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We analyze the rotational dynamics of six magnetic dipoles of identical strength at the vertices of a regular hexagon with a variable-strength dipole in the center. The seven dipoles spin freely about fixed axes that are perpendicular to the plane of the hexagon, with their dipole moments directed parallel to the plane. Equilibrium dipole orientations are calculated as a function of the relative strength of the central dipole. Small-amplitude perturbations about these equilibrium states are calculated in the absence of friction and are compared with analytical results in the limit of zero and infinite central dipole strength. Normal modes and frequencies are presented. Bifurcations are seen at two critical values of the central dipole strength, with bistability between these values. 

Abstract A major challenge in modern biology is understanding how the effects of short-term biological responses influence long-term evolutionary adaptation, defined as a genetically determined increase in fitness to novel environments. This is particularly important in globally important microbes experiencing rapid global change, due to their influence on food webs, biogeochemical cycles, and climate. Epigenetic modifications like methylation have been demonstrated to influence short-term plastic responses, which ultimately impact long-term adaptive responses to environmental change. However, there remains a paucity of empirical research examining long-term methylation dynamics during environmental adaptation in nonmodel, ecologically important microbes. Here, we show the first empirical evidence in a marine prokaryote for long-term m5C methylome modifications correlated with phenotypic adaptation to CO2, using a 7-year evolution experiment (1,000+ generations) with the biogeochemically important marine cyanobacterium Trichodesmium. We identify m5C methylated sites that rapidly changed in response to high (750 µatm) CO2 exposure and were maintained for at least 4.5 years of CO2 selection. After 7 years of CO2 selection, however, m5C methylation levels that initially responded to high-CO2 returned to ancestral, ambient CO2 levels. Concurrently, high-CO2 adapted growth and N2 fixation rates remained significantly higher than those of ambient CO2 adapted cell lines irrespective of CO2 concentration, a trend consistent with genetic assimilation theory. These data demonstrate the maintenance of CO2-responsive m5C methylation for 4.5 years alongside phenotypic adaptation before returning to ancestral methylation levels. These observations in a globally distributed marine prokaryote provide critical evolutionary insights into biogeochemically important traits under global change. 

The full scope and limitations of the catalytic acylative kinetic resolution of a range of tertiary heterocyclic alcohols (78 examples,sup to >200) is reported under operationally‐simple conditions, using low loadings of a commercially available Lewis basic isothiourea catalyst, HyperBTM (generally 1 mol %). The protocol is highly effective for the kinetic resolution of 3‐substituted 3‐hydroxyoxindole and α‐substituted α‐hydroxylactam derivatives bearing up to three potential recognition motifs at the stereogenic tertiary carbinol center. The full power of this methodology has been showcased through the synthesis of highly enantioenriched biologically‐active target compounds in both enantiomeric forms. To provide further insight into the reaction mechanism, a detailed kinetic analysis of this Lewis base‐catalyzed acylation of tertiary alcohols is reported using the variable time normalization analysis (VTNA) method.

 

Axially chiral phenols are attractive targets in organic synthesis. This motif is central to many natural products and widely used as precursors to, or directly, as chiral ligands and catalysts. Despite their utility few simple catalytic methods are available for their synthesis in high enantiopurity. Herein the atropselective acylation of a range of symmetric biaryl diols is investigated using isothiourea catalysis. Studies on a model biaryl diol substrate shows that the high product er observed in the process is a result of two successive enantioselective reactions consisting of an initial enantioselective desymmetrization coupled with a second chiroablative kinetic resolution. Extension of this process to a range of substrates, including a challenging tetraorthosubstituted biaryl diol, led to highly enantioenriched products (14 examples, up to 98:2 er), with either HyperBTM or BTM identified as the optimal catalyst depending upon the substitution pattern within the substrate. Computation has been used to understand the factors that lead to high enantiocontrol in this process, with maintenance of planarity to maximize a 1,5‐S⋅⋅⋅O interaction within the key acyl ammonium intermediate identified as the major feature that determines atropselective acylation and thus product enantioselectivity.

 

Axially chiral phenols are attractive targets in organic synthesis. This motif is central to many natural products and widely used as precursors to, or directly, as chiral ligands and catalysts. Despite their utility few simple catalytic methods are available for their synthesis in high enantiopurity. Herein the atropselective acylation of a range of symmetric biaryl diols is investigated using isothiourea catalysis. Studies on a model biaryl diol substrate shows that the high product er observed in the process is a result of two successive enantioselective reactions consisting of an initial enantioselective desymmetrization coupled with a second chiroablative kinetic resolution. Extension of this process to a range of substrates, including a challenging tetraorthosubstituted biaryl diol, led to highly enantioenriched products (14 examples, up to 98:2 er), with either HyperBTM or BTM identified as the optimal catalyst depending upon the substitution pattern within the substrate. Computation has been used to understand the factors that lead to high enantiocontrol in this process, with maintenance of planarity to maximize a 1,5‐S⋅⋅⋅O interaction within the key acyl ammonium intermediate identified as the major feature that determines atropselective acylation and thus product enantioselectivity.

 

A combination of experimental and computational studies have identified a C=O⋅⋅⋅isothiouronium interaction as key to efficient enantiodiscrimination in the kinetic resolution of tertiary heterocyclic alcohols bearing up to three potential recognition motifs at the stereogenic tertiary carbinol center. This discrimination was exploited in the isothiourea‐catalyzed acylative kinetic resolution of tertiary heterocyclic alcohols (38 examples,s factors up to >200). The reaction proceeds at low catalyst loadings (generally 1 mol %) with either isobutyric or acetic anhydride as the acylating agent under mild conditions.

 

A combination of experimental and computational studies have identified a C=O⋅⋅⋅isothiouronium interaction as key to efficient enantiodiscrimination in the kinetic resolution of tertiary heterocyclic alcohols bearing up to three potential recognition motifs at the stereogenic tertiary carbinol center. This discrimination was exploited in the isothiourea‐catalyzed acylative kinetic resolution of tertiary heterocyclic alcohols (38 examples,s factors up to >200). The reaction proceeds at low catalyst loadings (generally 1 mol %) with either isobutyric or acetic anhydride as the acylating agent under mild conditions.