Team:Paris/Modeling/Programs

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 Revision as of 15:53, 25 September 2008 (view source)Hugo (Talk | contribs) (→Introduction)← Older edit Latest revision as of 07:06, 30 October 2008 (view source) (27 intermediate revisions not shown) Line 1: Line 1: - {{Paris/Menu}} + {{:Team:Paris/MenuBackup}} - {|cellspacing="5" cellpadding="10" style="background:#649CD7; width: 965px;" + - |-valign="top" + - |style="background:#ffffff"| + ==Complexations Caracterisation== ==Complexations Caracterisation== Line 21: Line 18: ==Equilibrium of a Complex== ==Equilibrium of a Complex== - Know, if we imagine a given amount of ''Ai'' and ''Bi'', 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 : + Know, if we imagine a given amount of ''Ai'' and ''Bi'', 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 : -
[[Image:GammaC.jpg]]
+ [[Image:GammaC.jpg|center]] so that the equilibrium gives : so that the equilibrium gives : -
[[Image:EqGammaC.jpg]]
+ [[Image:EqGammaC.jpg|center]] with with -
[[Image:Keff.jpg]]
+ [[Image:Keff.jpg|center]] ==Hill Functions== ==Hill Functions== Line 39: Line 36: 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> [[Image:approxhill.jpg]] + [[Image:approxhill.jpg|center]] that we can easily solve with : that we can easily solve with : -
[[Image:reshill.jpg]]
+ [[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 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]]
+ [[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. 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]]
+ [[Image:hilldef.jpg|center]] ==Induction by a small molecule== ==Induction by a small molecule== ===Introduction=== ===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), which, bound to a protein, will act as a transcription factor. + 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 : 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 : Line 76: Line 73: * ''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) * ''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) * ''SMint'' binds to ''Prot'', with a cooperativity ''n'' and a dissociation constant ''Kd'', to form the complex ''Compl'' * ''SMint'' binds to ''Prot'', with a cooperativity ''n'' and a dissociation constant ''Kd'', to form the complex ''Compl'' - * The cell culture is in a chemostat : the cell population is N; The renewal rate (flow of the medium, in L.min-1) is ''rnw''; The Volume of a cell is ''Vint''; and of the outer medium is ''Vext'' ( = ''Volume(chemostat)'' - ''N'' * ''Vint'') + * 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 ''Vint''; and of the outer medium is ''Vext'' ( = ''Volume(chemostat)'' - ''N'' * ''Vint'') ===Induction for the caracterisations=== ===Induction for the caracterisations=== see [[Team:Paris/Modeling/estimation#How_to_control_the_concentration_of_the_transcription_factor_.3F| details on the induction]] see [[Team:Paris/Modeling/estimation#How_to_control_the_concentration_of_the_transcription_factor_.3F| details on the induction]] - In this experimental conditions, the input medium has a constant (controlled) concentration [''SM'']i. So, we can easily make the hypothesis that it impose the [''SM'']ext in the outer medium ; and, under the hypothesis of a "quick enough" diffusion, we get at steady-state : + In these experimental conditions, we assume that the external concentration of SM remains constant. So, at steady-state, +
[''SM'']ext = [''SM'']int
- [[Image:EquaConc.jpg|center]] + Then, considering the previous complexations equations, and by the equilibrium of ''Prot'' ( ''expr(prom)'' denotes protein production rate ), we have at steady states - + - Then, considering the previous complexations equations, and by the equilibrium of ''Prot'', we have at steady states + [[Image:ProtSt.jpg|center]] [[Image:ProtSt.jpg|center]] Line 97: Line 93: [[Image:ComplProt.jpg|center]] [[Image:ComplProt.jpg|center]] - Then, if one of the precious meloecules are ''transcription factor'' for a given promoter, by considering the previous Hill functions (with cooperativity ''m'' and dissociation constant ''KHILL'' and the amount of [''SM'']i, we have the following expressions : + These equations will be used to estimate ''Kd'' and ''n'', and then to estimate the relative amount of bound or free protein in the cell, for a given amount of ''SM''. - [[Image:Hillcompl.jpg|center]] - - - - [[Image:Hillprot.jpg|center]] - - - Finally, these expressions will allow us to evaluate values of ''Kd'' and ''KHILL'', with a classical least-square 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 ''Kd'' in the literature. Otherwise, that makes 4 parameters (''n'', ''m'', ''Kd'' et ''KHILL''). - - ===In the final system=== ==Finding Parameters== ==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 [[Team:Paris/Modeling/estimation#First_hypothesis|hypothesis]])), we can have an approximation of the corresponding ''Ceq''. 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 [[Team:Paris/Modeling/Parameters|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. )))))))) - - ((((( [[Image:Estimation.jpg|center]] ))))) - - (((((( The corresponding code can be found there : [[Team:Paris/Modeling/|Estimation of the parameters]] ))))))) |}
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Latest revision as of 07:06, 30 October 2008

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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 :

Then, since we guess that the only datas we will have are the quantities of A and B introduced (Ai and Bi), the only equations we will deal with is the following, entirely determining the concentration Ceq 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 Ai and Bi, 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
• Vint is the average volume of a cell (in L) ; Vext 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)
• SMint binds to Prot, with a cooperativity n and a dissociation constant Kd, 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 Vint; and of the outer medium is Vext ( = Volume(chemostat) - N * Vint)

Induction for the caracterisations

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 Kd and n, and then to estimate the relative amount of bound or free protein in the cell, for a given amount of SM.

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