Pooling designs for clone library screening in the inhibitor complex model

Department of Mathematics and Science

National Taiwan Normal University (Lin-

Kou)

Speaker: Fei-Huang Chang

This work joins with Huilan Chang, Frank K. Hwang.

2

Outline

Introduction Main tools : d-disjunct, (H:d)-disjunct matrix The classical model The complex model Conclusions

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Introduction

In the clone library screening problem, the goal is to identify a small set of positive clones from a large set of negative clones.

Group test is a test on a subset of clones Negative outcome: if all clones are negative Positive outcome: if exists at least one

positive clone

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Introduction

Pooling design : a sequence of group tests

A nonadaptive pooling design :all tests simultaneously

A pooling design is usually represented by a binary matrix M where rows are indexed by tests and columns by clones.

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A part of a pooling design

0

0

1

1010010

0100101

1011001Test 1

Test 2

Test 3

clone 1

outcome

positive

negative

The clone 4 must be a positive clone

clone 2 clone 3 clone 4 clone 5 clone 6 clone 7

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d-disjunct Matrix

A binary matrix is called d-disjunct if no column is covered by the union of any other d columns.

Theorem (Du and Hwang 2000) A d-disjunct matrix can identify all p positive clones. (p<=d).

Proof: For any negative clone c, there exists a row (test) which contains

c but none of all d positive clones. i.e. the outcome of the test is negative.

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1-inhibitor model

A third type of clones called inhibitors whose

existence may cancel the effect of positive clones.

1-inhibitor model :(proposed by Farach et al 1997)

a single inhibitor clone dictates the test outcome to be negative regardless of how many positive clones.

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1

0

0

1010010

0100101

1011001Test 1

Test 2

Test 3

clone 1

negative

clone 2 clone 3 clone 4 clone 5 clone 6 clone 7

Inhibitorclone

Negativeclones

Positiveclones

Theorem: (Hwang, Chang 2007)

A (d+s)-disjunct matrix can identify all p positive clones with at most s inhibitors.

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The general inhibitor model

k-inhibitor model : k inhibitor clones dictate the test outcome to be negative.

(k,g)-inhibitor model : k inhibitor clones cancel the effect of g positive clones.

The general inhibitor model : we don’t know the two parameters k and g for sure.

1-inhibitor model k-inhibitor model (k,g)-inhibitor modelthe general inhibitor model

1997 2003 2007

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The complex model

In some DNA screening environment, it takes a subset of clones, called a complex, to induce a positive outcome. We call such a model the complex model, as versus the clone model.

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Complex model without inhibitor

0

0

1

1010011

0101101

1011111Test 1

Test 2

Test 3

clone 1 clone 2 clone 3 clone 4 clone 5 clone 6 clone 7

Complex 1={clone 1, clone 2, clone 3} may be positiveComplex 2={clone 3, clone 4, clone 5} may be positiveComplex 3={clone 5, clone 7} may be positive

Complex 4={clone 1, clone 3, clone 4} must be negativeComplex 5={clone 1, clone 2, clone 7} must be negative

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(H:d)-disjunct matrix

For a complex X, we say a row (test) covers X if it has 1-entry in every column of X.

^X: denotes the set of tests covering X. (H:d)-disjunct :

.,, of nonebut

covering row a exists always there

,,, complexes 1any for if

1

0

10

d

d

XX

X

XXXd

d

i iXX10 )(^^ i.e.

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A general inhibitor complex model (r,d,s)

Let H denote the given set of complexes in the considered problem, then H can viewed as a hypergraph with clones as vertices and complexes as edges.

Denote the minimum number of rows in an (H:d)-disjunct matrix with n columns by t(H:d,n).

A general inhibitor complex model (r,d,s): r: a complex has at most r clones. d: at most d positive complexes. s: at most s inhibitor complexes.

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(Chang, Chang , Hwang 2010)

Theorem 1:

The number of rows in a nonadaptive pooling design for the (r,d,s) general inhibitor complex model is at least t(H:s,n).

Theorem 2:

An (H: d+s)-disjunct matrix can identify all positive complexes under the (r,d,s) general inhibitor complex model.

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Conclusions

We gave a nonadaptive pooling design to the most general model in such an environment with no need know the exact relation between inhibitors and positive complexes.

We gave a lower bound and an upper bound of tests in a nonadaptive pooling design for some inhibitor complex model.

In fact, we also gave a nonadaptive pooling design for the inhibitor complex model with error-tolerance ability and gave a more efficient pooling degign for the k-inhibitor complex model.

This design is an (H:d)-disjunct matrix whose construction has been studied in Dyachkov et al. (2001), Gao et al. (2006) and Wei (2004)

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References

Chang FH, Chang H, Hwang FK (2010) Pooling designs for clone library screening in the inhibitor complex model. J Comb Optim DOI 10.1007/s10878-009-9279-9 (online first)

Chang FH, Hwang FK (2007) The identification of positive clones in a general inhibitor model. J Comput Syst Sci 73:1090-1094

De Bonis A, Vaccare U (1998) Improved algorithms for group testing with inhibitors. Inf Process Lett 67:57-64

De Bonis A, Vaccare U (2003) Constructions of generalized superimposed codes with applications to group testing and conflict resolution in multiple access channels. Theor Comput Sci A 356:223-243

Du DZ, Hwang FK (2000) Combinational group testing and its applications, 2nd edn. World scientific, Singapore.

Du DZ, Hwang FK (2005) Identifying d positive clones in the presence of inhibitors. Int J Bioinform Res Appl 1:168-168

Gao H, Hwang FK, Thai M, Wu W, Znati T (2006) Construction of d(H)-disjunct matrix for group testing in hypergraphs. J Comb Optim 12: 297-301

Stinson DR, Wei R (2004) Generalized cover-free families. Discrete Math 279:463-477

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The End

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