Hen egg-white lysozyme (HEWL; Seikagaku, six times recrystallized) was used also for this study without further purification [20] [21] . Solutions of HEWL [90 mg/ml (6.3 mM)] and NaCl [15% (w/v) (2.56 M)] in a sodium acetate buffer (50 mM, pH = 4.6 ± 0.1), and the buffer were prepared separately as stock solutions. The solutions were passed through a filter with 0.2-μm pore. Just before FLS measurements, these solutions were mixed at the appropriate rate in a micro test tube (1.5 ml) at 20 ± 0.5˚C. A sample (~18 μl) for the FLS measurements was taken immediately from the test tube and injected into the sample cell.

The sample cell was composed of two microscope cover glass plates (BK-7, 18 mm × 32 mm, and 0.15 mm thick) and a silicon rubber sheet (~1.0 mm thick). These components were cleaned with a supersonic wave syringe using a neutral detergent and ethyl alcohol, respectively. The rubber sheet had a circular hole with a diameter of about 6 mm, and was sandwiched between the two cover glass plates. The mixed lysozyme solution was dropped into the circular space between the glass plates. This parallel plate cell allows for precise measurements of the forward light scattering at small angles from the sample solutions.

Figure 1 shows a schematic of the setup for F-SLS measurements and photographs of a scattered light around a transmitted light beam from crystallizing lysozyme solutions, which were projected onto a screen far from the solution cell. The slight light in the photograph (a), which extends around the white bright spot of the transmitted light in the center, is the forward scattered light at small angles. This forward light scattering is characteristic of a solution with pre-crystalline lysozyme aggregates and clusters. When precipitates or microcrystals of lysozyme had formed, ring-like diffractions are observed on the forward screen, as shown in the photograph (b), which are distinct from the small amounts of light due to small clusters.

(a) (b)

The apparatus for F-SLS measurements was constructed in house [20] [21] . The sample cell was mounted on the θ rotation stage of a θ - 2θ coupling stage with a common axis, which was controlled with a personal computer through a controller. A narrow laser beam (~0.7 mm in diameter, spread angle < 1.2 × 10⁻³ rad) from a diode-pumped solid state laser (λ = 473 nm, 25 mW) was directed onto the sample cell perpendicularly (at a fixed incident angle of θ = 0). The incident light was split with a half-mirror and the intensity was monitored using a photo-diode detector, as a reference light for estimation of the relative light scattering intensity. Light scattered from the sample solution was detected using a small solid photomultiplier detector mounted on the 2θ rotation stage. To measure the scattered light precisely at the scattering angle θ_s, a pair of slits (~2 mm in diameter) was positioned on the 2θ rotation stage at an appropriate distance in front of the detector. The detected scattered light at each angle was recorded on the computer through an analog-digital interface converter.

Light scattering of the buffer solutions was extremely weak and a constant for scattering angles except for in very small angular range (<2˚), so that the background scattering of the sample solutions was negligible. Furthermore, scattering pattern of lysozyme small aggregates <0.2 μm, which were passed through the 0.2-μm filter in the solution preparation, was similar to that of the buffer solutions and the scattering at the small angles can be involved in the background scattering. Thus we estimated the scattering light intensity I_s as the ratio of monitored incident light intensity, and obtained the SLS patterns for the sample solutions from measurements of scattering light intensities by scanning the 2θ rotation stage at a step angle of 0.1˚. The measured was corrected for the measurement angle by multiplying by a factor of, to obtain the accurate scattering angle distribution. In the light scattering measurements for lysozyme solutions before crystallization, the influence of multiple light scatterings can be ignored, because the transmitted light in the sample solutions is strong because of the lack of an absorption at the energy of the probe light (λ = 473 nm), while the scattering light intensity is weak, as shown in the photograph (a) of Figure 1. We obtained the scattering distribution in q space, I_s (q) from the SLS measurements. It is worth noting that the scattering measurements allow the estimation of, without considering the index of solvent, because the scattering light from the sample solution is refracted in the cell structure of solution (scatters)/glass plate/air and is measured on the air side through the plane glass cell. The scattering angle range of 2˚ < θ_s < 15˚ corresponds to the scattering wave vector of 0.46 μm⁻¹ < q < 3.47 μm⁻¹.

Measuring the forward light scattering, as displayed in the photographs, requires time of the order of minutes when using this experimental setup (Figure 1). For example, to measure the SLS in the scattering angles from 15˚ up to 2˚ at a scanning step of 0.1˚, required about 6 minutes. The addition of the electrolyte NaCl to the HEWL solutions induces aggregation because the electrolyte ions decrease the electrostatic potential screening (Debye-Hűckel screening [16] ), which results in weakening the repulsive forces between colloidal proteins. Since initial aggregation of lysozyme just after the addition of precipitant is quite drastic [13] [14] , the SLS for the HEWL solutions was measured about 30 min after mixing when they had reached a relatively stable state.

The crystallization conditions for lysozyme are summarized in Table 1. Two days after the HEWL solutions had been mixed with the precipitant NaCl, we judged whether lysozyme crystals had grown using a microscope (×40). Table 1 corresponds to a phase diagram that describes the relationship between concentrations of crystallizing protein and precipitant at a constant temperature and at a constant pH. The results in Table 1 indicate a dissolving area with no crystal growth of protein, a metastable area with crystal growth, and the boundary region. The boundary in lysozyme crystallization is observed, for example, at around 3.5% (w/v) NaCl in 20 mg/ml HEWL (denoted by a pair of plus and minus in Table 1). Figure 2 shows photographs of 30 mg/ml (2.1 mM) HEWL solutions in micro test tubes two days after the addition of 3.0% (w/v) or 3.5% (w/v) NaCl. No crystal growth is observed in the 3.0% (w/v) NaCl solution, while in the 3.5% (w/v) NaCl solution, many lysozyme crystals have grown (N > 100 in 180 μl), which are tetragonal or orthorhombic crystals < 0.2 mm³ in volume.

Many experiments on protein crystallization indicate that a metastable state between precipitant and dissoluble regions is advantageous for obtaining good quality protein crystals [1] [2] . It is very important in protein crystallization to discern the difference between protein aggregations in the dissolving and the metastable areas from pre-crystalline solutions. The crystallization boundary for the measured lysozyme solutions (Table 1) nearly corresponds to a solubility curve in the phase diagram, for example, as presented in [4] . It is worth noting that the crystallizing lysozyme solutions around the boundary are not situated on the liquid-liquid phase-separa- tion area. Information on the boundary of protein crystallization in the metastable region is useful for crystallization screening.

Figure 3 shows the log-log plots of I_s vs. q from the SLS profiles at small angles for (a) 20 mg/ml HEWL solu-

a. Symbols: +, Crystallization; −, No Crystallization; +−, Crystallization or not.

tions with 1.5% and 4.0% (w/v) NaCl and for (b) 30 mg/ml HEWL solutions with 1.5% and 3.5% (w/v) NaCl. The solution samples evaluated by F-SLS were in a state where microcrystal growth of lysozyme was unobservable using a microscope just after the F-SLS measurements. The addition of higher NaCl to the solutions at the same HEWL concentration exhibits a considerable increase in the F-SLS intensity. The enhanced F-SLS curve indicates formation of larger aggregates (clusters) of lysozyme, caused by the additional NaCl. The fluctuations in the SLS are not a noise signal, which are attributed to dynamics of lysozyme aggregates. The SLS log-log profiles of the higher NaCl addition can be fitted by a linear function with a negative slope giving a power law of. The power value α is 1.59 ± 0.06 for the 20 mg/ml solution with 4.0% (w/v) NaCl. The determination coefficient R² is 0.950, as obtained from least-square fitting. For the 30 mg/ml solution with 3.5% (w/v) NaCl, α = 1.66 ± 0.05 and R² is 0.955. The R² gives a measure of how well the data are fitted by a curve (0 ≤ R² ≤ 1).

The power-law SLS pattern indicates a fractal aggregation of lysozyme in the pre-crystalline solutions. As this is well fitted by a linear function, the power value α gives a fractal dimension D for the lysozyme fractal clusters, which gives information on the interior fractal structure [16] . In contrast with the higher NaCl solutions, the 1.5% (w/v) NaCl solutions exhibit no power law in the SLS curve. This indicates that the lysozyme aggregates form in the 1.5% (w/v) NaCl solutions are not fractal clusters. No crystals of lysozyme were grown in the 1.5% (w/v) NaCl solutions after several weeks, while in the higher NaCl solutions the growth of lysozyme crystals were confirmed after two days, as shown in Table 1.

Although we could not interpret the power law for the SLS patterns of the 1.5% (w/v) NaCl solutions in Figure 3, the accuracy of the linear fit of the SLS data can act as a defacto metric. To this end, we estimated the power value α for the measured SLS data of the pre-crystalline solutions and evaluate the fit with R² values. Typical results are shown in Figure 4. At NaCl concentrations lower than 3% (w/v) in the 20 and 30 mg/ml HEWL solutions, the SLS curves could not be fitted to a linear function (R² < 0.9, the broken line in Figure 4(b)). Thus we can confirm the formed aggregates are not fractal clusters in these conditions. Through this region, the α value increases and R² rises as the NaCl concentration increases. This behavior indicates that the added NaCl leads to lysozyme aggregation and the formation of fractal clusters.

In contrast, in the higher NaCl concentration region of >4% (w/v), the SLS patterns can be fitted to a power law (R² > 0.9, as shown in Figure 4(b)), which means the aggregates are fractal clusters. As is shown in Table 1, the crystallization boundary exists between 3% and 4% (w/v) NaCl in 20 and 30 mg/ml HEWL solutions. Comparing the data in Figure 4 with the crystallization conditions, reveals that on the higher NaCl concentration side

of the crystallization boundary, the fractal dimensions D (in this case, equal to α) of the lysozyme clusters are larger than those of the lower concentration area and exceed a value of 1.5 (the broken line in Figure 4(a)). The comparison proves that the pre-crystalline solutions where lysozyme crystals grow exhibit the formation of fractal clusters with fractal dimensions over 1.5.

The fractal dimension D is strongly related to the mass density of the fractal aggregate structure and the fractal aggregates can be characterized by the power-law dependence of an average local density at distance r from a given point [16] [17] :, where d is a spatial dimension. The structure factor obtained from static light scattering measurements of is directly connected to, and is given by the Fourier transform of [16] . Since the average local density expresses a spatial correlation between two arbitrary scattering sites in the clusters, the larger the correlation in the distances between scattering elements, such as particles forming fractal structures, the larger the D, which means that the structure of fractal clusters is denser.

The D of fractal clusters formed in the crystallizing lysozyme solutions is of the range of 1.5 ~ 1.8, as seen in Figure 4(a). The estimated D values are not significantly different from those reported by Georgalis et al. [13] [14] or Tanaka et al. [7] [15] . For typical fractal aggregates like colloidal silica, gold and polystyrene particles, two types of fractal models are as well known: diffusion-limited cluster aggregation (DLCA) with D ~ 1.8 and reaction-limited cluster aggregation (RLCA) with D ~ 2.1 [16] -[19] . The fractal dimensions of the lysozyme clusters formed in the pre-crystalline solutions progressing to crystallizations are closer to the value of DLCA than RLCA. The lysozyme clusters are well hydrated and contain a considerable amount of H₂O, unlike fractal clusters of colloidal metals, silica, and polystyrene particles. Water-soluble protein crystals like lysozyme are typically made up of over 60% water [1] . Therefore, the pre-crystalline protein clusters probably have a low mass density as compared with colloidal particles, meaning the D of the lysozyme clusters will be smaller. d/2 < D < d describes the range of D known to undergo colloidal aggregation, as determined from simulations and experiments [16] . In the formation of the aggregates d = 3, meaning the fractal dimension must be in the range, 1.5 < D < 3. This agrees with the boundary value of D = 1.5 found here for lysozyme fractal aggregates that lead to the formation of crystalline nuclei. These results indicate that high density fractal clusters of lysozyme with D > 1.5 are formed in solutions which will progress to crystallization.