## 2 Materials and Methods
### Optimization Design of the Nutrient Broth Components  
Modern experimental designs fall into three main study types: factorial, mixture, or custom. The latter is used when the experiment requires adjustments that cannot be accommodated by a standard mixture design, such as when the differences between the high and low of all the mixture components are not the same, or when mixture and process variables are both used in the same design. In order to optimize growth media in terms of biomass yield, the effect of six mixture components and one numeric factor was studied. Yeast extract (Merck KGaA, Darmstadt, Germany) and malt extract (Oxoid Ltd., Basingstoke, UK) were used as a protein source, while fructose (Schneekoppe GmbH, Buchholz, Germany) and soluble starch (Carl Roth GmbH, Karlsruhe, Germany) provided the carbohydrate source, and the latter also acted as a glucose substitute. We focused on fructose because it was reported that silver birch sap consists mostly of fructose (5.39% \(w/w\)) and, to a lesser extent, glucose (4.46% \(w/w\)) and sucrose (0.58% \(w/w\)) [17]. All growth media formulas were supplemented with equal amounts of magnesium sulfate heptahydrate (Merck KGaA, Darmstadt, Germany) and calcium chloride dihydrate (Honeywell GmbH, Seelze, Germany). Sterilization was performed at 121 \({}^{\circ}\)C for 15 min. The pH value of the growth media represented the single numeric factor, which was measured and entered into the experimental design after each media variant was sterilized, before being inoculated. The boundary values for each variable were chosen according to relevant literature data regarding submerged mushroom mycelium cultivation (Table 1). The software Design-Expert version 11.1.0.1 (Stat-Ease Inc., Minneapolis, MN, USA) was used to achieve the mathematical modeling and statistical analysis of the experiment. By working with the type of design space described so far and inputting the above constraints, the software generates model points (runs) that are chosen \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Factor** & **Symbol** & **Low Value (g/L)** & **High Value (g/L)** \\ \hline Malt extract & A & 1.10 & 2.30 \\ \hline Yeast extract & B & 2.00 & 4.00 \\ \hline Fructose & C & 11.00 & 19.00 \\ \hline Soluble starch & D & 11.00 & 19.00 \\ \hline MgSO\({}_{4}\) & E & 0.10 & 0.10 \\ \hline CaCl\({}_{2}\) & F & 0.05 & 0.05 \\ \hline pH & G & 4.97 & 5.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design constraints. limited to as few runs as possible. These included five replicate points and five lack-of-fit points, supplementing the 20 required model points (Table 2). To determine the appropriateness of a design involving mixture components, prediction-based metrics such as fraction of design space (FDS) statistics are employed. The FDS graph is useful for calculating the volume of the design space with a prediction variance equal to or less than a specified value. This custom volume's ratio to the total volume is the fraction of the design space. The response surface was drawn by predicting the mean outcome as a function of inputs over the region of experimentation (Figure 1). \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Run** & **Malt Extract (g/L)** & **Yeast Extract (g/L)** & **Fructose (g/L)** & **Soluble Starch (g/L)** & **MgSO\({}_{4}\) (g/L)** & **CaCl\({}_{2}\) (g/L)** & **pH** \\ \hline
2 &  
The custom quadratic \(\times\) linear model in which the terms A, B, C, and D, and interactions: AB, AC, AD, AG, BC, BD, BG, CD, CG, and DG, were evaluated by FDS statistics found that 80% of the design space was below a standard error mean of 2.399. By plotting the standard error of design and the different model values for the A, B, and G terms, the latter's effect was shown to be the main source of the standard error mean value (Figure 2). If the interactions related to the G term were not evaluated, the standard error mean for the same fraction of design space became 0.489.