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- | <TD COLSPAN=2> 2) Outside the cells, HSL is cululated with the same transport term tah we use in the previous equation. The degradation of HSL in the external medium and the dilution controled via the chemostat accounts for HSL external decrease. So the external HSL concentration is given by: </TD> | + | <TD COLSPAN=2> 2) Outside the cells, HSL is cumulated with the same transport term tah we use in the previous equation. The degradation of HSL in the external medium and the dilution controled via the chemostat accounts for HSL external decrease. So the external HSL concentration is given by: </TD> |
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Revision as of 23:41, 27 October 2008
Model Construction
Motivation
We are interested to understand the factors that account for an oscillating behaviour at population level. We build two models to study systems that are equipated with quorum sensing capabilities but that relay on different designs principles. The proposed models are:
- In the one model, we use a modular design. We consider that the core system is one of the modules of the system. In adittion, the other module is an two gene oscillator system presented in [Garcia-Ojalvo] that accounts for quorum sensing. We call this alternative the 'bimodular system'.
- In the other model, namely the 'unimodular system', we rewire the architecture of the core system to have the desired functionality of the system in a single circuit.
Both the bimodular and unimodular models describe events that happend not only at the cellular level (as in the core system) but also at the population level due to interactions needed bettwen a cell and its environment during quorum sensing.
In the following sections, we first describe the population modeling (the common part amoung our two proposed models) to then focus our attention to the characteristics that are exclusive to each of the modeling alternatives.
Description
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Common Dynamics: Chemostat |
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The variation of cells' concentration in the chemostat over time can be expressed in terms of a production (positive) term and degradation (negative) terms: |
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For the production term, we use a logistic equation to model cell growth, according to standard assumptions. 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 αcell and at high population density, the population reaches a maximum concentration, cmax.
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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 (Drenewal) and cell death (d).
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Common Dynamics: Quorum Sensing |
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In order to model the quorum sensing dynamics, we consider that: |
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1) Inside a cell, the HSL concentration increases proportionaly 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 beetwen the internal and external concentration of HSL). Thus, the equation for the internal HSL concentration is: |
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2) Outside the cells, HSL is cumulated with the same transport term tah we use in the previous equation. The degradation of HSL in the external medium and the dilution controled via the chemostat accounts for HSL external decrease. So the external HSL concentration is given by: |
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that is equivalent to: |
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where and |
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3) The activation of envZ depends of the concentration of HSL according to: |
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Bimodular System |
Unimodular System |
Description Left |
Description Right |
Left |
Right |
cont... |
cont |
Parameters Search
The following table sumarize our findings. The parameters' values that are used during the simulations. Most them are found in literature others are obtained from further analysis.
Parameters
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Chemostat
| Parameter
| Meaning
| Original Value
| Normalized Value
| Unit
| Source
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| αcell
| Growth rate
| 0.0198
| 1
| min-1
| wet-lab
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cmax
| Carrying capacity for cell growth
| 0.1
| 0.1
| µm3
| [3]
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Drenewal
| Dilution rate
| 0.00198
| 0.1
| min-1
| wet-lab ([3])
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d
| Death rate
| 0.0099
| 0.5
| min-1
| wet-lab
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Quorum Sensing
| Parameter
| Meaning
| Original Value
| Normalized Value
| Unit
| Source
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| γHSL
| Degradation rate
| 0.0053
| 0.2690
| min-1
| wet-lab
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γHSLext
| Degradation rate
| 0.0106
| 0.5380
| min-1
| [6]
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βHSL
| Production rate
| 0.3168
| 16
| min-1
| ∅
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η
| Diffusion rate
| 10
| 505
| min-1
| [2]
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nHSL
| Hill coefficient
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| 4
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| [3]
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θHSL
| Hill characteristic concentration for the second operator
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| 0.5
| c.u
| [3]
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