Topological Symmetries of Molecules Erica Flapan December 13, 2013 Workshop: Topological Structures in Computational Biology Institute for Mathematics and its Applications Erica Flapan Topological Symmetries of Molecules Molecular symmetries The symmetries of a molecule determine many important aspects of its behavior. For example, symmetry is useful for: • Predicting reactions • Crystallography • Spectroscopy • Quantum chemistry • Analyzing the electron structure of a molecule • Classifying molecules Erica Flapan Topological Symmetries of Molecules Living organisms Asymmetric molecules interact with one another like feet and shoes. Asymmetric objects Erica Flapan Topological Symmetries of Molecules Living organisms Asymmetric molecules interact with one another like feet and shoes. Asymmetric objects Amino acids, sugars, and other molecules in living organisms are asymmetric. DNA is different from its mirror image Hence we react differently to mirror forms of asymmetric molecules. Erica Flapan Topological Symmetries of Molecules Pharmaceuticals Some pharmaceuticals and their mirror images: • Ibuprofen is an anti-inflamatory, but its mirror form is inert. • Naproxen is an anti-inflamatory, but its mirror form is toxic. • Darvon is a pain killer, but its mirror form is the cough suppressant Novrad. Erica Flapan Topological Symmetries of Molecules Pharmaceuticals Some pharmaceuticals and their mirror images: • Ibuprofen is an anti-inflamatory, but its mirror form is inert. • Naproxen is an anti-inflamatory, but its mirror form is toxic. • Darvon is a pain killer, but its mirror form is the cough suppressant Novrad. Drugs are synthesized in a 50:50 mix of mirror forms. If a molecule has mirror image symmetry these are the same. Otherwise, the two forms may need to be separated to avoid dangerous side effects. Knowing whether a structure will have mirror symmetry is useful in drug design. Erica Flapan Topological Symmetries of Molecules Mirror image symmetry But what do we mean by mirror symmetry?. Definition: A molecule is said to be chemically chiral if it can not transform itself into its mirror image at room temperature. Otherwise, it is said to be chemically achiral. Note: This definition describes the behavior of a molecule not its topology or geometry. Erica Flapan Topological Symmetries of Molecules Mirror image symmetry But what do we mean by mirror symmetry?. Definition: A molecule is said to be chemically chiral if it can not transform itself into its mirror image at room temperature. Otherwise, it is said to be chemically achiral. Note: This definition describes the behavior of a molecule not its topology or geometry. Definition: A rigid object is said to be geometrically chiral if it cannot be superimposed on its mirror image. Otherwise, it is said to be geometrically achiral. Erica Flapan Topological Symmetries of Molecules Geometric vs.chemical achirality If an object can be rigidly superimposed on its mirror image, then it is chemically the same as it’s mirror image. Geometrically Chemically Achiral Achiral (the same as mirror (can transform itself image as a rigid object) into its mirror image) Erica Flapan Topological Symmetries of Molecules Geometric vs.chemical achirality If an object can be rigidly superimposed on its mirror image, then it is chemically the same as it’s mirror image. Geometrically Chemically Achiral Achiral (the same as mirror (can transform itself image as a rigid object) into its mirror image) Thus the set of geometrically achiral molecules is a subset of the set of chemically achiral molecules. chemically achiral ? But is there a chemically achiral molecule which is not geometrically achiral ? geometrically achiral Cl Cl C C H 3C CH 3 H H Erica Flapan Topological Symmetries of Molecules Geometric vs.chemical chirality This molecule (without the faces) was synthesized by Kurt Mislow in 1954 to show that geometric and chemical chirality are different. NO 2 O2N O O C C O O O2N { { left propellerNO 2 right propeller Propellers are behind the screen. They turn simultaneously. Erica Flapan Topological Symmetries of Molecules Geometric vs.chemical chirality This molecule (without the faces) was synthesized by Kurt Mislow in 1954 to show that geometric and chemical chirality are different. NO 2 O2N O O C C O O O2N { { left propellerNO 2 right propeller Propellers are behind the screen. They turn simultaneously. Left propeller has her left hand forward, right propeller has her right hand forward. So, as rigid structures, a right propeller is different from a left propeller. Erica Flapan Topological Symmetries of Molecules Molecule is chemically achiral Original Mirror form NO 2 O2N O2N O O NO 2 O O C C C C O O O O O2N NO 2 NO2 O2N Erica Flapan Topological Symmetries of Molecules Molecule is chemically achiral Original Mirror form NO 2 O2N O2N O O NO 2 O O C C C C O O O O O2N NO 2 NO2 O2N Mirror form is the same as original, except vertical and horizontal hexagons have switched places. Erica Flapan Topological Symmetries of Molecules Molecule is chemically achiral Original Mirror form NO 2 O2N O2N O O NO 2 O O C C C C O O O O O2N NO 2 NO2 O2N Mirror form is the same as original, except vertical and horizontal hexagons have switched places. Proof of chemical achirality Erica Flapan Topological Symmetries of Molecules Molecule is chemically achiral Original Mirror form NO 2 O2N O2N O O NO 2 O O C C C C O O O O O2N NO 2 NO2 O2N Mirror form is the same as original, except vertical and horizontal hexagons have switched places. Proof of chemical achirality • Rotate original molecule by 90◦ about a horizontal axis to get mirror form with propellers horizontal. Erica Flapan Topological Symmetries of Molecules Molecule is chemically achiral Original Mirror form NO 2 O2N O2N O O NO 2 O O C C C C O O O O O2N NO 2 NO2 O2N Mirror form is the same as original, except vertical and horizontal hexagons have switched places. Proof of chemical achirality • Rotate original molecule by 90◦ about a horizontal axis to get mirror form with propellers horizontal. • Rotate propellers back to vertical position to get mirror form. Erica Flapan Topological Symmetries of Molecules Molecule is geometrically chiral Original Mirror form NO 2 O2N O2N O O NO 2 O O C C C C O O O O O2N NO 2 NO2 O2N Proof of geometric chirality • Suppose molecule is rigid. Then propellers don’t rotate. • In original form, left propeller is parallel to adjacent hexagon. • In mirror form, left propeller is perpendicular to adjacent hexagon. • A left propeller cannot change into a right propeller. • As rigid objects, the original and mirror form are distinct. Erica Flapan Topological Symmetries of Molecules Geometric vs.chemical chirality This example shows that for non-rigid molecules, geometric chirality does not necessarily imply chemical chirality. chemically achiral NO 2 O2N O O C C O O O2N NO 2 Cl Cl geometrically C C achiral H 3C CH 3 H H Erica Flapan Topological Symmetries of Molecules Molecular rigidity and non-rigidity In fact, some molecules are rigid, some are flexible, and some have pieces that can rotate around certain bonds. Cl Cl Cl H O O O O O O Br C O O H 3 C O O H H O O O O O O H O O rigid flexible rotating propeller NO 2 O2N O O C C O O O2N NO 2 two simultaneous propellers Erica Flapan Topological Symmetries of Molecules Topological Chirality So no mathematical characterization of chemical chirality that works for all molecules is possible. The definition of geometric chirality treats all molecules as completely rigid, which is not correct. Now we will treat all molecules as completely flexible, which is also not correct. The truth is somewhere in the middle. Definition A molecule is said to be topologically achiral if, assuming complete flexibility, it is isotopic to its mirror image. Otherwise it is said to be topologically chiral. Erica Flapan Topological Symmetries of Molecules Topological vs chemical chirality If this molecule were flexible, we could grab the CO2H and push it to the left, while pulling the H to the right. CH 3 CH 3 C C H H CO 2 H HO 2C NH 2 NH 2 interchange mirror So it is topologically achiral. However, the molecule is rigid so it’s chemically and geometrically chiral. Topologically Chemically achiral achiral (the same as mirror image (the same as mirror as a flexible object) image experimentally) Erica Flapan Topological Symmetries of Molecules Topological chirality Topologically chiral Chemically chiral Geometrically chiral (different from mirror (different from mirror image as a flexible (different from mirror image experimentally) image as a rigid object) object) None of the reverse implications hold. Erica Flapan Topological Symmetries of Molecules Topological chirality Topologically chiral Chemically chiral Geometrically chiral (different from mirror (different from mirror image as a flexible (different from mirror image experimentally) image as a rigid object) object) None of the reverse implications hold. If we heat a molecule which is geometrically chiral but not topologically chiral, we can force it to change to its mirror form. Even if we heat a topologically chiral molecule, it will not change to its mirror form. Thus knowing whether or not a molecule is topologically chiral helps to predict its behavior. Erica Flapan Topological Symmetries of Molecules The first example In 1986, Jon Simon gave the first example of a topologically chiral molecule by proving that a molecular M¨obius ladder with three rungs is topologically chiral. O O O O O O O O O O O O O O O O O O We sketch Simon’s proof. Erica Flapan Topological Symmetries of Molecules Set-up We represent the molecule as a colored graph M3, distinguishing sides from the rungs, because chemically they are different.