System Improvements Description



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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Same environment
Using a chemostat
Same starting point, different synchronization mechanisms
HSL mediated simple oscillator (learn more...)
HSL mediated coupled oscillators (learn more...)

Modeling Alternatives

The proposed systems are:

HSL mediated coupled oscillators HSL mediated simple oscillator
Bimo.png Unimo.png
  • 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:
↓ 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:
that is equivalent to:
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.

Alternatives Description

HSL mediated coupled oscillators HSL mediated simple oscillator
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

Remarkably, almost all parameter values are available from experimental measurements in [1] and from the work of Garcia-Ojalvo 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:

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 see remarks

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]