A first foray into twinning – Doug Rees had worked on a twinned protein crystal as a student with Lipscomb, and had brought an interest in methods for dealing with twinned crystals when he arrived at UCLA. One of my first projects as a student with Doug was to figure out if there was a way to disentangle isomorphous replacement data in the presence of twinning, particularly if the twinning fraction was high enough to prevent ‘detwinning’ (i.e. recovering the true crystallographic intensities from the twinned observables). The underlying problem was that the typical equations upon which MIR is based — which relate the complex quantities FP, FPH, fh — were of no use if the values of the individual quantities were unknown. My first publication was a solution to this problem of phasing in the presence of (even perfect) twinning. The solution involved casting the ordinary phasing problem (e.g. phase circles) into four dimensions; phase circles became hyperspheres in 4-D. This was necessary to account for the mixing of the real and imaginary components of two different twin-related reflection. For example, an observed intensity gave the value for the sum of two separate crystallographic intensities, meaning that Iobs = A12 + B12 + A22 + B22 (where A1 is the real part of the structure factor for the first twin-related reflection, and so on); this is an equation for a hypersphere. Information from the heavy atom phasing model then gave additional hyperspheres, whose intersections eventually made it possible to extract out the correct real and imaginary parts of the two separate structure factors, despite the absence of direct measurement of the individual reflection intensities. An Acta A paper published in 1987 (Acta Cryst. A43, 30) provided the necessary theory and a synthetic test case. In the meantime, solving twinned crystal structures by molecular replacement turned out to be relatively routine — as a student, Matt Redinbo published the first molecular replacement structure from a perfectly twinned protein crystal (Acta Cryst. D49, 375 (1993)). In constrast, no highly twinned structures have been phased by MIR methods using the general solution described in the 1987 Acta paper. Experimental phasing (MIR or anomalous) has been successful in cases of lower twin fraction by correcting (‘detwinning’) the observed data, or in some cases of high twin fraction by essentially ignoring the twinning during the phasing stages and incorporating it only later in the refinement stage.

Twin fraction estimation – Numerous methods had been described for estimation the twin fraction, but a particularly simple one emerged from an analysis that involved taking a pair of reflection intensities and reducing them to an effectively normalized or unitless quantity by dividing their difference by their sum, giving a value between -1 and 1, or between 0 and 1 if the absolute value is taken. This parameter was called H. When the two observed reflections in question are related to each other by twinning (meaning both are derived from partial mixing of two separate twin-related reflection intensities), the statistical distribution for the parameter H is uniform, and has a simple dependence on the twin fraction, alpha. This led to a particularly simple way of examining diffraction data and estimating the twin fraction (Acta Cryst A44, 142 (1988)).

Local statistics for detecting twinning – As noted above, good methods were available for estimating the twin fraction in crystals suffering from partial twinning by evaluating the overall degree of similarity between pairs of twin-related reflections. In addition, numerous stastical analyses had been developed for detecting cases in which perfect twinning was involved. In perfect twinning, the twin-fraction is 1/2, causing erroneously high symmetry to appear in the diffraction pattern. This situation presents a different kind of challenge compared to partial twinning. In a case of perfect twinning, the problem is not to determine the twin fraction, but rather the problem is to recognize that the observed diffraction data suffer from high or perfect twinning by examining the overall intensity statistics, which deviate significantly from theoretical Wilson statistics. The complication is that sometimes other kinds of crystallographic issues interfere with the intensity statistics, and can defeat the traditional tests for twinning that simply look at the cumulative distribution of normalized intensities. One confounding problem is anisotropy, which tends to give more extreme distributions of intensities. To overcome this problem, a new equation was introduced that examines the differences between pairs of reflections near each other in reciprocal space; an equation similar to the one used for comparing twin-related intensities in the partial twinning test was adopted, but here the reflection pairs are nearby rather than twin-related. The statistics for this parameter (called L) relating local pairs of reflections were found to be simple for both the situation of normal (untwinned) data and perfectly twinned data. The use of the measure L therefore provides a test for perfect (or high) twinning that is robust to anisotropy (Padilla and Yeates, Acta D59, 1124-1130 (2003)). In addition, pseudo-centering in a crystal can lead to confounding effects similar to anisotropy, and the local test can overcome this situation as well if the reflections to be compared are chosen properly.