Under the lens

During practical experiments in the laboratory, one is often left with a large number of factors, which need to be tested in order to create a meaningful model. A model which is considered meaningful should be capable of relating experimental factors (explanatory variables) to a response variable (experimental outcome) via a process/system as shown in Figure 1.

The simplest linear response model is presented below: \begin{equation} y_{i}=\beta_{0}+\beta_{1}x_{1}+...+\beta_{n}x_{n}+\varepsilon_{i} , \varepsilon_{i} \sim N(0,\sigma^{2}I) \end{equation} Where $y_{i}$ is the response variable of the i’th observation, the $x_{1..n}$ are the explanatory variable of the i’th response, the $\beta_{0..n}$ are the regression coefficients and finally the residuals of the model $\varepsilon_{i}$ are considered to be normally distributed with a constant variance ($\sigma^{2}I$, with Ibeing the unity matrix) and a mean of 0.
This model will, for convenience, be written in matrix notation: \begin{equation} Y=X\beta+\varepsilon, \varepsilon \sim N(0,\sigma^{2}I) \end{equation} Where X is the design matrix of the model the size of k x N where k is the number of factors and N is the number of responses. Y is a vector of response variables of size 1 x Y, Y is a vector of regression coefficients of size 1 x N and finally, the \varepsilon is also a vector of the size of 1 x N containing the residuals for each response variable. The model is very important, as this is the foundation of the experimental design showing which factors should undergo testing and the selection is thus specifically related to the design matrix (see later).

To understand the importance of an experimental design one could look at previously employed methods. In traditional design of experiments (DOE) one would use the change “one factor at a time until no improvements can be achieved” principle (2). However, this technique does not take into account that of a possible interaction between factors. Why this is a problem can be illustrated via our experiment regarding the testing of compressive strength in the fungal bricks (indsæt link til experimentet-ikkeskrevet). Had we only taken one factor at a time and keeping everything else constant, the possible interaction of for instance different burning temperature and different burning time would not be identified, thus leaving out a significant part of the explained variance, resulting in a worse fit of the compressive strength, our response variable.

A general DOE which may provide a foundation of an experimental model (such as the linear model presented earlier) are often employed as a factorial design (4). When using a factorial design, one can with relative ease gain a solid modelling of the entire design space of an experiment. The design works by having multiple factors which are all considered to be either at a high level or low level, thus spanning over the largest design space possible. A simple example of such a design can be written as an ANOVA model: \begin{equation} y_{ijk}=\mu+\alpha_{i}+\beta_{j}+(\alpha\beta)_{ij}+\varepsilon_{ijk}, \varepsilon_{ijk} \sim N(0,\sigma^{2}I) \end{equation}

Microscopic view

One of our models focuses on the morphology at the hyphal level by simulating the movement of hyphal tips, branching rates, extension rates and the density levels during a growth period in two dimensions. All of the code scripts can be found on our GitHub repository.

The mycelium of a fungus consists of many interwoven hyphae, and the density depends on how many filaments there are in a given location. In figure 1 below, it is possible to see microscopic pictures of fungal mycelium. Three different levels of zoom illustrate how the network looks, where it can be observed how they interlink and how a fungal filament can branch into more.