Orientation-dependent handedness and chiral design

Chirality occupies a central role in fields ranging from biological self-assembly to the design of optical metamaterials. The definition of chirality, as given by Lord Kelvin, associates chirality with the lack of mirror symmetry: the inability to superpose an object on its mirror image. While this definition has guided the classification of chiral objects for over a century, the quantification of handed phenomena based on this definition has proven elusive, if not impossible, as manifest in the paradox of chiral connectedness. In this work, we put forward a quantification scheme in which the handedness of an object depends on the direction in which it is viewed. While consistent with familiar chiral notions, such as the right-hand rule, this framework allows objects to be simultaneously right and left handed. We demonstrate this orientation dependence in three different systems - a biomimetic elastic bilayer, a chiral propeller, and optical metamaterial - and find quantitative agreement with chirality pseudotensors whose form we explicitly compute. The use of this approach resolves the existing paradoxes and naturally enables the design of handed metamaterials from symmetry principles

A Self-Organizing Algorithm for Modeling Protein Loops

Protein loops, the flexible short segments connecting two stable secondary structural units in proteins, play a critical role in protein structure and function. Constructing chemically sensible conformations of protein loops that seamlessly bridge the gap between the anchor points without introducing any steric collisions remains an open challenge. A variety of algorithms have been developed to tackle the loop closure problem, ranging from inverse kinematics to knowledge-based approaches that utilize pre-existing fragments extracted from known protein structures. However, many of these approaches focus on the generation of conformations that mainly satisfy the fixed end point condition, leaving the steric constraints to be resolved in subsequent post-processing steps. In the present work, we describe a simple solution that simultaneously satisfies not only the end point and steric conditions, but also chirality and planarity constraints. Starting from random initial atomic coordinates, each individual conformation is generated independently by using a simple alternating scheme of pairwise distance adjustments of randomly chosen atoms, followed by fast geometric matching of the conformationally rigid components of the constituent amino acids. The method is conceptually simple, numerically stable and computationally efficient. Very importantly, additional constraints, such as those derived from NMR experiments, hydrogen bonds or salt bridges, can be incorporated into the algorithm in a straightforward and inexpensive way, making the method ideal for solving more complex multi-loop problems. The remarkable performance and robustness of the algorithm are demonstrated on a set of protein loops of length 4, 8, and 12 that have been used in previous studies

Estimation under group actions: recovering orbits from invariants

Motivated by geometric problems in signal processing, computer vision, and structural biology, we study a class of orbit recovery problems where we observe very noisy copies of an unknown signal, each acted upon by a random element of some group (such as Z/p or SO(3)). The goal is to recover the orbit of the signal under the group action in the high-noise regime. This generalizes problems of interest such as multi-reference alignment (MRA) and the reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain matching lower and upper bounds on the sample complexity of these problems in high generality, showing that the statistical difficulty is intricately determined by the invariant theory of the underlying symmetry group. In particular, we determine that for cryo-EM with noise variance $\sigma^2$ and uniform viewing directions, the number of samples required scales as $\sigma^6$. We match this bound with a novel algorithm for ab initio reconstruction in cryo-EM, based on invariant features of degree at most 3. We further discuss how to recover multiple molecular structures from heterogeneous cryo-EM samples.Comment: 54 pages. This version contains a number of new result

Generation of a virtual library of terpenes using graph theory, and its application in exploration of the mechanisms of terpene biosynthesis

Terpenes form a large group of organic compounds which have proven to be of use to many living organisms being used by plants for metabolism (Pichersky and Gershenzon, 1934; McGarvey and Croteau, 1995; Gershenzon and Dudareva, 2007), defence or as a means to attract pollinators and also used by humans in medical, pharmaceutical and food industry (Bicas, Dionísio and Pastore, 2009; Marmulla and Harder, 2014; Kandi et al., 2015). Following on literature methods to generate chemical libraries using graph theoretic techniques, complete libraries of all possible terpene isomers have been constructed with the goal of construction of derivative libraries of possible carbocation intermediates which are important in the elucidation of mechanisms in the biosynthesis of terpenes. Virtual library generation of monoterpenes was first achieved by generating graphs of order 7, 8, 9 and 10 using the Nauty and Traces suite. These were screened and processed with a set of collated Python scripts written to recognize the graphs in text format and translate them to molecules, minimizing through Tinker whilst discarding graphs that violate chemistry laws. As a result of the computational time required only order 7 and order 10 graphs were processed. Out of the 873 graphs generated from order seven, 353 were converted to molecules and from the 11,7 million produced from order 10 half were processed resulting in the production of 442928 compounds (repeats included). For screening, 55 366 compounds were docked in the active site of limonene synthase; of these 2355 ligands had a good Vina docking score with a binding energy of between -7.0 and -7.4 kcal.mol-1. When these best docked molecules were overlaid in the active site a map of possible ligand positions within the active site of limonene synthase was traced out

Topological sorting and self-assembly of knotted molecules: models and simulations

Knots are ubiquitous objects and decorative elements that have been studied since antiquity. During the centuries knots have become important not only for their mysterious and elegant aspects, but also for their practical relevance. Knots in ropes, for example, have always been useful for different practical applications, from climbing to sailing, from fishing to medicine. Chains that are sufficiently long or compactified are prone to develop knots. This is a "statistical necessity" that has been conjectured by Delbruck in 1962 and mathematically proved by Sumners and Whittington nearly 30 years later. In particular, they showed that for a self-avoiding polygon, the knotting probability tends to unity as the polygon length tends to infinity. This statistical necessity makes topological entanglement a genuine characteristic of polymeric systems. In case of linear polymer chains, knots can be untied by a suitable reptation of the polymer in space and therefore the entanglement is referred as physical knots. On the other hand, if the polymer ends are joined by a cyclisation reaction, the geometrical self-entanglement becomes trapped in the form of a proper mathematical knot, whose topology cannot be changed by any geometrical rearrangement of the polymer except by cutting it. Among polymers, double-stranded DNA (dsDNA) provides an ideal system to study the spontaneous occurrence of knots. In fact, differently from proteins and RNA, metric and topological properties of dsDNA are well captured by aspecific polymer models where only the polymer contour length, persistence length and thickness come into play. Studying knots in dsDNA is informative also to understand their biological implication. The presence of knots, in fact, severely affects several cellular processes, such as transcription and replication, with detrimental effects. Fortunately, cellular mechanisms have adopted countermeasures: there exist enzimes, namely topoisomerases, that are capable of simplifying the topological complexity of the DNA entanglement by favouring the selective cross-passage of pairs of DNA strands. The action of topoisomerases has been understood thanks to the topological profiling of DNA molecules realised with gel electrophoresis. This is the typical technique that permits to sort short DNA molecules by knot type. In particular, molecules are electrically driven through the obstacles of an agarose gel, where their mobility depends on the specific knot type. However, this technique can be used to profile only relatively short DNA molecules (10-15 kb). For longer ones, gel electrophoresis resolution would severely degrades, especially for knots with high number of crossings. This raises the problem of developing novel techniques that can be applied to characterise knot types in longer DNA molecules. Here, we will use molecular dynamics simulations and theoretical approaches to discuss the possibility to use spatially modulated nanochannels to sort ring polymer by their knot type. This approach permits, in principle, to separate polymers by their topological complexity, overcoming the aformentioned limits of gel electrophoresis. The spontaneous knotting of DNA is largely controlled by events where, for example, a loop is threaded by one termini; as a result both the complexity and size of the knots, as well as their location along the DNA contour, are stochastic. This is not the case for other types of biomolecules, particularly proteins, where the folding process towards the native state is tightly controlled by their chemical composition (primary sequence) via their intra-molecular interactions. As a result, proteins whose native state is knotted always feature the same knot type in the same sequence location. Mimicking such reproducible molecular knotting processes are, at least in part, the motivation of the ongoing quest of synthetic chemistry to create synthetic molecules tied in specific knot types. In this regard, chemists succeeded in controlling chemical reactions between small building blocks to assemble molecules with a priori desired topology. The chemists who developed this set of techniques, whose contribution opened up the way to a revolutionary chemistry, were awarded with the Chemistry Nobel Prize in 2016. Despite the high interest in the topic, up to recently, only a handful of different knot types have been synthesised. The reason is due to various challenging aspects of the synthesis process. These include the choice of the suitable building blocks, their correct spatial arrangement, and, above all, the selection of the designable target topology. Not every knot type, in fact, is necessarily expected to be equally designable in practice. In this thesis, we performed a computational and theoretical study to explore which designable molecular knots could be accessible for molecular synthesis with current experimental techniques

Defects of micropolar continua in Riemann-Cartan manifolds and its applications

We derive equations of motion and its solutions in the form of solitons from deformational energy functionals of a coupled system of microscopic and macroscopic deformations. Then criteria in constructing the chiral energy functional is specified to be included to obtain soliton-like solutions. We show various deformational measures, used in deriving the soliton solutions, can be written when both curvature and torsion are allowed, especially by means of microrotations and its derivatives. Classical compatibility conditions are re-interpreted leading to a universal process to derive a distinct set of compatibility conditions signifying a geometrical role of the Einstein tensor in Riemann-Cartan manifolds. Then we consider position-dependent axial configurations of the microrotations to construct intrinsically conserved currents. We show that associated charges can be written as integers under a finite energy requirement in connection with homotopic considerations. This further leads to a notion of topologically stable defects determined by invariant winding numbers for a given solution classification. Nematic liquid crystals are identified as a projective plane from a sphere hinted by the discrete symmetry in its directors. Order parameters are carefully defined to be used both in homotopic considerations and free energy expansion in the language of microcontinua. Micropolar continua are shown to be the general case of nematic liquid crystals in projective geometry, and in formulations of the order parameter, which is also the generalisation of the Higgs isovectors. Lastly we show that defect measures of pion fields description of the Skyrmions are related to the defect measures of the micropolar continua via correspondences between its underlying symmetries and compatibility conditions of vanishing curvature

An algorithm to enumerate all possible protein conformations verifying a set of distance constraints

International audienceBackground: The determination of protein structures satisfying distance constraints is an important problem in structural biology. Whereas the most common method currently employed is simulated annealing, there have been other methods previously proposed in the literature. Most of them, however, are designed to find one solution only. Results: In order to explore exhaustively the feasible conformational space, we propose here an interval Branch-and-Prune algorithm (iBP) to solve the Distance Geometry Problem (DGP) associated to protein structure determination. This algorithm is based on a discretization of the problem obtained by recursively constructing a search space having the structure of a tree, and by verifying whether the generated atomic positions are feasible or not by making use of pruning devices. The pruning devices used here are directly related to features of protein conformations. Conclusions: We described the new algorithm iBP to generate protein conformations satisfying distance constraints, that would potentially allows a systematic exploration of the conformational space. The algorithm iBP has been applied on three α-helical peptides

Physical organic studies of substituted norbornyl systems: aspects of mechanisms and chirality

Fenchone and camphor are essential natural products that are available optically pure and contribute to the chiral pool in asymmetric synthesis. Further, they are both derivatives of norbornane, a structure that undergoes a remarkable diversity of rearrangements in acidic conditions. This work explores two aspects of the camphor/fenchone derived systems. Firstly an attempt to clarify rearrangement mechanisms on a camphor system successfully via deuterium labelling and unsuccessfully via derivatization of fenchone (with rearrangement) to produce other 13C-labelled camphor substitutions, has resulted in confirmation of a theoretically proposed, highly concerted Wagner-Meerwein, 6,2 - hydride shift, Wagner-Meerwein rearrangement in competition with an associated 2,3-methide shift. Kinetics and activation parameters for many steps have been resolved in this rearrangement of the deuterium labelled camphor-derived tosylate system to two pairs of isotopomers. Further, the kinetics and formation of an unexpected pair of dimers encountered during the scheme for 13C labelling are investigated in detail. These dimers (forming during the initial stages of the synthetic scheme) are unusual in that they are not expected rotamers of each other, but diastereomers resulting from chirality of a sulfur atom in a sulfite moiety. A feasible mechanism of formation that matches the kinetics has been proposed in this unexpectedly complex system, and thermodynamic parameters have been determined. The second aspect of substituted norbornyl systems pertains to their chirality, and the influence of this chirality on reaction mixtures, with an aim to identify novel chiral micellar catalysts for use in heterogeneous reaction mixtures. Headway has been made towards the synthesis of the appropriate surfactants to be used in the construction of these micelles, but extensive molecular dynamics simulations have illustrated the feasibility of forming the stable chiral micelles in a dual-solvent system, and detail precisely the influence of chirality on surrounding media. These studies add important physical organic data as well as show the immense possibilities pertaining to substituted norbornane systems