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Contents 1 Complexations Caracterisation 2 Equilibrium of a Complex 3 Hill Functions 4 Induction by a small molecule 4.1 Introduction 4.2 The situation 4.3 Induction for the caracterisations 5 Finding Parameters

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 steady-state :

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 steady-state (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 steady-state 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 steady-state,

[SM]_ext = [SM]_int

Then, considering the previous complexations equations, and by the equilibrium of Prot ( expr(prom) denotes protein production rate ), we have at steady states

and by the definition of the dissociation constant

that leads to the following expression of the complexes and proteins :

These equations will be used to estimate K_d and n, and then to estimate the relative amount of bound or free protein in the cell, for a given amount of SM.

Finding Parameters

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