Given a transmembrane protein, we wish to find related ones by a database search. Due to the strongly hydrophobic amino acid composition of transmembrane domains, suboptimal results are obtained when general-purpose scoring matrices such as BLOSUM are used. Recently, a transmembrane-specific score matrix called PHAT was shown to perform much better than BLOSUM. In this article, we derive a transmembrane score matrix family, called SLIM, which has several distinguishing features. In contrast to currently used matrices, SLIM is non-symmetric. The asymmetry arises because different background compositions are assumed for the transmembrane query and the unknown database sequences. We describe the mathematical model behind SLIM in detail and show that SLIM outperforms PHAT both on simulated data and in a realistic setting. Since non-symmetric score matrices are a new concept in database search methods, we discuss some important theoretical and practical issues.