1. Computational Molecular Biology
Group Testing – Pooling Designs

2. Group Testing (GT) Definition: Each test is on a subset of items
Given n items with at most d positive ones
Identify all positive ones by the minimum number of tests
Each test is on a subset of items
Positive test outcome: there exists a positive item in the subset
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3. An Idea of GT _ _ _ _ _ _ _ _ _ _ _ + _ _ _ _ _ + Positive Negative
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4. Example 1 – Sequential Method 9
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5. Example 2 – Non-adaptive Method
P4 p5 p6 p p p
Non-adaptive group testing is called pooling design in biology
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6. Sequential and Non-adaptive
Sequential GT needs less number of tests, but longer time.
Non-adaptive GT needs more tests, but shorter time.
In molecular biology, non-adaptive GT is usually taken. Why?
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7. Because…
The same library is screened with many different probes. It is expensive to prepare a pool for testing first time. Once a pool is prepared, it can be screened many times with different probes.
Screening one pool at a time is expensive. Screening pools in parallel with same probe is cheaper.
There are constrains on pool sizes. If a pool contains too many different clones, then positive pools can become too dilute and could be mislabeled as negative pools.
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8. Pooling Designs Problem Definition Pool: a subset of clones
Given a set of n clones with at most d positive clones
Identify all positive clones with the minimum number of tests
Pool: a subset of clones
Positive pool: a pool contains at least one positive clone
Clones = Items
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9. Relation to Pooling Designs
clones
c1 c cj cn
p … 0 … 0 … 0 … 0 0
p … 0 … 0 … 0 …
pools
pi … 0 … 1 … 0 …
pt 0 0 … 0 … 0 … 0 …
txn tx1
M[i, j] = 1 iff the ith pool contains the jth clone
Decoding Algorithm:
Given M and V, identify all positive clones V
Testing
Mtxn =
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10. Observation Observation: All columns are distinct.
clones
c1 c2 c cj
p p p
pools
Observation: All columns are distinct.
To identify up to d positives, all unions of up to d columns should be distinct!
Union of d columns: Boolean sum of these d columns
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11. Challenges
Challenge 1: How to construct the binary matrix M such that:
Outputs of any union of d columns are distinct
Challenge 2: How to design a decoding algorithm with efficient time complexity [O(tn)]
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12. d-separable Matrix All unions of d columns are distinct. clones
c1 c2 c cj cn
p … 0 … 0 … 0 … 0 … 0 … 0 … 0
p … 0 … 0 … 0 … 0 … 0 … 0 … 0
p … 0 … 0 … 0 … 0 … 0 … 0 … 0
pools … 0 … 0 … 0 … 0 … 0 … 0 … 0 .
pi … 0 … 0 … 1 … 0 … 0 … 0 … 0
pt … 0 … 0 … 0 … 0 … 0 … 0 … 0
All unions of d columns are distinct.
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13. d-separable Matrix All unions of up to d columns are distinct.
clones
c1 c2 c cj cn
p … 0 … 0 … 0 … 0 … 0 … 0 … 0
p … 0 … 0 … 0 … 0 … 0 … 0 … 0
p … 0 … 0 … 0 … 0 … 0 … 0 … 0
pools … 0 … 0 … 0 … 0 … 0 … 0 … 0 .
pi … 0 … 0 … 1 … 0 … 0 … 0 … 0
pt … 0 … 0 … 0 … 0 … 0 … 0 … 0
All unions of up to d columns are distinct.
Decoding: O(nd)
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14. d-disjunct Matrix
Definition: A binary matrix Mtxn is a d-disjunct matrix (d < t) if:
The union of any d columns does not contain any other column
Example:
A 2-disjunct matrix M =
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15. d-disjunct Matrix (cont)
d-disjunct matrix can efficiently identify up to d positive clones. Why?
Theorem 1: All unions of d distinct columns are distinct (thus d-disjunct implies d-separable)
Theorem 2: The number of clones not in negative pools is always at most d
Corollary 1: The tests of negative outputs determine all negative clones
Decoding time complexity: O(tn)
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16. Proof of Theorem 2
Note that an item does not appearing in any negative pool iff its corresponding column is contained by the union of d positive columns
Therefore, the number of items not appearing in any negative pool is more than d iff there are at least a non-positive item whose column is contained by the d positive columns
But M is d-disjunct, hence Theorem 2 follows
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17. Decoding Algorithm Input: d-disjunct matrix M and output vector V
Output: All positive clones
for each clone c in n clones
if c is in a negative pool
remove c
return remaining clones
c1 c2 c3 c4 c5 c6 p P P P
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18. Fields
Field: is any set of elements that satisfies the field axioms for both addition and multiplication and is a division algebra
Eg: Compex, Rational, Real
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19. Division Algebra
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20. Finite Fields Finite Field:
is a field with a finite field order, i.e., number of elements.
The order of a finite field is always a prime or a prime power (power of a prime)
Eg: 16 = 2^4 is a prime power where 6, 15 are not
Eg: in GF(5), 4+3=7 is reduced to 2 modulo 5
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21. How to construct a d-disjunct matrix
Consider a finite field GF(q). Choose s, q, k satisfying:
Step 1: Construct matrix Asxn as follows:
for x from 0 to s -1
for each polynomials pj of degree k
A[x,pj] = pj(x)
p p pj pn 1
A =
x p2(x) pj(x)
s-1
First, consider a finite field of order q.
Construct the matrix A with s rows and n columns. Rows are indexed in the value of s where columns are indexed in the value of polynomials of degree k. The value of each cell in the matrix A is assigned as follows:
For each row x and colums pj, we assign the value pj(x) over the finite field.
Why n \le q^k, to make sure we have enough polynomial to associate with n items
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22. Algorithm (cont) Step 2: Construct matrix Btxn from Asxn as follows:
for x from 0 to s -1
for y from 0 to q -1
for each polynomials pj of degree k
if A[x,pj] = = y
B[(x,y),pj] = 1
else B[(x,y),pj] = 0
p p2 pj pn 1
A =
x p2(x) pj(x)
s-1
p2(x) ≠ y
p p pj pn
(0,0)
(0,1)
B =
(x,y)
(s-1,q-1)
pj(x) = y
Next, at the second step, we construct the matrix B from the matrix A. The matrix B has t rows and n columns. The columns are indexed in the polynomials of degree k. The rows are indexed in the ordered pairs in s values and in q values. The values of each cell in the matrix B is assigned as follows:
Let look at the row (x,y) and the column p2 in B. Then in the matrix A, if this cell is != to y, then …
We claim that B is d-disjunct and just use the simple decoding algorithm as we just presented to identify all the positive clones. 1
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23. Algorithm Analysis
Theorem 3: (Correctness) If kd ≤ s ≤ q, then Btxn is d-disjunct.
Theorem 4: The number of tests t obtained from this algorithm is t = qs = O(q2) where:
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24. Errors in Experiments False negative: False positive:
Pool contains some positive clones
But return the negative outcome
False positive:
Pool contains all negative clones
But return the positive outcome
Now, it is well known that there may exist some errors in biological experiments. The test may return some false negative or false positive results. In the false negative, the pool contains some positive clones. It should return a positive outcome. However, under testing errors, it return a negative outcome. Likewise, in the false positive, the pool contains all negative clones. So, how we can correct these errors?
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25. An e-Error Correcting Model
Definition:
Assume that there is at most e errors in testing
All positive clones can still be identified
Hamming distance: the Hamming distance of two column vectors is the number of different components between them
e-error-correcting: A matrix is said to be e-error-correcting if the Hamming distance of any two unions of d columns is at least 2e + 1
We call this an e-error correcting model. In this model, we assume that there is at most e errors in testing. After constructing the d-disjuct matrix and get the outcome vector which consists of at most e errors, the model is still able to correct these errors in order to identify all the positive clones.
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26. (d,e)-disjunct Matrix
Definition: An t × n binary matrix M is (d, e)-disjunct if for any one column j and any other d columns j1, j2, , jd, there exist e + 1 rows i0, i2, … , ie such that Miuj = 1 and Miujv = 0 for u = 0, 1,…, e and v = 1, 2, , d
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27. E-error Correcting
Theorem 5: For every (d,k)-disjunct matrix, the Hamming distance between any two unions of d columns is at least 2k + 2
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28. Theorem 6
Theorem 6: Suppose testing is based on a (d,e)-disjunct matrix. If the number of errors is at most e, then the number of negative pools containing a positive item is always smaller than the number of negative pools containing a negative item
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29. Proof of Theorem 6
Let i be a positive item, j be a negative item. Suppose #negative pools containing i = m. Then m pools must receive errors. Hence, there are at most e – m error tests turning negative outcome to positive outcome. Moreover, if no error exists, # negative pools containing j is at least e + 1 due to (d,e)-disjunct. Hence #negative pools containing j is at least (e+1)-(e-m) = m +1>m
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30. Decoding in e-error-correcting
Corollary: From Theorem 6, we see that to decode positives from testing based on (d,e)-disjuct matrix, we only need to compute the number of negative pools containing each item and select d smallest one. This runs in time O(nt)
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31. Decoding Algorithm with e Errors
T = empty set
for each clone ci (i = 1…n)
t(ci) = # negative pools containing ci
T = T t(ci)
end for
Let Td = set of d smallest t(ci) in T
return ci if t(ci) in Td
Time complexity: O(tn)
In this proposed method, for each clone, we count the number of negative pools containing this clone. Then we just select d smallest one. This decoding algorithm is able to correct all e-errors and find all positive clones because of the previous theorem that we have proved.
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