Description of the root classes in the model in R
Experimental design in R language
(Post processing of CRootBox)
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 system with a fixed distance. We assume that the growth rate and the density of mycorrhizea are high enougth so that the contact happens always between mycorrhizea and not between real roots of the plants.
To do that, we positionned the two root systems in the XZ-plane leading to the representation of the experimental montage. 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 an other translation, this time to both root systems, so that no segment has a negative x coordinate. This will greatly simplify the next step consisting in represent the 2D space by matrices. By ploting with a different color for the different root types, we possibly can check our assumption that contacts happen between mycorrhizea (remenber that mycorrhizea correspond to type 20 and type 50).
(graphique avec les 2 systèmes racinaires)
We divided the experimental space in a grid composed by cells characterized by a 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 short. For each root system we developped a matrix containing for each cell the number of root's segments it contains. After, we overlapped the two matrices in order to see if there is contact. If a same cell contains segments in the two matrices, we assume there is a mycorrhizea association. It is highly likely that two segments in the same cell are not really in contact but, when closed of each other, mycorrhizea can connect thanks to chemiotropism, justifying the assumption.
Compare the two matrices thus allows to find the cells where there is contact. From this information, we look in those cells for the youngest segment from each plant. The age of the older of the two correspond to the minimal time needed for what we consider as a contact. The minimum of the results obtained for the different cells is our estimation of the time needed for a mycorrhizea association between the two plants.
Modelling 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 plant have been made, let's call them A and B. First two dimensions are for the XZ space and a third dimension is for the time. The time is discretized so that we have one time layer per day. We consider one plant emetting a signal and the other plant able to receive the sigal. For the emitting plant, the array contains the number of emetting segments in each cell. We consider that all 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 contain more 1 if it correspond to an advanced time value.
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 three 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 mycorhize is located in (ti, xi, yi) and that the molecule concentration in (ti, xi, yi) is greater than 1 / 100,000 units of concentration / cm 2. 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 in the annexe 3.
Results
Roots architecture
Figure 1 = Root architecture
Mycorrhizae
In this section, we present some results of the estimation of the time needed for the establishment of a mycorrhizae relation between two plants.
Here, in all the cases, the distance between the two collars is set to 10.8 cm, corresponding to the distance set in the experiment. We used several root architectures from CRootBox with a different mycorrhizae growth rate.
The first representation below is for a mycorrhizae growth rate of 0.08 cm/day, wich seems a coherent value for the media used in the in vitro experiment. After the 30 days of the simulation, there is no contact between the two plants. If a response happens before, we will deduce that it is not due to mycorrhizae.