Team:BGU Israel/Model1.html

A Stochastic Birth-Death Model

It is possible to describe the behavior of a bacterial population by looking at a specific generation k of bacteria and examining its inter-division rate and death rate:



Where β is inter-division rate and μ is death rate.
A deterministic equation to describe such a population will look like this:



This deterministic approach assumes that the doubling time equals the inter-division time, meaning


The stochastic approach for the same issue suggests that there are different inter-division and death time distributions that will fit a specific population behavior, and not just an average time.

One can replace the use of the rates β and μ with the use of inter-division and death time distributions instead [1].

There is evidence [2][1] and it is fair to assume that the distributions for cell division and death times are similar to the Gamma distribution. Right after an individual cell has divided, in order for a second proliferation to happen, a set of events in the cell cycle need to occur, therefore proliferation probability is low right after the division, is rising as time advances, and is falling again asymmetrically as time goes by after avg. cell cycle time. The same can be assumed for death time distribution because the cell cycle can be considered a sensitive time with high death possibility.

In P.A.S.E 1, the concentration of cI protein directly affects those distributions, as after the cell is released, no more induction that produces cI occurs. As the protein’s concentration decreases, repression of Holin and Endolysin becomes less and less effective, and they are produced in higher rates, this effects the cell survivability, or in our model, the death time distribution is changing such that the cell is more likely to die before it proliferates.

Same can be said for P.A.S.E 2, with less and less tyrosine-tRNA-synthetase, crucial steps in the cell cycle can’t be completed, thus changing death time distribution in the same manner as explained for P.A.S.E 1.

The Gamma(a,b) distribution is defined by its parameters- ’a’ is shape (dimensionless), ’b’ is scale (minutes in our case). The Mean of the distribution equals a*b and represents the average inter division time or average death time.

In this model we chose distributions for inter-division and death times such that they will describe the behavior. Those distributions were initially configured to represent a regular genetically modified E.coli, inter division time distribution mean was set for a*b=35. We defined this initial setting to correspond to a P.A.S.E 1 with 100% CI concentration, or a P.A.S.E 2 with 100% tyrosine tRNA synthetase concentration (the state of the organism just after its release when it is fully induced). The algorithm, which is based on a Gillespie algorithm, advanced each individual in the population with time according to those distributions (a decision is being made each iteration for each cell if it will die or proliferate). With each update, individual protein concentration was updated as well.

As described in the overview, we defined the problematic parameters that we are modeling as the leakage of the system, and the strength of the mechanism influencing the cell. If the system is leaky, protein concentration will remain high and the cell will be able to survive. If the mechanism is not effective, protein concentration will have no effect on the cell’s viability.

So how it all manifested in our model?
When a bacterium is released and no more induction occurs, concentration will behave like this:


Where k is generation (k=1, [protein]=100%).

By simply defining different rates for ’Leakage’ we are changing the way protein concentration changes over time.
It is important to remember, control over the distribution as in Figure 1, is basically control over the shape parameter ’a’ , therefore we chose the death time shape parameter to be linearly depended on the concentration of protein as follows:



Mechanism strength was simply introduced by changing the ’slope’ and ’n’ parameters to represent a stronger influence of [protein] on ’a’.

Protein leakage was changed by changing the function that updates the concentration each iteration.

25 combinations of leakages and mechanism strengths were simulated with repeats, and means and deviation of different parameters were analyzed.

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References

[1] Lorenzo Pasotti, Nicolò Politi, Susanna Zucca, Maria Gabriella Cusella De Angelis, Paolo Magni PLoS One. 2012; 7(7): e39407. Published online 2012 July 20. View Source.