Reverse de Bruijn: Utilizing Reverse Peptide Synthesis to Cover All Amino Acid k-mers

Abstract

Peptide arrays measure the binding intensity of a specific protein to thousands of amino acid peptides. By using peptides that cover all k-mers, a comprehensive picture of the binding spectrum is obtained. Researchers would like to measure binding to the longest k-mer possible, but are constrained by the number of peptides that can fit into a single microarray. A key challenge is designing a minimum number of peptides that cover all k-mers. Here, we suggest a novel idea to reduce the length of the sequence covering all k-mers by utilizing a unique property of the peptide synthesis process. Since the synthesis can start from both ends of the peptide template, it is enough to cover each k-mer or its reverse, and use the same template twice: in forward and reverse. Then, the computational problem is to generate a minimum length sequence that for each k-mer either contains it or its reverse. We developed an algorithm ReverseCAKE to generate such a sequence. ReverseCAKE runs in time linear in the output size and is guaranteed to produce a sequence that is longer by at most $\varTheta (\sqrt{n}\log {n})$ characters compared to the optimum n. The obtained saving factor by ReverseCAKE approaches the theoretical lower bound as k increases. In addition, we formulated the problem as an integer linear program and empirically observed that the solutions obtained by ReverseCAKE are near-optimal. Through this work we enable more effective design of peptide microarrays.

Appendix

Fig. 4.

Results and performance of ReverseCAKE and the ILP solver. (A) Results of sequence lengths are portrayed as ratios to an original de Bruijn sequence ($|\varSigma |^k$). The lower bound is from Proposition 1. The dashed red line is at half. (B,C) Runtimes and maximum memory usage of the algorithms, respectively. Y-axis is on a log-scale. (Color figure online)

Table 1. Lengths of reverse de Bruijn sequences produced by ReverseCAKE and an ILP solver. The columns are organized as follows: (i) the alphabet, where aa stands for amino acid; (ii) the length of a regular de Bruijn sequence that does not exploit reverse peptide synthesis; (iii) the lower bound on RdB sequence length (Proposition 1); (iv–v) the lengths of the sequence computed by ReverseCAKE (Sect. 3.3) and an ILP solver that reached an optimal solution (Sect. 3.4); (vi) the saving factor is the ratio between an optimal RdB sequence and a de Bruijn sequence.

Table 2. Performance evaluation of ReverseCAKE and an ILP solver. The runtime and maximum memory usage are reported in seconds (sec) and kilobytes (KB).