Deterministic Model

Introduction

Our objective was to assemble a system that could respond to different concentrations of a specific contaminant in water. More specifically, the idea was to produce different amounts of a coloured pigment depending on the concentration of the contaminant present in water.

We worked with Arsenic knowing that an arsenic detoxification operon (Ars operon) exist in nature and more important, it exists as a biobricks.

To skip directly to our mathematical model click here.

Got Ideas?

For our biosensor we came up with three different designs each one motivated on improving the flaws detected while analysing the previous ones using mathematical models as our main tools. We describe briefly the first two designs, which were rejected, to emphasize the importance of mathematical model to gain insight on how our system works and as a feasibility study taking into considerations .

Arsenic promoter + ArsR + Reporter

First we tried to keep it as simple as we could. Figure 1 shows our first design which, in theory, produces different responses to different concentrations of arsenic in water. Figure 2 shows a simulation graphing the time series response of the Pars promoter-arsR for different concentrations of ars.

Figure 1: Our first biological design: Arsenic promoter repressed by ArsR wich is inhibit by Arsenic.

Althought we have a system that recognise and respond differently to different concentrations of arsenic since our reporter was meant to be a coloured pigment (the reasons for this are detailed in the project tab) scale tuning needs to be taken into account.The reporter is downstream the Pars promoter and the design shows little flexibility, i.e. few available parameters, to produce a discernable coloured output.

Another problem is that we aimed to use a reporter (unknown at that time) with a low degradation rate thus if the production of the pigment didn't stop at some point, accumulation of the pigment leading to eye saturation would become a problem. Although the model shows a steady state (figure 2) this accounts only for one bacteria with the degradation/dilution rate governed mainly by dilution.

Nothing else to do but to come up with a different idea for our design. Note that the model of the Pars promoter + ArsR is still usefull because is the main block of any design to detect Arsenic in water. For details on how we model this click here.

Toggle Switch + Quorum Sensing + Reporter

Figure 3: Graph of the nullclines of a toggle network with (a) bistability consequence of repressor cooperativity and (b) monostability consequence of no repressor cooperativity.

Our second design was based on the work done by the Grenoble Team 2010. They coupled a toggle switch with a quorum sensing circuit to produce a system that produces a coloured pigment if and only if a specific IPTG/Hg concentration ratio existed in the water. IPTG concentrations are determined previously so as to activate the system at a desired Hg concentration.

Two main problems (among others) arised from the mathematical analysis of this system:

• The minimum requierments to obtain bistability is that at least one of the inhibitors must show cooperativity greater than one. Still, bistability will rely on the strenght of the promoters.The lack of robustness of the system based on its dependence on cooperativity and promoter strenght made the idea of having a modular device capable of working with a set of promoters-inhibitor imposible to achieve.

• The initial setting makes the toggle harder to switch back and although the Grenoble Team proposed a possible solution they never effectivly tested it. This attempted against our desire of making a functional device and not a theoretical device.

Temporal control of reporter expression with an Incoherent-FFL

Parallel to the Toggle Switch + Quorum Sensing design we came up with an idea that could solve the problems of our first design: Pars + ArsR + Reporter.

We wanted a system that could:

1. respond differently to different concentrations of a specific contaminant.
2. stop in time to avoid the accumulation of our reporter. Keep in mind that our reporter was meant to be a coloured pigment with low degradation rate so you could see the response of the system with your naked eyes. We wished to avoid eye saturation of the pigment.

We knew that Pars + ArsR could satisfy our first requirement but we needed a kind of "timer" to meet our second requierment plus some sort of amplification system to calibrate the Arsenic promoter sensibility range with the "naked eye sensibility".

Based on Basu 2004 biological network, we designed our own which met our basic requirements. For a detailed description click here.

Mathematical Model

The more you know =★: Before starting to talk about our model, if you know nothing about mathematical models of gene networks we highly recommend Uri Alon's book: An Introduction to Systems Biology.

To describe the transcription rate of a gene under the influence of a transcription factor we use the family of gene input functions known as Hill functions:

Hill function for activator

<math alt="Square root of pi">\sqrt{\pi}</math>

Hill function for repressor

Hill function for repressor lifted by inducer

Hill function for repressor and activator

<math>ax^2 + bx + c = 0</math>

<math>ax^2 + bx + c = 0</math>

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Arsenic promoter + ArsR + Reporter