Research Topics in

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• Research Topics in
• Computational and
• Mathematical
• Biology
• Lecture 4A
• Dr. Eduardo Mendoza
• www.engg.upd.edu.ph/compbio
• Mathematics Department Physics
  Department
• University of the Philippines
  Ludwig-Maximilians-University
• Diliman Munich, Germany
• eduardom_at_math.upd.edu.ph
  Eduardo.Mendoza_at_physik.uni-muenchen.de

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Topics to be covered

• What is the topology of shape space?
• Basics of generalized topological spaces
• Closure spaces
• Recombination spaces
• Neighborhood spaces
• Subspaces
• Quotient spaces
• Product Spaces

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Schultes-Bartel Experiment and neutral networks

• Constructed in vitro LIG-HDV from 2 parent
  sequences LIG-P HDV-P (n88, diff catalytic
  activities)
• LIG-HDV compatible with both associated shapes
  (example for Intersection Theorem!)
• Independent optimization (thru mutation
  selection) yielded sequences LIG-4 and HDV-2 with
  full catalytic efficiency of parents (direct
  proof of existence of extended neutral networks,
  example of an RNA molecule which switches between
  two conformations)

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A picture of the neutral network
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Review Structure of sequence space

• S(n,A) sequences of length n over the alphabet A
• S(n) S(n,A) with A A,U,G,C
• Similarly for shapes Sh(n,A), Sh(n)
• Hamming graph structure (distance) on S(n,A)
  well interpreted as point-wise mutation this is
  the well-known hypercube

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What is the topology of shape space?

• Shape space has been traditionally been seen as
  a metric space
• Various metrics (or distance) functions have been
  defined, However none of these have a clear
  biological interpretation (vs. the case in
  sequence space)
• Starting with work of Fontana-Schuster FOSC98
  this metric approach has been seriously
  questioned
• The work of Stadler et al SSWF01 established a
  biologically relevant topological approach based
  on the concept of accessibility

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The concept of accessibility (1)
SSWF01
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The concept of accessibility (2)
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Non-symmetry of accessibility
FONT02
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Accessibility leads to generalized topological
spaces

• Using accessibility to define nearness of
  shapes leads even to a more general structure
  than a topology
• to a pre-topology (if only mutations are
  considered as operations in S(n))
• even just to a closure space (or equiv. a
  neighborhood space, if recombination is allowed)

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Closure, Interior and Neighborhood

• We will deal with closely related concepts

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Axioms for Generalized Topological (GT) Spaces
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Exercises

• What are the corresponding properties of the int
  function for (K0)-(K4)?
• Show the equivalence of the formulations of (K2)
  for the closure and neighborhood functions

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Example equivalence of (K4)
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GT-Terminology

• (X,cl) is
• an isotone space if (K0),(K1) hold
• a neighborhood space if (K0), (K1), (K2) hold
• a pretopological space if (K0), (K1), (K2), (K3)
  hold
• a topological space if (K0), (K1), (K2), (K3),
  (K4) hold

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Overview of space structures
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Neighborhoods of sets
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Continuous functions
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Biological examples for closure functions

• Mutation on sequence (or genotype) space
• cl (A) set of all mutations that can be
  obtained from A in a single step
• Homework 1 show that cl defines a pretopology on
  S(n), ie, (K0)-(K3) hold

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2. Recombination functions
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Recombination Space
(R,cl) in other words is a neighborhood
space. Homework 2. Prove Theorem 1
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Induced closure function on phenotype sets
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Subspaces of Neighborhood Spaces
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Product spaces
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Quotient Spaces
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LMUs Old Physics Building workplace for
Planck, Roentgen, Boltzmann, Wien, Sommerfeld,
von Laue, Gerlach, Heisenberg, Binnig,
Lets have a break!
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References