## 2 Materials and Methods
### Determination of Gallic Acid and Quercetin by HPLC-DAD  
In the first step, 10 g of frozen blackberry sample was poured into 100 mL of methanol and was sonicated in an ultrasonic bath (ultrasound-assisted extraction, UAE) for 30 min. The supernatants were filtered through a 0.45 \(\upmu\)m syringe filter. The analyses were operated on an Agilent 1260 HPLC system (Agilent, Santa Clara, CA, USA) with a photodiode array detector, with Betasil C\({}_{\text{18}}\) (150 mm \(\times\) 4.6 mm, 5 \(\upmu\)m) analytical column (Thermo Scientific(r), Waltham, MA, USA ). The mobile phase was a binary solvent mixture of A (0.1% formic acid in water) and B (methanol). The solvents used were filtered through a 0.45 \(\upmu\)m filter and degassed. The mobile phase flow rate was 0.6 mL-min\({}^{-1}\). The gradient program was: 0-2 min: 15% B; 2-7 min: 15-30% B; 11-15 min: 30-80% B; 15-20 min: 80-15% B (UV absorbance was monitored in the interval \(\uplambda=20\)-400 nm [34; 35]. The calibration curve was constructed by plotting standard solutions' peak area against the concentration (5-100 \(\upmu\)g/mL for gallic acid and 2.5-50 \(\upmu\)g/mL). The determination coefficient (R\({}^{2}\)) was 0.9918 for gallic acid and 0.9952 for quercetin. The LOD and LOQ were 0.85 and 2.17 \(\upmu\)g/mL for gallic acid and 0.17 and 0.32 \(\upmu\)g/mL for quercetin.  
Calculation of a Solvent Mixture Optimal Concentration for Anthocyanin Extraction Using the Hansen Solubility Parameters  
The theoretical optimal concentration of a solvent mixture (e.g., acetonitrile-water solution) could be calculated by using the \(\delta_{\mathrm{P}}\)-\(\delta_{\mathrm{H}}\) coordinate system. In this coordinate, the solvent-water solution is represented by a line, and the flavylium cation is shown by a point (P). The coordinate of the intersection of the perpendicular lines from the point P to the line will give the two Hansen parameter of the optimal solvent composition. The equation of the line for solvent-water mixture is given by:  
\[\delta^{\mathrm{s}}{}_{\mathrm{P}}=\mathrm{m}\cdot\delta^{\mathrm{s}}{}_{ \mathrm{H}}+\mathrm{n} \tag{5}\]  
where: \(\delta^{\mathrm{s}}{}_{\mathrm{P}}\)--the polar Hansen parameter of the solvent mixture; \(\delta^{\mathrm{s}}{}_{\mathrm{H}}\)--the hydrogen-bond Hansen parameter of the solvent mixture; m-slope; n-intercept. The numerical values for the slope and intercept are determined by linear regression.  
The point, representing the flavylium cation, is marked as P(\(\delta^{\mathrm{f}}{}_{\mathrm{H}}\),\(\delta^{\mathrm{f}}{}_{\mathrm{P}}\)).  
The coordinates of the closest point of the line to the P point, representing the optimal solvent composition (P*(\(\delta^{\mathrm{s}}{}_{\mathrm{H}}\),\(\delta^{\mathrm{s}}{}_{\mathrm{P}}\))), are given by:  
\[\delta^{\mathrm{s}}{}_{\mathrm{H}}=(\delta^{\mathrm{f}}{}_{\mathrm{H}}+\mathrm{ m}(\delta^{\mathrm{f}}{}_{\mathrm{P}}-\mathrm{n}))/(1+\mathrm{m}^{2}) \tag{6}\]  
and  
\[\delta^{\mathrm{s}}{}_{\mathrm{P}}=\mathrm{m}\cdot\delta^{\mathrm{s}}{}_{ \mathrm{H}}+\mathrm{n} \tag{7}\]  
As the dependency V%-\(\delta^{\mathrm{s}}{}_{\mathrm{H}}\) is also linear, given by V%-\(\delta^{\mathrm{s}}{}_{\mathrm{H}}+\mathrm{b}\) (where a-slope, b-intercept, both determined by linear regression), the optimal solvent composition could be expressed as:  
\[\mathrm{V}_{\mathrm{s}}\%=\mathrm{a}\cdot\delta^{\mathrm{s}}{}_{\mathrm{H}}+ \mathrm{b} \tag{8}\]