Folding RNA a confluence of biology, mathematics, and physics

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Folding RNA a confluence of biology,
mathematics, and physics

• A. Zee
• Institute for Theoretical Physics
• University of California
• Santa Barbara, CA, USA

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Parisi Fest, Rome 2008
From http//chimera.roma1.infn.it/GIORGIO/indexh
ome.htm
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• Unhappily, I have co-authored only one paper
  (thanks to Marc Mezard) with Giorgio Parisi
• M. Mezard, G. Parisi, A. Zee,
• Spectra of Euclidean Random Matrices, Nuclear
  Physics B559 (3), pp. 689-701, 1999

• But certainly, in my work with E. Brezin, M.
  Kardar, and others, I am constantly invoking and
  quoting Parisis classic work, such as the BIPZ
  paper of random matrix theory.

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Field Theorists Fold RNA

• An international collaboration with
• Henri Orland
• Matt Pillsbury
• Anthony Taylor
• Graziano Vernizzi
• Paolo Ribeca
• Michael Bon
• (started with a sabbatical year spent by H.
  Orland at KITP)
• We exploit topological concepts embedded in
  large N matrix field theory to attack an
  important problem in biology the folding of RNA

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Our parents worked hard to give us DNA, a message
written in an alphabet with 4 letters C, G, A, T
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Central Dogma of Biology Watson Crick
Information storage hard disk floppies to
give to your friends Function (enzyme, motor,
muscle,)
You dont want to mess with your hard disk, but
you can have as many copies of RNA as you want
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RNA
Much much much shorter than DNA

• A message written in a 4-letter alphabet, C, G,
  A, U
• Attraction between C and G between A and U
• Hydrogen bond saturates --gt once a C paired
  with a G, it does not pair with another G
• Think of beads on a long chain connected by
  rigid rods. Saturation of hydrogen bond means
  that all beads could be glued at most once.

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In each one of our cells, stretches of DNA open
up information is copied onto RNA
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RNA is single stranded, unlike DNA
Because of attraction between C and G between A
and U, once RNA floats free, it folds into a
definite shape Different message gives different
shape RNA folding problem Why is this
important?
Given the message what is its shape?
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Watson Crick thought that RNA is merely a
passive carrier of information from DNA to
protein (messenger RNA) Revenge of
RNA Important discovery of last 15 years or so
RNA plays a crucial enzymatic role Biochemistry
incredibly complicated, but to first
approximation shape of molecule (lock and
key) By the way, biologists working on the
origin of life now generally believe that it
started with an RNA world. DNA came later.
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Yeast tRNA
40 overlap with corresponding human
tRNA common metabolism
Mutant form of U
sequence space shape space
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Secondary Structure small subunit ribosomal RNA
C,G,A,U nucleotides
hairpin
cloverleaf
Escherichia coli
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Single strand Helix Hairpin Bulge Multiloop



PDB 283d PDB 405d PDB 1e4p PDB 1r7w PDB
1kh6
Examples of basic RNA secondary structure motifs
(spacefill view, 3d secondary structure,
secondary structure motifs)
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• glue
• planar as shown two hairpins
• but if glue 4 to 3, no longer planar
• Rigidity constraint chain not infinitely
  flexible, gluing 1 and 2 not allowed.
• If glue 5 and 1, gives kissing hairpins

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Statistical mechanics Gluing the nucleotide at
site i to the nucleotide at site j gives a
contribution of V_ij to the partition function V
contains the Boltzmann factor including the
rigidity constraint etc. V carries information
about the specific sequence we are studying.
Exists a literature on random sequences.
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Z 1 S Vij S Vij Vkl S Vik
Vjl S Vik Vjl Vmn S VVVV
These look like Feynman diagrams a quark
propagating along, emitting and absorbing gluons
Just list all L! possibility

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Mathematically, better to glue the two ends to
form a circle
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Mathematicians do not like to draw on a flat
piece of paper prefer manifolds without
boundary, compactify plane into sphere so no
boundary at infinity.
Put a hole in the sphere to represent RNA
chain punctured sphere
Sphere genus g o
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Physicists lines cross Mathematicians on what
kind of surface (what genus?) can we draw this
without the lines crossing?
Punctured Torus Torus (or doughnut, bagel) with
a hole put into it
Punctured Torus Torus genus g1
Different meaning in biology and mathematics!
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Genus
Mathematics (Topology) How many handles?
Better word than holes Biology
(Taxonomy) Group of species, e.g. Homo,
Pan A joke about a conference in mathematical
biology
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Two handles
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• Biologists called planar diagrams, i.e. diagrams
  that could be drawn on a sphere without crossing
  secondary structures.
• They call diagrams that could be drawn on a torus
  without crossing tertiary structures
• Experimentally, the relative importance of
  tertiary structures to secondary structures can
  be dialed by varying Mg ion concentration (has
  to do with screening due to Mg)

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Hairpin
PDB 1a51
H-pseudoknot
PDB 1a60
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More RNA pseudoknots
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Eulers Theorem
F V E 4 4 6

• F ?V ?E
• 4-2 1 3
• 2

? F ?V ?E 0 F V E 2 4 4 6
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Now theoretical physicists could exploit our
knowledge of random matrix theory and large N
quantum chromodynamics to formulate the RNA
folding problem and have fun! Random matrix
theory was invented by Wigner (Nobel laureate) in
the mid-1950s to study nuclear physics, but its
growth and ramifications over the last 50 years
have been beyond the wildest dreams of Wigner and
his friends, with implications and uses in
diverse fields, e.g. disordered materials in
condensed matter physics, pure mathematics,
operator algebra, M-theory (supposed to contain
string theory) more recently econophysics
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Hollywood already knows! The matrix secretly
rules the universe.
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Matrix theory representation of Z

Flavor or site
L matrices living in 1d
where
Color
Non-translation invariant Gaussian field theory
with non-trivial observable
Wick contraction reproduces Z as given
earlier Expansion in 1/N gives us a systematic
procedure for extracting secondary structure,
tertiary structure, etc.
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matrix
Now N explicit can do large N expansion
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Pillsbury, Orland, Zee, Phys. Rev. E72 showed
that there are 8 topologies of pseudoknots with
genus 1.
In an obvious pairing notation ABAB ABACBC ABCAB
C ABCADBCD 4 obtained by nesting as shown in
following figure
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There are only 8 types of irreducible pseudoknot
with genus g1
Quite common in RNA databases
Very rare
Not yet been reported
Encouragement to experimentalists! (e.g., Prof.
A. Mondragon, Northwestern University, Evanston
IL)
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Recent progress focus on massive Monte Carlo,
numerical work largely carried out by Graziano
Vernizzi et al. See Bon, Vernizzi, Orland,
Zee, q-bio.BM/0607032 (21 July 2006)

• Caution to particle and quantum field theorists
• In this paper LoopsColors-1 and
• Genus g (P-L)/2 (P is pairs)
• But concepts of reducibility, primitive diagrams,
  etc are same
• as in Dyson.

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More progress We (this means the young people!)
analyzed the two main databases for RNA

1. PSEUDOBASE

1. wwPDB (world wide protein database, it contains
   some RNA)

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PDB 1vou 70s ribosome from Escherichia Coli
primitive arc structure
Genus 7 Length 2825
Pseudoknot with genus 1, decorated with 6 simple
H- and K-pseudoknots
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• As a theoretical benchmark we solved
  analytically a simple
• model (V taken to be 1 except for the diagonal
  elements,
• non-zero as required to regulate the calculation)
• Vernizzi, Orland, Zee, Phys. Rev. Letters 94
  (2005)
• (generalized Laguerre polynomials vs. Kazakov
  method)
• Asymptotically, genus 0.25 L . For L2000, g
  500.
• Including steric constraints reduces g 280.
• (Vernizzi, Ribeca, Orland, Zee, Phys. Rev. E73
  (2006))
• Interestingly, Nature reduces the possible
  topology.
• (Our analysis shows that most structures built
  from very simple
• primitive blocks, with genii 1 or 2, nested
  inside more complex
• pseudoknots, of genus typically smaller than 8.)

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• Recent paper
• Bon, Vernizzi, Orland, Zee,
• Journal of Molecular Biology, vol. 379 (2008), p.
  900.
• explanatory and review written for biologists

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• An amusing footnote
• A classic problem in combinatorial optimization
  is the bipartite
• matching problem, known as the marriage problem
• Given happiness (energylt0) when m_i and w_j
  matched
• Minimize total energy
• Studied by Knuth, Nieuwenhuizen, Orland, Mezard
  Parisi,
• Y-C. Zhang, ..
• Obviously, the N 1 limit of our RNA folding
  representation!