Quantitative symmetry and chirality--a fast computational algorithm for large structures: proteins, macromolecules, nanotubes, and unit cells

Symmetry is one of the most fundamental properties of nature and is used to understand and investigate physical properties. Classically, symmetry is treated as a binary qualitative property, although other physical properties are quantitative. Using the continuous symmetry measure (CSM) methodology one can quantify symmetry and correlate it quantitatively to physical, chemical, and biological properties. The exact analytical procedures for calculating the CSM are computationally expensive and the calculation time grows rapidly as the structure contains more atoms. In this article, we present a new method for calculating the CSM and the related continuous chirality measure (CCM) for large systems. The new method is much faster than the full analytical procedures and it reduces the calculation time dependency from N! to N(2), where N is the number of atoms in the structure. We evaluate the cost of the applied approximations, estimate the error of the method, and show that deviations from the analytical solutions are within an error of 2%, and in many cases even less. The method is applicable at the moment for the cyclic symmetry point groups- C(i), C(s), C(n), and S(n), and therefore it can be used also for chirality measures, which are the minimal of the S(n) measures. We demonstrate the application of the method for large structures across chemistry: proteins, macromolecules, nanotubes, and large unit cells of crystals.     

The Story Behind the Link between Molecular Chirality and Crystal Shape

Here we describe the story behind the link between molecular chirality and macroscopic phenomena, the latter being a probe for the direct assignment of absolute configuration of chiral molecules. First, a brief tour of the history of molecular stereochemistry, starting with the classic experiment reported by Pasteur in 1848 on the separation of enantiomorphous crystals of a salt of tartaric acid, and his conclusion that the molecules of life are chiral of single-handedness. With time, this study raised, inter alia, two fundamental questions: the absolute configuration of chiral molecules and how a molecule of given configuration shapes the enantiomorphous morphology of its crystal. As for the first question, following the beginning of crystal structure determination by X-ray diffraction in 1912, it took almost 40 years before Bijvoet assigned molecular chirality through the esoteric method involving anomalous X-ray scattering. We have been able to address and link both questions through ‘everyday concepts of left and right’ (in the words of Jack Dunitz) by the use of ‘tailor-made’ auxiliaries. By such means, it proved possible to reveal, through morphology, etch patterns, epitaxy and symmetry reduction of both chiral and, paradoxically, centrosymmetric crystals, the basic chiral symmetry of the molecules of life, the α-amino acids and sugars.     

Analytical methods for calculating Continuous Symmetry Measures and the Chirality Measure

We provide analytical solutions of the Continuous Symmetry Measure (CSM) equation for several symmetry point-groups, and for the associated Continuous Chirality Measure (CCM), which are quantitative estimates of the degree of a symmetry-point group or chirality in a structure, respectively. We do it by solving analytically the problem of finding the minimal distance between the original structure and the result obtained by operating on it all of the operations of a specific G symmetry point group. Specifically, we provide solutions for the symmetry measures of all of the improper rotations point group symmetries, S(n), including the mirror (S(1), C(S)), inversion (S(2), C(i)) as well as the higher S(n)s (n > 2 is even) point group symmetries, for the rotational C(2) point group symmetry, for the higher rotational C(n) symmetries (n > 2), and finally for the C(nh) symmetry point group. The chirality measure is the minimal of all S(n) measures.     

Large-Scale Chirality Measures and General Symmetry Deficiency Measures for Functional Group Polyhedra of Proteins

Functional Group Polyhedra provide a simplified representation of the most essential spatial features of macromolecules, especially, of globular proteins. Since the functional group polyhedron model focuses on large scale features, the chirality and other symmetry deficiency measures of these molecules, when adapted to these polyhedra, should also be based on the characterization of large scale shape features. Two new approaches for the evaluation of such symmetry deficiency and chirality measures are presented.     

直链烷烃取代衍生物Wiener指数的简便计算方法

根据直链烷烃衍生物分子拓扑结构的特点,将直链烷烃衍生物拆分为由直链单元和取代基团几个部分构成,再根据每部分的拓扑结构特点给出相应的计算公式.从而提出了一个计算直链烷烃衍生物Wiener指数的简便方法,达到简化计算的目的.该方法简化了传统Wiener指数的计算方法,使Wiener指数的计算具有效率高、不易出错等优点,便于Wiener指数计算程序化,从而提高了Wiener指数的实用性.     

Symmetrizer: Algorithmic determination of point groups in nearly symmetric molecules

Symmetry is an extremely useful and powerful tool in computational chemistry, both for predicting the properties of molecules and for simplifying calculations. Although methods for determining the point groups of perfectly symmetric molecules are well‐known, finding the closest point group for a “nearly” symmetric molecule is far less studied, although it presents many useful applications. For this reason, we introduce Symmetrizer, an algorithm designed to determine a molecule's symmetry elements and closest matching point groups based on a user‐adjustable tolerance, and then to symmetrize that molecule to a given point group geometry. In contrast to conventional methods, Symmetrizer takes a bottom‐up approach to symmetry detection by locating all possible symmetry elements and uses this set to deduce the most probable point groups. We explain this approach in detail, and assess the flexibility, robustness, and efficiency of the algorithm with respect to various input parameters on several test molecules. We also demonstrate an application of Symmetrizer by interfacing it with the WebMO web‐based interface to computational chemistry packages as a showcase of its ease of integration. © 2012 Wiley Periodicals, Inc.     

Isometric groups and chirality of nonrigid molecules: a generalization of Kelvin's theorem

A generalization of Kelvin's theorem for the chirality of rigid molecules to nonrigid molecules is given. It is based on the concept of the isometric group of nonrigid systems which allows a formulation of the chirality criterion of nonrigid systems in close analogy with Kelvin's original formulation. A number of examples, in particular such as investigated by Mislow, are given.     

Continuous symmetry measures of irreducible representations: application to molecular orbitals

The formalism of continuous symmetry measures is extended to describe the extent to which a function, such as a molecular orbital, transforms under the symmetry operations of a given point group according to each irreducible representation of this group. For distorted molecules we are able to calculate the degree to which any molecular orbital transforms with respect to the irreducible representations of the pseudosymmetry group that is valid for a higher symmetry, idealized geometry, showing which irreducible representations participate in each molecular orbital upon symmetry loss in the geometry of the nuclear framework.     

Relationships between temperature,magnetic moment,and continuous symmetry measures in spin crossover complexes

A study of the changes in the symmetry of the metal coordination sphere that accompany spin crossover transitions has been carried out. Changes in symmetry have been calibrated by using the continuous symmetry measures approach. Different behaviors have been found for several families of complexes, depending on the nature of their ligands. The most remarkable changes take place for trischelate complexes, for which octahedricity, trigonal prismacity, chirality, bite angle, bond distance, tempeature, pressure, magnetic moment, and density are correlated.     

Lifting the mirror symmetry of metal surfaces: decoupling the electronic and physical manifestations of surface chirality

Naturally occurring metal surfaces possess planes of mirror symmetry on the nanometer-length scale. This mirror symmetry can be lifted and chirality "physically" conveyed onto a surface by adsorbing a chiral molecule. Until now, it has not been known whether the conveying of chirality is limited to just the physical structure or whether it goes deeper and permeates the electronic structure of the underlying surface. By using optically active second harmonic generation (OA-SHG), it is demonstrated that the adsorption of some, but not all, chiral molecules can reversibly, and without significant structural rearrangement, measurably lift the mirror symmetry of the surface electronic structure of a metal. It is proposed that the ability of a chiral molecule to place a significant "chiral perturbation" on the electronic structure of a surface is correlated to its adsorption geometry. The microscopic origins of the observed optical activity are also discussed in terms of classical models of chirality. The results of the study challenge current models of how chiral adsorbates induce enantioselectivity in the chemical/physical behavior of heterogeneous systems, which are based on geometric/stereochemical arguments, by suggesting that chiral electronic perturbations could play a role.     

Chemical applications of topology and group theory 13. Chirality and framework groups [1]

Pople has recently introduced the concept of a framework group to specify the full symmetry properties of a molecular structure. Furthermore, Pople has developed powerful algorithms for the use of framework groups to generate all distinguishable skeletons with a given number of sites. This paper studies the systematics of chirality arising from different framework groups. In this connection framework groups can be classified into four different types: linear, planar, achiral, and chiral. Chiral framework groups lead to chiral systems for any ligand partition including that with all ligands equivalent. Linear framework groups are never chiral even for the ligand partition with all ligands different. Planar framework groups are also never chiral since all sites are in the same plane, which therefore remains a symmetry plane for any ligand partition. However, the mirror symmetry of the molecular plane of a planar framework group can be destroyed by a process called polarization; this process can be viewed as the mathematical analogue of complexing a planar aromatic hydrocarbon to a transition metal. The chirality of four-, five-, and six-site framework groups is discussed in terms of the maximum symmetry ligand partitions resulting in removal of all of the symmetry elements corresponding to improper rotations S _n (including reflections S ₁ and inversions S ₂) from achiral and polarized planar framework groups. The Ruch-Schönhofer group theoretical algorithms for the calculation of chiral ligand partitions and pseudoscalar polynomials of lowest degree (“chirality functions”) are adapted for use with these framework groups. Other properties of framework groups relevant to a study of their chirality are also discussed: these include their transitivity (i.e. whether all sites are equivalent or not), their normality (i.e. whether proper rotations correspond to even permutations and improper rotations correspond to odd permutations), and the number of sites in their symmetry planes.     

Continuous chirality measures in transition metal chemistry

The definition of the continuous chirality measure(CCM) is provided and its applications are summarized in this tutorial review, with special emphasis on the field of transition metal complexes. The CCM approach, developed in recent years, provides a quantitative parameter that evaluates the degree of chirality of a given molecule. Many quantitative structural correlations with chirality have been identified for most of the important families of metal complexes. Our recent research has shown that one can associate the chirality measures with, e.g., enantioselectivity in asymmetric catalysis. We also explore a fragment approach to chirality in which we investigate which part of a molecule is responsible for the chirality-associated properties of a given family of compounds.     

Linearized embedding: A new metric matrix algorithm for calculating molecular conformations subject to geometric constraints

There are many methods in the literature for calculating conformations of a molecule subject to geometric constraints, such as those derived from two-dimensional NMR experiments. One of the most general ones is the EMBED algorithm, based on distance geometry, where all constraints except chirality are converted into upper and lower bounds on interatomic distances. Here we propose a variation on this where the molecule is assumed to have fixed bond lengths, vicinal bond angles and chiral centers; and these holonomic constraints are enforced separately from the experimental constraints by being built into the mathematical structure of the problem. The advantages of this approach are: (1) for molecules having large rigid groups of atoms, there are substantially fewer variables in the problem than all the atomic coordinates; (2) rigid groups achieve in the end more accurate local geometry (e.g., planar aromatic rings are truly planar, chiral centers always have their correct absolute chirality); (3) it is easier to detect inconsistencies between the holonomic and the experimental constraints; and (4) when generating a random sampling of conformers consistent with all constraints, the probability of achieving satisfactory structures tends to be greater.     

Nucleoside-assisted self-assembly of oligo(p-phenylenevinylene)s at liquid/solid interface: chirality and nanostructures

The formation of DNA nucleoside-assisted π-conjugated nanostructures was studied by means of scanning tunneling microscopy (STM) and force field simulations. Upon adsorption of the achiral oligo(p-phenylenevinylene) (OPV) derivative at the liquid/solid interface, racemic conglomerates with mirror related rosettes are formed. Addition of the DNA nucleosides D- and L-thymidine, which act as "chiral handles", has a major effect on the supramolecular structure and the expression of chirality of the achiral OPV molecules. The influence of these "chiral handles" on the expression of chirality is probed at two levels: monolayer symmetry and monolayer orientation with respect to the substrate. This was further explored by tuning the molar ratio of the building blocks. Molecular modeling simulations give an atomistic insight into the monolayer construction, as well as the energetics governing the assembly. Thymidine is able to direct the chirality and the pattern of OPV molecules on the surface, creating chiral lamellae of π-conjugated dimers.     

Identification of symmetries in molecules and complexes

An algorithm is presented that quickly detects local and global symmetries of single molecules and complexes. Based upon the Morgan Naming Algorithm, the algorithm involves traversing the molecule from a starting atom and building up a molecule name based upon the names of the atoms encountered along the traversal. Additional molecule names are generated from other starting atoms, and name-name matches are identified as corresponding to symmetry operations. A number of enhancements relative to prior methods yield increased efficiency and extended functionality. In particular, the present method detects not only global symmetries but also local symmetries associated with bond rotations as well as symmetries that are only apparent when alternate resonance forms are considered. Importantly, the present method works not only for single molecules but also for multimolecular complexes. As a consequence, it is well, and perhaps uniquely, suited to applications in supramolecular and host-guest chemistry. Applications include filtering out redundant conformations during conformational searching and free energy calculations; accelerating ligand-receptor docking calculations by reducing the sampling ranges of rotatable bonds linked to locally symmetric groups, such as phenyls; and automating the calculation of symmetry numbers for thermochemical applications.