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

3 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

4 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.

5 A part of a pooling design Test 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 5clone 6clone 7

6 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.

7 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.

8 Test 1 Test 2 Test 3 clone 1 negative clone 2 clone 3 clone 4 clone 5clone 6clone 7 Inhibitor clone Negative clones Positive clones Theorem: (Hwang, Chang 2007) A (d+s)-disjunct matrix can identify all p positive clones with at most s inhibitors.

9 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 modelk-inhibitor model (k,g)-inhibitor model the general inhibitor model

10 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.

11 Complex model without inhibitor Test 1 Test 2 Test 3 clone 1clone 2 clone 3 clone 4 clone 5clone 6clone 7 Complex 1={clone 1, clone 2, clone 3} may be positive Complex 2={clone 3, clone 4, clone 5} may be positive Complex 3={clone 5, clone 7} may be positive Complex 4={clone 1, clone 3, clone 4} must be negative Complex 5={clone 1, clone 2, clone 7} must be negative

12 (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 :

13 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.

14 (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.

15 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)

16 References Chang FH, Chang H, Hwang FK (2010) Pooling designs for clone library screening in the inhibitor complex model. J Comb Optim DOI /s (online first) Chang FH, Hwang FK (2007) The identification of positive clones in a general inhibitor model. J Comput Syst Sci 73: 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: 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: 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: Stinson DR, Wei R (2004) Generalized cover-free families. Discrete Math 279:

17 The End Thanks for your attention!