Computational Molecular Biology

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Computational Molecular Biology

• Pooling Designs Inhibitor Models

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An Inhibitor Model

• In sample spaces, exists some inhibitors
• Inhibitor anti-positive
• (Positives Inhibitor) Negative

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Inhibitor

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x

Negative
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An Example of Inhibitors
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Inhibitor Model

• Definition
• Given a sample with d positive clones, subject to
  at most r inhibitors
• Find a pooling design with a minimum number of
  tests to identify all the positive clones (also
  design a decoding algorithm with your pooling
  design)

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Inhibitors with Fault Tolerance Model

• Definition
• Given n clones with at most d positive clones and
  at most r inhibitors, subject to at most e
  testing errors
• Identify all positive items with less number of
  tests

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Preliminaries
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2-stages Algorithm
What is AI? The set AI should contains all the
inhibitors and no positives. Hence the set PN
contains all positives (and some negatives) but
no inhibitors
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2-stages Algorithm
At this stage, the problem become the
e-error-correcting problem.
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Non-adaptive Solution (1 stage)

1. P contains all positives
2. N contains all negatives
3. O contains all inhibitors and no positives

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Non-adaptive Solution
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Generalization

• The positive outcomes due to the combination
  effect of several items
• Items are molecules
• Depends on a complex subset of molecules
• Example complexes of Eukaryotic DNA
  transcription and RNA translation

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A Complex Model

• Definition
• Given n items and a collection of at most d
  positive subsets
• Identify all positive subsets with the minimum
  number of tests
• Pool set of subsets of items
• Positive pool Contains a positive subset

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What is Hypergraph H?

• H (V,E ) where
• V is a set of n vertices (items)
• E a set of m hyperedges Ej where Ej is a subsets
  of V
• Rank r max Ej s.t Ej inE

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Group Testing in Hypergraph H

• Definition
• Given H with at most d positive hyperedges
• Identify all positive hyperedges with the
  minimum number of tests
• Hyperedges suspect subsets
• Positive hyperedges positive subsets
• Positive pool contains a positive hyperedge
• Assume that Ei Ej

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d(H)-disjunct Matrix

• Definition
• M is a binary matrix with t rows and n columns
• For any d 1 edges E0, E1, , Ed of H, there
  exists a row containing E0 but not E1, , Ed
• Decoding Algorithm
• Remove all negatives edges from the negative
  pools
• Remaining edges are positive

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Construction Algorithms

• Consider a finite field GF(q). Choose k, s, and
  q
• Step 1
• for each v in V
• associate v with pv of degree k -1 over GF(q)

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A Proposed Algorithm

• Step 2 Construct matrix Asxm as follows
• for x from 0 to s -1 (rkd lts lt q)
• for each edge Ej inE
• Ax,Ej PE(x) pv(x) v in Ej
• E1 E2 Ej Em
• 0
• 1
• A
• x PE2(x) PEj(x)
• s-1

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A Proposed Algorithm

• Step 3 Construct matrix Btxn from Asxm as
  follows
• for x from 0 to s -1
• for each PEj(x)
• for each vertex v in V
• if pv(x) in PEj(x), then B(x, PEj(x)),v
  1
• else B(x, PEj(x)),v 0
• E1 E2 Ej Em
• 0
• 1
• A
• x PEj(x)
• s-1

v1 v2
vj vn (0,
PE0(0)) (0, PE1(0)) B (x,
PEj(x)) (s-1, PEm(s-1))
0
1
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Analysis

• Theorem If rd (k -1) 1 s q, then B is
  d(H)-disjunct

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Proof of d(H)-disjunct Matrix Construction

• Matrix A has this property
• For any d 1 columns C0, , Cd, there exists a
  row at which the entry of C0 does not contain the
  entry of Cj for j 1d
• Proof Using contradiction method. Assume that
  that row does not exist, then there exists a j
  (in 1d) such that entries of C0 contain
  corresponding entries of Cj at least r(k-1)1
  rows. Then PEj(x) is in PE0(x) for at least
  r(k-1)1 distinct values of x. This means that Ej
  is in E0

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Proof of d(H)-disjunct Matrix Construction (cont)

• Prove B is d(H)-disjunct
• Proof A has a row x such that the entry F in
  cell (x, E0) does not contain the entry at cell
  (x, Ej) for all j 1d. Then the row ltx,Fgt in B
  will contain E0 but not Ej for all j 1d