My talk at the Princeton FRG meeting in March 2011

1. Igor Rivin, Temple University and
IAS School of Mathematics

4.  Can’t say it better than the Wikipedia:
 Macromolecular docking is the computational
modeling of the structure of complexes formed by two
or more interacting biological macromolecules. Protein-
protein complexes are the most commonly attempted
targets of such modeling, followed by protein-nucleic
acid complexes.
 The ultimate goal of docking is the prediction of the
three dimensional structure of the macromolecular
complex of interest as it would occur in a living
organism. Docking itself only produces plausible
candidate structures. These candidates must be
ranked to predict what would occur in nature.

6.  For protein-protein docking, can model
molecules by three-dimensional
bodies, but since the interaction happens
along the surface, the question can be
asked in the context of the geometry of
surfaces.
 What surface do we look at?

7.  There are many kinds, two favorites are:
 Van der Waals surface

8.  And my favorite, solvent accessible
surface (an εneighborhood of the van der
Waals surface, where εis the van der
Waals radius of a water molecule.

10.  Given two surfaces S1 and S2, we want to
find a coarse quasi-isometry between as
big a subsurface of S1 and S2 as possible.
 Notice that since the realization in E3 of the
surfaces is floppy (the molecules are not
rigid), we are really trying to solve the
intrinsic problem.

11.  Sowe solve a simpler problem first (the
drunk under a streetlight paradigm).

12.  We forget the subsurface confusion, and
assume that both our surfaces are
homeomorphic to the sphere S2.
 And this quasi-isometry thing is a little too
puzzling, so will go to the conformal
category, which is a little easier to work
with (and clearly is closely related to the
metric category).

13.  As Boris Springborn had said, conformality
occurs for mysterious reasons (probably
because conformal maps are aesthetically
pleasing…)
 In some biological settings, conformality is
natural (regions all grow at the same rate).

14.  Given two metrics on S2, how do we find
the conformal map between them?
 Note that the metrics are still coarse, so
we can ignore the small-scale details.
 Here is our plan of attack:

15. 1. First, find a conformal map Φ between
the surface S1 and the round metric on
S2.
2. Second, find a conformal map ψ between
S2 and S2.
3. Match them up somehow.

16.  First,
find a nice triangulation of the
surface S1. To do this, sample a bunch of
points from S1. Compute the Delaunay
triangulation (dual of Voronoi
tesselation), with respect to that point set
so, crudely speaking, every point is
connected to points close to it.

17.  Then,construct a circle packing on the
round sphere S2, combinatorially
equivalent to the Delaunay triangulation
we have just constructed.

19.  Given a topological triangulation T of S2,
we can draw it in such a way, that a disk
can be centered at every vertex, and two
disks are tangent precisely when the
vertices are joined by an edge of T.
 The packing is then unique (up to Mobius
transformation).

20.  The circle packing theorem as stated on
the last slide was noticed by W. P.
Thurston in the late seventies, though it
follows from the work of Koebe on the
uniformization of circle domains (done
many years before the birth of
Thurston, around 1916).

21.  Thurston had observed that the circle
packing theorem was an immediate
corollary of the Andreev theorem on non-
obtuse angled polyhedra in three-
dimensional hyperbolic space: the circle
packing of a triangulation and its dual
together constitute a right-angled ideal
polyhedron in three-dimensional hyperbolic
space, the existence of which followed
from Andreev’s theorem.

23.  Idealpolyhedra are polyhedra with all
vertices on the sphere of infinity (ideal
boundary) of three-dimensional hyperbolic
space.

25.  The previous page has the logos of the
various versions of Mathematica, the ideal
icosahedron logo was created by Henry
Cejtin and Igor Rivin, and later modified by
Michael Trott.

26.  Andreev’s argument is non-algorithmic.
Thurston gave a procedure to construct a
circle packing, but did not give any
indication of convergence speed, or indeed
a proof that it converged (this was supplied
several years later by Al Marden and Burt
Rodin [1989, appeared in 1992])

27.  IR(1994) showed that the construction of
a convex ideal polyhedron with prescribed
dihedral angles is a convex optimization
problem, and thus admits a very fast
algorithm (quadratic in practice). The
algorithm is particularly fast for the special
case of circle packing.

28. A direct variational algorithm for circle
packing (no polyhedra) was constructed by
Yves Colin de Verdiere (1991) using a
different functional.
 The two functionals are Legendre
transforms of each other as shown by A.
Bobenko and B. Springborn (2004).

29.  Anothercharacterization/algorithm (IR,
around 2000?): Consider hyperideal
polyhedra (all vertices beyond infinity) with
prescribed combinatorics. The volume is a
concave function of the dihedral angles.
The function is improper, the maximum is
achieved when all of the interboundary
distances collapse, which means that it
gives you the circle packing.

30.  Lots of related work has been done (X.
Bao/F. Bonahon, J-M Schlenker,
Bobenko/Springborn).
 Cruel irony: volume is convex on ideal and
hyperideal polyhedra, but not for compact
polyhedra (if it were, hyperbolization would
be easy).

31.  The idea of using circle packings to
construct conformal mappings is also due
to Thurston, but the first proof that this
actually works is due to Burt Rodin and
Dennis Sullivan [1987] (without any
convergence rate estimate). Later, much
work on the subject was done by He and
Schramm [1993-1998]

32. I have never seen any comparison
between circle packing and the more
traditional methods of conformal mapping.
No question that CP looks cool, but how
good is the convergence speed?

33.  We have constructed an approximation to
a conformal mapping from S1 onto the
round sphere via a circle packing
scheme, and similarly from S2. These two
mappings give us two densities f1 and f2
on the round sphere S2. We try to find a
Mobius transformation of S2, which comes
closest to transforming f1 to f2.

34.  So first, let’s look for the best rotation.

35.  So,
let’s look at the problem in one
dimension less:

36.  Given two (positive) functions f and g on
the circle S1, find a rotation r, such that of f
– g r is as small as possible.
 Where by small, we mean in L2 norm,
since that is easier to analyze.

37.  Given two(finite) sets S and T (of the same
cardinality) of points on the circle, find the
rotation which minimizes the distance
between them (where the level 0 question
is: how do you define distance?)

38.  We have the Fourier transform, which is an
isometry,
 So that minimizing the L2 norm of f – g r is
the same as maximizing the real part of
the the scalar product <F(f), F(g r)>, which
is a trigonometric polynomial (in the
rotation angle).

39.  Do we really have to sample densely?
 No! The maximal value of a trig polynomial
p(x) is the smallest number Y, such that
q(x)=Y – p(x) is non-negative everywhere
on S1.
 A trig polynomial q(x) is non-negative if
there another trig polynomial w(x) such
that q(x)=|w(x)|2 (Fejer-Riesz
theorem), and that…

40. ( is a semidefiniteness condition, so finding
the best rotation is a semi-definite
program, so convex! And fast (OK, not
very slow).
 If we are willing to be sleazy, there is a
very fast, and very easy to implement
algorithm

41.  maxshift2[t1_, t2_] :=
Block[
{trans1 = Fourier[t1], trans2 =
Fourier[t2]}, InverseFourier[
trans1 Conjugate[trans2] +
Conjugate[Reverse[trans1]]
Reverse[trans2]]]
 bestrot[t1_, t2_] :=
Ordering[Abs[maxshift2[t1, t2]], -1] - 1

42.  So far, all we know is how to rotate…

43.  Fora measure μon the boundary at infinity
of a Gromov-hyperbolic space, we define
the conformal barycenter of μto be:

44.  Which is (surprise!) (geodesically) convex,
so computing the argmin reduces to
convex programming.
 But not easy convex programming, since
one needs to work with the Klein model of
hyperbolic space, and then the problem is
not actually convex. Still, this can be dealt
with.

45.  There are also dynamics based(identical)
algorithms due (independently) to J. Milnor
and W. Abikoff/Ye (2002?), but they do not
generalized to higher dimensions (at least
not obviously), since they use complex
analysis.

46.  Now, to find the Mobius transformation, we
compute the conformal barycenter of our
two functions, apply a (hyperbolic)
translation to map one to the other, then a
rotation, and we are done…

47.  Since we are done in one dimension lower
than we started.
 In three dimension everything works, but
the Fourier transformation step now
involves spherical harmonics, and Wigner
D-matrices, and it is possible that
convexity is lost (but maybe not, work in
progress…)

48.  All
this leaves many more questions than
we had answers, but that’s the way it
should be.