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Computational Molecular Biology. Pooling Designs – Inhibitor Models. An Inhibitor Model. In sample spaces, exists some inhibitors Inhibitor = anti-positive (Positives + Inhibitor) = Negative. _. _. _. _. _. Inhibitor. +. _. x. +. Negative. An Example of Inhibitors. - PowerPoint PPT Presentation
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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
_+__
___x
Inhibitor
Negative
+
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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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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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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)
kqnqskrd and1)1(
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Step 2: Construct matrix Asxm as follows:
for x from 0 to s -1 (rkd <=s < q)
for each edge Ej inE
A[x,Ej] = PE(x) = {pv(x) | v in Ej}
E1 E2 Ej Em
0
1
A =
x PE2(x) PEj(x)
s-1
A Proposed Algorithm
)}(),(),({)(
then },,{
3212
3212
xpxpxpxP
vvvE
vvvE
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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
A Proposed Algorithm
v1 v2 vj vn
(0, PE0(0))
(0, PE1(0))
B =
(x, PEj(x))
(s-1, PEm(s-1))
(x)P (x)pjE2v
0 1
(x)P (x)p Ejv j
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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 = 1…d
Proof: Using contradiction method. Assume that that row does not exist, then there exists a j (in 1…d) 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 = 1…d. Then the row <x,F> in B will contain E0 but not Ej for all j = 1…d
