Population Dynamics Model

A Population Dynamics Model can be fitted in our system. Theoretically, two approaches have been planned:

$1.1)$ $By$ $Probabilities$

When some new plasmids are genetically introduced into the cytoplasm of E.Coli bacteria, this doesn't garantee that the future copies will contain it. Indeed, these plasmids can be lost after cell division or replication, so it's interesting to study a model based on the different possibilities of plasmid combinations in bacteria, like in the studies of mutations in animals. A typical example of a similar way is found if we study the mutations of the eyes or hair color in a family, by vertical genes transfer. In this case, there's a horizontal genes transfer too, originated by the plasmids.

Figure m2 : A first example of bacteria generations at same acceptation plasmid rate for a Toxin (T), an Antitoxin (A), the two (T&A) or no plasmid (-). The number on the right of the letter G indicates the generation number.

A Probabilistic Model is useful because easily undertsood, but necessits some assumptions to make sense. We remark in [Fig m2] that we will converge to finally obtain bacteria without Toxin or Antitoxin. In this case, four kinds of plasmids are ingered at same rate by bacteria, other cases are described on the following link. This first modelling part shows that the impact of the antibiotics must be included in a realistic model, because without antibiotics, E.Coli bacteria would finish without the plasmids necessary to activate our Mighty Coli system.

$\Longrightarrow$ Prob page

$1.2)$ $By$ $Logistic$ $\small\&\normalsize$ $Lotka$ $Equations$

The Logistic Equation was initially introduced during the beginning of the XIXth Century, by the belgian mathematician P.F. Verhulst. Now, this equation is mainly used in Population Dynamics Models, especially in Biological Sciences. Mathematicians currently finish Ph.D thesis using this, and the analytical $Lotka$-$Volterra$ $model$ is directly associated with the Verhulst theory.

Another interesting model is obtained from Euler-Lotka equation to the $Leslie$ matrix $coefficients$. By identifying the caracteristic eigenvalues and eigenvectors analyse, we estimate the asymptotic stability (stable steady state) and the profile of the growth rate, as in [Fig.m3]. An advantage is that this model is yet discrete, so by computing it, we don't loose info of the background theory.

Figure m3 : Quantity of bacteria normalized by the maximal value in function of time, with the second bacterial growth evolution theory purposed. The population goes to an asymptotic stable state.

The idea is to compare these two models, applicated in our experimental growth conditions and to fit the better corresponding with our colonies, balanced with explicative biological arguments to interprete the choice. Other models exist, f.e. by Monod equation, but these are less consistent with our global and partial systems. This second modelling part shows that the bacterial population grows to converge until a constant quantity. Consequently, it is important to add bacterial food enough and to select the more productive bacteria, for example using a bioreator where Mighty Coli could work.

$\Longrightarrow$ Lotka page