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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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@@ Line 1: / Line 1: @@
-{{Paris/Menu}}
+{{:Team:Paris/MenuBackup}}
-==Finding Parameters==
+==Complexations Caracterisation==
-This program aims at fill [[Team:Paris/Modeling/Bibliography|this data bank]], from experimental data obtained by our wet lab.
-We are looking for constants involved in Hill functions : <br>
+The first hypothesis is that a complexation reaction is fully determined by the following :
-* Inducer x on py : formation rate of y = b*(x^n/(K^n + x^n)) <br>
+<center> [[Image:reacthill.jpg]] </center>
-* Inhibitor x on py : formation rate of y = b*((K^n)/(K^n + x^n)) <br>
+and that the rates k<sub>+</sub> et k<sub>-</sub> stay constant under all conditions. Then, the second hypothesis is that these equations are (kinetically speaking) ''elementary'' :
-With following chemical interpretations :
+<center> [[Image:kinhill.jpg]] </center>
+and at steady-state : <center> [[Image:sshill.jpg]] </center>
-* b = "quantity of DNA multiplied by the transduction rate" <br>
+Then, since we guess that the only datas we will have are the quantities of ''A'' and ''B'' introduced (''A<sub>i</sub>'' and ''B<sub>i</sub>''), the only equations we will deal with is the following, entirely determining the concentration ''C<sub>eq</sub>'' 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) :
-* (K^n) = "activation constant : dissociation constant of the fixation of x on py" <br>
-* n = "stoechiometric coef. of x in the fixation of x on py" <br>
-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.
+<center> [[Image:Eqnss.jpg]] </center>
-[[Image:Estimation.jpg]]
+==Equilibrium of a Complex==
+Know, if we imagine a given amount of ''A<sub>i</sub>'' and ''B<sub>i</sub>'', 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 &gamma;'', we get :
+[[Image:GammaC.jpg|center]]
+so that the equilibrium gives :
+[[Image:EqGammaC.jpg|center]]
+with
+[[Image:Keff.jpg|center]]
+==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
+[[Image:approxhill.jpg|center]]
+that we can easily solve with :
+[[Image:reshill.jpg|center]]
+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
+[[Image:changeK.jpg|center]]
+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.
+[[Image:hilldef.jpg|center]]
+==Induction by a small molecule==
+===Introduction===
+In certain steps of our [[Team:Paris/Project|system]], and in our [[Team:Paris/Modeling/estimation|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 :
+[[Image:CouplDiff.jpg|center]]
+where
+* ''permeability'' is the membrane permeability multiplied by the average external surface of a cell (in min<sup>-1</sup>)
+* ''N'' is the average cell population
+* ''V<sub>int</sub>'' is the average volume of a cell (in L) ; ''V<sub>ext</sub>'' 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'')
+* ''&gamma;'' 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<sub>int</sub>'' binds to ''Prot'', with a cooperativity ''n'' and a dissociation constant ''K<sub>d</sub>'', 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<sup>-1</sup>) is ''rnw''; The Volume of a cell is ''V<sub>int</sub>''; and of the outer medium is ''V<sub>ext</sub>'' ( = ''Volume(chemostat)'' - ''N'' * ''V<sub>int</sub>'')
+===Induction for the caracterisations===
+see [[Team:Paris/Modeling/estimation#How_to_control_the_concentration_of_the_transcription_factor_.3F| details on the induction]]
+In these experimental conditions, we assume that the external concentration of SM remains constant. So, at steady-state,
+<center>[''SM'']<sub>ext</sub> = [''SM'']<sub>int</sub></center>
+Then, considering the previous complexations equations, and by the equilibrium of ''Prot'' ( ''expr(prom)'' denotes protein production rate ), we have at steady states
+[[Image:ProtSt.jpg|center]]
+and by the definition of the dissociation constant
+[[Image:Kd.jpg|center]]
+that leads to the following expression of the complexes and proteins :
+[[Image:ComplProt.jpg|center]]
+These equations will be used to estimate ''K<sub>d</sub>'' 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==
-The corresponding code can be found there : [[Team:Paris/Modeling/|Estimation of the parameters]]
+|}<br style="clear:both" />