Team:Paris/Analysis/Construction2

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.

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

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:

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 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:      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:

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)      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.

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: