3D Reconstruction of Icosahedral Viruses from X-ray Scattering Data

拟议研究的目的是开发基于x射线散射测量的确定小球形病毒结构的新方法。这方面的研究是及时的，因为同步加速器可以测量质量优越的数据，但为了充分利用这些数据，需要新的方法。例如，将有可能在溶液中研究病毒动力学，例如成熟和分解；x射线束的高强度允许在大约100毫秒内记录溶液散射模式，并且数据的质量，就进行低角度散射测量的能力和探测器的灵敏度而言，非常高，因此可以从单个100毫秒的快照中重建低分辨率结构，因此可以跟踪结构的演变。这是电气工程学院的首席研究员和生物科学系的John E. Johnson教授之间的跨学科合作。它将电子工程方面的图像处理、应用数学分析和数值计算专业知识与生物科学方面的生物、生物物理学和x射线散射实验专业知识相结合。贯穿所有提出的工作的主要主题是需要利用小球形病毒中存在的二十面体对称性。首席研究员在二十面体谐波方面有了新的数学结果，基于这些结果的方法将允许以比过去更自然和更有计算吸引力的方式利用对称性的重建方法的发展。提出了三个方面的工作：从溶液x射线散射模式重建3D低分辨率病毒结构：提出了一个病毒模型，该模型精确地展示了二十面体对称。提出了一种迭代重构算法，这是对标准电子密度修正算法的重要推广。意义：由于不需要结晶，成功的重组将使病毒结构调查变得容易，成本低；将为基于晶体x射线衍射数据的高分辨率结构重建提供新的起点；最重要的是，这将使研究病毒在自然溶液环境中的成熟和分解成为可能。核酸包装和核酸-蛋白相互作用：小球形病毒为研究这些问题提供了一个优雅而相对简单的模型系统。然而，因为体积大；病毒中的核酸不表现出二十面体对称，在高分辨率x射线衍射实验中，蛋白质在原子分辨率下可见，而核酸则不可见。因此，为了从x射线衍射实验中最大限度地提取蛋白质和核酸的信息，有必要对蛋白质和核酸使用不同的模型。提出了该模型和利用低分辨率晶体x射线衍射数据估计模型参数的最小二乘算法。意义：成功重建将验证难以测量的低分辨率晶体x射线衍射数据，并提供基于x射线衍射而不是基于冷冻电镜数据的核酸分布及其与蛋白质衣壳相互作用的另一种观点。使用不完整晶体x射线衍射数据集进行细化：在许多前沿问题中，由于病毒数量有限、同步加速器时间有限或无法可靠地培养高质量晶体，数据集是不完整的。由于病毒粒子具有二十面体对称性，因此衍射数据也具有二十面体对称性，在最简单的情况下，这意味着数据是60倍冗余的。5倍冗余的数据通常足以完善结构。因此，大量的数据（例如，80%）可能会丢失，但应该仍然有可能改进结构。然而，标准的算法和软件要求记录一套完整的数据。因此，在60倍冗余的基础上，对不完整数据集进行插值，生成完整数据集。然而，目前的插值方法没有利用数据的二十面体对称性，因此不准确。我们建议开发和实现更复杂的插值方法，将利用二十面体对称。意义：这项工作的成功将允许从现有的以前不可用的数据集中解决新的结构。

The objective of the proposed research is the development of new methods for determining the structure of small spherical viruses based on x-ray scattering measurements. Research in this area is timely because synchrotrons allow the measurement of qualitatively superior data and new methods are needed in order to fully exploit the data. For instance, it will be possible to study virus dynamics, e.g., maturation and disassembly, in solution: the high intensity of the x-ray beam allows the recording of a solution scattering pattern in roughly 100 ms and the quality of the data, in terms of the ability to make low-angle scattering measurements and the sensitivity of the detectors, is very high so that a low-resolution structure can be reconstructed from a single 100 ms snap shot and therefore the evolution of the structure can be tracked. This is an interdisciplinary collaboration between the principal investigator, in the School of Electrical Engineering, and Professor John E. Johnson, in the Department of Biological Sciences. It combines image processing, applied mathematical analysis, and numerical computation expertise from the electrical engineering side with biological, biophysical, and x-ray scattering experimental expertise from the biological sciences side. The major theme that extends through all of the proposed work is the need to exploit the icosahedral symmetry present in small spherical viruses. The principal investigator has new mathematical results in icosahedral harmonics and methodology based on these results will allow the development of reconstruction methods which exploit the symmetry in a much more natural and computationally attractive fashion than has been possible in the past. Work is proposed in three areas: Reconstruction of 3D low-resolution viral structure from solution x-ray scattering patterns: A model of the virus is proposed which exactly exhibits the icosahedral symmetry. An iterative recon struction algori thm, which is an significant generalization of the standard electron density modification algorithm, is proposed. Significance: Successful reconstruction would allow easy low-cost surveys of virus structure since no crystallization is required; would provide new starting points for reconstructions of high-resolution structures based on crystal x-ray diffraction data; and, most importantly, would enable the study of virus maturation and disassembly in the natural solution environment. Nucleic acid packing and nucleic acid-protein interactions: Small spherical viruses provide an elegant and relatively simple model system for studying these problems. However, because the bulk; of the nucleic acid in a virus does not exhibit icosahedral symmetry, the protein is visible at atomic resolution in a high-resolution x- ray diffraction experiment while the nucleic acid is not. Therefore, in order to extract the maximum amount of information concerning both protein and nucleic acid from the x-ray diffraction experiment, it is necessary to use different models for the protein and nucleic acid. Such models and least-squares type algorithms for estimating the parameters in the models from low-resolution crystal x-ray diffraction data are proposed. Significance: Successful reconstruction will validate the low-resolution crystal x-ray diffraction data, which is difficult to measure, and provide an alternative view of the nucleic acid distribution and its interaction with the protein capsid based on x-ray diffraction rather than cryo electron microscopy data. Refinement using incomplete crystal x-ray diffraction data sets: In many cutting-edge problems the data set is incomplete due to a limited amount of virus, limited synchrotron time, or inability to reliably grow high-quality crystals. Because the virus particle has icosahedral symmetry, it follows that the diffraction data also has icosahedral symmetry which, in the simplest case, implies that the data is 60-fold redundant. Data that is 5-fold redundant is typically sufficient to refine a structure. Therefore, large amounts of the data (e.g., 80%) can be missing and it should still be possible to refine the structure. However, standard algorithms and software require that a complete set of data be recorded. Therefore, based on the 60-fold redundancy, the incomplete data set is interpolated to generate a complete data set. However, the current interpolation methods do not exploit the icosahedral symmetry of the data and are therefore inaccurate. We propose to develop and implement more sophisticated interpolation methods that will exploit the icosahedral symmetry. Significance: Success in this work will allow the solution of new structures from existing previously unusable data sets.