Learning Immune-Defectives Graph Through Group Tests

This paper deals with an abstraction of a unified problem of drug discovery and pathogen identification. Pathogen identification involves the identification of disease-causing biomolecules. Drug discovery involves finding chemical compounds, called lead compounds, that bind to pathogenic proteins and eventually inhibit the function of the protein. In this paper, the lead compounds are abstracted as inhibitors, pathogenic proteins as defectives, and the mixture of “ineffective” chemical compounds and non-pathogenic proteins as normal items. A defective could be immune to the presence of an inhibitor in a test. So, a test containing a defective is positive if it does not contain its “associated” inhibitor. The goal of this paper is to identify the defectives, inhibitors, and their “associations” with high probability, or in other words, learn the immune defectives graph (IDG) efficiently through group tests. We propose a probabilistic non-adaptive pooling design, a probabilistic two-stage adaptive pooling design, and decoding algorithms for learning the IDG. For the two-stage adaptive-pooling design, we show that the sample complexity of the number of tests required to guarantee recovery of the inhibitors, defectives, and their associations with high probability, i.e., the upper bound, exceeds the proposed lower bound by a logarithmic multiplicative factor in the number of items. To be precise, lower and upper bounds of Ω((r + d) log n + rd) and O(rd log n) tests, respectively, are identified for classifying r inhibitors and d defectives amongst n items, and identifying their associations. For the nonadaptive pooling design, we show that the upper bound (given by O((r + d)
²
log n) tests) exceeds the proposed lower bound (given by max{Q((r + d)logn + rd), Ω((r
²
/log r) log n), Ω(d
²
)} tests) by at most a logarithmic multiplicative factor in the number of items.