Simulation of the mycelium development

Fungal growth is initiated by $n$ number of spores, and a branch will start to extend from each of the spores added to the space. Following along one of these branches originating from a single spore, the hyphae will grow in a direction $\theta$ with a tip extension $r_{tip, i}$. A branching event, in which a new branch is formed from the first branch, can occur with a probability $q$. Tip extension rate is calculated by using the equation below, which considers growth kinetics for the fungi and the amount of substrate available. It essentially outputs the accelerated growth dependent on the amount of substrate available, where the accelerated growth equation depends on fungal kinetics and branch lengths (1, 3). \begin{equation} r_{tip, i} = \bigg(k_{tip,1} + k_{tip,2}\cdot \frac{l_{br,i}}{l_{br, i} + K_t}\bigg)\cdot\bigg(\frac{S}{S+K_s}\bigg) \end{equation} $k_{tip,1}$ is the initial tip extension rate of the branch and $k_{tip,2}$ is the difference between the maximum extension rate and $k_{tip,1}$ (1, 3). $K_t$ symbolizes the time it takes to reach half of the maximum extension rate. The length of branch $i$ described by $l_{br,i}$. $S$ is the substrate concentration and $K_S$ corresponds to the substrate concentration to reach half of the maximum growth level (4). As the starting coordinates of this simulation are $(x_0, y_0)$ and end coordinates $(x, y)$, the length of branch $i$ can be calculated as the distance between two points: \begin{equation} l_{br, i} = \sqrt{(x-x_0)^2 + (y-y_0)^2} \end{equation} By dividing the growth area into a grid of $w\cdot h$ areas, it is possible to investigate the uptake of substrate, hyphal movement and the development of biomass through the simulation.

At the start of the simulation, it can be assumed that the initial substrate concentration $S_0$ will be distributed evenly across the grid. When the spores are placed randomly, the branches start to develop by using the substrate available around the hyphal tip. So for each branch, it is checked which tip is in which grid and whether there is any substrate available. That means that one can track the substrate depletion in each grid, and when there is no more substrate in one of the areas of the grid, the hyphal tips located there can no longer grow and can be considered as inactive hyphae. If the total area around the mycelium no longer contains substrate, the growth will simply stop.

In the same way as tracking substrate depletion, biomass production can also be viewed in a grid. But instead of calculating a loss of substrate, each new branch extension is considered a gain in biomass or the density $d$. It should show the same mechanics as the substrate depletion, as these two are directly related to each other by the equation $l_{br,i}$.

As the fungi grow in size, the computational cost also increases. We were limited in running the simulation due to the power of our computers, thus resulting in us introducing two restrictions: the number of hyphae $M$ and the number of steps in the simulation $N$.