Model Construction
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 relying on different designs principles.
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Modeling Alternatives
The proposed systems are:
HSL mediated coupled oscillators
 HSL mediated simple oscillator



 In the first system, we use a modular design. Since the core system is a 'poor' oscillator we hope that by coupling it with a 'good' oscillator we could obtain sustained oscillations in the whole system. Thus, we consider that the core system is one of the modules of the system and that the other module is a two gene oscillator system presented in [2] 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 oscillators 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 specific to each of the modeling alternatives are presented.
Common Description
Logistic growth in chemostat:


Quorum sensing by HSL diffusion:


As in core system flhDC inhibited by EnvZ and fliA activated by FlhDC and FliA:


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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:


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 α_{cell} and at high population density, the population reaches a maximum concentration, c_{max}.

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 (D_{renewal}) 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:


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:


that is equivalent to:


where and

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.



Alternatives Description
HSL mediated coupled oscillators
 HSL mediated simple oscillator

Core system coupled with an oscillator:
 Modified core system that accounts for quorum sensing:



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Kinetic parameter values
Remarkably, almost all parameter values are available from experimental measurements in [1] and from the work of GarciaOjalvo in [2]. Additionally minimizing the number of parameters is possible by rescaling as done for the core system. Relevant values for the sole two missing parameters are found by exploiting a modularity assumption as described in the next section. The following table summarize our findings:
Parameters


Chemostat
 Parameter
 Meaning
 Original Value
 Normalized Value
 Unit
 Source

 α_{cell}
 Growth rate
 0.0198
 1
 min^{1}
 wetlab

c_{max}
 Carrying capacity for cell growth
 0.1
 0.1
 µm^{3}
 [3]

D_{renewal}
 Dilution rate
 0.00198
 0.1
 min^{1}
 wetlab ([3])

d
 Death rate
 0.0099
 0.5
 min^{1}
 wetlab


Quorum Sensing
 Parameter
 Meaning
 Original Value
 Normalized Value
 Unit
 Source

 γ_{HSL}
 Degradation rate
 0.0053
 0.2690
 min^{1}
 see remarks

γ_{HSLext}
 Degradation rate
 0.0106
 0.5380
 min^{1}
 [2]

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]

