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Our assumptions for modeling

Steady state flow from the nanotubes

A lot of our designs are based on a simple emitter cell - nanotubes - receiver cell principle. The basic models however can not take into account the nanotubes transfer. We decided to make separate models for particles transmission through the nanotubes and to assume that the number of signaling protein going through them reaches a "steady state" in the receiver cell. It can be reached when the flow through nanotubes into the cell exactly compensate protein degradation and dilution. The reason for such a choice is to simplify the interpretation of data. If we were using a non-steady flow of signaling protein, understanding the behaviour of our reporters would have been much more difficult.

The way we present our models is the following. We first model both emitting and receiving genes in the same cell to give us a control. We then see what would be the response of the receiver genes alone in a cell for different signaling protein steady states. We can then compare our models to reality and evaluate the influence of nanotubes on the response of the system.

Translation

We introduced the translation step in our models. This means each equation described above is transformed into two equations: one describing mRNA production and one describing the translation of this mRNA into a protein. We then need to introduce the protein production rate (s^-1). We also need to introduce a different degradation rate for each product. The first one will be for the degradation of mRNA associated with geneX and the second one with protein X itself. The equations for the auto-repression (other regulation systems are similar) are now:

This is justified notably because of our tRNA construction. This design relies directly on the translation process, we therefore need to model it if we want to compare models for different designs.

This hypothesis is inspired from the 2009 Aberdeen iGEM team modeling work.[1]

Delays for maturation

We also introduced a delay for protein production and maturation. Most models can ignore this but our experiments rely heavily on time measurements which means that for proteins with maturation time around 5 min (to compare with the cell division time 40 min) we need to take this into account. We chose to model this simply by adding a delay to the response time as it is shown in the following equations:

The delay we for mRNA production is significantly lower than the one for protein production: << .

No delays due to diffusion in receptor cell

Most of our designs rely on the following scheme: an emitter cell creates a signaling protein which is received through the nanotubes by a receiver cell which is then activated. However, one could argue that after entering the receiver cell, the signaling proteins spend a significant time diffusing in it before reaching the receptor gene construct. We discussed this at length before finally agreeing not to consider this as a delay for our models.

Several models showed us that diffusion, even of a single signaling molecule, happens too fast to add any delay to the response time of the system. We will discuss two of these models here.

Using diffusion equations

The first step is to study the general principles of diffusion and to apply them to a single molecule. We expected to estimate the order of magnitude for diffusion time of molecules with this model, not to have a precise understanding of the movement of molecules in a cell. Most of our experimental designs rely on time measurement to characterize the nanotubes, it was therefore crucial to see if diffusion time could add a significant delay to the response of receiver cells.

The principle of this model is quite simple. We use the statistical diffusion equation with a new normalization constant so that it describes the behaviour of one molecule. Rather than obtaining a concentration field, we end up with a distribution of the density of probability to find the molecule at a certain position and a certain time. We did not use any kind of boundary conditions, we therefore only model the "movements" of one molecule floating in an infinite water medium.
The equation of diffusion is the following:

Where c is the concentration of particles in the cell, function of (position) and t (time). D is the diffusion coefficient.

We decided to use as initial conditions a Dirac function centered on the origin of space. The solution for such an equation is:

This solution shows that through time, the density of probability "diffuses" in all directions equally. So we can take advantage of the spherical symetry of the problem.

(Note that this serie of graph is here only as an example. The units are arbitrary.)

What is of interest for us is the instant when we can consider that the probability of presence of the molecule is "the same" for every point in the cell. The cell here is an arbitrary boundary which has no direct impact on the model. In our approach, the cell is a sphere centered on the origin and with a diameter of 1 micrometer (roughly Subtilis length). We chose to consider that the probability is "the same" when the lowest probability within the cell is 95% of the highest probability in the cell (i.e. in the center).

We can then solve analytically the equation. Let us call the time when the probability is the same (at 95%) within the cell boundaries. This means that for concentration for x=cell radius is 0.95 times concentration for x=0.

You will find in the table below a list of such characteristic times for different molecules.

Using a stochastic model

This model is very similar to the one used in [2]. We consider that the particle diffusing in the cell is a random walker.

We want to see how long it takes for a particle of a given size to diffuse to any point of a cell. We use the following parameters:

• V volume of the cell (10^-18 m³)
• characteristic size of the particle (m)
• D diffusion coefficient of the particle (m².s^-1)

We divide the cytoplasm volume V into occupation sites for the walker. The characteristic time associated with the transition from one site to another is: [3]

If we have R walkers of this type, the probability that a molecule arrives at a given occupation site during the time interval is: . In our case we study only one molecule (worst case scenario) so let us assume R=1.

The average time that elapses before the arrival of a particle is: .

You will find below a table of time before arrival for different molecules.

Parameters

Finally, we made some assumptions for certain parameters that are used in most of our models. These assumptions are discussed at length in the parameters section.