## 2 Materials and Methods
### Experimental Setup  
Functionalized membranes were assembled in a cross-flow ultrafiltration cell system (schematic representation in Figure 1). The system was composed of a stainless-steel module, which contained the membrane, two pressure gauges (Wika, Klingenberg, Germany) to measure inlet and outlet pressure, a feed tank, and a peristaltic pump (Masterflex, Chongqing, China) to feed the protein solution to the membrane. Before ultrafiltration experiments, the membrane was characterized by measuring the pure water permeance (m\(\cdot\)Pa\({}^{-1}\cdot\)s\({}^{-1}\)) (Lp). The permeate flux as function of time at different transmembrane pressure (TMP) values was measured; the steady-state values of flux were then plotted versus TMP, and, from the slope of the straight line obtained, the pure water permeance was calculated from Equation (1).  
\[\mathrm{J}=Lp\cdot\mathrm{\Delta P} \tag{1}\]where J is the permeate flux (L-h-1-m-2), and \(\Delta\)P is the TMP (bar). The reason for measuring this parameter pertains to the need to check the initial membrane performance as reference for subsequent use of membrane after protein fractionation experiments. Furthermore, pure water permeance was necessary to evaluate hydraulic resistances, adsorption, and irreversible fouling caused by the different protein solutions with respect to the initial condition. The effect of protein solutions on membrane fouling at different TMP was evaluated by a resistance in series model described by Equation (2):  
\[\text{Rtot}=\text{Rm}+\text{Rirr}+\text{Rrev} \tag{2}\]  
where Rm is the hydraulic resistance of the membrane itself, Rirr is the hydraulic resistance due to irreversible fouling, Rrev is the hydraulic resistance due to reversible fouling (concentration polarization and reversible deposited material), and Rtot is the total resistance given by the sum of different contributions. The membrane hydraulic resistance Rm can be calculated by the following equation [29; 30]:  
\[\text{Rm}=\frac{1}{Lp\mu} \tag{3}\]  
Here, \(Lp\) is the pure water permeance (m-Pa-1-s-1) of virgin membrane, and \(\mu\) (Pa s) the viscosity of the solution. The hydraulic resistance due to irreversible fouling is given by the pure water permeance of the used membrane (that is, membrane after ultrafiltration test and rinsed with water). The total hydraulic resistance is calculated by the membrane permeance using the given working solution and the viscosity of that solution. The reversible hydraulic resistance is calculated by difference from Equation (2). The purity of the protein in the collected fraction is calculated by Equation (4):  
\[\text{Purity}=\frac{[C_{A}]}{[C_{A}]+[C_{B}]} \tag{4}\]  
Figure 1: Schematic representation of cross-flow ultrafiltration system used. PG: pressure gauges, PP: peristaltic pump, V: valve. The same system was used to work in recycle mode (recirculating the permeate) and in concentration mode.  
where \(C_{A}\) and \(C_{B}\) are the concentration of the protein \(A\) and \(B\) in a given stream/solution. The recovery factor is calculated by Equation (5):  
\[\text{R.F.}=\frac{mg_{AP}}{mg_{AF}} \tag{5}\]  
where \(mg_{AP}\) is the amount (mg) of the protein \(A\) in the permeate, and \(mg_{AF}\) is the amount of the same protein in the feed.