Insight into structural biophysics from solution X-ray scattering.

The current challenges of structural biophysics include determining the structure of large self-assembled complexes, resolving the structure of ensembles of complex structures and their mass fraction, and unraveling the dynamic pathways and mechanisms leading to the formation of complex structures from their subunits. Modern synchrotron solution X-ray scattering data enable simultaneous high-spatial and high-temporal structural data required to address the current challenges of structural biophysics. These data are complementary to crystallography, NMR, and cryo-TEM data. However, the analysis of solution scattering data is challenging; hence many different analysis tools, listed in the SAS Portal (http://smallangle.org/), were developed. In this review, we start by briefly summarizing classical X-ray scattering analyses providing insight into fundamental structural and interaction parameters. We then describe recent developments, integrating simulations, theory, and advanced X-ray scattering modeling, providing unique insights into the structure, energetics, and dynamics of self-assembled complexes. The structural information is essential for understanding the underlying physical chemistry principles leading to self-assembled supramolecular architectures and computational structural refinement.

Mechanism of the Initial Tubulin Nucleation Phase.

Tubulin nucleation is a highly frequent event in microtubule (MT) dynamics but is poorly understood. In this work, we characterized the structural changes during the initial nucleation phase of dynamic tubulin. Using size-exclusion chromatography-eluted tubulin dimers in an assembly buffer solution free of glycerol and tubulin aggregates enabled us to start from a well-defined initial thermodynamic ensemble of isolated dynamic tubulin dimers and short oligomers. Following a temperature increase, time-resolved X-ray scattering and cryo-transmission electron microscopy during the initial nucleation phase revealed an isodesmic assembly mechanism of one-dimensional (1D) tubulin oligomers (where dimers were added and/or removed one at a time), leading to sufficiently stable two-dimensional (2D) dynamic nanostructures, required for MT assembly. A substantial amount of tubulin octamers accumulated before two-dimensional lattices appeared. Under subcritical assembly conditions, we observed a slower isodesmic assembly mechanism, but the concentration of 1D oligomers was insufficient to form the multistranded 2D nucleus required for MT formation.

Mechanism of Tubulin Oligomers and Single-Ring Disassembly Catastrophe.

Cold tubulin dimers coexist with tubulin oligomers and single rings. These structures are involved in microtubule assembly; however, their dynamics are poorly understood. Using state-of-the-art solution synchrotron time-resolved small-angle X-ray scattering, we discovered a disassembly catastrophe (half-life of ∼0.1 s) of tubulin rings and oligomers upon dilution or addition of guanosine triphosphate. A slower disassembly (half-life of ∼38 s) was observed following an increase in temperature. Our analysis showed that the assembly and disassembly processes were consistent with an isodesmic mechanism, involving a sequence of reversible reactions in which dimers were rapidly added or removed one at a time, terminated by a 2 order-of-magnitude slower ring-closing/opening step. We revealed how assembly conditions varied the mass fraction of tubulin in each of the coexisting structures, the rate constants, and the standard Helmholtz free energies for closing a ring and for longitudinal dimer-dimer associations.

Structure and Energetics of GTP- and GDP-Tubulin Isodesmic Self-Association.

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 kBT (-8.0 ± 0.2 kcal mol-1) and -13.1 ± 0.5 kBT (-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.

D+: software for high-resolution hierarchical modeling of solution X-ray scattering from complex structures.

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/).

Structure, Assembly, and Disassembly of Tubulin Single Rings.

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, D+ . 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.

Structure of Dynamic, Taxol-Stabilized, and GMPPCP-Stabilized Microtubule.

Microtubule (MT) is made of αß-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.

Structure and Intermolecular Interactions between L-Type Straight Flagellar Filaments.

Bacterial mobility is powered by rotation of helical flagellar filaments driven by rotary motors. Flagellin isolated from the Salmonella Typhimurium SJW1660 strain, which differs by a point mutation from the wild-type strain, assembles into straight filaments in which flagellin monomers are arranged in a left-handed helix. Using small-angle x-ray scattering and osmotic stress methods, we investigated the structure of SJW1660 flagellar filaments as well as the intermolecular forces that govern their assembly into dense hexagonal bundles. The scattering data were fitted to models, which took into account the atomic structure of the flagellin subunits. The analysis revealed the exact helical arrangement and the super-helical twist of the flagellin subunits within the filaments. Under osmotic stress, the filaments formed two-dimensional hexagonal bundles. Monte Carlo simulations and continuum theories were used to analyze the scattering data from hexagonal arrays, revealing how the bundle bulk modulus and the deflection length of filaments in the bundles depend on the applied osmotic stress. Scattering data from aligned flagellar bundles confirmed the theoretically predicated structure-factor scattering peak line shape. Quantitative analysis of the measured equation of state of the bundles revealed the contributions of electrostatic, hydration, and elastic interactions to the intermolecular forces associated with bundling of straight semi-flexible flagellar filaments.

Reciprocal Grids: A Hierarchical Algorithm for Computing Solution X-ray Scattering Curves from Supramolecular Complexes at High Resolution.

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.

Transformation of taxol-stabilized microtubules into inverted tubulin tubules triggered by a tubulin conformation switch.

Bundles of taxol-stabilized microtubules (MTs)--hollow tubules comprised of assembled αß-tubulin heterodimers--spontaneously assemble above a critical concentration of tetravalent spermine and are stable over long times at room temperature. Here we report that at concentrations of spermine several-fold higher the MT bundles (B(MT)) quickly become unstable and undergo a shape transformation to bundles of inverted tubulin tubules (B(ITT)), the outside surface of which corresponds to the inner surface of the B(MT) tubules. Using transmission electron microscopy and synchrotron small-angle X-ray scattering, we quantitatively determined both the nature of the B(MT)-to-B(ITT) transformation pathway, which results from a spermine-triggered conformation switch from straight to curved in the constituent taxol-stabilized tubulin oligomers, and the structure of the B(ITT) phase, which is formed of tubules of helical tubulin oligomers. Inverted tubulin tubules provide a platform for studies requiring exposure and availability of the inside, luminal surface of MTs to MT-targeted drugs and MT-associated proteins.

RNA encapsidation by SV40-derived nanoparticles follows a rapid two-state mechanism.

Remarkably, uniform virus-like particles self-assemble in a process that appears to follow a rapid kinetic mechanism. The mechanisms by which spherical viruses assemble from hundreds of capsid proteins around nucleic acid, however, are yet unresolved. Using time-resolved small-angle X-ray scattering (TR-SAXS), we have been able to directly visualize SV40 VP1 pentamers encapsidating short RNA molecules (500mers). This assembly process yields T = 1 icosahedral particles comprised of 12 pentamers and one RNA molecule. The reaction is nearly one-third complete within 35 ms, following a two-state kinetic process with no detectable intermediates. Theoretical analysis of kinetics, using a master equation, shows that the assembly process nucleates at the RNA and continues by a cascade of elongation reactions in which one VP1 pentamer is added at a time, with a rate of approximately 10(9) M(-1) s(-1). The reaction is highly robust and faster than the predicted diffusion limit. The emerging molecular mechanism, which appears to be general to viruses that assemble around nucleic acids, implicates long-ranged electrostatic interactions. The model proposes that the growing nucleo-protein complex acts as an electrostatic antenna that attracts other capsid subunits for the encapsidation process.

Entropic attraction condenses like-charged interfaces composed of self-assembled molecules.

Like-charged solid interfaces repel and separate from one another as much as possible. Charged interfaces composed of self-assembled charged-molecules such as lipids or proteins are ubiquitous. The present study shows that although charged lipid-membranes are sufficiently rigid, in order to swell as much as possible, they deviate markedly from the behavior of typical like-charged solids when diluted below a critical concentration (ca. 15 wt %). Unexpectedly, they swell into lamellar structures with spacing that is up to four times shorter than the layers should assume (if filling the entire available space). This process is reversible with respect to changing the lipid concentration. Additionally, the research shows that, although the repulsion between charged interfaces increases with temperature, like-charged membranes, remarkably, condense with increasing temperature. This effect is also shown to be reversible. Our findings hold for a wide range of conditions including varying membrane charge density, bending rigidity, salt concentration, and conditions of typical living systems. We attribute the limited swelling and condensation of the net repulsive interfaces to their self-assembled character. Unlike solids, membranes can rearrange to gain an effective entropic attraction, which increases with temperature and compensates for the work required for condensing the bilayers. Our findings provide new insight into the thermodynamics and self-organization of like-charged interfaces composed of self-assembled molecules such as charged biomaterials and supramolecular assemblies that are widely found in synthetic and natural constructs.

Effect of temperature on the structure of charged membranes.

Interactions between charged and neutral self-assembled phospholipid membranes are well understood and take into account temperature dependence. Yet, the manner in which the structure of the membrane is affected by temperature was hardly studied. Here we study the effect of temperature on the thickness, area per lipid, and volume per lipid of charged membranes. Two types of membranes were studied: membranes composed of charged lipids and dipolar (neutral) membranes that adsorbed divalent cations and became charged. Small-angle X-ray scattering data demonstrate that the thickness of charged membranes decreases with temperature. Wide-angle X-ray scattering data show that the area per headgroup increases with temperature. Intrinsically charged membranes linearly thin with temperature, whereas neutral membranes that adsorb divalent ions and become charged show an exponential decrease of their thickness. The data indicate that, on average, the tails shorten as the temperature rises. We attribute this behavior to higher lipid tail entropy and to the weaker electrostatic screening of the charged headgroups, by their counterions, at elevated temperatures. The latter effect leads to stronger electrostatic repulsion between the charged headgroups that increases the area per headgroup and decreases the bilayer thickness.

Solution X-ray scattering form factors of supramolecular self-assembled structures.

In this paper, the analysis of several involved models, relevant for evaluating solution X-ray scattering form factors of supramolecular self-assembled structures, is presented. Different geometrical models are discussed, and the scattering form factors of several layers of those shapes are evaluated. The thickness and the electron density of each layer are parameters in those models. The models include Gaussian electron density profiles and/or uniform electron density profiles at each layer. Various forms of cuboid, layered, spherical, cylindrical, and helical structures are carefully treated. The orientation-averaged scattering intensities of those form factors are calculated. Similar classes of form factors are examined and compared, and their fit to scattering data of lipid bilayers, capsids of the Simian virus 40 virus-like particle and microtubule is discussed. A more detailed model of discrete helices composed of uniform spheres was derived and compared to solution X-ray scattering data of microtubules. Our analyses show that when high-resolution data are available the more detailed models with Gaussian electron density profiles or helical structures composed of spheres should be used to better capture all the elements in the scattering curves. The models presented in this paper may also be applied, with minor corrections, for the analysis of solution neutron scattering data.