Team:LCG-UNAM-Mexico:odes

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=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).

Contents

 * Populations
 * Description of the system    of nonlinear differential equations
 * Parameters
 * Assumptions
 * References

Populations
Population   

Description of the system of nonlinear differential equations
The system was solved using dde23 solver in Matlab [3].
 * 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.

Parameters
Parameter   

Table 2 contains the parameters used for the equations.

Results
The initial population was coded as a population vector [Susceptible Infected Phages] in Matlab. 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. 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).
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Assumptions
In this modelling approach we assume that: