1. Computational Molecular Biology
Pooling Designs – Inhibitor Models

2. An Inhibitor Model _ _ _ _ _ + _ x +
In sample spaces, exists some inhibitors
Inhibitor = anti-positive
(Positives + Inhibitor) = Negative _ _ _ _ _
Inhibitor +
More challenging, in the pooling designs, there exists another type of clones called Inhibitor. _ x +
Negative
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3. An Example of Inhibitors
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4. 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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5. 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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6. Preliminaries
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7. 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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8. 2-stages Algorithm
At this stage, the problem become the e-error-correcting problem.
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9. Non-adaptive Solution (1 stage)
P contains all positives
N contains all negatives
O contains all inhibitors and no positives
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10. Non-adaptive Solution
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11. 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
Now, let’s look at the pooling design problem at the different angle. Up until this point, we have seen that the pool is positive if it contains a positive item. However, under some scenarios, the positive outcome of a test dues to the combination effect of several items. For example, if items are molecules, the many biological processes depends on the present of a complex, which is a subset of molecules. The study of this general version is very important.
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12. A Complex Model Definition Pool: set of subsets of items
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
And we call this general version a Complex Model. In the complex model, given n items and a collection of at most d positive subsets, the goal is to identify all positive subsets with minimum number of test.
Here, in this context, the pool is a set of subsets of items. The pools is positive if it contains a positive subset
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13. 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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14. 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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15. d(H)-disjunct Matrix Definition: Decoding Algorithm:
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
How can we solve this problem? Can we extended the d-disjunct matrix for the classical problem to the complex model. Fortunately, the answer is yes, we call it d(H)-disjunct matrix. The d(H)-disjunct matrix is also a binary matrix, with n column representing the number of items and t rows, representing the number of pools.
Using the same method as in the classical model, we were able to prove that the number of edges is not in the negative pools are at most d. Hence, the decoding algorithm is simply just similar as before…
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16. 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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17. 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 Ej Em 1
A =
x PE2(x) PEj(x)
s-1
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18. 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 1
A =
x PEj(x)
s-1
v v vj vn
(0, PE0(0))
(0, PE1(0))
B =
(x, PEj(x))
(s-1, PEm(s-1)) 1
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19. Analysis Theorem: If rd (k -1) + 1≤ s ≤ q, then B is d(H)-disjunct
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20. 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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21. 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
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