Complexations Caracterisation
The first hypothesis is that a complexation reaction is fully determined by the following :
and that the rates k_{+} et k_{} stay constant under all conditions. Then, the second hypothesis is that these equations are (kinetically speaking) elementary :
and at steadystate :
Then, since we guess that the only datas we will have are the quantities of A and B introduced (A_{i} and B_{i}), the only equations we will deal with is the following, entirely determining the concentration C_{eq} at steadystate (at least if we take the smallest real root of the equation, it is useless to demonstrate the unicity, or even the existence, of such a solution) :
Equilibrium of a Complex
Know, if we imagine a given amount of A_{i} and B_{i}, that are calculated as their equilibrium without taking acount of their complexation (but, for instance, of other interactions, productions and disappearance), and that the produced complex C disappears along time with a degradation rate γ, we get :
so that the equilibrium gives :
with
Hill Functions
The previous system of complexation applied in particular to the association of the Promoters (P) and its Transcription Factor (TF).
Because the promoters on a "low copy plasmid" exists in the cell in ten exemplaries, in contrary to a protein, which, as long as it is produced (even weakly) exists in thousands of exemplaries, we assume can the quantities of TF and P are different by several orders of magnitude. Then, with the previous notations, if A, B and C stands respectively for TF, P and the complex TF><P, we will get
that we can easily solve with :
Depending to the order n (also called cooperativity, because it correponds to the possibilities of the transcription factor to binds in a group on the promoter), this function is a sigmoïd, known as the Hill function. The parameter K , called activation constant, is often replaced in the previous expression by the following notation
It simplifies the manipulations of the expression ; we can notice that K represents now the amount of TF needed to bind half of the total P in the cell.
Induction by a small molecule
Introduction
In certain steps of our system, and in our caracterisation plan, we use the diffusion of a small molecule (SM), that binds to a transcription factor.
We make the hypothesis of a simple, passive diffusion, that leads at the steadystate at equal amount of the small molecule, inside and outside the cells. The resulting equations are the following coupling :
where
 permeability is the membrane permeability multiplied by the average external surface of a cell (in min^{1})
 N is the average cell population
 V_{int} is the average volume of a cell (in L) ; V_{ext} is the volume of the culture medium outer the cells (in L)
The situation
The involvement of the previous process is, in our systems,under these conditions :
 Prot is a protein, produce by a promoter
 expr_prom is the given expression of this promoter (production rate of Prot)
 γ is the degradation/dilution rate of the proteins and complexes in the cell (especially due to cell division, we made so far the hypothesis that it is the same for all the proteins)
 SM_{int} binds to Prot, with a cooperativity n and a dissociation constant K_{d}, to form the complex Compl
 The cell culture is in a chemostat : the cell population is N; The renewal rate (flow of the medium, divided by the volume of the chemostat, in min^{1}) is rnw; The Volume of a cell is V_{int}; and of the outer medium is V_{ext} ( = Volume(chemostat)  N * V_{int})
Induction for the caracterisations
see details on the induction
In these experimental conditions, we assume that the external concentration of SM remains constant. So, at steadystate,
[SM]_{ext} = [SM]_{int}
Then, considering the previous complexations equations, and by the equilibrium of Prot ( expr(prom) denotes the production inside the cell ), we have at steady states
and by the definition of the dissociation constant
that leads to the following expression of the complexes and proteins :
Then, if one of the previously defined molecules are transcription factor for a given promoter, by considering the previous Hill functions (with cooperativity m and dissociation constant K_{HILL}^{m} ; and by reintroducing the same convention as for the hill dissociation constant K_{d} := K_{d}^{n}) and the amount of [SM]_{i}, we have the following expressions :
Finally, these expressions will allow us to evaluate values of K_{d} and K_{HILL}, with a classical leastsquare optimization calculus. The less parameters we are seeking, the most precise would be the results. We can use the second expression to avoid expr.prom and γ ; we could perhaps find n and K_{d} in the literature. Otherwise, that makes 4 parameters (n, m, K_{d} et K_{HILL}).
In the final system
The final system is supposed to evoluate in a chemostat. However, the input medium doesn't contain any small molecule (SM). It presence comes from a production inside the cell.
We reuse the previous notations :
 k_{+}; k_{}; K_{d}; n; SM; Prot; Compl are all related to the complexation of the small molecule and the protein
 expr(prom_{P}) : given production rate of the protein
 expr(prom_{SM}) : given production rate of the small molecule
 γ is the general dilution term; rnw is the "renewal rate"; η_{ext} et η_{ext} are related to the diffusion of the small molecule
Thus, the resulting equations are the following :
In this final system with four equations, we are looking for the two variables [Compl] et [SM]_{int}. However, even with all of our caracterisations, we do not know the permeability (see the diffusion coupling equations) of the cellular membrane to the small molecule, neither k_{+} (we can still evaluate the values of N, V_{int}, V_{ext}, γ and rnw). But, as the phenomenon described here only appears at the quorum sensing part of the system (see the synchronisation), we will study different range for those parameters, to see their impact on the oscillation/synchronisation (in particular, the delay between the end of the cycle and the negative feedback (see the system) depends on them).
Finding Parameters
Then, by solving the complexation equation for a given set of K et n (and, in term of production of protein, the b (or β) parameter (see hypothesis)), we can have an approximation of the corresponding C_{eq}. The idea, in order to find the bests parameters which correspond to our measurements, is to run a basical program of quadratic optimisation. The final error would tell us if our modelisation is consistent.
((((the following is not coresponding to the new model, but we will use the same frame to expose the corresponding program)))))
(((((((This program aims at fill this data bank, from experimental data obtained by our wet lab. )))))))
(((((( After some trials, we have obtained interesting results in terms of precision. We are now trying to test the robustness of the estimations in order to quantify the influence of biological variance in the results as well as the influence of the number of points available. ))))))))
((((( )))))
(((((( The corresponding code can be found there : Estimation of the parameters )))))))
