# Analysis of Diffracted Intensities from Finite Protein Crystals with Incomplete Unit Cells

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Construction of a Model for the Average Diffracted Intensity Distribution

#### 2.1.1. The Whole-Pattern Fitting Model

#### 2.1.2. Crystal Dimensions

#### 2.2. Simulations

#### 2.2.1. Crystal Surfaces

## 3. Results and Discussion

#### 3.1. Integration Analysis for Incomplete Unit Cells

#### 3.2. Whole-Pattern Fitting and Integration Analysis

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

SXC | serial X-ray crystallography |

XFELs | X-ray free-electron lasers |

microED | micro electron diffraction |

## Appendix A

- (1)
- Equal numbers of each sub-unit. The unit cell is defined by 1,2 packing order with $\langle {L}_{1}^{*}\left({q}_{x}\right){L}_{2}\left({q}_{x}\right)\rangle ={\sum}_{{N}_{x}}P\left({N}_{x}\right)\frac{{sin}^{2}({N}_{x}{q}_{x}a/2)}{{sin}^{2}({q}_{x}a/2)}$.
- (2)
- Equal numbers of each sub-unit. The unit cell is defined by 2,1 packing order with $\langle {L}_{1}^{*}\left({q}_{x}\right){L}_{2}\left({q}_{x}\right)\rangle ={\sum}_{{N}_{x}}P\left({N}_{x}\right)\frac{{sin}^{2}({N}_{x}{q}_{x}a/2)}{{sin}^{2}({q}_{x}a/2)}{e}^{i\left({q}_{x}a\right)}$.
- (3)
- ${N}_{2x}={N}_{x}+1$. The unit cell is not defined. For this case,$\langle {L}_{1}^{*}\left({q}_{x}\right){L}_{2}\left({q}_{x}\right)\rangle ={\sum}_{{N}_{x}}P\left({N}_{x}\right)\frac{sin({N}_{x}{q}_{x}a/2)sin(({N}_{x}+1){q}_{x}a/2)}{{sin}^{2}({q}_{x}a/2)}{e}^{i({q}_{x}a/2)}$.
- (4)
- ${N}_{2x}={N}_{x}-1$. The unit cell is not defined. For this case,$\langle {L}_{1}^{*}\left({q}_{x}\right){L}_{2}\left({q}_{x}\right)\rangle ={\sum}_{{N}_{x}}P\left({N}_{x}\right)\frac{sin({N}_{x}{q}_{x}a/2)sin(({N}_{x}-1){q}_{x}a/2)}{{sin}^{2}({q}_{x}a/2)}{e}^{-i({q}_{x}a/2)}$.

## References

- Chapman, H.N.; Fromme, P.; Barty, A.; White, T.A.; Kirian, R.A.; Aquila, A.; Hunter, M.S.; Schulz, J.; DePonte, D.P.; Weierstall, U.; et al. Femtosecond X-ray protein nanocrystallography. Nature
**2011**, 470, 73–77. [Google Scholar] [CrossRef] [PubMed] - Stellato, F.; Oberthür, D.; Liang, M.; Bean, R.; Gati, C.; Yefanov, O.; Barty, A.; Burkhardt, A.; Fischer, P.; Galli, L.; et al. Room-temperature macromolecular serial crystallography using synchrotron radiation. IUCrJ
**2014**, 1, 204–212. [Google Scholar] [CrossRef] [PubMed] - Gati, C.; Bourenkov, G.; Klinge, M.; Rehders, D.; Stellato, F.; Oberthür, D.; Yefanov, O.; Sommer, B.P.; Mogk, S.; Duszenko, M.; et al. Serial crystallography on in vivo grown microcrystals using synchrotron radiation. IUCrJ
**2014**, 1, 87–94. [Google Scholar] [CrossRef] [PubMed] - Shi, D.; Nannenga, B.L.; Iadanza, M.G.; Gonen, T. Three-dimensional electron crystallography of protein microcrystals. ELife
**2013**, 2, e01345. [Google Scholar] [CrossRef] [PubMed] - Nannenga, B.L.; Shi, D.; Leslie, A.G.W.; Gonen, T. High-resolution structure determination by continuous-rotation data collection in MicroED. Nat. Methods
**2014**, 11, 927–930. [Google Scholar] [CrossRef] [PubMed] - Shi, D.; Nannenga, B.L.; de la Cruz, M.J.; Liu, J.; Sawtelle, S.; Calero, G.; Reyes, F.E.; Hattne, J.; Gonen, T. The collection of MicroED data for macromolecular crystallography. Nat. Protoc.
**2016**, 11, 895–904. [Google Scholar] [CrossRef] [PubMed] - Subramanian, G.; Basu, S.; Liu, H.; Zuo, J.M.; Spence, J.C.H. Solving protein nanocrystals by cryo-EM diffraction: Multiple scattering artifacts. Ultramicroscopy
**2015**, 148, 87–93. [Google Scholar] [CrossRef] [PubMed] - Patterson, A.L. The Scherrer Formula for X-ray Particle Size Determination. Phys. Rev.
**1939**, 56, 978–982. [Google Scholar] [CrossRef] - Langford, J.I.; Wilson, A.J.C. Scherrer after sixty years: A survey and some new results in the determination of crystallite size. J. Appl. Crystallogr.
**1978**, 11, 102–113. [Google Scholar] [CrossRef] - Holzwarth, U.; Gibson, N. The Scherrer equation versus the ’Debye-Scherrer equation’. Nat. Nanotechnol.
**2011**, 6, 534. [Google Scholar] [CrossRef] [PubMed] - Yefanov, O.; Gati, C.; Bourenkov, G.; Kirian, R.A.; White, T.A.; Spence, J.C.H.; Chapman, H.N.; Barty, A. Mapping the continuous reciprocal space intensity distribution of X-ray serial crystallography. Philos. Trans. R. Soc. B
**2014**, 369, 20130333. [Google Scholar] [CrossRef] [PubMed] - Hunter, M.S.; DePonte, D.P.; Shapiro, D.A.; Kirian, R.A.; Wang, X.; Starodub, D.; Marchesini, S.; Weierstall, U.; Doak, R.B.; Spence, J.C.H.; et al. X-ray Diffraction from Membrane Protein Nanocrystals. Biophys. J.
**2011**, 100, 198–206. [Google Scholar] [CrossRef] [PubMed] - Kirian, R.A.; Bean, R.J.; Beyerlein, K.R.; Yefanov, O.M.; White, T.A.; Barty, A.; Chapman, H.N. Phasing coherently illuminated nanocrystals bounded by partial unit cells. Philos. Trans. R. Soc. B
**2014**, 369, 20130331. [Google Scholar] [CrossRef] [PubMed] - Spence, J.C.H.; Kirian, R.A.; Wang, X.; Weierstall, U.; Schmidt, K.E.; White, T.; Barty, A.; Chapman, H.N.; Marchesini, S.; Holton, J. Phasing of coherent femtosecond X-ray diffraction from size-varying nanocrystals. Opt. Express
**2011**, 19, 2866–2873. [Google Scholar] [CrossRef] [PubMed] - Fienup, J. Phase Retrieval Algorithms: A comparison. Appl. Opt.
**1982**, 21, 2758–2769. [Google Scholar] [CrossRef] [PubMed] - Miao, J.; Sayre, D.; Chapman, H.N. Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects. J. Opt. Soc. Am. A
**1998**, 15, 1662–1669. [Google Scholar] [CrossRef] - Kirian, R.A.; Bean, R.J.; Beyerlein, K.R.; Barthelmess, M.; Yoon, C.H.; Wang, F.; Capotondi, F.; Pedersoli, E.; Barty, A.; Chapman, H.N. Direct phasing of finite crystals illuminated with a free-electron laser. Phys. Rev. X
**2015**, 5, 011015. [Google Scholar] [CrossRef] - Liu, H.; Zatsepin, N.A.; Spence, J.C.H. Ab-initio phasing using nanocrystal shape transforms with incomplete unit cells. IUCrJ
**2013**, 1, 19–27. [Google Scholar] [CrossRef] [PubMed] - Chen, J.P.J.; Millane, R.P. Diffraction by nanocrystals. J. Opt. Soc. Am. A
**2013**, 30, 2627–2634. [Google Scholar] [CrossRef] [PubMed] - Chen, J.P.J.; Millane, R.P. Diffraction by nanocrystals II. J. Opt. Soc. Am. A
**2014**, 31, 1730–1737. [Google Scholar] [CrossRef] [PubMed] - Dilanian, R.A.; Williams, S.R.; Martin, A.V.; Streltsov, V.A.; Quiney, H.M. Whole-pattern fitting technique in serial femtosecond nanocrystallography. IUCrJ
**2016**, 3, 127–138. [Google Scholar] [CrossRef] [PubMed] - Rietveld, H.M. A profile refinement method for nuclear and magnetic structures. J. Appl. Crystallogr.
**1969**, 2, 65–71. [Google Scholar] [CrossRef] - Le Bail, A.; Duroy, H.; Fourquet, J.L. Ab-initio structure determination of LiSbWO6 by X-ray powder diffraction. Mater. Res. Bull.
**1988**, 23, 447–452. [Google Scholar] [CrossRef] - Rodriguez, J.A.; Ivanova, M.I.; Sawaya, M.R.; Cascio, D.; Reyes, F.E.; Shi, D.; Sangwan, S.; Guenther, E.L.; Johnson, L.M.; Zhang, M.; et al. Structure of the toxic core of α-synuclein from invisible crystals. Nature
**2015**, 525, 486–490. [Google Scholar] [CrossRef] [PubMed] - Sawaya, M.R.; Rodriguez, J.; Cascio, D.; Collazo, M.J.; Shi, D.; Reyes, F.E.; Hattne, J.; Gonen, T.; Eisenberg, D.S. Ab initio structure determination from prion nanocrystals at atomic resolution by MicroED. Proc. Natl. Acad. Sci. USA
**2016**, 113, 11232–11236. [Google Scholar] [CrossRef] [PubMed] - Guinier, A. X-ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies; W. H. Freeman and Company: San Francisco, CA, USA, 1963. [Google Scholar]
- Ino, T.; Minami, N. X-ray diffraction by small crystals. Acta Crystallogr. Sect. A
**1979**, 35, 163–170. [Google Scholar] - Bogan, M.J. X-ray free electron lasers motivate bioanalytical characterization of protein nanocrystals: Serial femtosecond crystallography. Anal. Chem.
**2013**, 85, 3464–3471. [Google Scholar] [CrossRef] [PubMed] - Yang, H.; Carney, P.J.; Chang, J.C.; Guo, Z.; Villanueva, J.M.; Stevens, J. Structure and receptor binding preferences of recombinant human A(H3N2) virus hemagglutinins. Virology
**2015**, 477, 18–31. [Google Scholar] [CrossRef] [PubMed] - Berman, H.M.; Westbrook, J.; Feng, Z.; Gilliland, G.; Bhat, T.N.; Weissig, H.; Shindyalov, I.N.; Bourne, P.E. The Protein Data Bank. Nucleic Acids Res.
**2000**, 28, 235–242. [Google Scholar] [CrossRef] [PubMed] - Vaguine, A.A.; Richelle, J.; Wodak, S.J. SFCHECK: A unified set of procedures for evaluating the quality of macromolecular structure-factor data and their agreement with the atomic model. Acta Crystallogr. D Biol. Crystallogr.
**1999**, 55, 191–205. [Google Scholar] [CrossRef] [PubMed] - Winn, M.D.; Ballard, C.C.; Cowtan, K.D.; Dodson, E.J.; Emsley, P.; Evans, P.R.; Keegan, R.M.; Krissinel, E.B.; Leslie, A.G.W.; McCoy, A.; et al. Overview of the CCP4 suite and current developments. Acta Crystallogr. D Biol. Crystallogr.
**2011**, 67, 235–242. [Google Scholar] [CrossRef] [PubMed] - Wilson, A.J.C. The probability distribution of X-ray intensities. Acta Crystallogr.
**1949**, 2, 318–321. [Google Scholar] [CrossRef] - Patterson, A.L. A Fourier Series Method for the Determination of the Components of Interatomic Distances in Crystals. Phys. Rev.
**1934**, 46, 372–376. [Google Scholar] [CrossRef] - Kirian, R.A.; Wang, X.; Weierstall, U.; Schmidt, K.E.; Spence, J.C.H.; Hunter, M.; Fromme, P.; White, T.; Chapman, H.N.; Holton, J. Femtosecond protein nanocrystallography-data analysis methods. Opt. Express
**2010**, 18, 5713–5723. [Google Scholar] [CrossRef] [PubMed] - Kirian, R.A.; White, T.A.; Holton, J.M.; Chapman, H.N.; Fromme, P.; Barty, A.; Lomb, L.; Aquila, A.; Maia, F.R.N.C.; Martin, A.V.; et al. Structure-factor analysis of femtosecond micro-diffraction patterns from protein nanocrystals. Acta Crystallogr. Sect. A
**2011**, 67, 131–140. [Google Scholar] [CrossRef] [PubMed] - Ayyer, K.; Yefanov, O.M.; Oberthür, D.; Roy-Chowdhury, S.; Galli, L.; Mariani, V.; Basu, S.; Coe, J.; Conrad, C.E.; Fromme, R.; et al. Macromolecular diffractive imaging using imperfect crystals. Nature
**2016**, 530, 202–206. [Google Scholar] [CrossRef] [PubMed] - Dilanian, R.A.; Streltsov, V.A.; Quiney, H.M.; Nugent, K.A. Continuous X-ray diffractive field in protein nanocrystallography. Acta Crystallogr. Sect. A
**2013**, 69, 108–118. [Google Scholar] [CrossRef] [PubMed] - Wolf, E. New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross spectra of steady-state sources. J. Opt. Soc. Am.
**1982**, 72, 343–351. [Google Scholar] [CrossRef] - Nave, C.; Sutton, G.; Evans, G.; Owen, R.; Rau, C.; Robinson, I.; Stuart, D.I. Imperfection and radiation damage in protein crystals studied with coherent radiation. J. Synchrotron Radiat.
**2016**, 23, 228–237. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Differences in one-dimensional diffracted intensity profiles (${R}_{diff}$ in Equation (26)) with respect to average one-dimensional unit cell dimensions, ${N}_{x}$. Results are shown from simulations of unit cell distributions merged from 1000 one-dimensional lattices with dimensions defined by Gaussian distributions with a standard deviation of 25% of the mean dimension, ${N}_{x}$, in red and with dimensions defined by lognormal distributions of mean, ${N}_{x}$, in blue.

**Figure 2.**Schematic drawing of an externally incomplete unit cell distribution. Dark green cells indicate a whole unit cell; light green cells indicate an incomplete cell. For simplicity, molecules are represented by triangular shapes. The boundary of the crystal bulk is indicated with dashed green lines and the boundary of the filling region for incomplete unit cells is indicated by dashed red lines. This region was limited to a width of a single unit cell in this study.

**Figure 3.**Accuracy of extracted structure factor amplitudes via the integration approach for varying integration regions from crystals of average dimensions of 15-by-5 unit cells and varying occupancy levels of surface unit cells. The integration region is shown as the distance from Bragg locations; the upper value approaches a distance of 40% of the reciprocal lattice spacing. The legend indicates the average percentage of occupancy levels for each set of simulated crystals.

**Figure 4.**Accuracy of extracted structure factor amplitudes via the integration approach for varying integration regions from crystals of average dimensions of 30-by-10 unit cells and varying occupancy levels of surface unit cells. The integration region is shown as the distance from Bragg locations; the upper value approaches a distance of 40% of the reciprocal lattice spacing. The legend indicates the average percentage of occupancy levels for each set of simulated crystals.

**Figure 5.**Accuracy of extracted structure factor amplitudes for varying occupancy levels (%) of incomplete unit cells on crystal surfaces of average dimensions of 15-by-5 unit cells. Blue points show the results of the modified whole-pattern fitting analysis outlined in this study; purple points show results excluding the correction factor, $\frac{1}{2}\left(1+{C}_{12}\left(\mathbf{q}\right)\right)$. Green points show the results of the integration of Bragg reflections. Integration results are shown for a single integration region for which the most accurate structure factor amplitudes were extracted.

**Figure 6.**Accuracy of extracted structure factor amplitudes for varying occupancy levels (%) of incomplete unit cells on crystal surfaces of average dimensions of 30-by-10 unit cells. Blue points show the results of the modified whole-pattern fitting analysis outlined in this study; purple points show results excluding the correction factor, $\frac{1}{2}\left(1+{C}_{12}\left(\mathbf{q}\right)\right)$. Green points show the results of the integration of Bragg reflections. Integration results are shown for a single integration region for which the most accurate structure factor amplitudes were extracted.

**Figure 7.**Quality of fit of the modeled distributions for varying occupancy levels (%) of incomplete unit cells on crystal surfaces of average dimensions of 30-by-10 unit cells. Blue points show the results of the modified whole-pattern fitting analysis outlined in this study and purple points show results excluding the correction factor, $\frac{1}{2}\left(1+{C}_{12}\left(\mathbf{q}\right)\right)$. All values have the minimum R${}_{fit}$ factor obtained, ${R}_{0}$, subtracted for scaling purposes.

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**MDPI and ACS Style**

Williams, S.R.; Dilanian, R.A.; Quiney, H.M.; Martin, A.V.
Analysis of Diffracted Intensities from Finite Protein Crystals with Incomplete Unit Cells. *Crystals* **2017**, *7*, 220.
https://doi.org/10.3390/cryst7070220

**AMA Style**

Williams SR, Dilanian RA, Quiney HM, Martin AV.
Analysis of Diffracted Intensities from Finite Protein Crystals with Incomplete Unit Cells. *Crystals*. 2017; 7(7):220.
https://doi.org/10.3390/cryst7070220

**Chicago/Turabian Style**

Williams, Sophie R., Ruben A. Dilanian, Harry M. Quiney, and Andrew V. Martin.
2017. "Analysis of Diffracted Intensities from Finite Protein Crystals with Incomplete Unit Cells" *Crystals* 7, no. 7: 220.
https://doi.org/10.3390/cryst7070220