research papers
Determination of
using Bayesian statistics on Bijvoet differences^{a}Bruker AXS, PO Box 811, 2600 AV Delft, The Netherlands, and ^{b}Bijvoet Center for Biomolecular Research, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands
^{*}Correspondence email: rob.hooft@brukeraxs.nl
A new probabilistic approach is introduced for the determination of the x parameter. The we obtain is often about half of the in the value of the Flack x parameter. The proposed formalism is suited in particular to determination from diffraction data of biologically active (pharmaceutical) compounds where the strongest signal often comes from oxygen. It is shown that a reliable assignment in such cases can be made on the basis of Cu Kα data, and in some cases even with carefully measured Mo Kα data.
of a compound which is known to be based on Bijvoetpair intensity differences. The new method provides relative probabilities for different models of the chiral composition of the structure. The outcome of this type of analysis can also be cast in the form of a new value, along with associated that resembles the value of the well known Flack1. Introduction
Bijvoet, Peerdeman and van Bommel were the first to demonstrate that the et al., 1951). Their method was based on the complex contributions to the atomic scattering factors that make the intensities of Friedelrelated reflections (or their symmetry equivalents) different. This difference in intensity (the `Bijvoet difference') depends both on the atom types present in the molecule and the wavelength of the radiation used (Flack & Shmueli, 2007). The concept of `absolute configuration' has since been generalized to `absolute structure' to include cases where the polarity of the structure rather than the is determined (Jones, 1984; Glazer & Stadnicka, 1989).
of a chiral molecule could be determined by Xray crystallography (BijvoetTraditionally Kα data collected on a diffractometer with a point detector for compounds containing atom types heavier than phosphorus. Currently, most smallmolecule structure determinations are based on data collected on diffractometers equipped with CCD detectors using Mo Kα radiation. The impact of this change is that often a more accurate, highly redundant and complete data set is obtained, which, however, often contains a weaker signal.
determination was based on analysis of CuThere exists a significant interest in the determination of the ). Unfortunately, many molecules of interest do not contain atoms heavier than sulfur. In the past, this problem was solved with the introduction of a heavier atom in the structure, e.g. with the addition of HBr (Spek, 1976). The current trend is to attempt determination on the native compound, even when no atoms heavier than oxygen are present.
of biologically active molecules (van der Helm & Hossain, 1987Over time a number of methods for the determination of the
have been proposed.The most straightforward way of establishing the R factor and test for the statistical significance of the Rfactor difference. The latter is commonly done with the Hamilton test (Hamilton, 1965).
of a small molecule is to refine both enantiomers separately, subsequently select the with the lowest crystallographicA much more sensitive method (Zachariasen, 1965; Engel, 1972) is to select a subset of reflections from the measured data that are most sensitive to the (relatively weak reflections with a large Bijvoet difference), and compare the calculated Bijvoet differences with the observed differences. Just by comparing the signs of these differences, the can often be established even if the difference in R factor is inconclusive. Although the can be determined using this method, it is not easy to quantify the degree of certainty of the assignment. Le Page et al. (1990) present a method to accompany an determined in this way by a calculation of the probability that the should be inverted. For this calculation use a binomial distribution. This method has not found widespread use, and therefore its performance is difficult to assess.
Another variation on this method is used by the Bijvoet program in the DIRDIF program suite (Beurskens et al., 1980; Beurskens et al., 1999). This program uses a weighted average of the signs of the Bijvoet difference (B). This method can be very successful, but it needs a carefully selected subset of Bijvoet pairs to be effective. The assignment using this calculation is accompanied by a but it is very hard to establish the statistical correctness of this value as it relies on distributions being Gaussian, and disregards the careful selection of the reflection subset. Also, no difference can be seen in this calculation between a racemic twin and a weak signal: both will result in smaller absolute values of B and larger standard uncertainties.
Rogers (1981) was the first to introduce a parameter that can be refined as part of the leastsquares This parameter encodes the `strength' and sign of the measured signal measured in units of , the imaginary component of the complex atomic scattering factor.
The Rogers parameter was soon superseded by the Flack x parameter (Flack, 1983). The Flack x parameter encodes the relative abundance of the two components in an The value of the Flack x parameter can be determined using a fullmatrix leastsquares procedure [e.g. with the TWIN/BASF instructions in SHELXL97 (Sheldrick, 1997)]. A reasonable estimate of the Flack x parameter can be obtained by determining the parameter separately; this is automatically performed for all noncentrosymmetric structures in the SHELX97 package. Since the Flack x parameter can correlate with the atomic coordinates, especially for structures in space groups that do not have a fixed origin, the estimate can be inaccurate if its value deviates significantly from zero (Flack & Bernardinelli, 2006; Flack et al., 2006).
Since the value of the Flack x parameter is the result of a leastsquares its can be derived from the covariance matrix. This can be used to quantify the degree of confidence in the proposed Flack & Bernardinelli (2000) discuss criteria for the reliability of the assignment based on the in the Flack x parameter value. Their analysis, starting from only the has to assume that the distribution is normal also in its tails. The paper does not distinguish between the probability of obtaining the given the observations and the probability of the observations given the The Bayesian prior relating the two probabilities is ignored. The Flack & Bernardinelli (2000) method does not result in a quantitative statement about the assignment.
Parsons & Flack (2004) recently introduced a variation of the of the Flack x parameter. Their method relies on the careful determination of a few selected Bijvoet differences [as the ratio (I_{+}I_{})/(I_{+}+I_{})] which can either be obtained directly from a good redundant data set or by carefully adding some extra observations. Parsons & Flack (2004) show that this method increases the sensitivity of the determination.
Dittrich et al. (2006) recently reported advances in determinations made using `invarioms'. Invarioms are aspherical scattering factors that take into account electron density deformations. Using invarioms instead of the normal spherical scattering factors can improve figures of merit as well as the standard uncertainties of all refined parameters. Experience has shown that by using invarioms, the in the value of the Flack x parameter can be significantly reduced, and that the calculated value of the Flack x parameter is frequently closer to 0.0.
In the next section, we introduce a new way to determine the x parameter.
The method applies Bayesian statistics to the Bijvoet differences. The result of this approach is a series of probabilities for different hypotheses of the The solution with the highest probability can be determined, and this can be used to map the results to a value with that can be directly compared with the value of the Flack2. Theory and methods
For each
of reflections, we can defineand
Here, F^{ 2}_{c} are calculated intensities and F^{ 2}_{o} are observed intensities. If we assume completely independent observations of the two reflection intensities,
Now, we can define a variable z as follows:
If the i.e. they do not carry a standard uncertainty) the probability distribution of z is a standard normal Gaussian^{1}
of the model for calculation of the structure factors is correct, and we assume the calculated intensities to be correct (Based on all pairs of observations, we can now calculate the probability of the measured data, given the fact that the y = 0, it will become clear later in the paper why this notation is chosen):
is correctly specified in the model (the correct is noted as the conditionIn statistics, Bayes' theorem for conditional probabilities specifies that
In our case, we can use this theorem to invert our probability given above:
This value cannot be computed, as p(observations) (the probability of obtaining the current observations) is unknown. But to be able to make the assignment, we would like to calculate the ratio of and the similar term for the opposite designated as . The term of p(observations) disappears in the calculation of this ratio:
And if no prior knowledge about the i.e. p(y = 0) = p(y = 1)],
exists [To be able to do this, we define a value q analogous to z:
This value represents z_{h} for the inverted structure. Now
and
Hence
If the correct
and wrong hypotheses are the only two possibilities for a certain structure, this would be sufficient. However, in practice a structure may be a twin consisting of two inverses (socalled inversion twins), and a more general probability model is desired to express this. The can be described as a linear combination of the two structure factors of the pure enantiomeric structures. For eachThis linear combination is analogous to the definition of the Flack x parameter (Flack, 1983). Since = , this equation can be simplified to
We refer to the variable as the `generalized x = 0), and for the wrong = −1.0 (and x = 1). With the help of this parameter, we can now introduce for each reflection h the function :
For the correct = 1.0 (andIt can be seen easily that z_{h} is equal to and q_{h} is equal to . Note that this computation is also allowed with physically impossible values of . With this generalization, the probability distribution becomes
and from Bayes' theorem,
We can now avoid the need to calculate p(observations) in two ways: we can either have a discrete set of possible hypotheses for the value of , or we can study the continuum of all possible values.
In the case of the discrete set ,,…,, we can normalize easily:
In most cases, all priors will be set to 1/n. Two useful discrete sets of hypotheses that can be treated this way are the twomembered set the absolute structure is either correct or wrong and the threemembered set the absolute structure can be correct or wrong, but the sample may also be a 50/50 inversion twin.
If we want to consider the whole continuum of possible values of , the normalization of the probability function is less meaningful. In the case of a continuum, only ratios of different probabilities should be used, and these ratios do not depend on the normalization. However, all of the probability numbers are exceedingly small. For numerical stability reasons, it is advisable to bring all relative probabilities that we want to use in calculations to a reasonable size. To achieve this we can simply divide by a high value of the probability function. For this goal, we chose to use with = . We call the `incompletely normalized' result p_{u}:
Since this approach is most useful if no prior knowledge is assumed at all (note that we always have prior information, namely , but here we explicitly choose to ignore this), we simplify it to
It is observed in practice that (in the second definition) is a reasonably well behaved Gaussianlike function. We can therefore calculate^{2} a quantity G:
Using this definition, G is the best approximation of for the structure based purely on the observations and not using any prior knowledge (not even the physical restriction that must be in the interval [−1, 1]). Since in our practical experience looks very much like a symmetric Gaussian distribution, G will also be very close to the most probable value of . Like the value of the Rogers parameter, the value of G will be close to 1.0 for structures for which the of the model is correct, and close to −1.0 for structures for which the of the model is incorrect. Continuing along this path, we can calculate the variance of the distribution using
This can be used to estimate an uncertainty in the obtained value of G.
The concept of the unrestricted G follows naturally from the comparison of the definitions of z and q. This is, however, a new concept. With a simple change of parameter expression we can cast our result in a form comparable with the Flack x parameter:
parameterand
With this definition, y behaves like the Flack x parameter in that it will have a value of 0.0 for the correct model, and 1.0 for the inverted model.
3. Test calculations
Table 1 lists several data sets that were collected on different instruments. Some of these data sets happened to be of interest at that time; others were specially collected to test the statistical methods introduced in this paper.
All of the structures have weak Kα radiation. The theoretical signal at = 0° was estimated for each of the data sets from
signals. Roughly half of the data sets were collected using MoBoth summations run over atom types i, N_{i} is the number of atoms of type i in the structure, f is the scattering factor of the atom type, and the imaginary part of the factor (Weiss et al., 2001). For Mo Kα radiation, = 0.0060, = 0.0033 and = 0.0016. For Cu Kα radiation, = 0.0322, = 0.018 and = 0.0091. is called the `signal'. This does assume a random distribution of atoms in the cell; locations of resonant scatterers close to symmetry elements can cause weakening of the signal. On the other hand, this formula can be a pessimistic guess since f will decrease for increasing diffraction angles , whereas the factor is nearly independent of the diffraction angle.
All structures were refined using SHELXL97. After the observed Bijvoet pairs listed in the FCF output file of SHELXL were used for an analysis of the value of y. Care was taken not to use FCF files produced by SHELXL run using the TWIN/BASF instructions, as in such a case the calculated structure factors already have the Flack x calculation embedded and this would invalidate the analysis. Where available, the assignment was crosschecked with prior information; in other cases the structure for which the value of y was closest to 0.0 was chosen. Results of the analyses are given in Table 2.

4. Results and discussion
The power of the introduced method comes from the fact that it weights each observed Bijvoet difference based on its expected accuracy directly, rather than relying on the weight of the reflection intensities. Calculating these proper weights for a leastsquares procedure is very difficult, but proper weighting can be rather easily accomplished with the derived
procedure instead of using leastsquares. Bijvoet differences can be much smaller than the residual differences between the observed and calculated intensities, and the calculated differences are accurate as long as the resonant scatterers have been accurately positioned.It is essential to measure Bijvoet pairs for the calculation of y, where the Flack x parameter can be determined even if the data set covers at least the corresponding to the with an added inversion centre.
We only tested our methods on data sets with close to 100% coverage of Bijvoet pairs.
When prior information is given, e.g. that the sample must be either the structure or its inverse, the method presented can be used to calculate probabilities of the two possible hypotheses [p2(ok) and p2(wrong)]. These probabilities can be surprisingly decisive, even when the signal is very weak. For the test data sets measured using Cu Kα radiation, the of all structures can be proven beyond reasonable doubt if it is assumed (prior knowledge) that the original compound was For the threehypotheses model where the additional possibility of a 50% cannot be ruled out, the distinction given by the probabilities [we call these p3(ok), p3(twin) and p3(wrong)] is less pronounced, but even in that case many of the determined values for the Cu Kα data sets would satisfy the most stringent pharmaceutical requirements. The least surprising results are obtained when the whole continuum of structure compositions must be considered. In this case, the estimate that is obtained as the value of y has a smaller than the value of the Flack x parameter. In most cases, the value of y is also closer to zero than the value of the Flack x parameter. Structures for which the continuum approach is required have not been studied in this paper. Using the continuum approach to solve the binary question (as is commonly done with the Flack x parameter in existing studies of bioactive compounds) is suboptimal. In contrast, the use of p2(ok) and p2(wrong) directly gives quantitative reliability information.
For some data sets, calculations were performed both on the correct and on the inverted model, refined in SHELXL. For the AMBI data set, the p3(wrong) value of 1.64 × 10^{−10} increases to p3(ok) = 2.3 × 10^{−10} when the inverted structure is refined. The small difference between these values shows that the inverted cannot absorb more than a small fraction of the signal into the other refined structural parameters. Comparing the equivalent numbers for the M006C data set, which has a weaker signal and which does not have a fixed origin, shows a similarly sized relative increase from p3(wrong) = 0.031 to p3(ok) = 0.043 for the inverted structure. The magnitude of this structural bias is largely insignificant for the determination of pharmaceutically active compounds. It may, however, be significant for accurate determination of the twin ratio of inversion twins; this has not been the subject of our study.
There are two assumptions in the derivation of the probabilistic model: firstly, that the
of the two reflections that form each Bijvoet pairs are independent; secondly, that the standard uncertainties of the individual reflections are accurate.Both of these conditions are necessary conditions for to follow a standard normal distribution. These assumptions can be verified by making a normal probability plot (Abrahams & Keve, 1971) from all values . Such normal probability plots, made for the data sets above, show that the observed distribution of for most data sets indeed follows a Gaussian distribution (the of the normal probability plot is 0.999) but with . Two possible reasons can be suggested. (i) The used scaling programs overestimate the errors in the reflection intensities. This is highly unlikely. (ii) The measurement error in the Bijvoet difference is smaller than could be expected if the two errors in the reflection intensities were independent. The errors are in fact positively correlated, and the error in the Bijvoet difference is really smaller.^{3}
The second hypothesis is most likely. Even without knowing the source of the smaller
it is possible to use the information obtained from the normal probability plot to scale the standard uncertainties in the Bijvoet differences, thereby obtaining a corrected . This correction scales down the standard uncertainties in in all but two of the cases that were examined for this paper.The validity of such a downscaling of the errors can be confirmed by studying the result for a group of independent structure determinations and determining the value Z = for each of them. If all standard uncertainties have been determined correctly, the values of Z from a random population of structure determinations should form a standard normal distribution. For the structure determinations using Mo Kα radiation given in this paper, the average absolute value of Z is 0.61 (the expected value is 0.85) and the root mean square (r.m.s.) value of Z is 0.67 (expected 1.0) (Table 3). These results suggest that the error is indeed systematically overestimated. After applying the slope from the normal probability plot to correct the estimated standard uncertainties in the observed Bijvoet differences, the average absolute value of Z is 0.66 and the r.m.s. value of Z is 0.72. These values are still smaller than the expected values. The current benchmark set is too small for this to be considered proof of the merits of the downscaling procedure.

4.1. Centrosymmetric structures
Flack & Bernardinelli (2006) and Flack et al. (2006) investigated the value of the Flack x parameter for a set of centrosymmetric structures that were refined in a noncentrosymmetric Looking at the definition of the Flack x parameter,
it can be clearly seen that for the correct model, x is indeterminate since the two terms F_{h}^{ calc} and F_{h}^{ calc} are equal. The determination of x in these cases is therefore based purely on the random incorrect differences between the two `halfstructures' in the In this light, it is at first sight surprising and discomforting that the values observed have such small standard uncertainties. It is clear that for the Flack x parameter the assumption that the offdiagonal elements of the covariance matrix may be ignored is wrong. The assumption that all other parameters have been determined correctly by the leastsquares has been violated.
We have attempted a noncentrosymmetric solution and et al., 2005) ourselves to investigate this effect further. For this structure the Flack x parameter is 0.56 (4) and the value of y is 0.45 (3). Both values are close to 0.5 with a relatively small A detailed analysis of the data set indicated that the small is due to a few reflections for which the differences between the two halfstructures create a significant Bijvoet difference while, as expected for a centrosymmetric structure, the value is statistically insignificant. Such pairs are normally indicative of by inversion.
of a centrosymmetric rutheniumcontaining compound (HotzeThe only statistical difference in
between a real and a wrongly refined centrosymmetrical structure is that the calculated Bijvoet differences are much smaller than for a normal noncentrosymmetric structure with the same elemental composition. This is due to the fact that the configuration of the atoms is almost centrosymmetric (with respect to a suitably chosen origin, the phases of many reflections are close to 0 and and the phases of the contributions are close to and ) and hence the contribution to the scattering factors only results in relatively small scattering amplitude differences. It is difficult to determine a reliable criterion for this effect.It appears then that the distinction between a true ADDSYM (Spek, 2003), followed by a detailed inspection of the weak reflections after in the suggested centrosymmetric space group.
and a noncentrosymmetrically refined centrosymmetric structure is best made in real space by a symmetrydetection procedure like5. Recommended procedure
Current versions of y to take into account. We therefore recommend to refine the structure including the Flack x parameter (e.g. use the TWIN/BASF instructions in SHELXL). The value of y can then be determined separately using a utility that explicitly calculates structure factors for the Bijvoet pairs (e.g. the Bijvoet Pairs option in PLATON).^{4}
programs cannot use the value ofThis procedure will account for any correlation between the structural parameters and the absolute structure.
6. Conclusions
A new probabilistic procedure was introduced that can be used to establish the
The procedure is especially suitable for biologically active compounds, which often do not contain atoms with a larger signal than that of oxygen.The only special requirement for the data collection procedure imposed by the new probabilistic calculation is that Bijvoet pairs should be present in the data set. In contrast, the determination of the Flack x parameter also works for data sets that have a coverage of 0%, although this is not recommended practice.
One of the results of the procedure is a value y, which can be directly compared with the value of the Flack x parameter. We observe for our test data sets that the in the value of y is roughly half of the in the value of x. The observed deviation from 0.0 is consistent with the These observations are comparable with the results obtained using invarioms but without the significant efforts associated with the calculation of invarioms.
The calculations also give explicit probabilities for the y and without the assumption that the distribution of y is Gaussian. The explicit probability of an assignment error makes our procedure suitable to regulate the probability of erroneous assignments in pharmaceuticals. The probability calculations can be based either on a model with two hypotheses for the two absolute structures or optionally take the chance of a racemate into account as a third hypothesis.
assignment, without referring to the value ofThe procedure was tested on a number of lightatom structures (no atoms with a stronger Kα radiation, a mixed result was obtained: some structures could receive quite a good assignment; most structures show at least a clear direction. For all data sets measured using Cu Kα radiation (resulting in a roughly five times larger signal) an excellent discrimination was obtained with the chance of error for most structures below 10^{−100}. Of course, for most of the Cu Kα structures the in the Flack x parameter is small enough for an unambiguous assignment.
signal than that of oxygen). For those data sets collected using MoThe current method offers an alternative method to look at the same experimental data as addressed by the Flack x approach.
7. Availability
An implementation of the described algorithm by one of the authors is available in his PLATON (Spek, 2003) program (http://www.cryst.chem.uu.nl/platon/pl000000.html ).
Footnotes
^{1}One referee remarked that this may not be completely true in practice, especially far from the mean. However, large deviations from the mean occur when the differences are much larger than the in which case the assignment should be obvious in any case. The practical Gaussian nature of the distribution and the usability of the calculated can also be verified by the normal probability plot analysis as mentioned later in the paper.
^{2}The integrals can be computed using a summation with a suitably small step size, where the bounds of are chosen such that at the bounds is insignificantly small.
^{3}The positive correlation could be caused by the fact that there are many Friedel pairs in our data sets. For a the diffraction geometry could be more similar than for general Bijvoet pairs. This could cause systematic errors to cancel. A fourcircle goniostat could be employed to extend these advantages. This is an interesting subject for a future study.
^{4}For a TWIN/BASF SHELXL will write calculated structure factors into the FCF file that take the inversion ratio into account. Such an FCF file can therefore not be used to calculate the value of y.
Acknowledgements
The authors wish to thank Anita Coetzee, Martin Lutz, George Sheldrick and Bill David for stimulating discussions before and during the 2005 Crystallographic Computing School in Siena, Italy, and Martin Lutz, Huub Kooijman and Trixie Wagner for testing the method on their data sets. The authors also express their appreciation to all referees of this paper for very valuable comments on the subject.
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