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Team ULB-Brussels, modelisation of our

project!

Sommaire 1. Introduction                                                                                                                   2. Modeling the competition experiment 2.1. Notations and mathematical model 2.2. Natural selection? 2.2.1. Neglecting the Microcin diffusion 2.2.2. Qualitative discussion of the solutions of (2) in the general case 2.2.3. Conclusion: no natural selection 3. New experiment and conclusion

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 different pathways. The idea of our project is to solve this optimization problem by putting in competition different populations with different orders of genes, so that the population(s) with the optimal order(s) is (are) naturally selected.

     As a proof of concept, we will try to optimize the order of the genes governing the production of two antibiotics: Microcin C7 and Microcin B17. The first one inhibits a tRNA synthetase (thus inhibits the cellular division process), and the second inhibits a gyrase (thus provokes the cellular apoptosis). We put in competition a bacterium producing Microcin B17 and another producing Microcin C7, each of them possessing a low resistance to the opposite antibiotic, and some biological techniques are used in order to allow randomly every possible order of genes in the antibiotics production gene cassettes of the offsprings. We might then expect that natural selection occurs, so that the optimal order(s) of genes finally 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 different orders of genes in the antibiotics production gene cassettes.

     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 the cellular apoptosis, while Microcin C inhibits the production of the protein complexes that allow the cellular division process. Thus notice 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 different 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 differential equation system (where X = B;C and i runs through all the different possible gene orders for the antibiotics production gene cassettes):