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/).

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The interaction between multivalent ions and lipid membranes with saturated tails and dipolar (net neutral) headgroups can lead to adsorption of the ions onto the membrane. The ions charge the membranes and contribute to electrostatic repulsion between them, in a similar manner to membranes containing charged lipids. Using solution X-ray scattering and the osmotic stress method, we measured and modeled the pressure-distance curves between partially charged membranes containing mixtures of charged (1,2-dilauroyl-sn-glycero-3-phospho-l-serine, DLPS) and dipolar (1,2-dilauroyl-sn-glycero-3-phosphocholine, DLPC) lipids over a wide range of membrane charge densities. We then compared these pressure-distance curves with those of DLPC membranes in the presence of 10 mM CaCl. Our data and modeling show that when low osmotic stress is applied to the DLPC bilayers, the membrane charge density is equivalent to that of a charged membrane containing ca. 4 mol % DLPS and 96 mol % DLPC. As the osmotic stress increased, the charge density of the DLPC membrane decreased and resembled that of a membrane containing ca. 1 mol % DLPS. These data are consistent with desorption of the calcium ions from the DLPC membrane with increasing osmotic stress.

Block copolymer guided assembly of nanoparticles leads to the formation of nanocomposites with periodic arrangement of nanoparticles, which are important for applications such as photonic devices and sensors. However, linear block copolymers offer limited control over the internal arrangement of nanoparticles inside their hosting domains. In contrast, bottlebrush block copolymers possess unique architectural attributes that enable additional ways to control the local organization of nanoparticles. In this work, we studied the coassembly of 8 and 13 nm gold nanoparticles with three bottlebrush block copolymers differing in the asymmetry of their graft lengths. Assembly was performed in ultraconfined films, where it occurs quasi-two-dimensionally. Our results indicate that graft asymmetry could be used as an additional tool to enhance nanoparticle ordering by forcing them to localize at the center of the domain regardless of their size. This behavior is analyzed in terms of the influence of the graft asymmetry on the average conformations of the blocks.