Paul Sudeshna, Friedman Alan M, Bailey-Kellogg Chris, Craig Bruce A
Department of Statistics, Purdue University, 250 North University Street, West Lafayette, IN 47907, USA.
J Appl Crystallogr. 2013 Apr 1;46(Pt 2):404-414. doi: 10.1107/S002188981300109X. Epub 2013 Mar 5.
The interatomic distance distribution, (), is a valuable tool for evaluating the structure of a molecule in solution and represents the maximum structural information that can be derived from solution scattering data without further assumptions. Most current instrumentation for scattering experiments (typically CCD detectors) generates a finely pixelated two-dimensional image. In contin-uation of the standard practice with earlier one-dimensional detectors, these images are typically reduced to a one-dimensional profile of scattering inten-sities, (), by circular averaging of the two-dimensional image. Indirect Fourier transformation methods are then used to reconstruct () from (). Substantial advantages in data analysis, however, could be achieved by directly estimating the () curve from the two-dimensional images. This article describes a Bayesian framework, using a Markov chain Monte Carlo method, for estimating the parameters of the indirect transform, and thus (), directly from the two-dimensional images. Using simulated detector images, it is demonstrated that this method yields () curves nearly identical to the reference (). Furthermore, an approach for evaluating spatially correlated errors (such as those that arise from a detector point spread function) is evaluated. Accounting for these errors further improves the precision of the () estimation. Experimental scattering data, where no ground truth reference () is available, are used to demonstrate that this method yields a scattering and detector model that more closely reflects the two-dimensional data, as judged by smaller residuals in cross-validation, than () obtained by indirect transformation of a one-dimensional profile. Finally, the method allows concurrent estimation of the beam center and , the longest interatomic distance in (), as part of the Bayesian Markov chain Monte Carlo method, reducing experimental effort and providing a well defined protocol for these parameters while also allowing estimation of the covariance among all parameters. This method provides parameter estimates of greater precision from the experimental data. The observed improvement in precision for the traditionally problematic is particularly noticeable.
原子间距离分布函数(P(r))是评估溶液中分子结构的重要工具，它代表了无需进一步假设就能从溶液散射数据中获得的最大结构信息。目前大多数用于散射实验的仪器（通常是电荷耦合器件探测器）会生成精细像素化的二维图像。延续早期一维探测器的标准做法，这些图像通常通过对二维图像进行圆形平均处理，简化为一维散射强度分布曲线(I(q))。然后使用间接傅里叶变换方法从(I(q))重建(P(r))。然而，通过直接从二维图像估计(P(r))曲线，在数据分析中可以获得显著优势。本文描述了一种贝叶斯框架，使用马尔可夫链蒙特卡罗方法，直接从二维图像估计间接变换的参数，从而得到(P(r))。通过模拟探测器图像表明，该方法得到的(P(r))曲线与参考曲线几乎相同。此外，还评估了一种用于评估空间相关误差（例如由探测器点扩散函数引起的误差）的方法。考虑这些误差进一步提高了(P(r))估计的精度。在没有真实参考(P(r))的情况下，使用实验散射数据证明，与通过一维轮廓间接变换获得的(P(r))相比，该方法产生的散射和探测器模型更能紧密反映二维数据，这可以通过交叉验证中较小的残差来判断。最后，作为贝叶斯马尔可夫链蒙特卡罗方法的一部分，该方法允许同时估计光束中心以及(P(r))中的最长原子间距离(D_{max})，减少了实验工作量，并为这些参数提供了明确的协议，同时还允许估计所有参数之间的协方差。该方法从实验数据中提供了精度更高的参数估计。对于传统上有问题的(D_{max})，观察到的精度提高尤为明显。
