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Objective

In order to evaluate the feasibility of our project, we developed a mathematical model based on kinetic equations to simulate our experimental design. We considered important to approach this problem mathematically in order to evaluate some issues. Firstly, we evaluated the effect of linear DNA degradation of the ORF when inserted in bacteria after eletroporation. Secondly, we estimated the amount of ORF that should be amplified by PCR in order to optimize the recombination. We compared the results obtained using two recombination protein: CRE and FLP. Finally, we discuss methodologies to improve our design using the standard biological parts.

Mathematical model

The model we developed was based on the one proposed by Ringrose et al . The authors introduced a model to describe a excision recombination reaction illustrated in Fig. . We used the parameters characterized by the authors in order to simulate our experimental design that consists in the circularization and insertion of the ORF in the plasmid. We also introduced a linear DNA degradation rate in the model in order to be more accurate in simulating in vivo process.

The first step when making a model based on kinetic equations is to determine the states or configurations of the system. In our context, we refer to S the linear ORF without any monomer bound and to S_a the linear ORF with one monomer bound, see Fig. . All four monomer sites (two sites per loxP) have the same affinity for the monomers, resulting in a symmetry in the system in terms of energy of association. Because of that, there is no need of distinguishing the site that the first monomer binds, referred as S_a. To represent the next state - the DNA bound by two monomers - we need to distinguish between two possibilities: there can be one monomer in each loxP, represented by S_aa, or two monomers in the same loxP, represented by S_ab. It is essential to distinguish these two states because the affinity of the monomers for the target site is different if there is already one monomer bound to the neighbor site. The following states representing the ligation of third and fourth monomer - referred as S₃ and S₄, respectively - have the same affinity and there is no need of distinguishing them.

The rate of change of each state over time was approached by using kinetic equations. To illustrate this process, we described the kinetic equation which refers to the change of the state S over time:

$\frac{d}{dt}[S] = k_{-1}[S_{a}] - k_{1}[S][M]$

where M represents the concentration of recombinase monomers and k₁ and k_− 1 represent the association and dissociation rate constant, respectively. As described in the above equation, there is only two possibilities of changing the concentration of the state S: it can increase (positive sign) if a molecule in the state S_a loses the monomer or it can decrease (negative sign) if a monomer binds the DNA.

Using gel mobility shift assays it is possible to estimate the affinity of the monomer for their target site, represented by the parameters k_− 1 and k₁, as described by Ringrose et al . They also estimate, using the same proceeding, the parameters k_− 2 and k₂ referring to the association and dissociation rate constant of the monomer for a target site when the neighbor site is already occupied by another monomer. Other parameters (k₃₄, k_− 34, k₅ and k_− 5) were determined by the authors comparing the simulated and in vitro recombination data, see Table . The entire recombination reaction is illustrated in the figure .

Our experimental design consists in a circularization of the ORF and its insertion in the plug and play plasmid. The whole process is illustrated in figure and the equations of our model are presented in the appendix.