Team:Paris/Analysis/Construction2

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{{Paris/Header|Model Construction}}
== Description ==
<!--why use a chemostat-->
* A chemostat is generally used to keep bacteria volume constant in the medium. The constant conditions provided by the chemostat help us to control bacteria growth rate.  

<!--which model of chemostat we choose-->
* We assume a logistic model to determine bacteria growth in the chemostat, in agreement with standar procedures (reference). This hypothesis implies that bacteria growth rate has to be proportional to the existing bacteria population size and to the amount of available resources in the medium.

<!--Autoinducer -->
* As discussed [[Team:Paris/Network_analysis_and_design/System_improvements|previously]], we choose to use quorum sensing as a way to improve the oscillating behaviour of our [[Team:Paris/Network_analysis_and_design/Core_system|core system]] and, at the same time, as a way to archeive population synchronization. When a cultive of bacteria is synchronized, it means that every single cell express in average the same genes in unison. As a result, a maximum level of fluorescense is obtained.

== Kinetics ==
* The  variation of cells' concentration in the chemostat over time can be expressed in terms of a production (positive) term and degradation (negative) terms:

<center>
{|WIDTH=70%
|[[Image:final_chemostat.png|center|500px]]
|-   
| &nbsp;&nbsp;&nbsp;&nbsp;For the production term, we use a logistic equation to model cell growth, according to standard assumptions [Garcia-Ojalvor--ref]. The behaiviour obtained is the following one: at low population density, the concentration of cells in the chemostat (c) increase exponentialy with a growth rate α<sub>cell</sub> and at high population density, the population reaches a maximum concentration, c<sub>max</sub>. 
|-
| &nbsp;&nbsp;&nbsp;&nbsp;For the degradation term, we consider that c decrease proportionaly to both a dilution phenomena cause by the renewal of the medium in the chemostat (D<sub>renewal</sub>) and cell death (d).
|}</center>

*To model the quorum sensing dynamics, we consider:
<center>
{|WIDTH=70%
|the production of HSL as:
|-   
| [[Image:final_HSL.png|center|400px]]
|-
| the transport of HSL is given by:
|-
| [[Image:HSLext_sum.png|center|500px]]
|-
| that is equivalent to:
|-
|style="text-align:center;" | [[Image:HSLext2.png|550px]]  
|-
|style="text-align:center;" | where &nbsp;&nbsp; [[Image:Hsl_average.png|120px]] &nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; [[Image:cell_number_volume.png|80px]]
|-
| and the activation of ''envZ'' depends of the concentration of HSL according to:
|-
| [[Image:Final_EnvZ.png|center|400px]]
|}
</center>


<!-- equations in code:
d[c]/dt=alpha_cell*[c]*(c_max-[c])/c_max-(D_renewal+death)*[c]
d[LasI]/dt=beta_FlhDC*theta_FlhDC^n_FlhDC/(theta_FlhDC^n_FlhDC+[TetR]^n_FlhDC)-gamma*[LasI]
d[TetR]/dt=beta_TetR*[HSL]^n_TetR/(theta_TetR^n_TetR+[HSL]^n_TetR)-gamma*[TetR]
d[HSL::ext]/dt=eta*v_cell*[c]*([HSL]-[HSL::ext])-(D_renewal+gamma_HSLext)*[HSL::ext]
d[HSL]/dt=beta_HSL*[LasI]+eta*[HSL::ext]-(gamma+gamma_HSLint+eta)*[HSL]
-->

== Parameters Search==

manly from literature but also from  [[Team:Paris/Network_analysis_and_design/System_improvements/Description/QS_Module|S0 analysis]].

<center>
{|
|- style="background: #649CD7;"
! colspan="8" style="background: #649CD7;" | Parameters
|- style="background: #ffffff;"
! colspan="8" | &nbsp;
|- style="background: #649CD7;" |
|style="background: #ffffff; text-align:center;" |'''Chemostat'''
|Parameter
|Meaning
|Original Value
|Normalized Value
|Unit
|Source 
|- style="background: #dddddd;"|    
| ! rowspan="5" style="background: #ffffff;" | figure / equations of Chemostat
|style="background: #D4E2EF;" | α<sub>cell</sub>
|style="background: #dddddd;"| Growth rate
| style="background: #dddddd;"|0.0198
| style="background: #dddddd;"|1
|style="background: #dddddd;"|min<sup>-1</sup>
|style="background: #dddddd;"| [[Team:Paris/Modeling/BOB#Remarks|wet-lab]] 
|- style="background: #dddddd;"|     
| style="background: #D4E2EF;"| c<sub>max</sub>
| style="background: #dddddd;"| Carrying capacity for cell growth
| style="background: #dddddd;"| 0.1
| style="background: #dddddd;"| 0.1
| style="background: #dddddd;"| µm<sup>3</sup>
| style="background: #dddddd;"| [[Team:Paris/Modeling/BOB#Bibliography|[3]]]
|- style="background: #dddddd;"|    
| style="background: #D4E2EF;"| D<sub>renewal</sub>
| style="background: #dddddd;"| Dilution rate
| style="background: #dddddd;"| 0.00198
| style="background: #dddddd;"| 0.1
| style="background: #dddddd;"| min<sup>-1</sup>
| style="background: #dddddd;"| [[Team:Paris/Modeling/BOB#Remarks|wet-lab]] ([[Team:Paris/Modeling/BOB#Bibliography|[3]]])
|- style="background: #dddddd;"|    
| style="background: #D4E2EF;"| d
| style="background: #dddddd;"| Death rate
| style="background: #dddddd;"| 0.0099
| style="background: #dddddd;"| 0.5
| style="background: #dddddd;"| min<sup>-1</sup>
| style="background: #dddddd;"|[[Team:Paris/Modeling/BOB#Remarks|wet-lab]]
|- style="background: #ffffff;"
! colspan="8" | &nbsp;
|- style="background: #649CD7;" |
|style="background: #ffffff; text-align:center;" |'''Quorum Sensing'''
|Parameter
|Meaning
|Original Value
|Normalized Value
|Unit
|Source 
|- style="background: #dddddd;"|    
| ! rowspan="6" style="background: #ffffff;" | figure / equations of Quorum Sensing
| style="background: #D4E2EF;" |γ<sub>HSL</sub> 
| style="background: #dddddd;"|Degradation rate
| style="background: #dddddd;"|0.0053
| style="background: #dddddd;"|0.2690
| style="background: #dddddd;"|min<sup>-1</sup>
| style="background: #dddddd;"|[[Team:Paris/Modeling/BOB#Remarks|wet-lab]] 
|- style="background: #dddddd;"| 
| style="background: #D4E2EF;"|γ<sub>HSL<sub>ext</sub></sub> <br> <br>
| style="background: #dddddd;"|Degradation rate
| style="background: #dddddd;"|0.0106
| style="background: #dddddd;"|0.5380
|style="background: #dddddd;"| min<sup>-1</sup>
| style="background: #dddddd;"|[[Team:Paris/Modeling/BOB#Bibliography|[6]]]
|- style="background: #dddddd;"| 
| style="background: #D4E2EF;" |β<sub>HSL</sub> 
| style="background: #dddddd;"|Production rate
| style="background: #dddddd;"|0.3168
| style="background: #dddddd;"|16
| style="background: #dddddd;"|min<sup>-1</sup>
| style="background: #dddddd;"|[[Team:Paris/Modeling/BOB#Bibliography|∅]]
|- style="background: #dddddd;"| 
| style="background: #D4E2EF;" |η
| style="background: #dddddd;"|Diffusion rate
| style="background: #dddddd;"|10
| style="background: #dddddd;"|505
| style="background: #dddddd;"|min<sup>-1</sup>
| style="background: #dddddd;"|[[Team:Paris/Modeling/BOB#Bibliography|[2]]]
|- style="background: #dddddd;"| 
| style="background: #D4E2EF;" |n<sub>HSL</sub>
| style="background: #dddddd;"|Hill coefficient
| style="background: #dddddd;"|&nbsp;
| style="background: #dddddd;"|4
| style="background: #dddddd;"|&nbsp;
| style="background: #dddddd;"|[[Team:Paris/Modeling/BOB#Bibliography|[3]]]
|- style="background: #dddddd;"| 
| style="background: #D4E2EF;" |θ<sub>HSL</sub>
| style="background: #dddddd;"|Hill characteristic concentration for the second operator
| style="background: #dddddd;"|&nbsp;
| style="background: #dddddd;"|0.5
| style="background: #dddddd;"|[[Team:Paris/Modeling/BOB#Remarks|c.u]]
| style="background: #dddddd;"|[[Team:Paris/Modeling/BOB#Bibliography|[3]]]
|}</center>