In many biochemical processes large biomolecular assemblies play important roles. X-ray scattering is a label-free bulk method that can probe the structure of large self-assembled complexes in solution. As we demonstrate in this paper, solution X-ray scattering can measure complex supramolecular assemblies at high sensitivity and resolution. At high resolution, however, data analysis of larger complexes is computationally demanding. We present an efficient method to compute the scattering curves from complex structures over a wide range of scattering angles. In our computational method, structures are defined as hierarchical trees in which repeating subunits are docked into their assembly symmetries, describing the manner subunits repeat in the structure (in other words, the locations and orientations of the repeating subunits). The amplitude of the assembly is calculated by computing the amplitudes of the basic subunits on 3D reciprocal-space grids, moving up in the hierarchy, calculating the grids of larger structures, and repeating this process for all the leaves and nodes of the tree. For very large structures, we developed a hybrid method that sums grids of smaller subunits in order to avoid numerical artifacts. We developed protocols for obtaining high-resolution solution X-ray scattering data from taxol-free microtubules at a wide range of scattering angles. We then validated our method by adequately modeling these high-resolution data. The higher speed and accuracy of our method, over existing methods, is demonstrated for smaller structures: short microtubule and tobacco mosaic virus. Our algorithm may be integrated into various structure prediction computational tools, simulations, and theoretical models, and provide means for testing their predicted structural model, by calculating the expected X-ray scattering curve and comparing with experimental data.

Single and double tubulin rings were studied under a range of conditions and during microtubule (MT) assembly and disassembly. Here, tubulin was purified from porcine brain and used without any further modifications or additives that promote ring assembly. The structure of single GDP-rich tubulin rings was determined by cryo-transmission electron microscopy and synchrotron solution X-ray scattering. The scattering curves were fitted to atomic models, using our state-of-the-art analysis software, . We found that there is a critical concentration for ring formation, which increased with GTP concentration with temperature. MT assembly or disassembly, induced by changes in temperature, was analyzed by time-resolved small-angle X-ray scattering. During MT assembly, the fraction of rings and unassembled dimers simultaneously decreased. During MT disassembly, the mass fraction of dimers increased. The increase in the concentration of rings was delayed until the fraction of dimers was sufficiently high. We verified that pure dimers, eluted via size-exclusion chromatography, could also form rings. Interestingly, X-ray radiation triggered tubulin ring disassembly. The concentration of disassembled rings versus exposure time followed a first-order kinetics. The disassembly rate constant and initial concentration were determined. X-ray radiation-triggered disassembly was used to determine the concentration of rings. We confirmed that following a temperature jump, the mass fraction of rings decreased and then stabilized at a constant value during the first stage of the MT assembly kinetics. This study sheds light on the most basic assembly and disassembly conditions for in vitro single GDP-rich tubulin rings and their relation to MT kinetics.

This paper presents the computer program D+ (https://scholars.huji.ac.il/ uriraviv/book/d-0), where the reciprocal-grid (RG) algorithm is implemented. D+ efficiently computes, at high-resolution, the X-ray scattering curves from complex structures that are isotropically distributed in random orientations in solution. Structures are defined in hierarchical trees in which subunits can be represented by geometric or atomic models. Repeating subunits can be docked into their assembly symmetries, describing their locations and orientations in space. The scattering amplitude of the entire structure can be calculated by computing the amplitudes of the basic subunits on 3D reciprocal-space grids, moving up in the hierarchy, calculating the RGs of the larger structures, and repeating this process for all the leaves and nodes of the tree. For very large structures (containing over 100 protein subunits), a hybrid method can be used to avoid numerical artifacts. In the hybrid method, only grids of smaller subunits are summed and used as subunits in a direct computation of the scattering amplitude. D+ can accurately analyze both small-and wide-angle solution X-ray scattering data. This article describes how D+ applies the RG algorithm, accounts for rotations and translations of subunits, processes atomic models, accounts for the contribution of the solvent as well as the solvation layer of complex structures in a scalable manner, writes and accesses RGs, interpolates between grid points, computes numerical integrals, enables the use of scripts to define complicated structures, applies fitting algorithms, accounts for several coexisting uncorrelated populations, and accelerates computations using GPUs. D+ may also account for different X-ray energies to analyze anomalous solution X-ray scattering data. An accessory tool that can identify repeating subunits in a Protein Data Bank file of a complex structure is provided. The tool can compute the orientation and translation of repeating subunits needed for exploiting the advantages of the RG algorithm in D+. A Python wrapper (https://scholars. huji.ac.il/uriraviv/book/python-api) is also available, enabling more advanced computations and integration of D+ with other computational tools. Finally, a large number of tests are presented. The results of D+ are compared with those of other programs when possible, and the use of D+ to analyze solution scattering data from dynamic microtubule structures with different protofilament number is demonstrated. D+ and its source code are freely available for academic users and developers (https://bitbucket.org/uriraviv/public-dplus/src/ master/).

Tubulin self-association is a critical process in microtubule dynamics. The early intermediate structures, energetics, and their regulation by fluxes of chemical energy, associated with guanosine triphosphate (GTP) hydrolysis, are poorly understood. We reconstituted an in vitro minimal model system, mimicking the key elements of the nontemplated tubulin assembly. To resolve the distribution of GTP- and guanosine diphosphate (GDP)-tubulin structures, at low temperatures (∼10 °C) and below the critical concentration for the microtubule assembly, we analyzed in-line size-exclusion chromatography–small-angle X-ray scattering (SEC-SAXS) chromatograms of GTP- and GDP-tubulin solutions. Both solutions rapidly attained steady state. The SEC-SAXS data were consistent with an isodesmic thermodynamic model of longitudinal tubulin self-association into 1D oligomers, terminated by the formation of tubulin single rings. The analysis showed that free dimers coexisted with tetramers and hexamers. Tubulin monomers and lateral association between dimers were not detected. The dimer–dimer longitudinal self-association standard Helmholtz free energies were −14.2 ± 0.4 k B T (−8.0 ± 0.2 kcal mol–1) and −13.1 ± 0.5 k B T (−7.4 ± 0.3 kcal mol–1) for GDP- and GTP-tubulin, respectively. We then determined the mass fractions of dimers, tetramers, and hexamers as a function of the total tubulin concentration. A small fraction of stable tubulin single rings, with a radius of 19.2 ± 0.2 nm, was detected in the GDP-tubulin solution. In the GTP-tubulin solution, this fraction was significantly lower. Cryo-TEM images and SEC-multiangle light-scattering analysis corroborated these findings. Our analyses provide an accurate structure–stability description of cold tubulin solutions.

Microtubule (MT) is made of αβ-tubulin heterodimers that dynamically assemble into a hollow nanotube composed of straight protofilaments. MT dynamics is facilitated by hydrolysis of guanosine-5'-triphosphate (GTP) and can be inhibited by either anticancer agents like taxol or the nonhydrolyzable GTP analogues like GMPPCP. Using high-resolution synchrotron X-ray scattering, we have measured and analyzed the scattering curves from solutions of dynamic MT (in other words, in the presence of excess GTP and free of dynamic-inhibiting agents) and examined the effect of two MT stabilizers: taxol and GMPPCP. Previously, we have analyzed the structure of dynamic MT by docking the atomic model of tubulin dimer onto a 3-start left handed helical lattice, derived from the PDB ID 3J6F . 3J6F corresponds to a MT with 14 protofilaments. In this paper, we took into account the possibility of having MT structures containing between 12 and 15 protofilaments. MTs with 12 protofilaments were never observed. We determined the radii, the pitch, and the distribution of protofilament number that best fit the scattering data from dynamic MT or stabilized MT by taxol or GMPPCP. We found that the protofilament number distribution shifted when the MT was stabilized. Taxol increased the mass fraction of MT with 13 protofilaments and decreased the mass fraction of MT with 14 protofilaments. GMPPCP reduced the mass fraction of MT with 15 protofilaments and increased the mass fraction of MT with 14 protofilaments. The pitch, however, remained unchanged regardless of whether the MT was dynamic or stabilized. Higher tubulin concentrations increased the fraction of dynamic MT with 14 protofilaments.