Stereochemical Topology
By
Krista Joslin
Carla Ranallo
Clyde Tedrick
Mark Herried
Steve Egle
Nathan Zimmer
Brian King
Paper Outline
I.
Stereochemical Topology
A) Molecular graphs as topological objects in space
B) Topological Chirality and Achirality
II.
Molecular Moebius Ladders
A) Description and background
B) Statement of Simon’s 1986 proof
III.
Topological Concepts and Machinery
A) Topological Spaces
B) Manifolds
C) Covering Spaces
i. 2-fold Coverings
ii. 2-fold Branched coverings
iii. General Covering Spaces
IV.
Conclusion
Topological Stereochemistry
We will first review the topic of topological stereochemistry. Stereochemistry is
the study of stereoisomers, which are compounds that have the same chemical formula
and the same connectivity but different arrangements of their atoms in a 3 – dimensional
space. Stereochemistry is also the study of synthesis, characterization and analysis of
molecular structures that are topologically nontrivial. We will examine when can or
cannot one embedded graph be “deformed” into another and the properties of embedded
graphs that are preserved by deformation.
We will now examine the graphical representation of a molecule. A graph, G =
(V, E) is a collection of vertices and edges, where V is the set of vertices and E the set of
edges. The graph can be represented in two forms; directed and undirected for two
vertices where all can flow in both directions.
We will next take a look at achirality. Let us consider a graph embedded in R3.
This graph is topologically achiral if it can be deformed into its mirror image. We can
define deformation by a way of bending, twisting and or rotating without breaking or
tearing the molecule. Another way of observing achirality is by observing symmetrical
elements. If the molecule or object has either a plane of symmetry or a center of
symmetry it is also said to be achiral. A real world example of an achiral organic
molecule is 2 – proponol.
Now let’s examine chirality. We can observe the definition of chirality as
meaning the opposite of chirality. The formal definition states a graph embedded in R3 is
topologically chiral if it is not identical or non-superimposable upon and cannot be
deformed into its mirror image. Once again we define deformation by a way of bending,
twisting and or rotating without breaking or tearing the molecule. We can consider our
hands to be chiral, because they cannot be deformed into their mirror image. A real
world example of a chiral organic molecule is 2 – butanol. Some other examples include
glucose and all other sugars, certain proteins, nucleic acids, DNA, as well as over half the
organic compounds found in common drugs.
Homeomorphism and ambient isotopy are the two mathematical models of
chirality. We first define homeomorphism as a function h: A → B, where h is a
homeomorphism if h is continuous and h has a continuous inverse. Homeomorphisms
can be either differentiable or piecewise linear functions.
We now take another a look at a second type of mathematical model which is
ambient isotopy. We define ambient isotopy as a set M that is a subset of Rn, in which A
and B are contained. A is ambient isotopic to B in M if there is a continuous function
F:M x I → M such that for each fixed t ∈ I the function F(x,T) is a homeomorphism,
F(x,0) = x for all x ∈, and F(A x {1}) = B. This in turn defines both achiral and chiral
molecules to be ambient isotopic to their mirror image. Topological achiral is defined
when there exists an orientation reversing homeomorphism of (R3, G) in an embedded
graph G ⊂ R3. If an orientation reversing homeomorphism does not exist then G is
topologically chiral. An orientation reversing homeomorphism is when h in R3 is
isotopic to the reflection map and orientation preserving is when h in R3 is isotopic to the
identity map. Two homeomorphisms are isotopic if one can be continuously deformed
into the other. Another important thing to remember is that every homeomorphism is
isotopic to either the identity map or to a reflection map, but not to both.
Mobius Ladder
A mobius ladder, Mn consists of a simple closed curve K with 2n vertices.
Together with n additional edges a1,…..an such that if the vertices on the curve K are
consecutively labeled 1, 2, ….n then the vertices of each edge ai are I and I + n. K is the
loop of the mobius ladder Mn and a,…..an are the rungs of Mn. An example of a mobius
ladder is catenane. An application of catenane is molecular memory for computers.
Random access data storage could be provided by rings of atoms. Researchers
who have developed a system of microscopic chemical switches that could form the basis
of tiny, fast, and cheap computers. This system could allow our computers to do things
that we could not even imagine now.
Molecular memory works by using a pulse of electricity. This pulse causes one
ring to flip or rotate around the other. This is how the switch is turned on. Putting the
electrons back switches it off. It is also very easy to see when the catenane is working. It
is green in the beginning and switches to maroon when it is working, you can even use
your eyes to detect it.
John Simon’s Theorem
John Simon’s theorem proves that embedded graphs representing the molecular
Mobius ladders with an odd number of rungs greater than two is necessarily topologically
chiral. In contrast, a mobius ladder with an even number of rungs has a topologically
achiral embedding. Simon used topological machinery to prove his theory. The concepts
he used were topological spaces and covering spaces.
Topological Concepts and Machinery
A topology on a set X is a collection T if the subsets if X have the following
properties, Fi, X are in T, the union of the elements of any sub collection of T is in T, and
the intersection of the elements of any finite sub collection of T is in T. A set X, for
which a topology T has been specified is called a topological space written (X,T). If we
have a topological space and U is a subset of X, U is in the topology, U is called an open
set. We can say a topological space is a set X together with a collection of subsets of X,
called open sets such that Fi, X are both open, arbitrary unions of open sets are open. Let
(X,T) be a topological space. A basis B is a collection of subsets X, (called basis
elements) such that:
(1) ∀ x X ∃ B
∩
∃ B3
∃ X B3 ⊂
∩
Manifolds
In essence, a manifold is a space that is locally like Rn, however lacking a
preferred system of coordinates. Furthermore, a manifold can have global topological
properties that distinguish it from the topological trivial Rn. Let M be a subset of Rp for
some p. A subset U of M is said to be open in M if U equals the intersection of M and V
where V is an open set in Rp. Let n be a natural number. We say that M is an n-manifold
if each point x of M is contained in an open set U of M that is either homeomorphic to Rn
or to the half-space R+n. For example, think of a cover of a baseball as if it were a hollow
sphere (which would be an example of a two-manifold), while the rubber or cork ball and
the twine make up the solid center (an example of a three-manifold). Both of the
manifolds are subsets of three space. We can imagine a baseball imbedded in three
space, and let T denote the surface area of a baseball. For any point x contained within T,
we can choose V to be a small open ball whose center falls on x. By definition since V is
open, the set U which is given by the intersection of sets T and V is also open. If you
choose a small enough radius for V, then the resulting U will yield a small slightly curved
disk whose interior is homeomorphic to the interior of a flat disk. A flat disk in turn is
homeomorphic to R2. We can similarly argue that a sold sphere is a three manifold.
Only in this case the interior points of the sphere are contained in an open set that is
homeomorphic to R3. Some examples of one-manifolds are line segments, lines, circles,
and the unions of these. Some examples of two-manifolds are mobius strips, annulus,
and the surface of sphere. Some examples of three-manifolds are three dimensional
spheres, three dimensional balls, and torus(doughnuts). Manifolds have generated so
much interest because they are easier to deal with than other subsets of Rn.
Covering Spaces
Let M be a subset of Rp for some p. A subset U of M is said to be open in M if
U=M∩V where V is an open set in Rp. Let n be a natural number. We say that M is an
n-manifold if each point x of M is contained in an open set U of M that is either
homeomorphic to Rn or to the half-space Rn + = {(x1, ……xn) Rn | xn > 0}. Let M be a
subset of Rn and let h:M M be a homeomorphism. Let r be a natural number. Then hr
is the homeomorphism obtained by performing h some number r times. If r is the
smallest number such that hr is the identity map, then we say h has an order of r. If there
is no such r then we say h does not have finite order. Let h: M M be an orientation
preserving homeomorphism of order two. Let M and N be three manifolds. Let p:M N
be a function, that is continuous and takes open sets to open sets. If p(x) = p(y) if and
only if either x = y or h(x) = y then p is said to be a projection map. Let M and N be
three manifolds, and let p:M M be a covering involution. Let p:M N be a projection
map. Let A denote the set of points x in M such that h(x) = x. If B = p(A) is a onemanifold then we say M is a twofold branch cover of N branched over B. If A is the
empty set then we say M is a twofold cover of N.
Twofold Branch Covers
Twofold Branch covers in simple terms consists of two functions. The first
function h allows movement between multiple manifolds along a fixed point and the
second function p wraps the multiple manifolds into one manifold making a two-to-one
ratio. The full definition of twofold branch covers are as follows:
Let M and N be three-manifolds, and let h: M -> M be an orientation-preserving
homeomorphism of order two. Let p: M -> N be a function such that p(x)=p(y) if and
only if either x = y or h(x) = y. Suppose that p is a continuous onto map that takes open
sets to open sets. Let A denote the set of points x in M such that h(x)=-x. If B = p(A) is a
one-manifold, then we say that M is a twofold branched cover of N branched over B.
To help explain the concept of twofold branch covers we will go through a two
dimensional example.
Let M denote a unit disk expressed in polar coordinates. We define h: M -> M by h(r, Θ)
= (r, Θ + 180°), so that h rotates M by 180°.
°)
θ)
θ + 180)
Now define p: M -> M by p(r, Θ) = (r, 2Θ). As in our previous examples, p(x) = p(y) if
and only if either x = y or h(x) = y. We let M1 be the surface obtained by cutting M open
along a single radius of the disk, and we let M2 be a copy of M1. By stretching M1 and M2
open, we can think of each as a half-disk, and the disk M is obtained by gluing these two
half-disks together.
M1
M2
M
Thus again h interchanges M1 and M2, and p sends each of M1 and M2 onto M. What
makes twofold branch covers different from twofold covers is the fact that in twofold
covers h fixes no points, but in this example h fixes the center point of M making it a
twofold branch cover. This is an example where M is a twofold branched cover of itself
with branch set the center point of M.
General Covering Spaces
General Covering Spaces are defined by letting P: E
B where P is the projection
map of E onto B is continuous and surjective. If every point b of B has a neighborhood U
that is evenly covered by P1 the P is called a covering map and E is said to be a covering
space of B. This covering space is can be thought of as identical pancakes, the projection
P of the pancakes are like if you stab your stacked pancakes with a single tined fork. If
you lift the pancake up you will still have onto projection of the E, the upper pancake and
B, the lower pancake.
This idea can also be handled by looking at a unit circle and a number line the
number line divided by π. By looking at the idea you can tell the mapping of E onto B.
As the number line deforms around the unit circle the numbers correlate with the
dividends of π. As the number line gets back to the beginning of the circle or 2π the
number line starts mapping onto itself this is called the covering involution. Every point
on the number line has a correlating point on the unit circle while also having a covering
involution onto itself.
Bibliography
Flapan, Erica. When topology Meets Chemistry.
Simon, Jonathon. “Molecular Graphs as Topological Objects in Space.”
Simon, Jonathon. “Topological Chirality of Certain Molecules.”
www.cosminvers.com