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Team:ULB-Brussels/Modeling
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1. Introduction
Since complex biological pathways are used in an industrial way in order to produce molecules of interest, it has become crucial to understand and, above all, optimize these pathways. However, biological systems are so complex that it is sometimes impossible to have a complete understanding of the reactions and mechanisms of the di�erent pathways. The idea of our project is to solve this optimization problem by using the integron platform { which represents a natural genetic optimiza- tion tool in bacteria { and putting in competition di�erent populations with di�erent gene orders, so that the population(s) with the optimal order(s) will be naturally selected with time.
As a proof of concept, we will try to optimize the order of the genes governing the production of two natural antibiotics: Microcin C7 and Microcin B17. The �rst one inhibits a tRNA synthetase (thus inhibits protein synthesis and, as a consequence, cell division), and the second inhibits a gyrase (thus provokes inhibition of DNA replication and eventually cell death). We might then expect that natural selection occurs, so that the optimal gene order(s) �nally emerge.
In the sequel, we model this competition experiment, and try to see in what sense and in what conditions natural selection could happen.
2. Modeling the competition experiment
In the following, we write Microcins B and C for Microcins B17 and C7, respectively. Further, the bacterial populations producing these antibiotics will be denoted by Bi and Cj , respectively, where the indices i and j run through all di�erent gene cassette orders.
We consider the experiment where all these populations are put in competition together. In our
model, for the sake of simplicity, we will simply consider that Microcin B causes the production of
some protein complexes that provoke cell death (bactericidal), while Microcin C inhibits cell division
of the protein complexes that allow the cellular division process. (bacteriostatic). Thus note that the quantities AXi and DXi have no biological meaning, but are
used phenomenologically to better describe the situation.
2.1. Notations and mathematical model
The study of the di�erent populations will be accomplished through the time evolution of the following dynamical quantities. Notice that subscribed letters will designate the given population, while
superscripted letters will stand for the corresponding antibiotics.
Using these constants and dynamical variables, we can describe the biological competition experiment by the following di�erential equation system (where X = B;C and i runs through all the
di�erent possible gene orders for the antibiotics production gene cassettes):
This general model can be simpli�ed if we suppose that populations Xi are completely immune to Microcin X, which is a totally reasonable assumption. Further, since bacteria and Microcins have
half-lives that are much larger than the experiment time, we may neglect the corresponding terms.
If we also neglect the saturation e�ect in the population growth (which is natural if the experiment
is carried out in exponential phase), we then get the following simpler system:
2.2. Natural selection?
We could hope that putting together bacteria with all the di�erent gene orders leads to a natural
selection regime, meaning that the subpopulations with the best o�ensive and/or defensive charac-
teristics
4. Appendix
4.1. Determining analytically the asymptotical behaviour of the solutions of (1)
4.2. Parameter estimation
