Research Topics in

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• Research Topics in
• Computational and
• Mathematical
• Biology
• Lecture 5A
• 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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Answer for exercise/homework
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Schedule for rest of course

• Feb 17 Mace Eclevia Factorization for finite
  pretopogical spaces
• Feb 24 A. Balbuena Conserved Structures in RNA
  viruses
• Mar 3 H. Adorna RNA and formal languages
• Mar 10 RNA Shape Space closing discussion
• Mar 17 no meeting

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Topics to be covered

• Factorization of isotone spaces
• Biological motivation (mathematical theory of
  biological characters)
• Quick review products and quotients of isotone
  spaces
• The Factorization Theorem for isotone spaces
• Local Factorization
• Uniqueness of factorization

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Mathematical Theory of Biological Characters

• Lectures based on following papers/book
• STST01 B. Stadler, P. Stadler, G. Wagner, W.
  Fontana The topology of the possible formal
  spaces underlying patterns of change, J. Theor.
  Biol. 213 (2001)
• WAS02 G. Wagner, P. Stadler Quasi-Independence,
  Homology and the Unity of Type a Topological
  Theory of Characters, SFI preprint, Feb 02
• STAD02 P. Stadler Genotype-Phenotype Map, SFI
  preprint
• IMKL00 W. Imrich, P. Klavzar Product Graphs
  Structure and Recognition, Wiley 2000

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Biological motivation (1)

• Phenotype refers to the physical, organizational
  and behavioral expression of an organism during
  its lifetime
• Genotype refers to a heritable repository of
  information that instructs the production of
  molecules whose interactions, in conjunction with
  the environment, generate and maintain the
  phenotype
• Simplest example genotype RNA sequence,
  phenotype RNA shape
• What are the properties of the genotype-phenotype
  map? (RNA the folding map)

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RNA Genotype-Phenotype Map
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Biological motivation (2)

• Biological character a feature or property of
  the phenotype (eg morphological, genetic,)
• Several concepts for sameness /difference of
  characters
• historical homology (Darwin), biological homology
  (Wagner)
• Quasi-independence (Lewontin)
• ?Need for formal (mathematical) concept to
  clarify
• Stadler et al approach use topological
  properties of phenotype space to achieve this

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Review Product spaces
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Factorization

• Isomorphism bijective, bi-continuous
• In this case pair of projections x ? (pr1(x),
  pr2(x)) is the isomorphism

We will later use the concept of prime factor to
define a (primitive) character of a phenotype x
in X.
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Review Quotient Spaces
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Orthogonal Partitions
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Example RNA Shapes
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Factorization in pictures (1)
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Factorization in pictures (2)
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Factorization in pictures (3)
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The Factorization Theorem
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Proof strategy (substitute isotone space for
pretopology in text below), actual proof is 6
pages long and quite technical in STST01
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Proof in Lemma steps (1)
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Proof in Lemma steps (2) (substitute isotone
space for pretopology in text below)
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Local Factorization (1)
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Local Factorization (2)
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Uniqueness of factorization

• In general factorization into prime factors is
  not unique
• Example will given next week (after the strong
  product of directed graphs is explained)
• Preview factorization is unique for finite
  connected pretopological spaces

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LMUs Old Physics Building workplace for
Planck, Roentgen, Boltzmann, Wien, Sommerfeld,
von Laue, Gerlach, Heisenberg, Binnig,
Thats all, folks!