applications, 2, 35-74
[3] L.F. Shampine and S. Thompson, Solving DDEs in MATLAB.
[4] Heineman, R., Springman, R., Bull, J. (2008). Optimal Foraging by Bacteriophages through Host Avoidance. The American Naturalist, 171(4), E149-E157.
[5] 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.
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