![]() ![]() ![]() Consequently, the absolute number of photons in a spot (which determines the maximum possible signal-to-noise ratio) depends on exactly where the spot falls on the detector surface. The intensity of a Bragg spot is not simply the square of the structure factor, but depends on several other factors including exposure time, crystal volume and the geometry of diffraction. Only recently has it become clearly established that radiation damage at cryogenic temperatures is proportional to dose (Henderson, 1990 ▶ Gonzalez & Nave, 1994 ▶ Glaeser et al., 2000 ▶ Sliz et al., 2003 ▶ Leiros et al., 2006 ▶ Owen et al., 2006 ▶ Garman & McSweeney, 2007 ▶ Garman & Nave, 2009 ▶ Holton, 2009 ▶) and this understanding enabled the present work. ![]() The resulting orbital shapes (Slater, 1929 ▶) led directly to the cross-sections needed to compute absorption effects in the 1960s and steady improvements continue to this day (Hubbell, 2006 ▶). For example, Darwin’s variable ‘ f’ required the development of quantum theory to explain its observed value (Debye, 1915 ▶, 1988 ▶). The formula for the integrated intensity of a spot was introduced by Darwin (1914 ▶), but much subsequent work was required to fill out the original theory. All other sources of noise, including background scattering, are neglected until the discussion in § 3.2. For simplicity, in the present work we consider the X-ray detector and indeed the entire diffractometer to be an ideal device subject only to the shot noise of the net spot photons themselves (the square root of the number of counts). Counting detectors such as multi-wire (Cork et al., 1974 ▶) and pixel arrays (Kraft et al., 2009 ▶) do not have this kind of noise and the optimal data-collection strategy with these detectors is different (Xuong et al., 1985 ▶ Schulze-Briese et al., 2007 ▶). For example, the time-honored practice of recording the three-dimensional diffraction pattern on as few images as possible was not simply an effort to save money on film, but to minimize noise intrinsic to the detection process such as ‘fog’ on the film or the read-out circuit of a charge-coupled device (CCD). Here, we endeavor to keep the theory general and independent of the limitations of current diffraction hardware. The International Tables for Crystallography (Wilson & Prince, 1999 ▶) contain most of the critical pieces of the puzzle assembled here and the original references are spread out over nearly a century of literature. But is there a theoretical limit? The work presented here establishes a firm theoretical framework for computing the absolute signal available from very small macromolecular crystals and every effort is made to explicitly and unambiguously spell out the definitions and derivations. The last 15 years have seen many experimental estimates of how small a protein crystal can be and still yield a complete data set (Gonzalez & Nave, 1994 ▶ Glaeser et al., 2000 ▶ Teng & Moffat, 2000 ▶, 2002 ▶ Facciotti et al., 2003 ▶ Sliz et al., 2003 ▶ Li et al., 2004 ▶ Nelson et al., 2005 ▶ Sawaya et al., 2007 ▶ Coulibaly et al., 2007 ▶ Standfuss et al., 2007 ▶ Moukhametzianov et al., 2008 ▶ reviewed by Holton, 2009 ▶) and this size has been decreasing as technology improves. These results suggest that reduction of background photons and diffraction spot size on the detector are the principal paths to improving crystallographic data quality beyond current limits. These represent 15-fold to 700-fold less scattering power than the smallest experimentally determined crystal size to date, but the gap was shown to be consistent with the background scattering level of the relevant experiment. Taking the net photon count in a spot as the only source of noise, a complete data set with a signal-to-noise ratio of 2 at 2 Å resolution was predicted to be attainable from a perfect lysozyme crystal sphere 1.2 µm in diameter and two different models of photoelectron escape reduced this to 0.5 or 0.34 µm. The influences of molecular weight, solvent content, Wilson B factor, X-ray wavelength and attenuation on scattering power and dose were all included. In this work, classic intensity formulae were united with an empirical spot-fading model in order to calculate the diameter of a spherical crystal that will scatter the required number of photons per spot at a desired resolution over the radiation-damage-limited lifetime. ![]()
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