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Population counter : Modelling

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Introduction

We aim at setting up a biological device counting random events occuring in a microfluidic device (see figure 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 roles:

• 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;
• 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:

• 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, 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.
• Let's call R the constant concentration (how "constant" is it?} of LuxR in a cell. Some of it binded with AHL and we called it LuxR*, so this gives:
  
  The reaction between LuxR and AHL gives a chemical equilibrium determined by the constant k_reac:
  
  so at the equilibrium:
  
  which gives:
  
  
• Now the concentration of AHL in the room decreases as the microfluidic device evacuates it, which gives birth to a diffusion term in the equation, and increases as it is produced by the cells since AHL can go throuh the cell membrane. For now, we shall assume the diffusion of AHL inside the device is instantaneous. Following the paper "A synchronized quorum of genetic clocks" by Hasty and al., we define the intracellular concentration of AHL, [AHL_i], and the external one, [AHL_e].
  
  The first term says that if the cell produces LuxI, then it produces AHL at a certain rate. The second term concerns the exchange between the cell and the medium, and is proportional to the difference of concentration.
  
  where k_diff is the constant of diffusion, related to parameters of the fluid and of the device (such as the surface of the section of the path and others to precise), k_membrane is the same as previously, but is now compensated by d, the cell density, so that the exchanges between cells and the medium is balanced. A note on diffusion further.
  Here we consider only the case where the concentration of AHL is homogeneous, ie the concentration inside and outside the cell are the same. This is justified since the caracteristic time of diffusion of AHL between the outside and the inside of a cell is short compared k_I. The contrary would lead to a discussion we started in paragraph \ref{modif_model}. We end up with this equation:
  
  

The synthesis of LuxI.

Above, we introduced directly the concentration of LuxI without giving more details. However, we need to model the production of LuxI to calibrate the counter. This means modelling how the bacteria evolve in the device and how they react to arabinose.

• Arabinose is modeled simply in the model with a diffusion equation:
  
  with the boundary condition at the border between the device and the tunnel corresponding to a positive value if a pulse is being sent, and 0 otherwise. Thus we get the profile of arabinose independently of enerything else, except the pulse itself.
• The device may have several sizes, however one device will not ever host more than 2.10⁵ cells, which is approximately the ration between the volume of a device and the volume of a cell. Moreover, considering both populations, switched and unswitched, we will consider the bacteria are in the exponential phase, and that they can be evacuated through the path to the tunnel. Therefore, we need a model for the behaviour of the bacteria (movement and birth/death) that allows us to deal with a few cells \footnote{at the beginning, when we only have a few recombinated cells.} up to more than 10⁵ cells. An individually-centered model would be computationnaly inefficient, while a continuous model would perform wrong computations for small numbers of bacteria. I have implemented something relying on a binomial distribution: among n people, we pick a random number of them to move, or give birth; this random number being ruled by a probability p proportional to the gradient of concentration of bacteria. Discretising time and space, one can draw at every step time a random binomial number\footnote{X being non-negative if the first argument in the binomial law below is non-negative, negative otherwise}:
  
  where...

Algorithm and simulations

For now, we use a simple spatial discretisation scheme, assuming lateral diffusion is instantaneous\footnote{this may make simulations inaccurate and thus yields a piece of the program to be improved}, so that diffusion is only considered along one axis (the one from the tunnel to the top of the device (see the first figure).
Discuss the existence of two phases, the critical parameters being the diffusion coefficient of AHL and the time between two pulses.