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.

Dipolar lipid membranes may adsorb multivalent ions. The binding constant depends on the type of lipid and ions. In this paper, we focus on the adsorption of calcium ions onto 1,2-dilauroylphosphatidylcholine (DLPC) membrane. Using small-angle-X-ray scattering we found that at ambient room temperature ca. 0.6 mM CaCl2 is a critical concentration at which calcium ions adsorbed to 30 mg/mL (ca. 48 mM) DLPC membrane. We then determined the structure of the lamellar phases formed at CaCl2 concentrations below and above the critical concentration and characterized the effect of temperature and incubation time on the adsorption process. Our findings suggest that calcium adsorption to DLPC membranes requires an initial nucleation phase.

When aqueous salt solutions contain multivalent ions (like Ca 2+ or Mg 2+ ), strong correlation effects may lead to ionbridging, net attraction, and tight-coupling between like-charged interfaces. To examine the effects of surface charge density, temperature, salt type, and salt concentration on the structures of tightly coupled charged interfaces, we have used mixed lipid membranes, containing either saturated or unsaturated tails in the presence of multivalent ions. We discovered that tightly coupled membrane lamellar phases, dominated by attractive interactions, coexisted with weakly coupled lamellar phases, dominated by repulsive interactions. To control the membrane charge density, we mixed lipids with negatively charged headgroups, DLPS and DOPS, with their zwitterionic analogue having the same tails, DLPC and DOPC, respectively. Using solution X-ray scattering we measured the lamellar repeat distance, D, at different ion concentrations, temperatures, and membrane charge densities. The multivalent ions tightly coupled the mixed lipid bilayers whose charged lipid molar fraction was between 0.1 and 1. The repeat distance of the tightly coupled phase was about 4 nm for the DLPS/DLPC mixtures and about 5 nm for the DOPS/DOPC mixtures. In this phase, the repeat distance slightly increased with increasing temperature and decreased with increasing charge density. When the molar fraction of charged lipid was 0.1 or 0.25, a less tightly coupled phase coexisted with the tightly coupled phase. The weakly coupled lamellar phase had significantly larger D values, although they were consistently shorter than the D values in monovalent salt solutions with similar screening lengths.

The interaction between lipid membranes and ions is associated with a range of key physiological processes. Most earlier studies have focused on the interaction of lipids with cations, while the specific effects of the anions have been largely overlooked. Owing to dissolved atmospheric carbon dioxide, bicarbonate is an important ubiquitous anion in aqueous media. In this paper, we report on the effect of bicarbonate anions on the interactions between dipolar lipid membranes in the presence of previously adsorbed calcium cations. Using a combination of solution X-ray scattering, osmotic stress, and molecular dynamics simulations, we followed the interactions between 1,2-didodecanoyl- sn -glycero-3-phosphocholine (DLPC) lipid membranes that were dialyzed against CaCl 2 solutions in the presence and absence of bicarbonate anions. Calcium cations adsorbed onto DLPC membranes, charge them, and lead to their swelling. In the presence of bicarbonate anions, however, the calcium cations can tightly couple one dipolar DLPC membrane to the other and form a highly condensed and dehydrated lamellar phase with a repeat distance of 3.45 ± 0.02 nm. Similar tight condensation and dehydration has only been observed between charged membranes in the presence of multivalent counterions. Bridging between bilayers by calcium bicarbonate complexes induced this arrangement. Furthermore, in this condensed phase, lipid molecules and adsorbed ions were arranged in a two-dimensional oblique lattice.