TY - JOUR
AU - Fujiyoshi,, Yoshinori
AB - Abstract Atomic scattering factors for electrons are strongly affected by the charge status of the scattering atoms. The difference in scattering factors for charged and neutral atoms is most pronounced in the resolution range below 5 Å. As a result of the negative scattering factors of negatively charged atoms in the low-resolution range, charged glutamate or aspartate residues produce weaker densities in electron crystallographic maps than their neutral forms. Such charge effects were indeed observed in an experimental map of bacteriorhodopsin. Here we present mathematical simulations of this charge effect on electron crystallographic density maps that corroborate the experimental results. For the simulations, we first evaluated the errors introduced by approximating atomic scattering factors for neutral and charged atoms by Gaussians. The simulations then showed that the effect of a polarized pair of oxygen and hydrogen atoms on the density (polarization effect) was much smaller than that expected from the individual charged atoms (charge effect), due to charge compensation. Still, density maps obtained by electron crystallography are expected to show slightly elongated features toward the positively charged atoms. electron crystallography, scattering factor, density map, charge effect, polarization effect, bacteriorhodopsin Introduction Maps obtained by X-ray diffraction are known as electron density maps, because they reflect the distribution of the electrons in the analyzed molecules. Different from X-ray crystallography, maps obtained by electron crystallography reflect the electrostatic potential of the atoms in the molecules, also known as the shielded Coulomb potential. A potentially useful feature of electron scattering is that the atomic scattering factors are strongly affected by the charge status of the scattering atoms in the specimen. Especially in the resolution range below 5 Å, the atomic scattering factor for a charged atom is very different from that of its neutral form. Thus, the charge status might be visualized by electron crystallography, while it is difficult by X-ray crystallography because of the limited signal-to-noise ratio by the difference of only one electron. The visualization could have tremendous impact on understanding the mechanism of pumps and transporters, since many of them are coupled with ion movement inside the protein. In addition, many enzymatic activities are related to electron and proton transfer and, therefore, the charge visualization is important also to understand the enzymatic functions. In our previous structural analysis of bacteriorhodopsin, the possibility of using electron crystallography to differentiate between charged and neutral residues was discussed. This was based on the weaker appearance of density for charged aspartate and glutamate residues when low-resolution data were included in the calculation of the experimental (FO) map using observed amplitudes [1]. The model was refined using data only in the resolution range from 8 to 3 Å to keep the charge effects to the minimum. After the refinement, a difference (FO – FC) map was calculated, in which the low-resolution data were included. For the calculation of the FC map, scattering factors for neutral atoms were used for all atoms in the model. If there were charge effects in the FO map, these should then be revealed in the difference map. Furthermore, the best R- and R (free)-factors were obtained, when scattering factors of 40–45% charged atoms for carbon and oxygen atoms of the peptide bond carbonyls in bacteriorhodopsin were used in the calculations [2]. In the FO – FC maps using scattering factors for charged atoms, some peaks were present next to some of the acidic residues, which indicate the differences between the calculation using the scattering factors for charged atoms and the experimental data. One objective of this paper is to evaluate the possible source of the differences, for example, errors in the approximation of scattering factors for charged atoms quantitatively. Here, we simulated electron crystallographic maps by approximating scattering factors for neutral and charged atoms by Gaussians and a constant. We discuss the observed charge effects in the resulting density maps, which support our previous qualitative observations in the experimental map of bacteriorhodopsin. Methods Fitting of atomic scattering factors for neutral atoms The table in Section 2.4 “Scattering Factors for the Diffraction of Electrons by Crystalline Solids” (Doyle and Cowley) of the International Tables for X-ray Crystallography Vol. IV [3] provides numerical values for the atomic scattering factors of electrons. In many refinement programs and crystallographic software packages, such as X-PLOR [4], CNS [5] and REFMAC [6] of the CCP4 programs [7], atomic scattering factors are described by several Gaussians and a constant. The Gaussian is well suited to fit atomic scattering factors because its shape resembles that of scattering factors for neutral atoms. The Fourier transform of a Gaussian is well characterized and is also a Gaussian. We thus used the program SCATTER developed by Ceska [8] to approximate atomic scattering factors as four Gaussians and a constant: (1) Scattering factors at a given accelerating voltage were calculated by multiplying the values provided in the International Tables for X-ray Crystallography by ⁠, where β = v/c (v, velocity of electrons; c, velocity of light), to account for relativistic effects. For neutral atoms, the Gaussian coefficients and the constant used to approximate scattering factors for 120 keV electrons have previously been published [8]. We first converted the coefficients for 120 keV electrons to those for 300 keV electrons, because our high-resolution structural study of bacteriorhodopsin [1] was performed with data collected with a 300 keV microscope [9]. Gaussian coefficients and a constant at different accelerating voltages can easily be calculated using the following equations: (2) (3) (4) The values for β at different accelerating voltages are provided in Table 2.4.6C in the International Tables for X-ray Crystallography Vol. IV (e.g., β = 0.58667 at 120 keV, β = 0.77652 at 300 keV). Coefficients adjusted for 300 keV electrons were used as starting values for the program SCATTER to fit the data in the International Tables for X-ray Crystallography in the resolution range up to 1 Å. Gaussian coefficients and constants for scattering factors for 300 keV electrons for neutral atoms refined by SCATTER are listed in Table 1. The resulting fit shows errors of less than 1% over the entire fitted resolution range (∞–1 Å) for all atoms (the maximum error was 0.98% for H at a resolution of 1.004 Å). Scattering factors for 300 keV electrons over the fitted resolution range using the determined coefficients are shown in Fig. 1a. Atomic potential ρ(s) (Fig. 1b) was calculated by the Fourier transformation of the atomic scattering factor f(k) (where k = 2sinθ/λ) according to the following equations, which also take the B-factor into account: (5) (6) Fig. 1.  Open in new tabDownload slide Fitting of scattering factors for neutral atoms and their Fourier transforms (atomic potential). The number in brackets next to the name of the atom is the atomic number (Z). (a) Fitting of scattering factors. The data plotted in this figure are from the International Tables for X-ray Crystallography Vol. IV. The fitted lines were calculated by summing the Gaussians and constants listed in Table 1. No B-factor was applied. (b) Atomic potential. The lines were calculated according to Eq. (6) using the Gaussian coefficients and constants listed in Table 1. In Fig. 1b, the same symbols were used as in Fig. 1a. The B-factor was set to 20 Å2. Fig. 1.  Open in new tabDownload slide Fitting of scattering factors for neutral atoms and their Fourier transforms (atomic potential). The number in brackets next to the name of the atom is the atomic number (Z). (a) Fitting of scattering factors. The data plotted in this figure are from the International Tables for X-ray Crystallography Vol. IV. The fitted lines were calculated by summing the Gaussians and constants listed in Table 1. No B-factor was applied. (b) Atomic potential. The lines were calculated according to Eq. (6) using the Gaussian coefficients and constants listed in Table 1. In Fig. 1b, the same symbols were used as in Fig. 1a. The B-factor was set to 20 Å2. Table 1.  Gaussian coefficients and constants for scattering factors of neutral atoms and their errors (300 keV, resolution range: ∞-1.0 Å) . . . . . . . . . . (max) . (max) . . a1 . b1 . a2 . b2 . a3 . b3 . a4 . b4 . c . Error average (max) . Error (%) average (max) . H 0.1297 46.8016 0.2112 15.9941 0.2311 15.9003 0.3039 2.3250 −0.0393 0.0007 0.21 (0.0016 at 1.79 Å) (0.98 at 1.00Å) C 0.3952 57.2981 1.6383 21.5154 1.5133  6.5505 0.2279 1.7733 −0.2074 0.0004 0.02 (0.0009 at 1.39 Å) (0.08 at 1.39 Å) N 0.3298 42.6954 1.3152 16.7093 1.5101  5.1343 0.2977 0.3550 −0.0571 0.0004 0.02 (0.0009 at 1.19 Å) (0.09 at 1.19 Å) O 0.0809 45.4657 0.6058 21.9993 1.5891  7.8305 1.2088 0.8786 −0.3366 0.0004 0.02 (0.0009 at 5.00 Å) (0.06 at 1.43 Å) P 0.7370 76.5599 3.0716 32.2621 3.0952 12.2991 1.4714 2.5098 −0.3349 0.0004 0.01 (0.0008 at 3.33 Å) (0.04 at 1.14 Å) S 0.9514 55.0850 3.5797 21.3364 2.7300  6.4516 0.4257 1.4133 −0.5039 0.0004 0.01 (0.0012 at 3.33 Å) (0.07 at 1.11 Å) . . . . . . . . . . (max) . (max) . . a1 . b1 . a2 . b2 . a3 . b3 . a4 . b4 . c . Error average (max) . Error (%) average (max) . H 0.1297 46.8016 0.2112 15.9941 0.2311 15.9003 0.3039 2.3250 −0.0393 0.0007 0.21 (0.0016 at 1.79 Å) (0.98 at 1.00Å) C 0.3952 57.2981 1.6383 21.5154 1.5133  6.5505 0.2279 1.7733 −0.2074 0.0004 0.02 (0.0009 at 1.39 Å) (0.08 at 1.39 Å) N 0.3298 42.6954 1.3152 16.7093 1.5101  5.1343 0.2977 0.3550 −0.0571 0.0004 0.02 (0.0009 at 1.19 Å) (0.09 at 1.19 Å) O 0.0809 45.4657 0.6058 21.9993 1.5891  7.8305 1.2088 0.8786 −0.3366 0.0004 0.02 (0.0009 at 5.00 Å) (0.06 at 1.43 Å) P 0.7370 76.5599 3.0716 32.2621 3.0952 12.2991 1.4714 2.5098 −0.3349 0.0004 0.01 (0.0008 at 3.33 Å) (0.04 at 1.14 Å) S 0.9514 55.0850 3.5797 21.3364 2.7300  6.4516 0.4257 1.4133 −0.5039 0.0004 0.01 (0.0012 at 3.33 Å) (0.07 at 1.11 Å) Open in new tab Table 1.  Gaussian coefficients and constants for scattering factors of neutral atoms and their errors (300 keV, resolution range: ∞-1.0 Å) . . . . . . . . . . (max) . (max) . . a1 . b1 . a2 . b2 . a3 . b3 . a4 . b4 . c . Error average (max) . Error (%) average (max) . H 0.1297 46.8016 0.2112 15.9941 0.2311 15.9003 0.3039 2.3250 −0.0393 0.0007 0.21 (0.0016 at 1.79 Å) (0.98 at 1.00Å) C 0.3952 57.2981 1.6383 21.5154 1.5133  6.5505 0.2279 1.7733 −0.2074 0.0004 0.02 (0.0009 at 1.39 Å) (0.08 at 1.39 Å) N 0.3298 42.6954 1.3152 16.7093 1.5101  5.1343 0.2977 0.3550 −0.0571 0.0004 0.02 (0.0009 at 1.19 Å) (0.09 at 1.19 Å) O 0.0809 45.4657 0.6058 21.9993 1.5891  7.8305 1.2088 0.8786 −0.3366 0.0004 0.02 (0.0009 at 5.00 Å) (0.06 at 1.43 Å) P 0.7370 76.5599 3.0716 32.2621 3.0952 12.2991 1.4714 2.5098 −0.3349 0.0004 0.01 (0.0008 at 3.33 Å) (0.04 at 1.14 Å) S 0.9514 55.0850 3.5797 21.3364 2.7300  6.4516 0.4257 1.4133 −0.5039 0.0004 0.01 (0.0012 at 3.33 Å) (0.07 at 1.11 Å) . . . . . . . . . . (max) . (max) . . a1 . b1 . a2 . b2 . a3 . b3 . a4 . b4 . c . Error average (max) . Error (%) average (max) . H 0.1297 46.8016 0.2112 15.9941 0.2311 15.9003 0.3039 2.3250 −0.0393 0.0007 0.21 (0.0016 at 1.79 Å) (0.98 at 1.00Å) C 0.3952 57.2981 1.6383 21.5154 1.5133  6.5505 0.2279 1.7733 −0.2074 0.0004 0.02 (0.0009 at 1.39 Å) (0.08 at 1.39 Å) N 0.3298 42.6954 1.3152 16.7093 1.5101  5.1343 0.2977 0.3550 −0.0571 0.0004 0.02 (0.0009 at 1.19 Å) (0.09 at 1.19 Å) O 0.0809 45.4657 0.6058 21.9993 1.5891  7.8305 1.2088 0.8786 −0.3366 0.0004 0.02 (0.0009 at 5.00 Å) (0.06 at 1.43 Å) P 0.7370 76.5599 3.0716 32.2621 3.0952 12.2991 1.4714 2.5098 −0.3349 0.0004 0.01 (0.0008 at 3.33 Å) (0.04 at 1.14 Å) S 0.9514 55.0850 3.5797 21.3364 2.7300  6.4516 0.4257 1.4133 −0.5039 0.0004 0.01 (0.0012 at 3.33 Å) (0.07 at 1.11 Å) Open in new tab Fitting of atomic scattering factors for charged atoms We also calculated the Gaussian coefficients and constants for the charged atoms H+ and O− (Table 2). For H+, we used the scattering factors calculated by Kakitani et al. (to be published elsewhere). The values for O− are provided in the International Tables for X-ray Crystallography, but information for the low-resolution range (lower than 12.5 Å) is missing. The values for the few spots below this resolution were estimated from other negatively charged atoms I−, Cl−, and Br−, for which data are available to very low resolution in the International Tables for X-ray Crystallography. The way we estimated the values for the affected spots (shown by filled squares in Fig. 2a1) is described in the Appendix. Atomic scattering factors for charged atoms deviate dramatically from those for neutral atoms at a resolution below 5 Å, and it proved difficult to fit the entire resolution range with only four Gaussians. We therefore focused on fitting atomic scattering factors in the resolution range between 50 and 2.5 Å (Fig. 2a). Despite limiting the resolution range, the fitting errors were about 100 times larger for charged atoms than those for neutral atoms [The average error ranges from 0.0004 to 0.0007 for neutral atoms (Table 1) and from 0.0161 to 0.1238 for charged atoms (Table 2)]. However, the errors are still only in the range of a few percent for most resolutions (average error for H+ was 2.90% and that for O− was 1.12%). They should provide sufficient accuracy for the simulations presented here, because the FC differences between charged and neutral status are much larger than these differences as in Table 3. The errors become particularly large at resolutions where the scattering factors are small. For H+ this occurs in the high-resolution range (8.42% at 3.17 Å) and for O− around a resolution of 5.6 Å (5.97% at 5.56 Å), where the value of the scattering factor for O− goes from negative to positive. As shown in Eq. (5), the larger (bi + B) gives the sharper Gaussian. In tabulations of Gaussian coefficients (Tables 1, 2 and 4), the b values are listed in order of decreasing values (b1 > b2 > b3 > b4), which means that the first coefficients ai and bi (usually only a1 and b1) have the strongest effect on low-resolution features of the potential for charged atoms. For example as shown in Fig. 2a1 “Gaussian 1 for H+” (a1 = 812.7, b1 = 14278) mostly contributes to describe the atomic scattering factor of H+ compared with “Gaussian 2” (a2 = 183.1, b2 = 3462) or “Gaussian 3” (a3 = 47.9, b3 = 783) around 50 Å. The slopes in the curves describing the changes in atomic scattering factors are still getting sharper for charged atoms at a resolution below 50 Å (Fig. 2a1). If these data are included in the fitting, Gaussians with larger b values also have to be included in the calculations. Atomic potential for charged atoms (Fig. 2b) were also calculated by Fourier transformation of the atomic scattering factors according to Eqs. (5) and (6). Fig. 2.  Open in new tabDownload slide Fitting of scattering factors for H+ and O− and their Fourier transforms (atomic potential). Neutral atoms, H and O are also shown for comparison. (a1) Fitting of scattering factors for H+ and O− in the resolution range from 50 and 5.0 Å. Gaussians 1, 2, and 3 used for fitting of H+ are also shown. (a2) Fitting of scattering factors for H+ and O− in the resolution range from 50 and 2.5 Å. Gaussians 3, 4 and constant used for fitting of H+ are also shown. The data for H+ plotted in Fig. 2a1 and a2 were provided by T. Kakitani. The data for O− represented by filled circles in Fig. 2a1 and a2 were from the International Tables for X-ray Crystallography Vol. IV. *The data for O− represented by filled squares in Fig. 2a1 were estimated as described in the Appendix. (b) Atomic potential. In Fig. 2b, the same symbols were used as in Fig. 2a1 and a2. The B-factor was set to 20 Å2. Fig. 2.  Open in new tabDownload slide Fitting of scattering factors for H+ and O− and their Fourier transforms (atomic potential). Neutral atoms, H and O are also shown for comparison. (a1) Fitting of scattering factors for H+ and O− in the resolution range from 50 and 5.0 Å. Gaussians 1, 2, and 3 used for fitting of H+ are also shown. (a2) Fitting of scattering factors for H+ and O− in the resolution range from 50 and 2.5 Å. Gaussians 3, 4 and constant used for fitting of H+ are also shown. The data for H+ plotted in Fig. 2a1 and a2 were provided by T. Kakitani. The data for O− represented by filled circles in Fig. 2a1 and a2 were from the International Tables for X-ray Crystallography Vol. IV. *The data for O− represented by filled squares in Fig. 2a1 were estimated as described in the Appendix. (b) Atomic potential. In Fig. 2b, the same symbols were used as in Fig. 2a1 and a2. The B-factor was set to 20 Å2. Table 2.  Gaussian coefficients and constants for scattering factors of charged atoms and their errors (300 keV, resolution range: 50–2.5 Å) . a1 . b1 . a2 . b2 . . Error average . Error (%) average . . a3 . b3 . a4 . b4 . c . (max) . (max) . H+ −812.7002 14278.1309 −183.0536 3461.7549 0.1238 2.90 − 47.9442   783.2073 − 10.2941  123.3596 0.9183 (0.4713 at 49.02 Å) (8.42 at 3.17 Å) O− −631.0347  9194.4463 −103.0768 2409.6470 0.0161 1.12  −34.9994   785.6382  −12.0366  209.2285 2.1194 (0.0548 at 2.94 Å) (5.97 at 5.56 Å) H+1/2 −406.0906 14151.8887  −90.7672 3377.4961 0.0573 2.47  −23.2847   744.1006   −4.9473  109.4291 0.6636 (0.1701 at 19.69 Å) (6.71 at 2.98 Å) H+1/3 −270.4297 14078.5449  −60.1676 3328.5566 0.0401 2.15  −15.2479   721.5629   −3.2715  100.3644 0.5849 (0.1213 at 19.69 Å) (5.52 at 2.98 Å) O−1/2  −37.7416 12620.5059 −284.0005 7977.5732 0.0420 12.21  −45.5501  1599.1283  −10.9237  323.9079 2.2234 (0.0933 at 3.33 Å) (209.18 at 7.14 Å)a . a1 . b1 . a2 . b2 . . Error average . Error (%) average . . a3 . b3 . a4 . b4 . c . (max) . (max) . H+ −812.7002 14278.1309 −183.0536 3461.7549 0.1238 2.90 − 47.9442   783.2073 − 10.2941  123.3596 0.9183 (0.4713 at 49.02 Å) (8.42 at 3.17 Å) O− −631.0347  9194.4463 −103.0768 2409.6470 0.0161 1.12  −34.9994   785.6382  −12.0366  209.2285 2.1194 (0.0548 at 2.94 Å) (5.97 at 5.56 Å) H+1/2 −406.0906 14151.8887  −90.7672 3377.4961 0.0573 2.47  −23.2847   744.1006   −4.9473  109.4291 0.6636 (0.1701 at 19.69 Å) (6.71 at 2.98 Å) H+1/3 −270.4297 14078.5449  −60.1676 3328.5566 0.0401 2.15  −15.2479   721.5629   −3.2715  100.3644 0.5849 (0.1213 at 19.69 Å) (5.52 at 2.98 Å) O−1/2  −37.7416 12620.5059 −284.0005 7977.5732 0.0420 12.21  −45.5501  1599.1283  −10.9237  323.9079 2.2234 (0.0933 at 3.33 Å) (209.18 at 7.14 Å)a aThe second largest error percentage is 4.04% at 3.33 Å. Open in new tab Table 2.  Gaussian coefficients and constants for scattering factors of charged atoms and their errors (300 keV, resolution range: 50–2.5 Å) . a1 . b1 . a2 . b2 . . Error average . Error (%) average . . a3 . b3 . a4 . b4 . c . (max) . (max) . H+ −812.7002 14278.1309 −183.0536 3461.7549 0.1238 2.90 − 47.9442   783.2073 − 10.2941  123.3596 0.9183 (0.4713 at 49.02 Å) (8.42 at 3.17 Å) O− −631.0347  9194.4463 −103.0768 2409.6470 0.0161 1.12  −34.9994   785.6382  −12.0366  209.2285 2.1194 (0.0548 at 2.94 Å) (5.97 at 5.56 Å) H+1/2 −406.0906 14151.8887  −90.7672 3377.4961 0.0573 2.47  −23.2847   744.1006   −4.9473  109.4291 0.6636 (0.1701 at 19.69 Å) (6.71 at 2.98 Å) H+1/3 −270.4297 14078.5449  −60.1676 3328.5566 0.0401 2.15  −15.2479   721.5629   −3.2715  100.3644 0.5849 (0.1213 at 19.69 Å) (5.52 at 2.98 Å) O−1/2  −37.7416 12620.5059 −284.0005 7977.5732 0.0420 12.21  −45.5501  1599.1283  −10.9237  323.9079 2.2234 (0.0933 at 3.33 Å) (209.18 at 7.14 Å)a . a1 . b1 . a2 . b2 . . Error average . Error (%) average . . a3 . b3 . a4 . b4 . c . (max) . (max) . H+ −812.7002 14278.1309 −183.0536 3461.7549 0.1238 2.90 − 47.9442   783.2073 − 10.2941  123.3596 0.9183 (0.4713 at 49.02 Å) (8.42 at 3.17 Å) O− −631.0347  9194.4463 −103.0768 2409.6470 0.0161 1.12  −34.9994   785.6382  −12.0366  209.2285 2.1194 (0.0548 at 2.94 Å) (5.97 at 5.56 Å) H+1/2 −406.0906 14151.8887  −90.7672 3377.4961 0.0573 2.47  −23.2847   744.1006   −4.9473  109.4291 0.6636 (0.1701 at 19.69 Å) (6.71 at 2.98 Å) H+1/3 −270.4297 14078.5449  −60.1676 3328.5566 0.0401 2.15  −15.2479   721.5629   −3.2715  100.3644 0.5849 (0.1213 at 19.69 Å) (5.52 at 2.98 Å) O−1/2  −37.7416 12620.5059 −284.0005 7977.5732 0.0420 12.21  −45.5501  1599.1283  −10.9237  323.9079 2.2234 (0.0933 at 3.33 Å) (209.18 at 7.14 Å)a aThe second largest error percentage is 4.04% at 3.33 Å. Open in new tab Table 3.  The difference of the calculated FC values between charged and neutral residuesa Resolution range . Number of reflections . Average of FC(E204−) . Standard deviation of FC(E204−) . Minimum of FC(E204−) . Maximum of FC(E204−) . Average of FC(E204−)/FC(Neutral) . Standard deviation of FC(E204−)/FC(Neutral) . Minimum of FC(E204−)/FC(Neutral) . Maximum of FC(E204−)/FC(Neutral) . 50–30 Å    7 72.32 61.06 13.64 177.09 1.08 0.46 0.47 1.86 30–20 Å   24 37.85 22.46  6.47  87.61 1.17 0.34 0.83 2.26 20–10 Å  206 21.11 11.97  1.52  62.99 1.03 0.15 0.62 1.82 10–5 Å 1615 14.65  7.72  0.50  76.00 1.01 0.04 0.52 1.25  5–3 Å 6178 10.07  6.14  0.20  52.28 1.01 0.01 0.96 1.17 50–3 Å 8030 11.41  7.75  0.20 177.09 1.01 0.04 0.47 2.26 Resolution range . Number of reflections . Average of FC(E204−) . Standard deviation of FC(E204−) . Minimum of FC(E204−) . Maximum of FC(E204−) . Average of FC(E204−)/FC(Neutral) . Standard deviation of FC(E204−)/FC(Neutral) . Minimum of FC(E204−)/FC(Neutral) . Maximum of FC(E204−)/FC(Neutral) . 50–30 Å    7 72.32 61.06 13.64 177.09 1.08 0.46 0.47 1.86 30–20 Å   24 37.85 22.46  6.47  87.61 1.17 0.34 0.83 2.26 20–10 Å  206 21.11 11.97  1.52  62.99 1.03 0.15 0.62 1.82 10–5 Å 1615 14.65  7.72  0.50  76.00 1.01 0.04 0.52 1.25  5–3 Å 6178 10.07  6.14  0.20  52.28 1.01 0.01 0.96 1.17 50–3 Å 8030 11.41  7.75  0.20 177.09 1.01 0.04 0.47 2.26 Resolution range . Number of reflections . Average of FC(K129+) . Standard deviation of FC(K129+) . Minimum of FC(K129+) . Maximum of FC(K129+) . Average of FC(K129+)/ FC(Neutral) . Standard deviation of FC(K129+)/FC(Neutral) . Minimum of FC(K129+)/FC(Neutral) . Maximum of FC(K129+)/FC(Neutral) . 50–30 Å    7 81.98 66.21 15.44 202.81 1.31 0.74 0.53 2.52 30–20 Å   24 37.35 23.33  5.05  82.85 1.10 0.25 0.79 1.76 20–10 Å  206 20.85 11.71  2.71  64.21 1.03 0.16 0.59 2.10 10–5 Å 1615 14.52  7.62  0.50  74.69 1.00 0.04 0.63 1.50  5–3 Å 6178  9.98  6.08  0.06  51.62 1.00 0.04 0.23 2.43 50–3 Å 8030 11.32  7.79  0.06 202.81 1.00 0.05 0.23 2.52 Resolution range . Number of reflections . Average of FC(K129+) . Standard deviation of FC(K129+) . Minimum of FC(K129+) . Maximum of FC(K129+) . Average of FC(K129+)/ FC(Neutral) . Standard deviation of FC(K129+)/FC(Neutral) . Minimum of FC(K129+)/FC(Neutral) . Maximum of FC(K129+)/FC(Neutral) . 50–30 Å    7 81.98 66.21 15.44 202.81 1.31 0.74 0.53 2.52 30–20 Å   24 37.35 23.33  5.05  82.85 1.10 0.25 0.79 1.76 20–10 Å  206 20.85 11.71  2.71  64.21 1.03 0.16 0.59 2.10 10–5 Å 1615 14.52  7.62  0.50  74.69 1.00 0.04 0.63 1.50  5–3 Å 6178  9.98  6.08  0.06  51.62 1.00 0.04 0.23 2.43 50–3 Å 8030 11.32  7.79  0.06 202.81 1.00 0.05 0.23 2.52 aOnly the charge status of either E204 (upper table) or K129 (lower table) differs in a bacteriorhodopsin molecule in this simulation Open in new tab Table 3.  The difference of the calculated FC values between charged and neutral residuesa Resolution range . Number of reflections . Average of FC(E204−) . Standard deviation of FC(E204−) . Minimum of FC(E204−) . Maximum of FC(E204−) . Average of FC(E204−)/FC(Neutral) . Standard deviation of FC(E204−)/FC(Neutral) . Minimum of FC(E204−)/FC(Neutral) . Maximum of FC(E204−)/FC(Neutral) . 50–30 Å    7 72.32 61.06 13.64 177.09 1.08 0.46 0.47 1.86 30–20 Å   24 37.85 22.46  6.47  87.61 1.17 0.34 0.83 2.26 20–10 Å  206 21.11 11.97  1.52  62.99 1.03 0.15 0.62 1.82 10–5 Å 1615 14.65  7.72  0.50  76.00 1.01 0.04 0.52 1.25  5–3 Å 6178 10.07  6.14  0.20  52.28 1.01 0.01 0.96 1.17 50–3 Å 8030 11.41  7.75  0.20 177.09 1.01 0.04 0.47 2.26 Resolution range . Number of reflections . Average of FC(E204−) . Standard deviation of FC(E204−) . Minimum of FC(E204−) . Maximum of FC(E204−) . Average of FC(E204−)/FC(Neutral) . Standard deviation of FC(E204−)/FC(Neutral) . Minimum of FC(E204−)/FC(Neutral) . Maximum of FC(E204−)/FC(Neutral) . 50–30 Å    7 72.32 61.06 13.64 177.09 1.08 0.46 0.47 1.86 30–20 Å   24 37.85 22.46  6.47  87.61 1.17 0.34 0.83 2.26 20–10 Å  206 21.11 11.97  1.52  62.99 1.03 0.15 0.62 1.82 10–5 Å 1615 14.65  7.72  0.50  76.00 1.01 0.04 0.52 1.25  5–3 Å 6178 10.07  6.14  0.20  52.28 1.01 0.01 0.96 1.17 50–3 Å 8030 11.41  7.75  0.20 177.09 1.01 0.04 0.47 2.26 Resolution range . Number of reflections . Average of FC(K129+) . Standard deviation of FC(K129+) . Minimum of FC(K129+) . Maximum of FC(K129+) . Average of FC(K129+)/ FC(Neutral) . Standard deviation of FC(K129+)/FC(Neutral) . Minimum of FC(K129+)/FC(Neutral) . Maximum of FC(K129+)/FC(Neutral) . 50–30 Å    7 81.98 66.21 15.44 202.81 1.31 0.74 0.53 2.52 30–20 Å   24 37.35 23.33  5.05  82.85 1.10 0.25 0.79 1.76 20–10 Å  206 20.85 11.71  2.71  64.21 1.03 0.16 0.59 2.10 10–5 Å 1615 14.52  7.62  0.50  74.69 1.00 0.04 0.63 1.50  5–3 Å 6178  9.98  6.08  0.06  51.62 1.00 0.04 0.23 2.43 50–3 Å 8030 11.32  7.79  0.06 202.81 1.00 0.05 0.23 2.52 Resolution range . Number of reflections . Average of FC(K129+) . Standard deviation of FC(K129+) . Minimum of FC(K129+) . Maximum of FC(K129+) . Average of FC(K129+)/ FC(Neutral) . Standard deviation of FC(K129+)/FC(Neutral) . Minimum of FC(K129+)/FC(Neutral) . Maximum of FC(K129+)/FC(Neutral) . 50–30 Å    7 81.98 66.21 15.44 202.81 1.31 0.74 0.53 2.52 30–20 Å   24 37.35 23.33  5.05  82.85 1.10 0.25 0.79 1.76 20–10 Å  206 20.85 11.71  2.71  64.21 1.03 0.16 0.59 2.10 10–5 Å 1615 14.52  7.62  0.50  74.69 1.00 0.04 0.63 1.50  5–3 Å 6178  9.98  6.08  0.06  51.62 1.00 0.04 0.23 2.43 50–3 Å 8030 11.32  7.79  0.06 202.81 1.00 0.05 0.23 2.52 aOnly the charge status of either E204 (upper table) or K129 (lower table) differs in a bacteriorhodopsin molecule in this simulation Open in new tab Table 4.  Gaussian coefficients and constants for scattering factors of “charge differences” and their errors (300 keV, resolution range: 50–2.5 Å), and errors obtained by fitting charged atoms with eight Gaussians . a1 . b1 . a2 . b2 . . Error average . Error (%) average . . a3 . b3 . a4 . b4 . c . (max) . (max) . H+ − H −814.1870 14318.3643  −183.8043 3490.0320 0.1017 4.31  −48.4433   797.7763  −10.3269  131.2453 0.4217 (0.2917 at 19.69 Å) (13.44 at 3.08 Å) O− − O −621.7397  6873.5820  −50.4690  921.3515 0.0316 1.10  −11.5625   232.4147   −2.3098   68.9554 0.0180 (0.1888 at 10.00 Å) (3.30 at 2.50 Å) H+ H + (H+ − H)a 0.1019 2.84 (0.2905 at 19.69 Å) (8.11 at 3.08 Å) O− O + (O− − O)a 0.0318 1.83 (0.1895 at 10.00 Å) (21.91 at 5.56 Å) . a1 . b1 . a2 . b2 . . Error average . Error (%) average . . a3 . b3 . a4 . b4 . c . (max) . (max) . H+ − H −814.1870 14318.3643  −183.8043 3490.0320 0.1017 4.31  −48.4433   797.7763  −10.3269  131.2453 0.4217 (0.2917 at 19.69 Å) (13.44 at 3.08 Å) O− − O −621.7397  6873.5820  −50.4690  921.3515 0.0316 1.10  −11.5625   232.4147   −2.3098   68.9554 0.0180 (0.1888 at 10.00 Å) (3.30 at 2.50 Å) H+ H + (H+ − H)a 0.1019 2.84 (0.2905 at 19.69 Å) (8.11 at 3.08 Å) O− O + (O− − O)a 0.0318 1.83 (0.1895 at 10.00 Å) (21.91 at 5.56 Å) aEight Gaussians and two constants are used. The four Gaussians and constant given in Table 1 were used for the contribution by the neutral atoms and the four Gaussians and constant given in the upper rows of Table 4 were used for the contribution of the charge difference. Open in new tab Table 4.  Gaussian coefficients and constants for scattering factors of “charge differences” and their errors (300 keV, resolution range: 50–2.5 Å), and errors obtained by fitting charged atoms with eight Gaussians . a1 . b1 . a2 . b2 . . Error average . Error (%) average . . a3 . b3 . a4 . b4 . c . (max) . (max) . H+ − H −814.1870 14318.3643  −183.8043 3490.0320 0.1017 4.31  −48.4433   797.7763  −10.3269  131.2453 0.4217 (0.2917 at 19.69 Å) (13.44 at 3.08 Å) O− − O −621.7397  6873.5820  −50.4690  921.3515 0.0316 1.10  −11.5625   232.4147   −2.3098   68.9554 0.0180 (0.1888 at 10.00 Å) (3.30 at 2.50 Å) H+ H + (H+ − H)a 0.1019 2.84 (0.2905 at 19.69 Å) (8.11 at 3.08 Å) O− O + (O− − O)a 0.0318 1.83 (0.1895 at 10.00 Å) (21.91 at 5.56 Å) . a1 . b1 . a2 . b2 . . Error average . Error (%) average . . a3 . b3 . a4 . b4 . c . (max) . (max) . H+ − H −814.1870 14318.3643  −183.8043 3490.0320 0.1017 4.31  −48.4433   797.7763  −10.3269  131.2453 0.4217 (0.2917 at 19.69 Å) (13.44 at 3.08 Å) O− − O −621.7397  6873.5820  −50.4690  921.3515 0.0316 1.10  −11.5625   232.4147   −2.3098   68.9554 0.0180 (0.1888 at 10.00 Å) (3.30 at 2.50 Å) H+ H + (H+ − H)a 0.1019 2.84 (0.2905 at 19.69 Å) (8.11 at 3.08 Å) O− O + (O− − O)a 0.0318 1.83 (0.1895 at 10.00 Å) (21.91 at 5.56 Å) aEight Gaussians and two constants are used. The four Gaussians and constant given in Table 1 were used for the contribution by the neutral atoms and the four Gaussians and constant given in the upper rows of Table 4 were used for the contribution of the charge difference. Open in new tab In addition, Gaussian coefficients and constants were calculated for partially charged atoms after combining scattering factors of neutral and charged atoms linearly. Details are provided in the Appendix, and the Gaussian coefficients and constants used to describe partially charged atoms are also listed in Table 2. Fitting of atomic scattering factors for charge difference As Fourier transformation is linear, it is convenient to fit the difference between atomic scattering factors of charged and neutral atoms by the summation of Gaussians rather than to simulate difference maps between charged and neutral residues. The difference map between the experimental map of a molecule containing charged residues and the map calculated from the atomic model assuming that all atoms are neutral can then be simulated by the Fourier transform of the difference in atomic scattering factors between the charged and neutral atoms. In other words, instead of fitting the atomic scattering factors of the charged atoms, the scattering factors were separated into contributions from the neutral atoms and from the “charge difference”, and each were fitted separately by four Gaussians and a constant according to the following equations: (7) For the contributions of the neutral atoms, the Gaussian coefficients and constants listed in Table 1 were used. The scattering factors for charge difference, difference of scattering factors between charged and neutral atom, were fitted with four Gaussians and a constant in the resolution range from 50 to 2.5 Å as shown in Table 4. The fitting of the atomic scattering factors obtained by summation of all eight Gaussians and two constants showed the same degree of error as the fitting with four Gaussians and a constant (fitting errors by the summation of eight Gaussians and two constants for neutral atoms and charge differences for H+ and O− were 0.1019 and 0.0318 as shown in Table 4, while the fitting errors by four Gaussians and a constant for H+ and O− were 0.1238 and 0.0161 as shown in Table 2). Calculation of FC and density map For simulation of neutral and charged residues, the entire model of bacteriorhodopsin, which includes more than 220 residues and eight lipids, was used. As a result, the standard deviations (σ value) of the calculated maps remain almost constant, even if one of the residues in the protein was simulated with a different charge, and we could use the same contour levels for visualization of the maps. In these calculations, scattering factors for neutral atoms were used except for the charged atoms of the simulated residue. Glutamate residue (E204) or lysine residue (K129) was used for the simulation of charged status. For the simulation of neutral (protonated) glutamate (Fig. 3a1 and a3), the Gaussian coefficients and constant for O in Table 1 were applied to atoms OE1 and OE2 in the carboxyl group of the glutamate in X-PLOR. To simulate charged (deprotonated) glutamate (Fig. 3a2 and a4), both oxygen atoms in the carboxyl group were assumed to be charged evenly. Therefore, the Gaussian coefficients and constant for O−1/2 in Table 2 were applied to atoms OE1 and OE2. To simulate the lysine in its neutral (deprotonated) state (Fig. 3b1 and b3), the Gaussian coefficients and constant for H in Table 1 were applied to the two hydrogen atoms HZ1 and HZ2 in the amino group of the lysine. HZ3, the place for the third hydrogen atom in the amino group, was left empty assuming a lone electron pair on the nitrogen atom NZ of the amino group. For charged (protonated) lysine (Fig. 3b2 and b4), all three hydrogen atoms in the amino group were assumed to be charged evenly. Therefore, the Gaussian coefficients and constant for H+1/3 in Table 2 were applied to hydrogen atoms HZ1, HZ2 and HZ3 in X-PLOR. The calculated amplitudes (FC) were scaled to the experimental amplitudes of bacteriorhodopsin extracted from diffraction patterns. The FC values calculated for the neutral and charged status were compared and the differences were summarized in Table 3. The FC maps produced by X-PLOR were converted with the program MAPMAN [10] to the format suitable for display in the program O [11]. The volumes of the residues were compared using the histograms produced by MAPMAN. Fig. 3.  Open in new tabDownload slide Simulated density maps (FC maps) for neutral and charged glutamate and lysine residues. The density contoured at 1σ is shown in sky blue and the density at 2.5σ in magenta, where σ is the standard deviation from the mean density of the map. (a1) Neutral glutamate (Gaussian coefficients and constant for O given in Table 1 were applied to OE1 and OE2) using data only between 6 and 3 Å for the calculation. (a2) Charged glutamate (Gaussian coefficients and constant for O−1/2 given in Table 2 were applied to OE1 and OE2) using data only between 6 and 3 Å for the calculation. (a3) Neutral glutamate. The same scattering factors as in Fig. 3a1 were used but data in the resolution range from 50 to 3 Å were included in the calculation. (a4) Charged glutamate. The same scattering factors as in Fig. 3a2 were used but data in the resolution range from 50 to 3 Å were included in the calculation. (b1) Neutral lysine (Gaussian coefficients and constant for H given in Table 1 were applied to HZ1 and HZ2, with HZ3 assumed to be empty) using data only between 6 and 3 Å for the calculation. (b2) Charged lysine (Gaussian coefficients and constant for H+1/3 given in Table 2 were applied to HZ1, HZ2 and HZ3) using data only between 6 and 3 Å for the calculation. (b3) Neutral lysine. The same scattering factors as in Fig. 3b1 were used but data in the resolution range from 50 to 3 Å were included in the calculation. (b4) Charged lysine. The same scattering factors as in Fig. 3b2 were used but data in the resolution range from 50 to 3 Å were included in the calculation. Fig. 3.  Open in new tabDownload slide Simulated density maps (FC maps) for neutral and charged glutamate and lysine residues. The density contoured at 1σ is shown in sky blue and the density at 2.5σ in magenta, where σ is the standard deviation from the mean density of the map. (a1) Neutral glutamate (Gaussian coefficients and constant for O given in Table 1 were applied to OE1 and OE2) using data only between 6 and 3 Å for the calculation. (a2) Charged glutamate (Gaussian coefficients and constant for O−1/2 given in Table 2 were applied to OE1 and OE2) using data only between 6 and 3 Å for the calculation. (a3) Neutral glutamate. The same scattering factors as in Fig. 3a1 were used but data in the resolution range from 50 to 3 Å were included in the calculation. (a4) Charged glutamate. The same scattering factors as in Fig. 3a2 were used but data in the resolution range from 50 to 3 Å were included in the calculation. (b1) Neutral lysine (Gaussian coefficients and constant for H given in Table 1 were applied to HZ1 and HZ2, with HZ3 assumed to be empty) using data only between 6 and 3 Å for the calculation. (b2) Charged lysine (Gaussian coefficients and constant for H+1/3 given in Table 2 were applied to HZ1, HZ2 and HZ3) using data only between 6 and 3 Å for the calculation. (b3) Neutral lysine. The same scattering factors as in Fig. 3b1 were used but data in the resolution range from 50 to 3 Å were included in the calculation. (b4) Charged lysine. The same scattering factors as in Fig. 3b2 were used but data in the resolution range from 50 to 3 Å were included in the calculation. Results Simulations of neutral and charged residues For both acidic and basic residues, we simulated four situations, namely charged and neutral states of the residues and calculations with and without the inclusion of low-resolution data. We chose the glutamate residue E204 from the model of bacteriorhodopsin (pdb accession code 2AT9 [2]) as an example of an acidic residue. When the low-resolution data were omitted from the calculation of the density, i.e., only data from 6 to 3 Å were included, the charge had little effect (Fig. 3a1 and a2). The charged glutamate (Fig. 3a2) showed only slightly weaker density than the neutral one (Fig. 3a1), and the volume above the 1σ contouring level (colored in blue in the density maps) was reduced by only 2%. When the low-resolution data were included in the calculations of the density, i.e., data from 50 to 3 Å, the charge effect was significant. The density of the charged glutamate residue (Fig. 3a4) became much weaker than that of the neutral residue (Fig. 3a3), and the volume above the 1σ contouring level in the density map was reduced by 45%. The density above the 2.5σ contouring level (colored in magenta in the density maps) disappeared around the carboxyl group of the charged glutamate, and the O atoms in the carboxyl group of the charged glutamate were no longer represented by density above the 1σ contouring level (Fig. 3a4). We then chose the lysine residue K129 from the bacteriorhodopsin model as an example for a basic residue. If the density was calculated using only data between 6 and 3 Å (no low-resolution data), the charged lysine residue (Fig. 3b2) showed a slightly larger density than the neutral one (Fig. 3b1). The volume above the 1σ contouring level (colored in blue in the density maps) was increased by 11%. If the density was calculated using the data from 50 and 3 Å, the density of the charged lysine residue (Fig. 3b4) became substantially stronger than the neutral one (Fig. 3b3). The volume above the 1σ contouring level (colored in blue in the density maps) was increased by 82%. For these simulations, the glutamate and lysine residues were chosen from the atomic model of bacteriorhodopsin without considering the true charge status of these residues. The FC values calculated for charged and neutral status showed large difference especially in the low-resolution range as in Table 3. For example, the standard deviation of the FC ratio between reflections calculated assuming charged E204 and neutral E204 in the bacteriorhodopsin molecule was about 0.15 in the resolution range between 10 and 20 Å. Polarization effect In real molecules, charged residues are usually not isolated but compensated by opposite charges. To study the effect of such charge compensations, we studied a pair of hydrogen and oxygen atoms at a distance of 1 Å, placing the hydrogen atom at position −0.5 Å and the oxygen atom at position +0.5 Å. In the simulation shown in Fig. 4a, the hydrogen and oxygen atoms in the pair were assumed to be 100% polarized. To simulate the difference map between the polarized and nonpolarized situations, we calculated the sum of the “charge effects”, Fourier transforms of the scattering factors describing the charge differences of the two atoms. The “charge effects” for both the positive and negative charge are very large and have a very similar shape, but they have opposite signs. When summed, the two effects thus compensate each other to a large extent (Fig. 4a). Nevertheless, in the model we used the positions of the centers of the two charge effect curves are offset by 1 Å, resulting in a small remaining deviation. We call this the “polarization effect”, which is much smaller than the uncompensated charge effects. In Fig. 4b, we explain the potential of the polarized atoms as the summation of the potential of the neutral atoms and the “polarization effect”. Figure 5 shows the influence of the polarization effect depicted in Fig. 4 on the three-dimensional density map. The hydrogen and oxygen atoms are arranged horizontally, with the hydrogen atom on the left side and the oxygen atom on the right side. This atom pair is shown as a short stick in the center of Fig. 5a, b and c. The density for the neutral atoms is contoured in green at the 1σ level in Fig. 5a and c, and the density for the polarized atoms is contoured in blue at the 1σ level in Fig. 5b and c. The density obtained with a polarized atom pair (Fig. 5b) is enlarged in the area where it includes the positive density introduced by the polarization effect. The volume above the 1σ contouring level in the density map around the polarized atoms is increased by 26%. The center of density was shifted towards the side of the positively charged atom by about 0.1 Å (Fig. 5c), as a result of the slope of density due to the included polarization effect. Fig. 4.  Open in new tabDownload slide Polarization effect. (a) Effect of polarization between fully polarized H+ and O− atoms at a distance of 1 Å. The scattering factors used to describe the charge differences for H and O are listed in Table 4. The charge effects, potentials corresponding to the charge differences, were calculated by Fourier transformation of the scattering factors. Polarization effect was calculated by summing the two charge effects. The B-factor was set to 20 Å2. (b) Shift of the atom positions and enlargement of the density caused by the polarization effect. The potential of “H+ + O−” was calculated by adding the potential of “H + O” and “polarization effect.” Fig. 4.  Open in new tabDownload slide Polarization effect. (a) Effect of polarization between fully polarized H+ and O− atoms at a distance of 1 Å. The scattering factors used to describe the charge differences for H and O are listed in Table 4. The charge effects, potentials corresponding to the charge differences, were calculated by Fourier transformation of the scattering factors. Polarization effect was calculated by summing the two charge effects. The B-factor was set to 20 Å2. (b) Shift of the atom positions and enlargement of the density caused by the polarization effect. The potential of “H+ + O−” was calculated by adding the potential of “H + O” and “polarization effect.” Fig. 5.  Open in new tabDownload slide Influence of the polarization effect on the FC map. In the figures, hydrogen and oxygen atoms are spaced horizontally by 1 Å with the hydrogen atom on the left side and the oxygen atom on the right side. The atom pair is shown as a short stick model. This simulation was done in the environment of the entire protein. (a) The density map colored in green was calculated assuming that both atoms are neutral with data in the resolution range between 5 and 2 Å. (b) The density map colored in blue was calculated assuming that the atom pair are fully polarized with data in the resolution range between 5 and 2 Å. (c) Enlarged view of a superimposition of the density maps for neutral and polarized atoms shown in Fig. 5a and b. The center of the density shown in blue (polarized atoms) is shifted by about 0.1 Å compared to the center of the density shown in green (neutral atoms). Fig. 5.  Open in new tabDownload slide Influence of the polarization effect on the FC map. In the figures, hydrogen and oxygen atoms are spaced horizontally by 1 Å with the hydrogen atom on the left side and the oxygen atom on the right side. The atom pair is shown as a short stick model. This simulation was done in the environment of the entire protein. (a) The density map colored in green was calculated assuming that both atoms are neutral with data in the resolution range between 5 and 2 Å. (b) The density map colored in blue was calculated assuming that the atom pair are fully polarized with data in the resolution range between 5 and 2 Å. (c) Enlarged view of a superimposition of the density maps for neutral and polarized atoms shown in Fig. 5a and b. The center of the density shown in blue (polarized atoms) is shifted by about 0.1 Å compared to the center of the density shown in green (neutral atoms). Discussion We calculated crude approximations for the scattering factors in the low-resolution range for O− and other partially charged residues (listed in the Appendix). The way in which we approximated the scattering factors is not accurate and they should therefore be recalculated in the future using quantum theory. Although Gaussians are an appropriate tool to fit scattering factors for neutral atoms, they are not ideal to fit scattering factors for charged atoms. To accurately fit scattering factors for charged atoms over a wider resolution range with smaller errors, it may be necessary to use either more Gaussians or to develop a different mathematical description. When the difference map between the experimental map and the calculated map from the atomic model assuming all atoms to be neutral is calculated, the “polarization effect” will remain in the difference map if polarized atoms are present in the molecule. The influence of the polarization effect can be seen in the experimental map of polarized atoms when the low-resolution data are included in the calculations. When low-resolution data are omitted, the atomic positions will remain in place, but if the low-resolution data are included the polarization effect will result in a shift of the peak positions in the map. In addition, shifts in peak positions and changes in volumes of density features may even occur for the atoms surrounding a charged atom as long as they are within a few angstroms of each other. The difference of the calculated FC values for neutral and charged residues should be large enough to distinguish them in the experimental data as in Table 3. However, the approximation of the scattering factors by charged atoms is very crude, because it is known that the charge compensation due to the environment affects the scattering amplitudes significantly [12]. In real molecules, all charged residues are thought to be shielded either by counter ions, oppositely charged residues or a hydrogen bond network of waters surrounding them. Because of this shielding the charge effect will not be as pronounced in electron crystallographic density maps of real molecules as those seen in our simulations of individual charged residues presented here. Thus, more sophisticated approach to calculate the scattering amplitudes might be necessary to visualize the charge distribution from the experimental data. The true extent of the influence of charges on electron crystallographic density maps will become clearer when more experimental high-resolution data are available. Concluding remarks We simulated the influence of the “charge effect” and the “polarization effect” on three-dimensional maps obtained by electron crystallography. The quality of the experimental data obtained by electron crystallography is continually improving. For example, higher resolution than 2 Å is realistic in structure analysis of membrane proteins by electron crystallography [13]. Comparisons between experimental density maps and simulated density maps may thus become a powerful tool to provide further insight into the charge environment of amino acid residues in biological molecules. Appendix Estimation of the scattering factor for O− at very low resolution Data for the scattering factor for O− below a resolution of 12.5 Å are missing in the International Tables for X-ray Crystallography. The “charge difference” is different for each negatively charged atom, but at very low resolution it converges to the same value as shown in Fig. 6 with the resolution-dependent change approaching linearity. Therefore, at very low resolution the ratio between the scattering factors describing the charge difference of each atom and that of iodine (I) was approximated as being constant. The ratios were calculated at a resolution of 12.5 Å using the equations: (8) (9) These ratios were used to calculate the atomic scattering factor for O− in the resolution range below 12.5 Å using the equation: (10) To assess the accuracy of the method we used to approximate the atomic scattering factor for O− at resolutions below 12.5 Å, we used the same method to approximate the atomic scattering factor of Cl− (using the ratio RCl/Br) and compared the result with the data provided for Cl− in the International Tables for X-ray Crystallography. (11) (12) The error was within 0.2%, confirming that the method chosen to estimate the scattering factor for O− was sufficiently accurate for the simulation presented in this paper. Fig. 6.  Open in new tabDownload slide The ratio of charge differences between various atoms and I. The resolution range between 50 and 5 Å is shown. The lower the resolution, the closer become the ratios to 1. The open circles and dotted line for (O− – O)/(I− – I) represent the ratios calculated using the values for O− estimated from the ratios RO/Cl and RO/Br as explained in the text. Fig. 6.  Open in new tabDownload slide The ratio of charge differences between various atoms and I. The resolution range between 50 and 5 Å is shown. The lower the resolution, the closer become the ratios to 1. The open circles and dotted line for (O− – O)/(I− – I) represent the ratios calculated using the values for O− estimated from the ratios RO/Cl and RO/Br as explained in the text. Estimation of the scattering factor for partially charged atoms We also determined Gaussian coefficients and constants to approximate scattering factors for partially charged atoms after combining scattering factors for neutral and charged atoms linearly. For example, to calculate scattering factors for O−1/2, the atomic scattering factors for O and O− were first averaged and then four Gaussians and a constant were fitted to describe the averaged data. The average errors were 0.0573 and 0.0420 for H+1/2 and O−1/2, comparable to the errors obtained in the fitting of H+ and O−, which were 0.1238 and 0.0161 (Table 2). The maximum error percentage for O−1/2 was very high at a resolution 7.14 Å (209.2%), because at this resolution the value of the scattering factor for O−1/2 came close to 0. In the remaining resolution range, the error percentage was within a few percents (the next largest error percentage was 4.04% at 3.33 Å). Funding Kazato Research Foundation (to T.H.); Grants-in-Aid for Specially Promoted Research; Grant-in-Aid for 21st Century COE Research Kyoto University (A2), Japan Science and Technology Agency (JST), and Japan New Energy and Industrial Technology Development Organization (NEDO). We thank Dr. T. Kakitani for providing scattering factors of H+ and Dr. Y. Kimura for discussions at an early stage of this project. We acknowledge Dr. T. Walz for his critical reading. References 1 Kimura Y , Vassylyev D G , Miyazawa A , Kidera A , Matsushima M , Mitsuoka K , Murata K , Hirai T , Fujiyoshi Y . Surface of bacteriorhodopsin revealed by high-resolution electron crystallography , Nature , 1997 , vol. 389 (pg. 206 - 211 ) Google Scholar Crossref Search ADS PubMed WorldCat 2 Mitsuoka K , Hirai T , Murata K , Miyazawa A , Kidera A , Kimura Y , Fujiyoshi Y . The structure of bacteriorhodopsin at 3.0 Å resolution based on electron crystallography: implication of the charge distribution , J. Mol. 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Lipid-protein interactions in double-layered two-dimensional AQP0 crystals , Nature , 2005 , vol. 438 (pg. 633 - 638 ) Google Scholar Crossref Search ADS PubMed WorldCat Author notes 3 Present address: Structural Physiology Research Group, RIKEN SPring-8 Center, Harima Institute, 1-1-1 Kouto, Sayo-cho, Sayo, Hyogo 679-5148, Japan 4 Present address: Japan Biological Information Research Center (JBIRC), National Institute of Advanced Industrial Science and Technology (AIST), 2-41-6 Aomi, Koto-ku, Tokyo 135-0064, Japan 5 Present address: Graduate School of Integrated Science, Yokohama City University, Tsurumi, Yokohama 230-0045, Japan © The Author 2007. Published by Oxford University Press on behalf of Japanese Society of Microscopy. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org Oxford University Press
TI - Simulation of charge effects on density maps obtained by high-resolution electron crystallography
JF - Journal of Electron Microscopy
DO - 10.1093/jmicro/dfm019
DA - 2007-08-01
UR - https://www.deepdyve.com/lp/oxford-university-press/simulation-of-charge-effects-on-density-maps-obtained-by-high-S1ydNPPfY4
SP - 131
EP - 140
VL - 56
IS - 4
DP - DeepDyve
ER -