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Symmetry
Volume 8
Issue 1
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Symmetry, Volume 8, Issue 1 (January 2016) – 6 articles

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674 KiB  
Editorial
Acknowledgement to Reviewers of Symmetry in 2015
by Symmetry Editorial Office
Symmetry 2016, 8(1), 6; https://doi.org/10.3390/sym8010006 - 21 Jan 2016
Viewed by 3526
Abstract
The editors of Symmetry would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2015. [...] Full article
913 KiB  
Article
Symmetry in Sphere-Based Assembly Configuration Spaces
by Meera Sitharam, Andrew Vince, Menghan Wang and Miklós Bóna
Symmetry 2016, 8(1), 5; https://doi.org/10.3390/sym8010005 - 21 Jan 2016
Cited by 4 | Viewed by 5376
Abstract
Many remarkably robust, rapid and spontaneous self-assembly phenomena occurring in nature can be modeled geometrically, starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and [...] Read more.
Many remarkably robust, rapid and spontaneous self-assembly phenomena occurring in nature can be modeled geometrically, starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and bunches could be identical, as well, the underlying symmetry groups could be of large order that grows with the number of participating spheres and bunches. Thus, understanding symmetries and associated isomorphism classes of microstates that correspond to various types of macrostates can significantly increase efficiency and accuracy, i.e., reduce the notorious complexity of computing entropy and free energy, as well as paths and kinetics, in high dimensional configuration spaces. In addition, a precise understanding of symmetries is crucial for giving provable guarantees of algorithmic accuracy and efficiency, as well as accuracy vs. efficiency trade-offs in such computations. In particular, this may aid in predicting crucial assembly-driving interactions. This is a primarily expository paper that develops a novel, original framework for dealing with symmetries in configuration spaces of assembling spheres, with the following goals. (1) We give new, formal definitions of various concepts relevant to the sphere-based assembly setting that occur in previous work and, in turn, formal definitions of their relevant symmetry groups leading to the main theorem concerning their symmetries. These previously-developed concepts include, for example: (i) assembly configuration spaces; (ii) stratification of assembly configuration space into configurational regions defined by active constraint graphs; (iii) paths through the configurational regions; and (iv) coarse assembly pathways. (2) We then demonstrate the new symmetry concepts to compute the sizes and numbers of orbits in two example settings appearing in previous work. (3) Finally, we give formal statements of a variety of open problems and challenges using the new conceptual definitions. Full article
(This article belongs to the Special Issue Rigidity and Symmetry)
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Figure 1
<p>Two isomorphic bunches of five spheres.</p> Full article ">Figure 2
<p>The assembly configuration <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>1</mn> </msub> </math> consists of three isomorphic bunches. <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>2</mn> </msub> </math> is obtained from <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>1</mn> </msub> </math> with a strict congruence; <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>3</mn> </msub> </math> is obtained from <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>1</mn> </msub> </math> with a strict permuted congruence; and <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>4</mn> </msub> </math> is obtained from <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>1</mn> </msub> </math> with a strict isomorphism that is neither a strict congruence, nor a strict permuted congruence, nor a strict order preserving isomorphism.</p> Full article ">Figure 3
<p>Four assembly configurations obtained by applying <math display="inline"> <mrow> <mi mathvariant="italic">Waut</mi> <msub> <mspace width="-0.166667em"/> <mi mathvariant="script">A</mi> </msub> </mrow> </math> on the assembly configuration <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>1</mn> </msub> </math>. <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>2</mn> </msub> </math> is obtained from <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>1</mn> </msub> </math> with a congruence, while <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>3</mn> </msub> </math> is obtained from <math display="inline"> <msub> <mi mathvariant="script">B</mi> <mn>1</mn> </msub> </math> with a strict order-preserving isomorphism.</p> Full article ">Figure 4
<p>All non-isomorphic active constraint graphs with 12 edges of an assembly system of six bunches that are identical singleton spheres. The label on top is automatically generated by EASAL and specifies the orbit number of the shown active constraint graph.</p> Full article ">Figure 5
<p>An assembly configuration whose automorphism group is strictly contained in that of the corresponding active constraint graph. Here, the bunches are singleton spheres, and bunches of the same color have the same <math display="inline"> <mi mathvariant="script">C</mi> </math>, <span class="html-italic">r</span> and <span class="html-italic">δ</span>.</p> Full article ">Figure 6
<p>Any assembly configuration corresponding to the active constraint graph <span class="html-italic">G</span> has its strict congruence group strictly contained in <math display="inline"> <mrow> <msub> <mi mathvariant="italic">stab</mi> <mrow> <mi mathvariant="italic">Waut</mi> <msub> <mspace width="-0.166667em"/> <mi mathvariant="script">A</mi> </msub> </mrow> </msub> <mi>G</mi> </mrow> </math>. Here, the bunches are singleton spheres, and bunches of the same color have the same <math display="inline"> <mi mathvariant="script">C</mi> </math>, <span class="html-italic">r</span> and <span class="html-italic">δ</span>.</p> Full article ">Figure 7
<p>A fundamental region of the stratification for the assembly configuration space of the assembly configurations in <a href="#symmetry-08-00005-f004" class="html-fig">Figure 4</a> of six bunches, with each bunch being a singleton sphere and all bunches identical. Therefore, <math display="inline"> <mrow> <mi mathvariant="italic">Waut</mi> <msub> <mspace width="-0.166667em"/> <mi mathvariant="script">A</mi> </msub> </mrow> </math> is the complete symmetric group of the permutations of six elements, <math display="inline"> <msub> <mi>S</mi> <mn>6</mn> </msub> </math>. Each node shown is an orbit representative of an active constraint region corresponding to an active constraint graph. The grey part is those active constraint graphs (orbit representatives) whose corresponding constraint regions are empty. The example active constraint graph representatives on the right have arrows pointing to their regions in the stratification. The labels in the circles are unimportant: they are automatically generated and specify an orbit of an active constraint graph (example shown on the right).</p> Full article ">Figure 8
<p>The neighbors of one active constraint graph in the Hasse diagram of the stratification for the assembly system in <a href="#symmetry-08-00005-f004" class="html-fig">Figure 4</a>.</p> Full article ">Figure 9
<p>Klein 4-group acting on <math display="inline"> <msub> <mi mathvariant="script">T</mi> <mn>4</mn> </msub> </math>. See Example 3.</p> Full article ">Figure 10
<p>Partial Hasse diagram for the lattice of subgroups of the icosahedral group.</p> Full article ">Figure 11
<p>The values of the Möbius function of the subgroup lattice of <math display="inline"> <msub> <mi>G</mi> <mn>60</mn> </msub> </math>.</p> Full article ">
1810 KiB  
Review
Synthesis of Chiral Cyclic Carbonates via Kinetic Resolution of Racemic Epoxides and Carbon Dioxide
by Xiao Wu, José A. Castro-Osma and Michael North
Symmetry 2016, 8(1), 4; https://doi.org/10.3390/sym8010004 - 14 Jan 2016
Cited by 23 | Viewed by 8803
Abstract
The catalytic synthesis of cyclic carbonates using carbon dioxide as a C1-building block is a highly active area of research. Here, we review the catalytic production of enantiomerically enriched cyclic carbonates via kinetic resolution of racemic epoxides catalysed by metal-containing catalyst systems. Full article
(This article belongs to the Special Issue Asymmetric Catalysis)
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Figure 1

Figure 1
<p>Possible mechanisms for the coupling reaction of carbon dioxide and epoxides.</p> Full article ">Figure 2
<p>Cobalt(III) salen based catalyst systems for the synthesis of optically active cyclic carbonates.</p> Full article ">Figure 3
<p>Cobalt(III) and cobalt(II) salen based catalyst systems for the synthesis of optically active cyclic carbonates.</p> Full article ">Figure 4
<p>Bifunctional aluminium(III) and chromium(III) salen complexes <b>9</b> and <b>10</b> for the synthesis of optically active cyclic carbonates.</p> Full article ">
10117 KiB  
Article
Organocatalytic Asymmetric α-Chlorination of 1,3-Dicarbonyl Compounds Catalyzed by 2-Aminobenzimidazole Derivatives
by Daniel Serrano Sánchez, Alejandro Baeza and Diego A. Alonso
Symmetry 2016, 8(1), 3; https://doi.org/10.3390/sym8010003 - 13 Jan 2016
Cited by 11 | Viewed by 6683
Abstract
Bifunctional chiral 2-aminobenzimidazole derivatives 1 and 2 catalyze the enantioselective stereodivergent α-chlorination of β-ketoesters and 1,3-diketone derivatives with up to 50% ee using N-chlorosuccinimide (NCS) or 2,3,4,4,5,6-hexachloro-2,5-cyclohexadien-1-one as electrophilic chlorine sources. Full article
(This article belongs to the Special Issue Asymmetric Catalysis)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Chiral 2-aminobenzimidazole-catalyzed reactions.</p> Full article ">Figure 2
<p>Plausible mechanism for the asymmetric chlorination using Method A.</p> Full article ">Figure 3
<p>Plausible mechanism for the asymmetric chlorination using Method B.</p> Full article ">
1345 KiB  
Article
Natural Abundance Isotopic Chirality in the Reagents of the Soai Reaction
by Béla Barabás, Róbert Kurdi and Gyula Pályi
Symmetry 2016, 8(1), 2; https://doi.org/10.3390/sym8010002 - 8 Jan 2016
Cited by 13 | Viewed by 5654
Abstract
Isotopic chirality influences sensitively the enantiomeric outcome of the Soai asymmetric autocatalysis. Therefore magnitude and eventual effects of isotopic chirality caused by natural abundance isotopic substitution (H, C, O, Zn) in the reagents of the Soai reaction were analyzed by combinatorics and probability [...] Read more.
Isotopic chirality influences sensitively the enantiomeric outcome of the Soai asymmetric autocatalysis. Therefore magnitude and eventual effects of isotopic chirality caused by natural abundance isotopic substitution (H, C, O, Zn) in the reagents of the Soai reaction were analyzed by combinatorics and probability calculations. Expectable enantiomeric excesses were calculated by the Pars–Mills equation. It has been found that the chiral isotopic species formed by substitution in the otherwise achiral reagents provide enantiomeric excess (e.e.) levels that are higher than the sensitivity threshold of the Soai autocatalysis towards chiral induction. Consequently, possible chiral induction exerted by these e.e. values should be taken into account in considerations regarding the molecular events and the mechanism of the chiral induction in the Soai reaction. Full article
(This article belongs to the Special Issue Asymmetric Catalysis)
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Figure 1
<p>The Soai reaction [<a href="#B37-symmetry-08-00002" class="html-bibr">37</a>,<a href="#B38-symmetry-08-00002" class="html-bibr">38</a>,<a href="#B39-symmetry-08-00002" class="html-bibr">39</a>,<a href="#B41-symmetry-08-00002" class="html-bibr">41</a>,<a href="#B42-symmetry-08-00002" class="html-bibr">42</a>] (with the most efficient substrate).</p> Full article ">Figure 2
<p>Schematic structure of the dimeric intermediate of the Soai reaction, according to Schiaffino and Ercolani [<a href="#B44-symmetry-08-00002" class="html-bibr">44</a>] (<b>S-E4</b>).</p> Full article ">Figure 3
<p>(<b>A</b>) Schematic structure of a tetrameric intermediate of the Soai reaction according to Schiaffino and Ercolani [<a href="#B44-symmetry-08-00002" class="html-bibr">44</a>] (<b>S-E9</b>) and (<b>B</b>) schematic structure of the O<sub>4</sub>Zn<sub>4</sub> “twisted cube” fragment in compound <b>S-E9</b>.</p> Full article ">
2637 KiB  
Article
Structural Properties and Biological Prediction of ({[(1E)-3-(1H-Imidazol-1-yl)-1-phenylpropylidene] amino}oxy)(4-methylphenyl)methanone: An In Silico Approach
by Maha S. Almutairi, Devarasu Manimaran, Issac Hubert Joe, Ola A. Saleh and Mohamed I. Attia
Symmetry 2016, 8(1), 1; https://doi.org/10.3390/sym8010001 - 28 Dec 2015
Cited by 2 | Viewed by 4370
Abstract
Bioactive molecules are playing essential role in the field of drug discovery and various pharmaceutical applications. Vibrational spectral investigations of the anti-Candida agent ({[(1E)-3-(1H-imidazol-1-yl)-1-phenylpropylidene]amino}oxy)(4-methylphenyl)methanone ((1E)-IPMM) have been recorded and analyzed to understand its structural geometry, inter- [...] Read more.
Bioactive molecules are playing essential role in the field of drug discovery and various pharmaceutical applications. Vibrational spectral investigations of the anti-Candida agent ({[(1E)-3-(1H-imidazol-1-yl)-1-phenylpropylidene]amino}oxy)(4-methylphenyl)methanone ((1E)-IPMM) have been recorded and analyzed to understand its structural geometry, inter- and intra-molecular interactions. The equilibrium geometry, harmonic vibrational wavenumber, natural bond orbital (NBO) and Frontier orbital energy analyses have been carried out with the help of density functional theory with B3LYP/6-311++G(d,p) level of theory. The detailed vibrational assignments for the title molecule were performed on the basis of potential energy distribution analysis in order to unambiguously predict its modes. The calculated wavenumbers had good agreement with the experimental values. NBO analysis has confirmed the intramolecular charge transfer interactions. The predicted docking binding energy gave insight into the possible biological activity of the title molecule. Full article
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Figure 1

Figure 1
<p>Optimized structure of (1E)-IPMM.</p> Full article ">Figure 2
<p>(<b>a</b>) Simulated (<b>b</b>) Experimental FT-IR spectrum of (1<span class="html-italic">E</span>)-IPMM.</p> Full article ">Figure 3
<p>(<b>a</b>) Simulated (<b>b</b>) Experimental FT-Raman spectrum of (1<span class="html-italic">E</span>)-IPMM.</p> Full article ">Figure 4
<p>(<b>a</b>) HOMO; (<b>b</b>) LUMO plots of (1<span class="html-italic">E</span>)-IPMM.</p> Full article ">Figure 5
<p>Binding pose of (1<span class="html-italic">E</span>)-IPMM with amino acid residues.</p> Full article ">Figure 6
<p>Synthetic strategy to prepare the target molecule <b>3</b>. Reagents and conditions: (i) HN(CH<sub>3</sub>)<sub>2</sub>·HCl, (CH<sub>2</sub>O)<span class="html-italic"><sub>n</sub></span>, conc. HCl, ethanol, reflux, 2 h; (ii) imidazole, water, reflux, 5 h; (iii) H<sub>2</sub>NOH·HCl, KOH, ethanol, reflux, 18 h; (iv) 4-methylbenzoic acid, EDCI·HCl, DMAP, DCM, rt, 18 h.</p> Full article ">Figure 7
Full article ">Figure 8
Full article ">
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