Team:ULB-Brussels/Modeling

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Revision as of 20:01, 26 September 2012

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 using the integron platform { which represents a natural genetic optimiza- tion tool in bacteria { and putting in competition different populations with different 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 first 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) 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 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 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):           

     This general model can be simplified 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 effect 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 different gene orders leads to a natural selection regime, meaning that the subpopulations with the best offensive and/or defensive charac- teristics ( ) will be those with the best chances of reproduction, thus leading to the emergence of the bacteria with an optimal gene order.

     However, we will see in this section that natural selection can only happen on the basis of the immunity properties of the different subpopulations, so that, within subpopulations Bi (resp. Ci), the most immune ones will emerge, - which is unfortunately not interesting for us as we initially aimed to select the most productive ones. We first rigorously prove this fact in a simplified context, and then discuss numerical simulations of our model in the general case.

2.2.1. Neglecting the Microcin diffusion

     In this paragraph, we neglect the diffusion effects of the Microcins: in other words, we stop distinguishing between Microcins inside and outside bacteria. 2.2.2 Qualitative discussion of the solutions of (2) in the general case

     We now discuss the impact of different production and immunity rates on the time evolution of the solutions of general system (2). Since finding solutions analytically seems beyond possibilities, we use Mathematica 8 to solve the system numerically (with biologically reasonable values of the parameters) in the special case when there are only two different gene orders per type (Bi, Ci, i = 1; 2): see Figures 1-8 for a few numerical simulations of the relationship between populations B1, B2 (blue and red), as well as C1 and C2 (green and yellow). All populations have the same initial value.

    &nbspIn Figures 1-7, both populations B (resp. C) have the same immunity but different production rates T . We immediately observe that both populations B (resp. C) have exactly the same behaviour, so that absolutely no difference can be seen on the figures (on each graph, only two double curves can be seen: red/blue for the two populations B and green/yellow for the two populations C). More precisely, numerical simulations show that both populations B (resp. C) behave as one single population with production rate

4. Appendix

4.1. Determining analytically the asymptotical behaviour of the solutions of (1)

4.2. Parameter estimation