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
My T. Thai
[email protected]
2
An Example of Inhibitors
My T. Thai
[email protected]
3
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)
My T. Thai
[email protected]
4
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
My T. Thai
[email protected]
5
Preliminaries
My T. Thai
[email protected]
6
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
My T. Thai
[email protected]
7
2-stages Algorithm
At this stage, the problem become the e-errorcorrecting problem.
My T. Thai
[email protected]
8
Non-adaptive Solution (1 stage)
1. P contains all positives
2. N contains all negatives
3. O contains all inhibitors and no positives
My T. Thai
[email protected]
9
Non-adaptive Solution
My T. Thai
[email protected]
10
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
My T. Thai
[email protected]
11
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
My T. Thai
[email protected]
12
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 }
My T. Thai
[email protected]
13
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
My T. Thai
[email protected]
14
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
My T. Thai
[email protected]
15
Construction Algorithms
Consider a finite field GF(q). Choose k, s, and q:
rd (k  1)  1  s  q and n  q k
Step 1:
for each v in V
associate v with pv of degree k -1 over GF(q)
My T. Thai
[email protected]
16
A Proposed Algorithm
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
0
1
A=

Ej
E2  {v1 , v2 , v3} then
PE2 ( x)  { pv1 ( x), pv2 ( x), pv3 ( x)}

x

Em
PE2(x)
PEj(x)

s-1
My T. Thai
[email protected]
17
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
p v2 (x)  PE j (x)
E1
0
1
A=
E2

Ej

v1
Em
vj

vn
(0, PE1(0))
B
PEj(x)

=
(x, PEj(x))

s-1

(0, PE0(0))

x
v2
p v j (x)  PEj (x)
0
1

(s-1, PEm(s-1))
My T. Thai
[email protected]
18
Analysis
 Theorem: If rd (k -1) + 1≤ s ≤ q, then B is
d(H)-disjunct
My T. Thai
[email protected]
19
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(k1)+1 distinct values of x. This means that Ej is
in E0
My T. Thai
[email protected]
20
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
My T. Thai
[email protected]
21