Analysis of oscillatory rocking curve by dynamical diffraction in protein crystals

Significance

The observation of dynamical diffraction in protein crystals is an interesting topic because dynamical diffraction typically occurs in high-quality crystals such as those of silicon. However, to our knowledge, a report of protein crystals showing clear dynamical diffraction does not exist. Here, we present an observation of the oscillatory profile of rocking curves for protein crystals such as glucose isomerase. The oscillatory profiles are in good agreement with those predicted using the dynamical theory of diffraction. It is demonstrated that dynamical diffraction occurs in protein crystals as well. These results indicate the need for a dynamical diffraction model in protein-crystal structure analysis and model refinement that, to our knowledge, has never been used in conventional structural analysis.

Abstract

High-quality protein crystals meant for structural analysis by X-ray diffraction have been grown by various methods. The observation of dynamical diffraction in protein crystals is an interesting topic because dynamical diffraction generally occurs in perfect crystals such as Si crystals. However, to our knowledge, there is no report yet on protein crystals showing clear dynamical diffraction. We wonder whether the perfection of protein crystals might still be low compared with that of high-quality Si crystals. Here, we present observations of the oscillatory profile of rocking curves for protein crystals such as glucose isomerase crystals. The oscillatory profiles are in good agreement with those predicted by the dynamical theory of diffraction. We demonstrate that dynamical diffraction occurs even in protein crystals. This suggests the possibility of the use of dynamical diffraction for the determination of the structure and charge density of proteins.

The observation of dynamical X-ray diffraction in protein crystals is an interesting topic for the assessment of crystal perfection and the structural analysis of proteins. Generally, there are two principal theories—kinematical and dynamical—associated with X-ray diffraction in crystals (1–4). The kinematical theory treats the scattering from each volume element in the crystal sample as being independent of other elements. Therefore, kinematical diffraction commonly occurs in small crystals or low-quality crystals with defects such as dislocations. On the other hand, the dynamical theory takes into account multiple scattering within the crystal and is generally used whenever diffraction from a large perfect crystal is considered. Even in small crystals, dynamical diffraction is often encountered while measuring a strong reflection, which is usually a low-order reflection associated with a large structure factor. Thus, dynamical diffraction is an indicator of the perfection of crystals. Dynamical diffraction can appear in high-quality covalent crystals as the Pendellösung fringes in wedge-shaped crystals during X-ray topography (5, 6), the oscillatory profiles of rocking curves (7–10) in Si crystals, the maximum reflecting power as a function of the crystal thickness in Ge crystals (11), and the contrasts of stacking-fault fringes in diamond crystals (12).

High-quality protein crystals meant for structural analysis by X-ray diffraction have been grown by various methods including gel and microgravity growth (13). However, there is no report showing clear dynamical diffraction in protein crystals yet. We wonder whether the perfection of protein crystals is still low compared with that of high-quality Si crystals. Protein crystallographers assume based on kinematical theory that the diffraction intensity is proportional to the magnitude of the structure factor squared (14). Some X-ray topographic experiments for hen egg-white lysozyme (HEWL) (15), ferritin (16), and glucose isomerase (GI) crystals (17) showed fringe contrasts similar to Pendellösung fringes in part of the crystal. A recent report also showed beadlike or oscillatory contrasts along dislocation images in X-ray topographs (18). These topographic images are similar to typical images of dynamical diffraction (4, 19). From these observations, it is expected that dynamical diffraction might occur in protein crystals, although the related studies do not provide a quantitative explanation to support this premise. On the other hand, rocking curve measurements have been performed for several protein crystals (20–23). However, there is no report on the observation of the oscillatory profile associated with dynamical diffraction. In this study, we present observations of the oscillatory profiles of rocking curves in protein crystals such as GI crystals. The oscillatory profiles are in good agreement with those predicted by the dynamical theory of diffraction. It is demonstrated that dynamical diffraction, which is usually seen in high-quality Si crystals, occurs even in protein crystals.

Results and Discussion

Fig. 1 A and B show an optical micrograph of a typical GI crystal and the corresponding schematic figure, respectively. The digital X-ray topographs and rocking curves of such GI crystals of different sizes were measured by using a high-resolution charge-coupled device (CCD) camera. Fig. 1C shows a typical digital X-ray topograph for the same crystal as that in Fig. 1A. As seen in Fig. 1C, no dislocations are visible in the crystal. This means that dislocation-free GI crystals of high quality are obtained in this experiment. Additionally, it is confirmed that no clear change in the X-ray topographic image is observed before and after the rocking curve measurement (Fig. S1). Even in the high-resolution X-ray topographs taken with X-ray films, no crystal defects such as dislocations are observed after the rocking curve measurement (Fig. S2). These results suggest that the X-ray exposure for the rocking curve measurement in this experiment gives rise to no significant damage such as the generation of dislocations. Moreover, it should be noted that fringe contrasts similar to Pendellösung fringes exhibiting a high crystal quality, as reported previously (17, 18), are clearly observed at the tapered or wedgelike edges of the crystals in the X-ray topographs. The fringe contrasts seem to depend on the angular dispersion of the beam and the beam coherence (9). Thus, dislocation-free GI crystals of high quality are retained even after the rocking curve measurements in this experiment.

Fig. 1.

Typical GI crystal viewed from the crystallographic direction perpendicular to the $\left(0\overline{1}1\right)$ plane. (A) Optical micrograph, (B) corresponding schematic prepared with VESTA software (36), and (C) digital X-ray topograph with the CCD camera in BL20B at KEK-PF. No line contrasts corresponding to dislocations are observed in the X-ray topograph, whereas fringe contrasts similar to Pendellösung fringes exhibiting a high crystal quality seem to be observed at the tapered or wedgelike edges of the crystal.

Fig. 2 shows a typical rocking curve profile of a GI crystal taken with the 011 reflection. The thickness of the GI crystal is 199 µm. The horizontal axis is called the W scale (4), which is the parameter representing the deviation from the diffraction condition given by

$$W=\frac{2\mathrm{\Lambda }\sin {\theta }_B}{\lambda }\left({\theta }_B-\theta \right),$$ [1]

where

$$\mathrm{\Lambda }=\xi =\frac{\pi }{r_e}\frac{V_c}{\lambda }\frac{\cos {\theta }_B}{\left|F\right|}.$$ [2]

Here, $\mathrm{\Lambda }$ and $\xi$ are the periods of the Pendellösung fringes and the extinction distance in the Laue (transmission) case, respectively; ${\theta }_B$ is the Bragg angle, $\lambda$ is the wavelength of the incident beam, $V_c$ is the volume of the unit cell, $r_e$ is the classical electron radius (2.82 × 10⁻¹⁵ m), and $F$ is the structure factor. Also, $W$ for the 011 reflection of the GI crystal in Fig. 2 is obtained from Eq. 1 with ${\theta }_B$ = 0.480°, $\lambda$ = 1.2 Å, and $\mathrm{\Lambda }=$ 621 µm, which is calculated from Eq. 2 using $V_c$ = 9.63 × 10⁻²⁵ m³ and $\left|F\right|$ = 14,380 [Protein Data Bank (PDB) ID: 1mnz] (24).

Fig. 2.

Typical rocking curve for the 011 reflection of a GI crystal with a thickness of 199 µm, taken with an incident beam with a wavelength of 1.2 Å in BL20B at KEK-PF. In A and B, the intensities of the same rocking curves are shown on linear and logarithmic scales, respectively.

It should be noted that a fine structure corresponding to oscillation is clearly observed on the wings of the rocking curve. To further explain the oscillation, the intensity of the rocking curve shown on a linear scale in Fig. 2A is redrawn on a logarithmic scale, as shown in Fig. 2B. The oscillatory profile is more clearly observed with the logarithmic scale. Similar oscillatory rocking curves are also obtained from any circular areas with a diameter of 96.75 µm in the crystal, as shown in Fig. 3. However, the corresponding Bragg angle ${\theta }_B$ is slightly shifted from one end part of the crystal to the other end by only ∼0.001°, which is very small and comparable to the angular resolution limit of the apparatus. This means that the crystal includes a small distortion overall despite the fact that it is dislocation-free as mentioned above. As a result, even the small distortion perturbs the rocking curve profile of the whole crystal. Thus, clear oscillatory rocking curves are obtained from not the overall area but smaller areas with a diameter of 96.75 µm in the crystal as mentioned above. The series of X-ray topographic images associated with the oscillatory rocking curve are shown in Movie S1.

Fig. 3.

(A) Selected circular areas with a diameter of 96.75 µm (15 pixels) on a typical crystal image for rocking curves. (B–F) The rocking curve profiles are reconstructed from the reflected intensities in the selected areas. Similar oscillatory profiles of rocking curves are obtained from different sites.

It is well known from the dynamical theory of diffraction that rocking curves exhibit such oscillatory profiles for perfect crystals (1–4). Therefore, it is suggested that the oscillatory profiles observed in the present experiments can be attributed to the dynamical effect of perfect crystals. This is the real observation of the dynamical effect in the rocking curves of protein crystals. To our knowledge, such a dynamical effect has been observed in only high-quality crystals such as Si crystals so far. This implies that it is possible to grow high-quality crystals similar to Si crystals using protein crystals with huge and complex molecules, although the perfect region is smaller than that in Si crystals.

According to the dynamical theory of diffraction with no absorption (4), the intrinsic rocking curve profile in the symmetric Laue case is given by

$$\frac{I_g^n}{I_o}=\frac{{\sin }^2\left(\pi H\sqrt{W^2+1}/\mathrm{\Lambda }\right)}{W^2+1},$$ [3]

where $H$ is the crystal thickness. Both $W$ and $\mathrm{\Lambda }$ in Eq. 3 are expressed as a function of the wavelength of the incident beam $\lambda$, according to Eqs. 1 and 2, respectively. From Eq. 3, it is found that the period of oscillation in the rocking curve depends on the wavelength of the incident beam. Therefore, to demonstrate the dynamical effect for the oscillatory profiles as suggested above, the rocking curves for the same GI crystal were measured using incident beams of different wavelengths. Fig. 4A shows the rocking curves of 011 reflections for the same GI crystals, taken with different wavelengths of 1.0, 1.2, and 1.4 Å, where the crystal thickness is 199 µm. As seen in Fig. 4A, the period of oscillation decreases with the increasing wavelength of the incident beam. The wavelength dependence of the period of oscillation in the rocking curves is in good agreement with that predicted from Eq. 3. This supports the premise that the oscillatory profile measured in the present experiments can be attributed to the dynamical effect of diffraction.

Fig. 4.

Rocking curves for 011 reflections of the same GI crystals with a thickness of 199 µm, taken with incident beams with different wavelengths of 1.0, 1.2, and 1.4 Å, in BL20B at KEK-PF. (A) Measured rocking curves, (B) theoretical rocking curves, and (C) modified theoretical rocking curves with the average one.

Moreover, theoretical rocking curves based on the dynamical theory of diffraction with no absorption were calculated from Eq. 3 and compared with the measured values as mentioned above. The calculated curves for wavelengths of 1.0, 1.2, and 1.4 Å are shown in Fig. 4B. As seen in Fig. 4B, the periods of oscillations are in good agreement with the measured values, although the accuracy of the curve fitting is poor.

To improve the curve fitting, the average curve with no absorption (4) is added along with a ratio to Eq. 3, where the average curve is given by

$$\frac{{\overline{I}}_g^n}{I_o}=\frac{1}{2\left(W^2+1\right)}.$$ [4]

Therefore, the measured rocking curves were fitted with the modified theoretical curves given as follows:

$$I=r\frac{I_g^n}{I_o}+\left(1-r\right)\frac{{\overline{I}}_g^n}{I_o},$$ [5]

where $r$ is the ratio of $I_g^n/I_o$ to ${\overline{I}}_g^n/I_o$ with $0\le r\le 1$. The ratio, $1-r$, of the average curve corresponds to the degree of smearing or background in the oscillatory curve, which originates from the resolution limit due to the angular divergence of the beam (8, 9). Note that the oscillatory periods of the calculated curves are independent of the corrections using the average curves. The calculated curves are shown in Fig. 4C, where the values of $r$ for $\lambda$ = 1.0 Å, $\lambda$ = 1.2 Å, and $\lambda$ = 1.4 Å are 0.35, 0.33, and 0.38, respectively. As seen in Fig. 4C, the modified theoretical curves are in good agreement with the measured curves. Thus, it is concluded that dynamical diffraction occurs even in protein crystals such as GI crystals.

Nevertheless, it is known that kinematical diffraction occurs when a crystal is sufficiently thin, irrespective of whether the crystal is perfect or not. Therefore, one might expect that kinematical diffraction predominantly occurs for the thin crystal with a thickness of 199 µm, which is only one third of Pendellösung thickness. However, the rocking curve profile for the 199-µm-thick crystal predicted from kinematical theory is quite different from the measured rocking curve, as shown in Fig. 5. Namely, the full width at half maximum (FWHM) obtained by using kinematical theory is two orders smaller than that of the measured rocking curve, as presented in Table S1. Thus, the measured rocking curve obtained in this work is not explained by kinematical diffraction.

Fig. 5.

Comparison of the measured rocking curve with the theoretical rocking curve by dynamical and kinematical theory for the 011 reflection of the same GI crystals with a thickness of 199 µm, taken with an incident beam with a wavelength of 1.2 Å in BL20B at KEK-PF.

Furthermore, rocking curves were also measured for GI crystals of different thicknesses. According to Eq. 3, the period of oscillation in the rocking curves decreases as the thickness of the crystals increases. Fig. 6A shows the rocking curve profiles of 011 reflections for GI crystals with different thicknesses of 824, 362, 260, and 199 µm. As seen in the digital X-ray topographs in Fig. S3, these crystals are also as dislocation free and high quality as the GI crystal shown in Fig. 1. Additionally, from the X-ray topographic images after the rocking curve measurements, it is also confirmed that these crystals suffer no significant damage such as the generation of dislocations due to X-ray exposure for the rocking curve measurements, as for the GI crystal shown in Figs. S1 and S2.

Fig. 6.

Rocking curves for the 011 reflections of GI crystals with different thicknesses of 824, 362, 260, and 199 µm, taken with an incident beam with a wavelength of 1.2 Å in BL20B at KEK-PF. (A) Measured rocking curves, (B) modified theoretical rocking curves with no absorption, and (C) modified theoretical rocking curves with absorption.

As shown in Fig. 6A, the period of oscillation on the wing of the rocking curve profile appears to decrease with increasing crystal thickness. Actually, the oscillation would be invisible for crystals with a thickness greater than 824 µm. The invisibility of the oscillation is probably attributed to the smaller period of oscillation below the resolution limit due to the angular divergence of the beam. This result is also consistent with that predicted from Eq. 3 according to the dynamical theory of diffraction.

Fitting by Eq. 5 was also performed for the rocking curves of GI crystals with different thicknesses shown in Fig. 6A. The fitting curves are shown in Fig. 6B, where the values of $r$ for thicknesses of 824, 362, 260, and 199 µm are 0, 0.02, 0.14, and 0.38, respectively. By comparing Fig. 6 A and B, it appears that the accuracy of the fitting becomes increasingly poor with increasing crystal thickness. The deviation between the theoretical and measured curves is large for the largest GI crystal with a thickness of 824 µm. In particular, the calculated line width is much larger than the measured width. This can be ascribed to the effect of absorption by the crystals. Therefore, the theoretical curves were further calculated according to the dynamical theory with absorption (4) in the symmetric Laue case using the expression

$$\frac{I_g^a}{I_o}=\frac{\exp \left(-\mu H/\cos {\theta }_B\right)}{W^2+1}\left\{{\sin }^2\left(\frac{\pi H\sqrt{W^2+1}}{\mathrm{\Lambda }}\right)+{\text{sinh}}^2\left(\frac{\chi \pi H\sqrt{W^2+1}}{\mathrm{\Lambda }}\right)\right\}.$$ [6]

Note that $\mu$ is the linear absorption coefficient, and $\chi =\chi ″/\chi \prime$, where $\chi \prime$ and $\chi ″$ are the real and imaginary parts, respectively, of the electric susceptibility. As was done for the fitting of curves with no absorption in Eq. 5 mentioned above, the average curve with absorption (4) is added along with a ratio to Eq. 6, where the average curve with absorption is given by

$$\frac{{\overline{I}}_g^a}{I_o}=\frac{1}{4\left(W^2+1\right)}\left[\exp \left\{\frac{-\mu H}{\cos {\theta }_B}\left(1-\frac{\epsilon }{\sqrt{W^2+1}}\right)\right\}+\exp \left\{\frac{-\mu H}{\cos {\theta }_B}\left(1+\frac{\epsilon }{\sqrt{W^2+1}}\right)\right\}\right],$$ [7]

where $\epsilon$ is the dielectric constant.

The calculated curves with absorption are shown in Fig. 6C. The fitting parameters were $\mu$ = 0.18 mm⁻¹, $\chi$ = 0.001, and $\epsilon$ = 23. The value of the linear absorption coefficient is in good agreement with 0.23 mm⁻¹ with $\lambda$ = 1.2 Å, which is estimated from the mass absorption coefficients and atomic densities of GI crystals containing no intracrystalline water (25). The slight discrepancy in the values might be attributed to the presence of intracrystalline water in our sample. Additionally, the value of the dielectric constant is also consistent with the typical values of 3–40 for protein crystals, as reported previously (26–28). On the other hand, the value of $\chi$ is of the same order as that estimated from $\chi \prime \left(=-F{r}_e{\lambda }^2/\pi V_c\right)$ and the imaginary part $\chi ″$ and is lower by one order of magnitude than that of hard Si crystals (3). Therefore, these fitting parameters are reasonable values for protein crystals.

As seen in Fig. 6C, the accuracy of the fitting is largely improved, especially for the largest GI crystals with a thickness of 824 µm. This implies that the rocking curves for the large GI crystals can be also explained by the dynamical theory of diffraction with absorption. Thus, dynamical diffraction occurs for low-order reflections with large structure factors even in protein crystals.

Recently, it was reported that ultrahigh resolution (high-order reflection) crystallographic analysis with high-quality protein crystals and a high-energy beam could provide an accurate structure and charge density of proteins (29). In particular, the accurate analysis of the electron distribution in proteins becomes increasingly important since the characteristics in proteins are predominantly influenced by valence electrons. Such analysis performed with a high-energy beam can be based on the kinematical theory of diffraction (30). However, the analysis of the diffraction intensities of low-order reflections is often poor even when the high-order reflection intensities are well resolved (31). The accurate analysis of the diffraction intensities of lower-order reflections is a prerequisite, especially for the evaluation of the valence electron distribution. The analysis of the dynamical diffraction for low-order reflections shown in this work might be useful for obtaining a more accurate structure and charge density of proteins in the future.

Additionally, in protein-crystal structure analysis and model refinement, there is the basic mystery of why the agreement between the observed and calculated values of the structure factor, $F_{\text{obs}}$ and $F_{\text{calc}}$ (or between $I_{\text{obs}}$ and $I_{\text{calc}}$), is never as good as in the intensity measurements. There are candidate reasons for this of which the need for a dynamical diffraction model or a combined kinematical and dynamical diffraction model is one. The other reasons are the need for the wider adoption of ensemble refinement or more simply the need for the community and the PDB to adopt refereeing of structure articles with the underlying diffraction data. Our results propose the need for a dynamical diffraction model in protein-crystal structure analysis and model refinement, which, to our knowledge, has never been used in conventional structural analysis so far.

Conclusion

We have shown the oscillatory profiles of rocking curves corresponding to low-order reflections with a large structure factor for GI crystals. It was shown that the change in the oscillatory profile with the wavelength of the incident beam and the thickness of the crystal is well explained by the dynamical theory of diffraction. From these results, it was concluded that dynamical diffraction similar to that seen in high-quality Si crystals occurs even in protein crystals. This suggests the possibility of the use of dynamical diffraction for the determination of the structure and charge density of proteins.

Materials and Methods

Crystal Growth.

The solution of GI from Streptomyces rubiginosus was purchased from Hampton Research Corp. and used without further purification. The crystallization conditions were the same as those mentioned in previous reports (17, 18). The GI crystals were grown using macroseeds. Their thicknesses ranged from ∼200 to 800 µm, where they were measured using an optical microscope. The crystal belonged to the orthorhombic space group of I222 with lattice constants of $a$ = 93.88 Å, $b$ = 99.64 Å, and $c$ = 102.90 Å and contained two tetrameric molecules per unit cell (25). The crystals were bounded by the $\left\{110\right\}$, $\left\{101\right\}$, and $\left\{011\right\}$ crystallographic faces, as shown in Fig. 1B. For X-ray diffraction experiments such as X-ray topography and rocking curve measurements, the grown GI crystals were sealed in an acrylic cell, as has been reported previously (17, 18).

Rocking Curve Measurement.

The rocking curve measurements with X-ray topography were performed at room temperature in BL20B at the Photon Factory (PF) of the High Energy Accelerator Research Organization (KEK). The rocking curves were obtained in the Laue geometry configuration. A two-crystal monochromator consisting of a Si$\left(111\right)$ crystal was placed 11 m from the source and was used to select the X-ray wavelengths of interest. The monochromatic beams of $\text{λ}$ = 1.0, 1.2, and 1.4 Å without focusing were selected as the incident beams in this work. Note that the beam intensity is considerably low compared with conventional beam intensities at beam lines used for protein-structure analysis at synchrotron radiation facilities.

The incident beam with a size of 3 × 5 mm² covering an entire crystal sample was introduced almost perpendicular to the $\left(0\overline{1}1\right)\text{or}\left(101\right)$ plane of the crystal sample, which was mounted on a precision goniometer and was rotated with a high-resolution angular step [minimum angular step width: 0.19 arcsec (5.3 × 10⁻⁵°)] around the exact Bragg angle of the reflected wave. A schematic of the experimental setup with crystal mounting is shown in Fig. S4. The crystal thickness was almost perpendicular to the beam path. The scattering plane was perpendicular to the polarization direction of the incident beam. Under such conditions, the reflected images of the entire crystal corresponding to the angular steps were collected using a high-spatial-resolution, 2D, digital CCD camera (Photonic Science X-RAY FDI 1.00:1, effective pixel size: 6.45 × 6.45 µm²) with exposure times of ∼4 min. The reflected images from the CCD camera correspond to digital X-ray topographs. The conventional high-resolution X-ray topographs were obtained by using X-ray films with exposure times of ∼2 min in place of the CCD camera.

The dose of a 4-min exposure for the acquisition of a series of reflected images as mentioned above is ∼0.05 kGy, which is considerably lower than the conventional dose of 500 kGy used for protein-structure analysis at room temperature at synchrotron radiation facilities (20, 32, 33). However, a longer exposure of about 120 min even with a low beam intensity caused radiation damage that led to a change in the rocking curves. This means that the rocking curve profile is sensitive to the crystal quality and radiation damage. To avoid radiation damage, short exposure times of less than 20 min for each crystal sample, corresponding to a dose of less than 0.24 kGy, were used in the X-ray diffraction experiments.

X-ray rocking curve profiles were reconstructed from the reflected intensities in a selected circular area with a diameter of 96.75 µm (15 pixels) of the crystal image that corresponds to the effective beam spot size. Note that the size was determined by the effective size, which is sufficient to delineate the rocking curve using the reflected intensities. Similar rocking curve measurements were also carried out in BL38B1 at SPring-8, in which the X-ray topographic system for protein crystals is under construction.

Angular Resolution of the Optics.

To measure the rocking curves of the crystal samples, the angular resolution of the optics must be less than or equal to the intrinsic FWHM of the rocking curve predicted by dynamical theory. The angular resolution of the optics, $\phi$, which is the instrumental resolution function, can be estimated by using the following equation obtained from the DuMond diagram (34) and the geometry of the experimental system:

$$\phi =\left(1-\frac{\tan {\theta }_2}{\tan {\theta }_1}\right)\text{min}\left[\mathrm{\Delta }\theta ,{\sigma }_{y^{\text{'}}}\right]+\frac{\tan {\theta }_2}{\tan {\theta }_1}\omega ,$$ [8]

where

$$\mathrm{\Delta }\theta =2{\tan }^{-1}\left(\frac{2{\sigma }_y\sqrt{2\ln 2}+s}{2L}\right),$$ [9]

$$\omega =\frac{2}{\pi }\frac{r_e}{V_c}\frac{\left|F\right|{\lambda }^2}{\sin 2{\theta }_B}.$$ [10]

Here, ${\theta }_1$ and ${\theta }_2$ are the Bragg angles for the monochromator crystal and crystal sample, respectively; $\mathrm{\Delta }\theta$ is the beam divergence estimated from the geometry of the experimental system, ${\sigma }_{y^{\text{'}}}$ is the vertical beam divergence, and $\omega$ is the intrinsic FWHM of the rocking curve. In addition, ${\sigma }_y$ is the vertical beam size, $s$ is the slit size, and $L$ is the distance between the light source and the X-ray detector, which is the CCD camera.

At BL20B in the PF, $\mathrm{\Delta }\theta$ was calculated to be 3.39 arcsec (9.41 × 10⁻⁴°) using Eq. 9 with $L$ = 14.35 m; ${\sigma }_y$ = 0.059 mm, as shown in the facility status of the PF (35); and $s$ = 96.75 µm, which is the effective spot size mentioned above. For the 111 reflection of the Si crystal used as the monochromator in BL20B in the PF, $\omega$ was calculated to be 3.99 arcsec (1.11 × 10⁻³°) from Eq. 10 with $V_c$ = 1.60 × 10⁻²⁸ m³, $\left|F\right|$ = 45.0465 (Crystallography Open Database: 4507226), and ${\theta }_B$= 11.03° with $\lambda$ = 1.2 Å. Using these values of $\mathrm{\Delta }\theta$ and $\omega$, the angular resolution of the optics $\phi$ was evaluated to be 2.54 arcsec (7.06 × 10⁻⁴°) from Eq. 8 with ${\theta }_1$ = 11.03°, ${\theta }_2$ = 0.480° with $\lambda$ = 1.2 Å, and ${\sigma }_{y^{\text{'}}}$ = 2.48 arcsec (6.88 × 10⁻⁴°), as shown in the facility status of the PF (35).

The intrinsic FWHM of the rocking curve of the 011 reflection for GI crystals was also estimated from Eq. 10 with $\left|F\right|$ = 14,380 (PDB ID: 1mnz) (24), $V_c$ = 9.63 × 10⁻²⁵ m³, and ${\theta }_B$ = 0.480° with $\lambda$ = 1.2 Å. As a result, the FWHM was calculated to be 4.75 arcsec (1.32 × 10⁻³°). This value was larger than $\phi$ corresponding to the angular resolution of the optics as estimated above. This implies that the intrinsic FWHM for GI crystals can be well resolved in BL20B.

On the other hand, the intrinsic FWHM of the rocking curve for tetragonal HEWL crystals, which are often used as model protein crystals, was calculated to be 0.42 arcsec (1.17 × 10⁻⁴°) from Eq. 8 for a typical 440 reflection with an incident beam with a wavelength of 1.2 Å. The value of the intrinsic rocking width was smaller than $\phi =$ 2.81 arcsec (7.80 × 10⁻⁴°) of the angular resolution of the optics for tetragonal HEWL crystals in BL20B as mentioned above. In this case, resolving the intrinsic rocking curve profiles for tetragonal HEWL crystals might be difficult. Therefore, GI crystals would be more suitable for the study of the dynamical diffraction of protein crystals in BL20B.