Population counter : Modelling

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Introduction

We aim at setting up a biological device counting random events occuring in a microfluidic device (figure \ref{fig:1}). These events are recombinations in the cells of the device, stimulated by pulses of arabinose coming from the tunnel. Once a certain number of pulses have been triggered, we expect our device to start shining. First, we shall describe the dynamics of the population of bacteria in our device as well as its recombinations, which further lead to the rise of the concentration of produced AHL. AHL is a molecule such that, once a certain concentration threshold is reached, make all the cells start producing GFP, alerting us of the end of the experiment. Determining the role of a few parameters in a model we shall describe in the following should allow us to count how many recombinations (so how many events) had to happen before the whole device starts shining.

The process

Here we will give the bigger picture, discussing how every component of the process reacts and interacts with the other molecules. First, we send a pulse of arabinose through the tunnel. The bacteria are then expected to react in a way that is described in the next section. Those bacteria that recombinated start producing LuxI, which in turns promotes the production of AHL. This AHL will play two role:

\item it can bind with LuxR in the cell. LuxR is a protein produced so that its concentration inside the cell will be considered as constant. We call it LuxR$^f$ as long as it is free from AHL, LuxR$^*$ otherwise;
\item it can cross the cell membrane to bind with LuxR of other bacteria in the device.
Every time a pulse of arabinose is injected, the production of AHL increases. Once a certain level of AHL has been reached, all the cells produce GFP, which is the event putting an end to the counting.

Model

We can develop the information we gave in a system of equations:

\item In a first approach, we will consider that the dynamics of production of LuxI is represented by a step function: $[LuxI]$ is $0$ before the recombination, and $1$ after. If the time between the two states cannot be neglected\footnote{which is probably the case, since it is roughly 20 minutes, close to the expected time of a cell recombinating, and larger than the diffusion significant times.} compared to the average time between two recombinations, we may reconsider this simple model.
\item Let's call $R$ the constant concentration\footnote{how ``constant" is it?} of LuxR in a cell. Some of it binded with AHL and we called it LuxR*, so this gives: