Martin Cramer Pedersen

Small-angle scattering gives direct structural information about a membrane protein inside a lipid environment.

Monomeric bacteriorhodopsin (bR) reconstituted into POPC/POPG-containing nanodiscs was investigated by combined small-angle neutron and X-ray scattering. A novel hybrid approach to small-angle scattering data analysis was developed. In combination, these provided direct structural insight into membrane-protein localization in the nanodisc and into the protein-lipid interactions. It was found that bR is laterally decentred in the plane of the disc and is slightly tilted in the phospholipid bilayer. The thickness of the bilayer is reduced in response to the incorporation of bR. The observed tilt of bR is in good accordance with previously performed theoretical predictions and computer simulations based on the bR crystal structure. The result is a significant and essential step on the way to developing a general small-angle scattering-based method for determining the low-resolution structures of membrane proteins in physiologically relevant environments.

WillItFit: a framework for fitting of constrained models to small-angle scattering data

A software framework for analysis of small-angle scattering data is presented. On the basis of molecular constraints and prior knowledge of the chemical composition of the sample, the software is capable of simultaneously fitting small-angle X-ray and neutron scattering data to analytical or semi-analytical models of biomacromolecules. The software features various fitting routines along with the possibility of incorporating instrumental resolution effects on the fit. Finally, trust region estimation, based on the profile likelihood strategy, is implemented. The algorithms and models are written in C, whereas the user interface is written in Python. Parallelization is implemented using the OpenMP extensions to C. The source code is available for free upon request or via the associated code repository. The software runs on Linux, Windows and OSX and is available as an open-source initiative published under the General Publishing License.

Small-Angle X-Ray Scattering of the Cholesterol Incorporation into Human ApoA1-POPC Discoidal Particles

Structural and functional aspects of high-density lipoproteins have been studied for over half a century. Due to the plasticity of this highly complex system, new aspects continue to be discovered. Here, we present a structural study of the human Apolipoprotein A1 (ApoA1) and investigate the role of its N-terminal domain, the so-called globular domain of ApoA1, in discoidal complexes with phospholipids and increasing amounts of cholesterol. Using a combination of solution-based small-angle x-ray scattering (SAXS) and molecular constrained data modeling, we show that the ApoA1-1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC)-based particles are disk shaped with an elliptical cross section and composed by a central lipid bilayer surrounded by two stabilizing ApoA1 proteins. This structure is very similar to the particles formed in the so-called nanodisc system, which is based on N-terminal truncated ApoA1 protein. Although it is commonly agreed that the nanodisc is plain disk shaped, several more advanced structures have been proposed for the full-length ApoA1 in combination with POPC and cholesterol. This prompted us to make a detailed comparative study of the ApoA1 and nanodisc systems upon cholesterol uptake. Based on the presented SAXS analysis it is found that the N-terminal domains of ApoA1-POPC-cholesterol particles are not globular but instead an integrated part of the protein belt stabilizing the particles. Upon incorporation of increasing amounts of cholesterol, the presence of the N-terminal domain allows the bilayer thickness to increase while maintaining an overall flat bilayer structure. This is contrasted by the energetically more strained and less favorable lens shape required to fit the SAXS data from the N-terminal truncated nanodisc system upon cholesterol incorporation. This suggests that the N-terminal domain of ApoA1 actively participates in the stabilization of the ApoA1-POPC-cholesterol discoidal particle and allows for a more optimal lipid packing upon cholesterol uptake.

PET/CT Based In Vivo Evaluation of 64Cu Labelled Nanodiscs in Tumor Bearing Mice.

64Cu radiolabelled nanodiscs based on the 11 α-helix MSP1E3D1 protein and 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine lipids were, for the first time, followed in vivo by positron emission tomography for evaluating the biodistribution of nanodiscs. A cancer tumor bearing mouse model was used for the investigations, and it was found that the approximately 13 nm nanodiscs, due to their size, permeate deeply into cancer tissue. This makes them promising candidates for both drug delivery purposes and as advanced imaging agents. For the radiolabelling, a simple approach for 64Cu radiolabelling of proteins via a chelating agent, DOTA, was developed. The reaction was performed at sufficiently mild conditions to be compatible with labelling of the protein part of a lipid-protein particle while fully conserving the particle structure including the amphipathic protein fold.

Quantification of the information in small-angle scattering data

Small-angle X-ray and neutron scattering have become increasingly popular owing to improvements in instrumentation and developments in data analysis, sample handling and sample preparation. For some time, it has been suggested that a more systematic approach to the quantification of the information content in small-angle scattering data would allow for a more optimal experiment planning and a more reliable data analysis. In the present article, it is shown how ray-tracing techniques in combination with a statistically rigorous data analysis provide an appropriate platform for such a systematic quantification of the information content in scattering data. As examples of applications, it is shown how the exposure time at different instrumental settings or contrast situations can be optimally prioritized in an experiment. Also, the gain in information by combining small-angle X-ray and neutron scattering is assessed. While solution small-angle scattering data of proteins and protein–lipid complexes are used as examples in the present case study, the approach is generalizable to a wide range of other samples and experimental techniques. The source code for the algorithms and ray-tracing components developed for this study has been made available on-line.

Tiling patterns from ABC star molecules: 3-colored foams?

We present coarse-grained simulations of the self-assembly of 3-armed ABC star polyphiles. In systems of star polyphiles with two arms of equal length the simulations corroborate and expand previous findings from related miktoarm star terpolymer systems on the formation of patterns containing columnar domains whose sections are 2D planar tilings. However, the systematic variation of face topologies as the length of the third (unequal) arm is varied differs from earlier findings regarding the compositional dependence. We explore 2D 3-colored foams to establish the optimal patterns based on interfacial energy alone. A generic construction algorithm is described that accounts for all observed 2D tiling patterns and suggests other patterns likely to be found beyond the range of the simulations reported here. Patterns resulting from this algorithm are relaxed using Surface Evolver calculations to form 2D foams with minimal interfacial length as a function of composition. This allows us to estimate the interfacial enthalpic contributions to the free energy of related star molecular assemblies assuming strong segregation. We compare the resulting phase sequence with a number of theoretical results from particle-based simulations and field theory, allowing us to tease out relative enthalpic and entropic contributions as a function of the chain lengths making up the star molecules. Our results indicate that a richer polymorphism is to be expected in systems not dominated by chain entropy. Further, analysis of corresponding planar tiling patterns suggests that related two-periodic columnar structures are unlikely hypothetical phases in 4-arm star polyphile melts in the absence of sufficient arm configurational freedom for minor domains to form lens-shaped di-gons, which require higher molecular weight polymeric arms. Finally, we discuss the possibility of forming a complex tiling pattern that is a quasi-crystalline approximant for 3-arm star polyphiles with unequal arm lengths.

Invisible detergents for structure determination of membrane proteins by small-angle neutron scattering

A novel and generally applicable method for determining structures of membrane proteins in solution via small‐angle neutron scattering (SANS) is presented. Common detergents for solubilizing membrane proteins were synthesized in isotope‐substituted versions for utilizing the intrinsic neutron scattering length difference between hydrogen and deuterium. Individual hydrogen/deuterium levels of the detergent head and tail groups were achieved such that the formed micelles became effectively invisible in heavy water (D2O) when investigated by neutrons. This way, only the signal from the membrane protein remained in the SANS data. We demonstrate that the method is not only generally applicable on five very different membrane proteins but also reveals subtle structural details about the sarco/endoplasmatic reticulum Ca2+ ATPase (SERCA). In all, the synthesis of isotope‐substituted detergents makes solution structure determination of membrane proteins by SANS and subsequent data analysis available to nonspecialists.

Hyperbolic crystallography of two-periodic surfaces and associated structures

This paper describes the families of the simplest, two-periodic constant mean curvature surfaces, the genus-two HCB and SQL surfaces, and their isometries. All the discrete groups that contain the translations of the genus-two surfaces embedded in Euclidean three-space modulo the translation lattice are derived and enumerated. Using this information, the subgroup lattice graphs are constructed, which contain all of the group-subgroup relations of the aforementioned quotient groups. The resulting groups represent the two-dimensional representations of subperiodic layer groups with square and hexagonal supergroups, allowing exhaustive enumeration of tilings and associated patterns on these surfaces. Two examples are given: a two-periodic [3,7]-tiling with hyperbolic orbifold symbol {\sf {2223}} and a {\sf {22222}} surface decoration.

Polyhedra and packings from hyperbolic honeycombs

Significance The simplest 2D regular honeycombs are familiar patterns, found in an extraordinary range of natural and designed systems. They include tessellations of the plane by squares, hexagons, and equilateral triangles. Regular triangular honeycombs also form on the sphere; they are the triangular Platonic polyhedra: the tetrahedron, octahedron, and icosahedron. Regular hyperbolic honeycombs adopt an infinite variety of topologies; these must be distorted to be situated in 3D space and are thus frustrated. We construct minimally frustrated realizations of the simplest hyperbolic honeycombs. We derive more than 80 embeddings of 2D hyperbolic honeycombs in Euclidean 3 space, forming 3-periodic infinite polyhedra with cubic symmetry. All embeddings are “minimally frustrated,” formed by removing just enough isometries of the (regular, but unphysical) 2D hyperbolic honeycombs {3,7}, {3,8}, {3,9}, {3,10}, and {3,12} to allow embeddings in Euclidean 3 space. Nearly all of these triangulated “simplicial polyhedra” have symmetrically identical vertices, and most are chiral. The most symmetric examples include 10 infinite “deltahedra,” with equilateral triangular faces, 6 of which were previously unknown and some of which can be described as packings of Platonic deltahedra. We describe also related cubic crystalline packings of equal hyperbolic discs in 3 space that are frustrated analogues of optimally dense hyperbolic disc packings. The 10-coordinated packings are the least “loosened” Euclidean embeddings, although frustration swells all of the hyperbolic disc packings to give less dense arrays than the flat penny-packing even though their unfrustrated analogues in H2 are denser.

Surface embeddings of the Klein and the Möbius–Kantor graphs

This paper describes an invariant representation for finite graphs embedded on orientable tori of arbitrary genus, with working examples of embeddings of the Möbius-Kantor graph on the torus, the genus-2 bitorus and the genus-3 tritorus, as well as the two-dimensional, 7-valent Klein graph on the tritorus (and its dual: the 3-valent Klein graph). The genus-2 and -3 embeddings describe quotient graphs of 2- and 3-periodic reticulations of hyperbolic surfaces. This invariant is used to identify infinite nets related to the Möbius-Kantor and 7-valent Klein graphs.