Team:Paris/Analysis/Construction2

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Model Construction


Contents

Introduction

In this section, we investigate possible improvements of the core system. Our objective is twofold : the system has to provide sustained oscillations and these oscillations should be synchronized amongst a population of cells. To this aim we explore designs inspired by quorum sensing and model in a chemostat cell growth and species diffusion outside cells. We consider two models relaying on different designs principles.

↓ To joint our quest for alternatives click here! ↑


Same environment
Chemostat.png
Using a chemostat
Same starting point
Core system0.png
The core system
CoreSystemKeep3.png
We keep 3 interactions
Same quorum sensing mechanism
CoreSystemQS.png
LasI indirectly activates envZ via HSL
Alternative One Alternative Two
CoreSystemEnvZx2.png CoreSystemQS.png
Two copies of envZ No modifications, i.e. a single copy of envZ
CoreSystemEnvZx2Activation.png CoreSystemKeep3Activation.png
FlhDC and Flia active one of the copies of envZ FlhDC and Flia active lasI
CoreSystemEnvZx2ActivationInhibition.png CoreSystemKeep3ActivationNoInhibition.png
EnvZ inhibits the expression of lasI EnvZ does not inhibit the expression of lasI
TwoModules.png SingleModule.png
This system ends up with 2 modules This system consists of a single module

Modeling Alternatives

The proposed models are:

HSL mediated coupled oscillators HSL mediated simple oscillator
Bimo.png Unimo.png
  • In the first system, we use a modular design. We consider that the core system is one of the modules of the system and that the other module is a three gene oscillator system presented in [Garcia-Ojalvo] that accounts for quorum sensing. We call this alternative the 'HSL mediated coupled oscillators'.
  • In the second system, namely the 'HSL mediated simple oscillator', we rewire the architecture of the core system to introduce delay via HSL export in the environment in a single circuit.

Both the coupled and simple oscilltors describe events that happen not only at the cellular level (as in the core system) but also at the population level due to interactions needed between a cell and its environment.

In the following sections, we first describe the common part among the two proposed models and then focus our attention to the alternatives description where the characteristics that are exclusive to each of the modeling alternatives are presented.

Common Description

Use of chemostat:
DescriptionCommonDynamicsPart1.png
Use of quorum sensing:
DescriptionCommonDynamicsPart2.png
Use of some interactions from the core system:
DescriptionCommonDynamicsPart3.png
↓ read more on common dynamics... ↑


Common Dynamics: Chemostat
The variation of cells' concentration in the chemostat over time can be expressed in terms of a production (positive) term and degradation (negative) terms:
Final chemostat.png
    For the production term, we use a logistic equation to model cell growth, according to standard assumptions. The behaviour obtained is the following one: at low population density, the concentration of cells in the chemostat (c) increase exponentially with a growth rate αcell and at high population density, the population reaches a maximum concentration, cmax.
    For the degradation term, we consider that c decrease proportionally to both a dilution phenomena cause by the renewal of the medium in the chemostat (Drenewal) and cell death (d).
Common Dynamics: Quorum Sensing
In order to model the quorum sensing dynamics, we consider that:
     1) Inside a cell, the HSL concentration increases proportionally to the concentration of LasI and decreases according to both a degradation term (proportional to the internal HSL concentration) and a transport term (proportional to the difference between the internal and external concentration of HSL). Thus, the equation for the internal HSL concentration is:
Final HSL.png
     2) Outside the cells, HSL is accumulated with the same transport term that we use in the previous equation. The degradation of HSL in the external medium and the dilution controlled via the chemostat accounts for HSL external decrease. So the external HSL concentration is given by:
HSLext.png
that is equivalent to:
HSLext2.png
where    Hsl average.png    and     Cell number volume.png
Common Network Dynamics: FlhDC and Flia
FlhDC and Flia are regulated in the same way in both systems. FlhDC is produced under the influence of EnvZ via an inhibition. Flia is regulated for its self and FlhDC.
FlhDC-FliaEq.png

Alternatives Description

Bimodular System Unimodular System
Core system coupled with an oscillator: Modified core system that accounts for quorum sensing:
SumaryBiMo.png SumaryUniMo.png
↓ read more on alternatives description... ↑


HSL mediated coupled oscillators HSL mediated simple oscillator
     a) The expression of lasI is under the control of the same promotor that used for FlhDC.      a) The expression of lasI is regulated by FlhDC and Flia (as in the core system)
LasIEqInBIMOdularSys.png LasIEqInUNIModularSys.png
     b) The expression of envZ depends on both the activation from FlhDC and Flia (as in the core system) and the concentration of HSL present in the cell.      b) The expression of envZ depends only on the concentration of HSL present in the cell.
EnvZInBIMOdularSys.png EnvZInUNIModularSys.png

Kinetic parameter values

The following table sumarize our findings. Theses parameters values are used during the simulations. Most them are found in literature while three are obtained from further analysis.

Parameters
 
Chemostat Parameter Meaning Original Value Normalized Value Unit Source
αcell Growth rate 0.0198 1 min-1 wet-lab
cmax Carrying capacity for cell growth 0.1 0.1 µm3 [3]
Drenewal Dilution rate 0.00198 0.1 min-1 wet-lab ([3])
d Death rate 0.0099 0.5 min-1 wet-lab
 
Quorum Sensing Parameter Meaning Original Value Normalized Value Unit Source
γHSL Degradation rate 0.0053 0.2690 min-1 wet-lab
γHSLext

Degradation rate 0.0106 0.5380 min-1 [2]
βHSL Production rate 0.3168 16 min-1
η Diffusion rate 10 505 min-1 [2]
nHSL Hill coefficient   4   [3]
θHSL Hill characteristic concentration for the second operator   0.5 c.u [3]