Team:BGU Israel/Model1.html
From 2013.igem.org
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.
![](https://static.igem.org/mediawiki/2013/c/c6/BGU_gamma.png)
![](https://static.igem.org/mediawiki/2013/f/f5/BGU_eq4.png)
![](https://static.igem.org/mediawiki/2013/e/ee/BGU_eq5.png)
![](https://static.igem.org/mediawiki/2013/thumb/d/dd/BGU_shape-strength.png/800px-BGU_shape-strength.png)
References
[1] Stochastic Models of Lymphocyte Proliferation and Death. Zilman A, Ganusov VV, Perelson AS - PLoS ONE (2010). View Source. [2] J Theor Biol. 2004 Aug 21;229(4):455-76. A general mathematical framework to model generation structure in a population of asynchronously dividing cells. León K, Faro J, Carneiro J.View Source.