Material and Methods 

Quantification of root architecture

Description experimental setup

Thanks to the American in-vitro trial, it was possible to create a photo suite of the roots of each plant in their culture medium for each day until the end of the experiment. We analyzed these images in ImageJ, software used to process/analyze them. In particular, we used SmartRoot, a semi-automatic image analysis software, considered an ImageJ plugin, allowing to quantify root growth and other parameters of plant architecture.

Quantification of root traits (SmartRoot)

With SmartRoot software, one image is processed at a defined time. We delimited the primary root on the image and SmartRoot makes a trace over the entire length of this root. It is possible to correct the traces that were incorrectly carried out. We have also demarcated the secondary roots and SmartRoot continues the route for this secondary root. This route can then be attached to the primary root line. 
The entire route can then be imported into another image. We imported this set in the photo made the next day and continue the plots on the imported set. By performing this step on each photo for the duration of the experiment, it is possible to get a root list with their root type, length and insertion distance on the primary root if it is a secondary root. 
All these data were processed on R and Excel to obtain the growth rate of the primary and secondary roots as well as the distance between the insertion points of the secondary roots.
The length between the last branching, the length between the first branching and the beginning of the basal root and the maximum length of the roots were arbitrarily chosen to resemble the plants used in the American experimentation.

Modeling root and mycorrhizae architecture

Description of CRootBox

CRootBox is a software that aims to simulate different types of root architecture. To do this, CRootBox creates a primary root segment with a specific type assigned. This type gives at the segment different characteristics: growth rate, length, basal length, tropism, diameter, distance between secondary roots, etc. Therefore, this root can produce secondary root segments to which it can also be assigned a different type and so on. It is also possible to specify the number of primary roots of the plant (in this model, we will assume there are 3 primary roots). 
CRootBox will then create a sequence of segments over a defined simulation time creating a file where each segment is referenced with different information: its starting point in (xi, yi, zi), its arrival point, age, type, etc.

Description of the root classes in the model in R

In this model, we have defined 4 classes of roots that CRootBox will use to create the architecture of roots and mycorrhizae (Annex I).
The first type is attributed to the primary root: its growth rate characteristics and its distance between secondary roots were calculated using SmartRoot.
The second type is attributed to the secondary roots where the growth rate was determined using SmartRoot. We considered that the secondary roots do not make branching, which corresponds to the reality until the growth time of the plants tested is not too long. The other parameters needed for the first type and the second one have been arbitrarily chosen to be similar to the American experimentation.
The third type is attributed to the mycorrhizae. Indeed, from CRootBox, it is not possible to create two separate objects developing at the same time in a defined space. It is therefore impossible to simulate the growth of root and mycorrhizal at the same time in CRootBox , if they are considered as two different entities. To circumvent this problem, we considered the mycorrhizae as roots that can grow on any root type. In order to be plausible, we attributed to the third type a plagiotropism allowing mycorrhizae to develop in a mainly horizontal plane. The growth rate and branching distance were determined at the Louvain-la-Neuve mycology laboratory in Petri dish.  We recognize that the rate of growth of mycorrhizae can vary greatly depending on the environment. We have therefore produced a script to carry out a simulation suite for a fixed growth rate that will increase for each simulation (Annex II). This script could be used with other scripts performed in this work to determine a time panel so that two different plant mycorrhizae can touch and send a signal. 
The fourth type is attributed to the ramifications of mycorrhizae that have exactly the same characteristics as type 3 except that we consider that they do not branch to avoid overloading the amount of information that CRootBox needs to process in order not to significantly slow down the simulation.

Experimental design in R language

(Post processing of CRootBox, script is available in Annex III) 
Using the simulation results of the roots architecture implemented with mycorrhizae from CRootBox program, we were able to estimate with R the contact time between the two radical systems with a fixed distance.  We assume that the growth rate and the density of mycorrhizae are enough high so that the contact happens always between mycorrhizae and not between real roots of the plants.
For doing that, we positioned the two root systems in the XZ-plane leading to the representation of the experimental design. After loading the files, we applied a translation along the X axis to one of the two root systems. This translation corresponds to the distance between the two collars of the plants. Notice that we applied another translation, this time for both root systems, we avoid that segment could have a negative x coordinate. The minimal x coordinate is thus 0 and the maximal z coordinate is also 0. This will greatly simplify the next step consisting of the representation of the 2D space by matrices. By plotting with a different color for the different root types, we could check our assumption that contacts happen between mycorrhizae (mycorrhizae correspond to type 20 and type 50).
We divided the experimental space in a grid composed of cells characterized by a specific size depending on the distance between the collars of the two plants divided by 10. The idea is to increase the precision when the distance between the plants is shorter. For each root system, we developed a matrix containing for each cell the number of root's segments placed inside. After, we overlapped the two matrices in order to see if there is contact. If the same cell contains segments in both matrices, we assume there is a mycorrhizal association. It is probable that two segments in the same cell are not really in contact but, when closed off each other, mycorrhizae can connect thanks to chemotropism, justifying this assumption.
Compare the two matrices thus allows to find the cells where there is contact. From this information, we looked in those cells to find the youngest segment from each plant. The age of the older of the two corresponds to the minimal time needed for what we consider a contact. The minimum of the results obtained for the different cells is our estimation of the time needed for a mycorrhizal association between the two plants.

Modeling signal transfer in the environment

Describe the diffusion model

We kept the space discretization into cells like seen before. 3D array for each of both plants have been made, let's call them A and B. First two dimensions correspond to the XZ space and the third dimension correspond at the time. This is discretized so that we have a one-time layer per day. We considered one plant emitting a signal and the other plant able to receive the signal. For the emitting plant, the array contains the number of emitting segments in each cell. We consider that all the root segments exudate if they are at most 3 days old. For the receiving plant, the cells contain a 1 if there is at least one root segment, and a 0 if there is not. A layer contains more 1 if it corresponds to an advanced time value. The code to model A and B is in Annex IV.
To verify that a chemical-type signal can be sent from a root of plant A to the root of plant B a two-dimensional numerical diffusion and reaction model has been created. The numerical model has been solved according to the explicit finite difference method on the software RStudio©. The model includes 3-dimensional arrays, first two are A, B and the third one called C represent the concentration of signaling molecules. C characterizes the concentration in molecules signals in every point of the space and with each step of time. The signal is transmitted when the term product of the matrices B and C is non-zero, which is equivalent to saying that the signal is transmitted when a receiving root or a receiving mycorrhizae is located in (ti, xi, yi) and that the molecule concentration in (ti, xi, yi) is greater than 1 / 100,000 units of concentration/cm2. Since the type of exudate is unknown, it has been arbitrarily considered at first approach that a root emits a unit of molecular concentration per day per square centimeter and that the quantity required for detecting the signal must be at least 1/100 000 times the concentration emitted. If the emission rate and the concentration required for detecting the compound are known, the model can take it into account for more realistic modeling. The code of the model is reported in annex V.