Team:Paris/Analysis/Construction2

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Model Construction

Description

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.

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.

As discussed previously, we choose to use quorum sensing as a way to improve the oscillating behaviour of our 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:

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_cell and at high population density, the population reaches a maximum concentration, c_max.

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_renewal) and cell death (d).

To model the quorum sensing dynamics, we consider:

the production of HSL as:

the transport of HSL is given by:

that is equivalent to:

where

and

;

and the activation of envZ depends of the concentration of HSL according to:

500px

Parameters Search

manly from literature but also from S0 analysis.

Parameters

Chemostat	Parameter	Meaning	Original Value	Normalized Value	Unit	Source
figure / equations of Chemostat	α_cell	Growth rate	0.0198	1	min^-1	wet-lab
	c_max	Carrying capacity for cell growth	0.1	0.1	µm³	[3]
	D_renewal	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
figure / equations of Quorum Sensing	γ_HSL	Degradation rate	0.0053	0.2690	min^-1	wet-lab
	γ_{HSL_ext}	Degradation rate	0.0106	0.5380	min^-1	[6]
	β_HSL	Production rate	0.3168	16	min^-1	∅
	η	Diffusion rate	10	505	min^-1	[2]
	n_HSL	Hill coefficient		4		[3]
	θ_HSL	Hill characteristic concentration for the second operator		0.5	c.u	[3]

@@ Line 36: / Line 36: @@
 | that is equivalent to:
 |-
-|style="text-align:center;" | [[Image:HSLext2.png|550px]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; where &nbsp;&nbsp; [[Image:Hsl_average.png|100px]] &nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; [[Image:cell_number_volume.png|100px]];
+|style="text-align:center;" | [[Image:HSLext2.png|600px]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; where &nbsp;&nbsp; [[Image:Hsl_average.png|100px]] &nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; [[Image:cell_number_volume.png|100px]];
 |-
 | and the activation of ''envZ'' depends of the concentration of HSL according to: