Application of Bayesian analysis to indirect Fourier transformation in small-angle scattering

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1 Journal of Applied Crystallography ISSN Editor: Gernot Kostorz Application of Bayesian analysis to indirect Fourier transformation in small-angle scattering Bente Vestergaard and Steen Hansen Copyright International Union of Crystallography Author(s) of this paper may load this reprint on their own web site provided that this cover page is retained. Republication of this article or its storage in electronic databases or the like is not permitted without prior permission in writing from the IUCr. J. Appl. Cryst. (2006). 39, Vestergaard and Hansen ffl Bayesian analysis

2 Journal of Applied Crystallography ISSN Received 23 May 2006 Accepted 1 September 2006 Application of Bayesian analysis to indirect Fourier transformation in small-angle scattering Bente Vestergaard a and Steen Hansen b * a Department of Medicinal Chemistry, Danish University of Pharmaceutical Sciences, Universitetsparken 2, DK-2100 Copenhagen, Denmark, and b Department of Natural Sciences, The Royal Veterinary and Agricultural University, Thorvaldsensvej 40, DK-1871 FRB C, Denmark. Correspondence slh@kvl.dk # 2006 International Union of Crystallography Printed in Great Britain all rights reserved Using Bayesian analysis for indirect Fourier transformation (IFT) of data from small-angle scattering (SAS) leads to probability distributions for parameters describing the experimental data. This quantification may provide extra information about the scattering system. The shape of the probability distribution for the maximum diameter of the scatterer may contain information about e.g. the heterogeneity of the scattering sample. The information content in the experimental data can be quantified as an effective number of parameters which can be determined from the data. The applicability of the Bayesian approach to IFT in SAS is demonstrated using simulated as well as experimental data. 1. Introduction Indirect Fourier transformation (IFT) has been a standard technique for analysis of small-angle scattering (SAS) data for almost three decades. Many different approaches to IFT in SAS have been suggested during this period, e.g. by Glatter (1977), Moore (1980), Glatter & Müller (1982), Svergun et al. (1988), Hansen & Pedersen (1991) and Svergun (1992), but the most frequently used method for IFT in SAS is that of Glatter (1977). The method of Glatter imposes a smoothness criterion upon the distribution to be estimated, giving higher preferences to smoother solutions. This is in good agreement with the prior knowledge that most SAS experiments have very low resolution. The method of Glatter requires an estimate of the noise level of the experimental data as well an estimate of the maximum dimension of the scatterer. Furthermore, in state-ofthe-art SAS analysis programs, such as ATSAS2.1 (Konarev et al., 2006), these estimates are not fully objective, as the regularization parameter (determining the noise level) is incorporated (GNOM; Svergun, 1992) and the maximum dimension of the scatterer has to be estimated by the user. Using Bayesian analysis, these two parameters can be estimated (Hansen, 2000) and the probability distributions for the parameters can be calculated. As no input, except for the experimental data, is required from the user, the Bayesian approach removes ambiguities caused by different subjective choices by different users for this form of IFT. In the present manuscript, we demonstrate how the Bayesian approach may provide additional information about the scattering system. The advanced methods for three-dimensional modelling, such as ATSAS2.1, only intensify the need for a refined primary data analysis and an objective characterization/parametrization of the scattering data before continuing to modelling. Interesting applications of these parameters are also found in protein structure modelling (Zheng & Doniach, 2005; Bernadó et al., 2005). For a system consisting of two different scatterers, the probability distribution for the maximum diameter is sensitive to changes in the number ratio of the scatterers. This may be useful e.g. for sensitive detection of impurities or early detection of aggregates or degradation products, which may be relevant for stabilization and/or formulation of biological samples. It is shown that a variation of the number ratio of two different scatterers is consistently reflected in the ratio of the respective peaks in the probability distribution for the maximum diameter of the sample. It is thus possible to detect and describe two populations of scatterers in solution using objective criteria, which has not previously been done by IFT. Furthermore, the information content of given experimental data can be quantified in a logical and intuitively appealing manner using the Bayesian IFT. Examples of this are given and compared with the conventional number of Shannon channels (Shannon & Weaver, 1949) and useful applications are suggested. 2. Theory 2.1. Small-angle scattering In small-angle scattering, the intensity I is measured as a function of the length of the scattering vector q = 4 sinðþ=, where is the wavelength of the radiation and is half the scattering angle. For scattering from a dilute solution of monodisperse molecules of maximum dimension d, the J. Appl. Cryst. (2006). 39, doi: /s

3 intensity can be written in terms of the distance distribution function p(r) (see Glatter, 1982): Z d IðqÞ ¼4 0 pðrþ sinðqrþ qr Approximating the distance distribution function p(r) by p = ð p 1 ;...; p N Þ and measuring the intensity at a given q i, Iðq i Þ¼ XN j¼1 A ij p j þ e i ; where e i is the noise at data point i and the matrix A is given by A ij =4r sinðq i r j Þ=ðq i r j Þ, where r = r j r j 1. The aim of the indirect Fourier transformation is to restore p, which by virtue of the Fourier transform contains the full information present in the scattering profile. For uniform scattering density of the molecule, the distance distribution function is proportional to the probability distribution for the distance between two arbitrary scattering points within the molecule. For non-uniform scattering density, the distance distribution may have negative regions (if the scattering density of some region of the scatterer is less than the scattering density of the solvent). For non-dilute systems, the generalized indirect Fourier transformation (GIFT) extension of IFT was introduced by Brunner-Popela & Glatter (1997). In GIFT, interparticle effects are taken into account by including a structure factor in the calculations. Using IFT, high concentrations may give negative regions in the distance distribution function around the maximum size of the scatterer (see e.g. Glatter, 1982). From a relation similar to equation (1), the size distribution of a polydisperse sample can be estimated by IFTof SAS if the form factor of the scatterers is known IFT in SAS The method of Glatter (1977) is similar to the general method of Tikhonov & Arsenin (1977). Both methods estimate a function of interest by minimizing a new functional written as a weighted sum of the chi-square, 2, and a regularization functional, S: S þ 2 : dr: The 2 is defined in the conventional manner, i.e. 2 ¼ XM i¼1 ð1þ ð2þ ð3þ ½I m ðq i Þ Iðq i ÞŠ 2 ; ð4þ where I m ðq i Þ is the measured intensity and i is the standard deviation of the noise at data point i. For the choice of regularization functional, several forms for S exist. For the present calculations, the expression S = R p 00 ðxþ 2 dx has been selected, giving preference to smooth distance distribution functions p(r). Assuming p(0) = p(d) = 0, this regularization expression takes the discrete form 2 i S ¼ XN 1 p j ðp j 1 þ p jþ1 Þ 2 þ p2 1 þ 1 2 p2 N; ð5þ j¼2 which is very similar to Glatter s original smoothness constraint and gives similar results. An error estimate for the (average) distribution of interest is provided from the individual probabilities of all the solutions calculated (each solution corresponding to a specific choice of noise level and maximum dimension of the scatterer). This method appears to give more realistic error estimates (Hansen, 2000), especially in the region of the maximum diameter d, than other methods for IFT (e.g. Svergun & Pedersen, 1994) Estimation of the maximum diameter of the scatterer. The weight (or Lagrange multiplier) and the maximum diameter d of the scatterer are both necessary for the IFT and can be estimated using Bayesian analysis. As described by Hansen (2000), the posterior probability P for ð; dþ after data have been measured can be written h i NðN 1=2 Pð; dþ ¼ 1 expð S 2 =2Þ 2 þ 1Þ ð6þ det 1=2 ða þ 1 BÞ using the regularization from equation (5), writing A = rrs and B = rr 2 =2 (the Hessian). In equation (6), both matrices as well as ð S 2 =2Þ have to be evaluated at the point p where ð S 2 =2Þ takes its maximum value. In a Bayesian framework, the posterior probability is also referred to as the evidence. For a recent overview of Bayesian inference in physics, see Dose (2003) Information in SAS The sampling theorem (Shannon & Weaver, 1949) states that a continuous scattering curve I(q) from an object of maximum diameter d is fully represented by its values in a set of points (Shannon channels) at q n = n=d, where n = ð1;...; 1Þ. The number of Shannon channels N s necessary to represent the intensity I(q) in the interval [q min, q max ] is given by N s ¼ dðq max q min Þ=: Consequently, N s corresponds to the maximum number of free (independent) parameters that it is possible to determine from the experimental data. Moore (1980) showed that the information in reciprocal space can be related to real space by expressing the scattering data and the corresponding distribution in real space as two series having identical coefficients, but different basis functions. The first Shannon channel I(q 1 )atq 1 = =d in reciprocal space then determines the overall dimension of the corresponding Fourier representation of the data in real space. Hence for estimation of the maximum diameter of the scatterer, the first data point q min should be measured at q min =d. Adding the higher channels improves the resolution of the data by adding finer details to p(r). ð7þ 798 Vestergaard and Hansen Bayesian analysis J. Appl. Cryst. (2006). 39,

4 Oversampling the data (i.e. using q <=d) corresponds to measuring the extra Shannon channels within [q min, q max ] from the scattering of an object larger than d. Using regularization for IFT in SAS may introduce correlations between the coefficients of Moore. For these cases, the criterion q min <=d is relaxed and the exact relation between q min and d is dependent upon the regularization functional as well as the Lagrange multiplier. Examples of this are given below. The actual number of parameters that it is possible to determine from measured data in a given q range is dependent upon the noise level of the experiment and the instrumental smearing. Furthermore, the value of q max which enters into equation (7) is rarely uniquely defined due to large noise levels at large q values. As a more realistic measure of the information content in the data, the number of good parameters N g (e.g. Gull, 1989; Müller et al., 1996) has been suggested using regularization by the maximum entropy method: N g ¼ XN j¼1 j þ j : ð8þ Here j are the eigenvalues of B and is the Lagrange multiplier of equation (3). By this equation, N g counts the number of eigenvalues which are large compared with the Lagrange multiplier, balancing the information in the data (eigenvalues for B) against the weight of the regularizing functional or prior (eigenvalues for A). For entropy regularization, A = I, where I is the unity matrix. Hence equation (8) gives the number of directions in parameter space, which are determined well for the given noise level. Expressing the information content of the experimental data through equation (8) removes ambiguities due to the choice of q max as very noisy data do not contribute to N g. Overfitting the data will reduce the Lagrange multiplier and reduce the eigenvalues of A + B towards those of B, increasing N g above the number of Shannon channels N s. Underfitting the data to a higher chi-square will increase the Lagrange multiplier. This leads to a lower value for N g calculated from equation (8) (consistent with a stronger correlation of the Shannon channels and reduction of information in the experimental data). For the general case, the denominator of equation (8) has to be replaced by the eigenvalues of the matrix A + B (see e.g. MacKay, 1992). Figure 1 Estimation of p(r) from simulated scattering data from a sphere of diameter 20 nm. (a) Error bars: simulated scattering intensity. Full line: fit. (b) Estimated distance distribution function with error bars. (c) Evidence P(; d) scaled to 1. (d) P(d) corresponding to (c). J. Appl. Cryst. (2006). 39, Vestergaard and Hansen Bayesian analysis 799

5 Table 1 Estimation of the number of good parameters N g. q min (nm 1 ) q max (nm 1 ) M b N s N g Figure 2 Full line: N g as a function of absolute noise level a using simulated data for a sphere with d = 20 nm as shown in Fig. 1. The q range was [0.1, 1.0] nm 1. Dotted lines: (top) N g = 6.99 corresponding to the number of Shannon channels for d = 24.4 nm; and (lower) N g = 5.73 corresponding to the number of Shannon channels for d = 20 nm. Figure 3 (a) Simulated scattering data from an ellipsoid of diameters The use of the eigenvalues of B for experimental design and analysis of reflectivity data has been suggested by Sivia & Webster (1998). (13, 13, 23) nm in the interval [0.15, 2.0] nm 1. 3% relative noise and absolute noise were added to the data. (b) N g as a function of extra background added to the simulated data in (a). 3. Results 3.1. Estimation of N g For comparing N s and N g calculated by equations (7) and (8), respectively, simulated scattering data from a sphere of diameter 20 nm were used. The data were simulated in the q interval [0, 1.5] nm 1 using M = 100 points. An absolute noise of i = ai(0) + bi(q i ) was added, using various values for a and b (absolute and relative noise, respectively). Furthermore, the data were truncated at various values for q min and q max. In Fig. 1 the simulated data are shown along with the results using [q min, q max ] = [0.2, 1.5] nm 1 and a relative noise of b = 0.05 and a =10 4 (to avoid strong influence of the minima in the scattering profile). The distance distribution function shown in Fig. 1(b) was calculated as the mean of distributions, each corresponding to a specific choice of ð; dþ, and the error bars were calculated from the spread of these distributions. The probability distribution (evidence) for Pð; dþ is shown in Fig. 1(c) and integrating over leaves P(d) as shown in Fig. 1(d). Table 1 shows the corresponding values of N g and N s calculated for various levels of relative noise b and truncations. In Fig. 2, the variation of N g is shown for simulated data from a sphere of diameter 20 nm for [q min, q max ] = [0.1, 1.0] nm 1 as a function of the absolute noise level a using a fixed relative noise of b = The horizontal lines in Fig. 2 show the number of Shannon channels, N s = 5.73, corresponding to d = 20 nm, and N 2 = 6.99, corresponding to d = 24.4 nm, the latter being the maximum dimension used for the estimation of p(r). For simulated scattering data from an ellipsoid of diameters (13, 13, 23) nm, shown in Fig. 3(a), the variation of N g was tested by adding an extra (constant) background. The result is shown in Fig. 3(b), N g having a minimum close to the correct value for the background Estimation of d To test the sensitivity of the Bayesian estimation of d, scattering data from mixtures of two different scatterers were simulated. 800 Vestergaard and Hansen Bayesian analysis J. Appl. Cryst. (2006). 39,

6 Fig. 4(a) shows the simulated scattering data from various mixtures of two spheres. The diameters of the spheres were 18 and 23 nm, respectively, and the ratio I 2 (0)/I 1 (0) (index 2 referring to the larger spheres) was varied in the interval [0.01, 0.20]. The data were simulated in the q interval [0.1, 2.0] nm 1 using 95 data points, and an absolute noise of a =10 4 as well as a relative noise b = 0.02 were added. For I 2 (0)/I 1 (0) = 0.20, the estimated distance distribution function p(r) is shown in Fig. 4(b). For a lower fraction I 2 (0)/I 1 (0) = 0.04, the evidence Pð; dþ is shown in Fig. 4(c). Finally, in Fig. 4(d) the evidence for the maximum diameter d is shown for all six different ratios of I 2 (0)/I 1 (0) as indicated in the figure. As the fraction of larger spheres is increased from 1% to 20%, the number of good parameters N g increases from 9.88 to (as the effective d is increased). The numbers of Shannon channels corresponding to d = 18 nm and d =23nm are N s = and N s = 13.91, respectively. Fig. 5 shows the results of a similar analysis of mixtures of ellipsoids of revolution. The diameters of the ellipsoids were (10, 10, 18) nm and (13, 13, 23) nm, respectively. The data were simulated in the interval [0, 3.0] nm 1 using M = 100 points. An absolute noise of a = and a relative noise of b = 0.03 were added to the simulated data. For low ratios of I 2 (0)/I 1 (0) (index 2 referring to the larger ellipsoids) in the interval [1.5, 1.9], the data were truncated at q = 0.10 nm 1, and the estimate of the evidence for d is shown in Fig. 5(c). For truncation at a larger q =0.15nm 1, the corresponding results are shown in Fig. 5(c). The results illustrate that for the higher q min, it is necessary to use higher ratios for the ellipsoids (in the interval [2.0, 3.0]) to obtain similar ratios for the two peaks in P(d). Finally, in Fig. 6 experimental SAXS data (Vestergaard et al,, unpublished) are shown. The experimental data were measured at beamline X33, HASYLAB, EMBL-Hamburg, at protein concentrations between 2 and 8 mg ml 1 using 3 min exposures. One protein sample consisted of molecules with an expected maximum diameter d of 18 nm; the other consisted of a mixture of molecules with d values of 18 and 23 nm, respectively. Fig. 6(b) shows the distance distribution functions for the two data sets, while in Fig. 6(c) the corresponding estimates of d for the experimental data are shown. Fig. 6(d) shows the results using a 1:1 mixture [i.e. I 2 (0)/ I 1 (0) = 1] of the two data files for two different q intervals. For the experimental data, N g = 7.5 and 12.7 for the data from the smaller and larger scatterer, respectively; the corresponding numbers of Shannon channels were 18.9 and 22.8 (calculated from the estimates of d). Figure 4 Estimation of p(r) from simulated scattering data from spheres of diameters 18 nm and 23 nm in different ratios as described in the text. (a) Error bars: simulated scattering intensity. Full line: fit. (b) Estimated distance distribution function with error bars from data having 20% large spheres. (c) Evidence P(; d) (scaled to 1) from data having 4% large spheres. (d) P(d) from scattering data of spheres in different ratios (the percentage of larger spheres is indicated near the respective peaks). J. Appl. Cryst. (2006). 39, Vestergaard and Hansen Bayesian analysis 801

7 Figure 5 Estimation of p(r) from simulated scattering data from ellipsoids of diameters (10, 10, 18) nm and (13, 13, 23) nm in different ratios as described in the text. (a) Error bars: simulated scattering intensity from the 2:1 mixture. Full line: fit. (b) Estimated distance distribution function from data shown in (a). (c) Evidence P(d) from data having q min =0.10nm 1. Full line: ratio 1.5:1. Dashed line: ratio 1.7:1. Dotted line: ratio 1.9:1. (d) Evidence P(d) from data having q min =0.15nm 1. Full line: ratio 2:1. Dashed line: ratio 2.5:1. Dotted line: ratio 3:1. 4. Discussion 4.1. Estimation of N g Table 1 and Fig. 2 indicate that the number of good parameters N g calculated for various conditions is sensible compared with the Shannon estimates N s. The values for N g, also for the experimental data, are in good agreement with the observation that the number of free parameters rarely exceeds for SAS experiments (e.g. Svergun, 1999). Deducing N g good parameters from the data corresponds to reducing the number of degrees of freedom for the 2 by N g, which is similar to the conventional reduction of the number of degrees of freedom for the 2 by the number of fitting parameters. Fitting the true information in the data invariably leads to fitting of some of the noise as well (see MacKay, 1992). Writing the reduced chi-square 2 r for M data points leads to 2 r ¼ M M þ N g : In addition to fitting some of the noise, the estimate of p(r) may extend beyond the true maximum dimension d for the ð9þ scatterer (providing information about a larger region of direct space) which may also make N g exceed N s. For the simulated examples, these effects are seen for the lowest noise levels. As M is reduced (the last example in Table 1), 2 r is also reduced according to equation (9), which may lead to a slight increase in N g. Using IFT, the noise level of the experiment usually has to be estimated from the data, and binning of the data points beyond that which is necessary for practical purposes (storage, CPU time, etc.) should be avoided. For background subtraction, Fig. 3 indicates that N g may provide a coarse estimate of a flat background. This agrees with the fact than an erroneous background is often detected from the IFT when pð0þ 6¼ 0. The extra information in I(q), due to erroneous background subtraction and leading to the extra structure in p(r) at small r values, is consistent with a larger value for N g. This may be relevant for the analysis of SAXS results from weakly scattering molecules, such as proteins, or for automated data reduction. Using Bayesian methods, an unknown background may also be estimated by maximizing the evidence, assuming the background to be an additional unknown parameter. This leads to a result similar to that which is shown here. 802 Vestergaard and Hansen Bayesian analysis J. Appl. Cryst. (2006). 39,

8 Figure 6 Estimation of p(r) for experimental scattering data. (a) Points: merged scattering data resulting from measurements at 2, 4 and 6 mg ml 1. Upper curve: mixture of large and small scatterers, respectively. Lower curve: sample of small scatterers. Full lines: corresponding fits. (b) Estimated distance distribution functions from data shown in (a). Upper and lower curves as in (a). (c) Evidence P(d) from data having q min =0.15nm 1. Full line: corresponding to lower curve in (a). Dotted line: corresponding to upper curve in (a). (d) Evidence P(d) from 1:1 mixture of data sets shown in (a). Full line: estimate using q interval [0.15, 2] nm 1. Dotted line: estimate using q interval [0.2, 0.3] nm 1. For SAXS experiments, the parameter N g may be used to find the optimal exposure time t of a given biological system. A sudden increase in dn g /dt (which may be calculated in real time) may indicate deterioration of the sample due to X-ray exposure, and a low value for dn g /dt may indicate suboptimal use of exposure time. For SANS experiments, it is often relevant to relax the resolution to obtain sufficient flux. Through simulation/ calculation of N g for the experimental set up, it may be possible to give objective criteria for the optimal experimental settings Estimation of d Regarding estimation of the maximum dimension, it should be noted that it is frequently a relatively small fraction of p(r) from which d has to be deduced. For example, for a homogeneous sphere of diameter 23 nm, less than 1% of the area of p(r) (the intra-particle distances) is contained in the interval [21, 23] nm. For an ellipsoid of dimensions (13, 13, 23) nm, only about 0.02% of p(r) is contained in [21, 23] nm, which makes the correct estimate of d very difficult, even for a monodisperse sample. Consequently, the estimation of d is especially difficult for elongated scatterers and an underestimation of d is to be expected. This phenomenom is even more pronounced for complex mixtures, as exemplified by the mixture of spheres shown in Fig. 4 where the estimate of d for the larger scatterer is systematically smaller for smaller volume fractions of the large scatterer, but closer to the real value as the volume fraction increases. The corresponding change in N g is in good agreement with what should be expected from this example; as the larger dimension becomes present in a larger fraction, N g is increased. The peaks in P(d) in Fig. 3(d) are well separated. A similar result is shown in Fig. 5 using a mixture of ellipsoids of two sizes. As it is more difficult to estimate the correct maximum dimension in this case, the larger ellipsoid has to be present in a larger fraction, which is to be expected from the considerations mentioned above. The influence of the measured q range is seen by comparing Figs. 5(c) and 5 (d): as q min is increased, the relative contribution to the data from the larger ellipsoid is decreased and consequently the ratio of the ellipsoids has to be increased to make both peaks appear in the evidence P(d). This is also the case for the experimental data. Here, the two species present in the solutes have different overall shapes, thus representing a more complex mixture than two different J. Appl. Cryst. (2006). 39, Vestergaard and Hansen Bayesian analysis 803

9 bodies of similar geometry but with varying radii (Fig. 6c). Again it is apparent from Fig. 6(d) that the relative heights of the two main peaks in P(d) are influenced by the truncation at q min. Therefore, unless the geometries of the scatterers are known well a priori, the presence of two peaks in the evidence P(d) should only be used as an indication for polydispersity/ degradation of the sample. The source code (Fortran77) for the program used for the present calculations is available from the correspondence author. 5. Conclusion It has been shown that Bayesian analysis applied to IFT in SAS using the conventional method of smoothness regularization may give additional information using the number of good parameters N g and the probability distribution for the maximum diameter of the scatterer. We have shown that two different scatterers can be detected by the probability distribution for the maximum diameter and that the number of good parameters calculated from experimental data may be useful e.g. for background subtraction. The extra information may be useful for experiment design as well as for data analysis. References Bernadó, P., Blanchard, L., Timmins, P., Marion, D., Ruigrok, R. W. H. & Blackledge, M. (2005). Proc. Natl Acad. Sci. USA, 102, Brunner-Popela, J. & Glatter, O. (1997). J. Appl. Cryst. 30, Dose, V. (2003). Rep. Prog. Phys. 66, Hansen, S. (2000). J. Appl. Cryst. 33, Hansen, S. & Pedersen, J. S. (1991). J. Appl. Cryst. 24, Glatter, O. (1977). J. Appl. Cryst. 10, Glatter, O. (1982). Small-Angle X-ray Scattering, edited by O. Glatter & O. Kratky. London: Academic Press. Glatter, O. & Müller, K. (1982). Makromol. Chem. 183, Gull, S. F. (1989). Maximum-Entropy and Bayesian Methods, edited by J. Skilling, pp Dordrecht: Kluwer Academic Publishers. Konarev, P. V., Petoukhov, M. V., Volkov, V. V. & Svergun, D. I. (2006). J. Appl. Cryst. 39, MacKay, D. J. C. (1992). Maximum Entropy and Bayesian Methods, Seattle, 1991, edited by C. R. Smith, G. J. Erickson & P. O. Neudorfer, pp Dordrecht: Kluwer Academic Publishers. Moore, P. B. (1980). J. Appl. Cryst. 13, Müller, J. J., Hansen, S. & Pürschel, H.-V. (1996). J. Appl. Cryst. 29, Shannon, C. E. & Weaver, W. (1949). The Mathematical Theory of Communication. Urbana: University of Illinois Press. Sivia, D. S. & Webster, J. R. P. (1998). Physica B, 248, Svergun, D. I. (1992). J. Appl. Cryst. 25, Svergun, D. I. (1999). Biophys. J. 76, Svergun, D. I. & Pedersen, J. S. (1994). J. Appl. Cryst. 27, Svergun, D. I., Semenyuk, A. V. & Feigin, L. A. (1988). Acta Cryst. A44, Tikhonov, A. N. & Arsenin, V. Ya. (1977). Solution of Ill-Posed Problems. New York: Wiley. Vestergaard, B., Jorgensen, A., Christiansen, C., Sigurdskjold, B., Svergun, D. I., Gajhede, M. & Svensson, B. (2004). Unpublished data. Zheng, W. & Doniach, S. (2005). Protein Eng. Des. Select. 18, Vestergaard and Hansen Bayesian analysis J. Appl. Cryst. (2006). 39,