Calculation of the Flux Density Function for Protein Crystals from Small Scale Settling and Filtration Experiments

Development and engineering of protein crystals regarding mechanical stability and crystallizability occurs on a small scale. Later in the process chain of industrial production however, filtration properties are important to separate the crystals from mother liquor. Many protein crystals are sensitive to mechanical stress which is why it is important to know the filtration behavior early on. In this study we analyze settling and filtration behavior of isometric, rod-like and needle shaped lysozyme and rod-like alcohol dehydrogenase (ADH) crystals on a small scale using an optical analytical centrifuge. Needle shaped lysozyme and rod-like ADH crystals show compressible material behavior. With the results from settling and filtration experiments the flux density function is calculated and modeled which can be used to describe the whole settling and permeation process in dependency of the solids volume fraction. This is also an issue for simulations of industrial processes.


| INTRODUCTION
The improvement of bulk protein crystallization for purification or formulation is subject of current research. Crystalline proteins are advantageous compared to dissolved or amorphous solids because of their extended shelf life or modified drug release properties. 1,2 The implementation of a crystallization step in the downstream process chain for protein purification and formulation is discussed in literature by Hubbuch et al. 3 and Hekmat et al. 4 In recent research either the proteins were modified to improve or enable crystallization, or the crystals itself were altered to enhance mechanical stability. [5][6][7] Nowotny et al. 5 and Hermann et al. 6 identified crucial amino acids for crystallization in the sequence of the protein alcohol dehydrogenase from Lactobacillus brevis in a rational crystal engineering approach using a molecular dynamics model. They confirmed their findings with experimental results for three different mutants with improved crystallizability. Grob et al. 8 demonstrated the applicability of technical crystallization in downstream processing with the protein alcohol dehydrogenase. Kubiak et al. 7 used cross-linking of crystals, which drastically enhanced the crystal's ability to withstand mechanical stress. Kubiak et al. 9 reported anisotropic mechanical properties for cross-linked halohydrin dehalogenase crystals and investigated the effect of cross-linking time on crystals hardness and Young's modulus. Crystal breakage at pressures below 1 Â 10 5 Pa for native lysozyme crystals, which negatively affects the filtration behavior of such particle systems has been previously reported by Cornehl et al. 10,11 Filtration is a common procedure to separate solids from a liquid for further processing. 12 Hence, filtration is also suitable to separate protein crystals from saturated mother liquor. Cake forming filtration processes are well characterized and several models describing filtration parameters and cake properties may be found in literature. [13][14][15][16][17][18][19] Existing traditional filtration setups are changing to meet the challenges in pharmaceutical applications (integrated compact filtration devices 20,21 ) or to reduce the amount of sample required in early product development. Performing standard laboratory experiments to characterize filtration properties, for example, according to VDI 2762, 22 usually requires liters of suspension or fluid. Those amounts are not available in early product development like protein or crystal modification. This research is conducted on a much smaller scale ranging from micro-to milliliters. Investigating filtration properties of protein crystals on an early stage of process development therefore requires a much smaller laboratory setup. Radel et al. 23 introduced a downscaled filtration setup for an optical laboratory centrifuge which is based on the setup published by Loginov et al. 24 for investigating filtration properties on a small scale.
To model the settling and filtration process, the flux density function can be used. The flux density function for settling processes was introduced by Kynch 25 and amended by Tiller 26 for compressible sediments. He et al. 27 analyzed and modeled the hindered settling of disk shaped colloids and investigated the influence of particle orientation on settling velocity. Calculating values for the flux density function for solids volume fractions above the gel point from permeability data of filter cakes or sediments was added by Bürger et al. 28 The gel point describes the transition from a suspension to a particulate network.
The flux density function is only dependent on the solids volume fraction and describes the settling and permeation properties of a particle system. In numerical simulations of settling or filtration processes, the flux density function combined with the compressible yield stress are useful input parameters to model the whole separation process.

| Particle system
This study considers three different lysozyme crystal systems (isometric, rod-like and needle shaped) and the commercially not available crystal system alcohol dehydrogenase (ADH) from L. brevis. Precipitated hen egg-white lysozyme is obtained from Ovobest (Germany), the ADH enzyme is kindly provided by Prof. Weuster-Botz from the institute of biochemical engineering at the Technical University of Munich. Exemplary microscopy images of the crystal systems can be seen in Figure 1.
The crystallized proteins for needle shaped lysozyme, rod-like lysozyme and ADH are kindly provided by Prof. Kind from the institute of thermal process engineering at Karlsruhe Institute of Technology. Those crystals are produced using a low temperature vacuum evaporation process described in detail by Barros Groß and Kind. 29,30 Whereas isometric and ADH crystals grow at 297 K, the rod-like lysozyme crystals are produced at 308 K. Needle shaped lysozyme crystals require additional sodium chloride at a mass fraction of at least 0:03 in the crystallization buffer. Otherwise, the procedure is analogous to the rod-like crystals at a temperature of 297 K.
The production of isometric lysozyme crystals is achieved with displacement crystallization and hence requires two stock solutions. Stock solution A is a 25 mmol acetic acid buffer at pH 4:0. Stock solution B contains 80 g L À1 NaCl additionally. 100 g L À1 lysozyme are dissolved in 125 mL stock solution A and gently stirred with a blade stirrer at 350 rpm. A membrane pump adds 125 mL stock solution B to the mixture at a rate of 1 mL min À1 . The whole crystallization process takes about 16 h to reduce the supersaturation and to reach equilibrium.
Additionally, for the analysis of filtration behavior, the particle and supernatant properties density and viscosity must be known.
Those parameters are shown in Table 1. The solid density of crystalline lysozyme is obtained from literature. 31 In house measurements with Mettler Toledo Densito 30PX yield the liquid densities of the continuous, particle free phases using the oscillating tube method according to EN ISO 15212-1. 32 The solid density of ADH crystals is measured with gas pycnometer MP 1305 (Micromeritics GmbH, Germany). 33 Liquid viscosity is obtained with falling sphere viscometry.  which obviously is only a rough characterization for elongated particles like needles or rods. However, it is still possible to compare the different particle systems relative to each other. Error bars in Figure 2 indicate the minimum and maximum value of three independent repeated measurements.

| Settling experiments
The settling velocity of particle collectives depends on the solids volume fraction ϕ, which is defined as with V s , V T , V p as the solids volume, total volume and volume of the void. ϵ is the porosity which is commonly used in the characterization of particulate networks. For low solids volume fractions the particles settle individually depending on density (ρ) and diameter (x). In the case of spherical particles, Reynolds numbers below 0:25 and Newtonian fluids the velocity equals which is the Stokes law. 12 With increasing solids volume fraction, the settling regime changes from single particle over swarm to zone settling. In the last regime, most particles have the same settling velocity independent of their diameter. Typically, a sharp transition of clear liquid to suspension, the separation front, develops. 34,12 In broad sized distributions coarse particles might still settle faster than the remaining particles which leads to underestimation of the mean set-  F I G U R E 2 Cumulative particle size distributions of the crystalline particles obtained with laser diffraction Because the settling velocity in the centrifugal field is higher than in earth gravity (g), we have to divide the obtained settling velocity from the centrifuge by the relative centrifugal acceleration with r 0 and r 1 as the initial and final distance of the separation front from the centrifuge center and ω as angular velocity. To

| Filtration experiments
Filtration is a common technique to separate solids from liquids. In dead end cake filtration the particles are held back by a porous filter medium, like a porous membrane or a filter cloth while the filtrate passes through the filter medium. The retained particles on the filter medium first form a thin layer and build particle bridges, then a filter cake. This filter cake may retain particles even smaller than the pores of the filter medium. Often, the chosen pore size of the filter medium is larger than the particles to avoid clogging and to keep the filter medium resistance (R m ) low. The risk of clogged pores is maximal when the particle sizes are in a similar range as the pore sizes because then the particles fit into the pores. 12 In this study, however, the chosen pore size is much smaller than the particles to avoid filtrate pollution and clogging. This allows us to assume that all particles are retained at the cost of a higher filter medium resistance.
Darcy's law connects the pressure drop Δp, filter area (A m ), volume flow rate ( _ V) and the resistance against permeation (R) with each other. For constant pressure filtration Darcy's law follows The resistance is typically split into the resistance of the filter medium and resistance of the filter cake (R c ). The resistance of the filter cake changes with its height (h). Thus, for better comparability, the resistance is normalized with either its height or its mass The reciprocal value of the height specific resistance is the height specific permeability (K).
If the material is incompressible, resistance and solids volume fraction are constant. In the case of compressible material behavior those parameters increase with higher pressure. Compressible material behavior can occur due to particle rearrangement, destruction of card house structures in the case of elongated particles, elastic and plastic deformation or breakage. 13 To model resistance and solids volume fraction for compressible filter cakes Tiller and Kwon 15 proposed the equations Equations (9) and (10)  ln connects the distance from the centrifuge center of the membrane r m , the filter cake top r c and the liquid height above the filter cake h l with material and process properties. From this equation the integral height specific resistance of the filter cake is obtained. For the detailed working principle of the cuvette and data analysis please refer to Radel et al. 23 and Loginov et al. 36 In centrifugal filtration the pressure on the filter medium does not remain constant, but decreases with descending fluid height. Hence, all filtration related pressures mentioned are mean pressures during the filtration process.

| Flux density function
For obtaining the flux density function (f bk ) of protein crystals, centrifugal settling and filtration as described in the previous section is used.
In the case of solids volume fractions below the gel point can be seen in Figure 3, which also qualitatively shows the effect of the solids volume fraction on the settling velocity.
Measuring and tracing the separation front with an optical centrifuge allows to speed the measurement up and to substantially reduce the amount of required sample volume compared to using settling columns. This however means we have to alter Equation (12) to Using the resistance of a sediment or filter cake, values for flux density function above the gel point are calculated with where α h is a function of ϕ. Determining the mean solids volume fraction and mean height specific filtration resistance with the filtration cell described in Radel et al. 23   The corresponding values for the exponent N ranges from 0 for the isometric crystals to 0:981 for the lysozyme needles. This also shows the large differences in filter cake compressibility for the different particle systems.
However, to characterize compressibility and to calculate the flux density function, the height specific resistance on the right-hand side of Figure 5 is more suitable because it can be directly inserted into Equation (14). As can be seen, the gap between ADH and the lyso-

| Filter cake height
The filter cake height of the crystalline systems is directly available from the measurements with the filtration cell because the cake position is always visible and provides a characteristic drop from high to almost zero transmission in the transmission profiles. In Figure 6 the normalized filter cake heights over filtration pressure for the investigated crystal systems are shown. As can be seen from Figure 6, the tendency towards slightly smaller cake heights with increasing pressure is observed for rod-like lysozyme crystals. The same is true for rod-like ADH crystals but the tendency is much more distinct. The The data points also scatter more at lower pressures because the particles tend to partially smear the glass window of the filtration cell, resulting in a more skewed drop in transmittance and thus increasing the uncertainty of the filter cake position.

| Filtration pressure
The knowledge of cake height allows to calculate the corresponding integral solids volume fraction in the filter cake. The filtration pressure vs. solids volume fraction is depicted in Figure 7. For the more compressible systems (needle lysozyme and ADH) the curves show a slow increase of pressure with higher solids volume fractions. In the case of the more incompressible particle systems (isometric and rod-like lysozyme) the increase in pressure is much steeper. For an ideal incompressible particle system the pressure would jump from zero to infinity at the gel point.
The calculated fits according to Equation (9) show good agreement with measurement results. For the isometric crystals the extrapolated values of the dash dotted curve might not be completely accurate. In general, extrapolating is critical due to the exponential nature of the pressure on solids volume fraction dependency. Also, particle systems typically have a maximum solids volume fraction ϕ max at which further compression is impossible. This is currently not reflected in the applied model. From those results it becomes evident that the particle shape of the four crystal systems has a high influence on filterability. The best filtration properties exhibit isometric and rod-like lysozyme crystals because they have low filtration resistance and thus enable fast filtration. Also, those particle systems have high solids volume fractions which means less residual moisture in the filter cake and hence better dewatering properties. ADH and needle shaped lysozyme crystals on the other hand show compressible material behavior and have high resistances compared to the other two particle systems. The solids volume fraction at lower pressures is much smaller for those particle systems which means that more fluid remains in the pores of the filter cake. Thermal drying of filter cakes with crystalline systems is difficult because the dissolved protein in the pore fluid crystallizes when the fluid evaporates and thus blocks the pores needed to transport the evaporated gas.
The comparison of Figures 2 and 5 shows a big influence of particle shape, size and distribution on filter cake resistance and compressibility. The high compressibility of needle shaped lysozyme and rod-like ADH possibly follows from the formation of card house structures that recompress and compact at higher pressure. This leads to the sharp increase in cake resistance. For the needle shaped lysozyme, moreover, particle breakage is suspected at higher pressures, which can also occur with ADH. In the case of isometric and rod-like lysozyme, there is no or very little compressibility. The differences between rod-like lysozyme and ADH in compressibility behavior are due to two reasons.
First, the particle size distribution of ADH is broader and somewhat shifted to smaller crystals, which favors compression.
Second, for technical reasons, the initial solids volume fraction for lysozyme is much higher than for ADH, which may lead to a different cake structure.

| Settling
For two of the discussed crystal systems settling velocities in earth gravity (rod-like lysozyme) and in the centrifugal field (needle shaped lysozyme) have been determined. The isometric crystal suspension did not show a clear separation front so that the results are inconclusive.
The available sample volume for ADH did not allow a thorough analysis of settling behavior. To make the results from centrifugal settling experiments comparable with the other particle system, the velocities have been rec- with v C being the settling velocity in the centrifugal field and v g being the settling velocity in earth gravity. The definition of Z is given in Equation (4). The decrease of settling velocities in both particle systems is due to hindered settling in a particle collective. At low solids volume fractions single particle settling as described by Stokes is thus much faster than hindered settling at higher solids volume fractions.
The maximum solids volume fraction, which could be achieved with concentration of the particle systems, is for the rod-like crystals with 0:146 almost four times bigger than for the needle shaped crystals with 0:038. This also reflects in the big gap of solids volume fraction for those both crystal shapes in Figure 7. The gel point for lysozyme needles must hence be much lower than for rod-like lysozyme crystals.
Also, the total drop in settling velocity is for needle shaped lysozyme crystals with a factor of 161 much larger than for the rod-like crystals where the settling velocity drops at a factor of about 7:5. The influence of solids volume fraction on settling velocity is thus for the lysozyme needles much more dominant than for the rod-like crystals. It is also interesting to see that with settling experiments the solids volume fraction of lysozyme needles reaches 0:038 but in filtration this particle system is compressed to a solids volume fraction of 0:274 (see Figure 7). This corresponds to a 7:2-fold increase in solids volume fraction. This can be explained by the destruction of card house structures in the filter cake to a more compact cake and also by crystal breakage. Cornehl et al. 11 have analyzed lysozyme crystal breakage. They found that even at low pressures in typical filtration processes, crystal breakage occurs which worsens filtration and dewatering properties.
The solid line in Figure 8 right shows a modified fit based on Richardson and Zaki 34 (see Equation (3)). Since the model of A possible explanation is the transition from unstructured, randomly orientated rods at low solid volume concentrations towards an orientated particle collective due to steric hindering at higher solids volume fractions. He et al. 27 found and modeled a similar effect of particle orientation on settling velocity for colloidal disk-shaped particles.

| Flux density function
With the results from the previous sections, especially with the calculated fit models from Richardson and Zaki 34 and Tiller and Kwon, 15 the flux density function can be calculated as follows. Figure 9 shows the complete flux density function for the needle shaped particle system. The gel point is not exactly known but F I G U R E 9 Flux density function for needle shaped lysozyme particles F I G U R E 1 0 Flux density function for needle shaped lysozyme and ADH at solids volume fractions above the gel point expected to lie somewhere between 0:03 and 0:06. The dashed line in Figure 9 is the calculated flux density function according to the modified Richardson and Zaki model (first segment of Equation (16)).
The dash dotted line is calculated based on permeation data with the models of Tiller. The last five data points in Figure 9 are obtained with permeation experiments, the other data points at lower solids volume fractions are determined with settling experiments. Figure 10 shows the flux density function for solids volume fractions above the gel point for needle shaped lysozyme and ADH crystals calculated from permeation data. As can be seen the flux density of ADH crystals is much smaller than the flux density of needle shaped lysozyme which is due to the lower filter cake resistance of ADH. The model describes the measured data well except for the second data point of the needle shaped lysozyme.
The flux density function allows to model the settling and permeation behavior of a particle system only in dependency of the solids volume fraction. This is especially useful for processes in which both effects play a significant role. Also in simulation models, for instance in Bürger and Concha, 37 the flux density function is used as an input parameter for the simulation of settling processes.

| CONCLUSION
The settling and permeation behavior for different protein crystal systems can be analyzed on a small scale using a 3D printed filtration cell and an analytical optical centrifuge. The crystal shape has a big influence on the filtration and settling properties. Being able to characterize filtration and settling on a small scale allows testing different crystal shapes or crystal modifications early in process development.
Using the flux density function, the whole settling and permeation process can be modeled in dependency of the solids volume fraction.
For solids volume fractions below the gel point this requires the ability to observe a distinct separation front. This is not possible for all particle systems, but for example in the case of the needle shaped lysozyme crystals. Above the gel point the flux density function can be obtained with permeation experiments on a small scale with only about 300 μL of supernatant per experiment.
As a first step, the flux density function determined on a small scale allows to describe the sedimentation and filtration behavior of different protein crystal systems. The method presented in this article is not limited to proteins but can also be applied to other particle systems that are only available in small quantities during product development. As an outlook, simulations using the flux density function are conceivable, such as filtration in a chamber filter press. The flux density function and the compression resistance serve as input parameters, for example, in adapted models of Bürger et al. 38 and Garrido et al. 39 In material systems where particle breakage occurs in the filter cake, the change in flow resistance is also represented by the flux density function. Open Access funding enabled and organized by Projekt DEAL.

CONFLICT OF INTEREST
The authors declare that there is no conflict of interest to disclose. v velocity (m s À1 ) x particle diameter (μm) The data that support the findings of this study are available from the corresponding author upon reasonable request.