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Introduction to the Peptide Binding Problem of Computational Immunology: New Results

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Abstract

We attempt to establish geometrical methods for amino acid sequences. To measure the similarities of these sequences, a kernel on strings is defined using only the sequence structure and a good amino acid substitution matrix (e.g. BLOSUM62). The kernel is used in learning machines to predict binding affinities of peptides to human leukocyte antigen DR (HLA-DR) molecules. On both fixed allele (Nielsen and Lund in BMC Bioinform. 10:296, 2009) and pan-allele (Nielsen et al. in Immunome Res. 6(1):9, 2010) benchmark databases, our algorithm achieves the state-of-the-art performance. The kernel is also used to define a distance on an HLA-DR allele set based on which a clustering analysis precisely recovers the serotype classifications assigned by WHO (Holdsworth et al. in Tissue Antigens 73(2):95–170, 2009; Marsh et al. in Tissue Antigens 75(4):291–455, 2010). These results suggest that our kernel relates well the sequence structure of both peptides and HLA-DR molecules to their biological functions, and that it offers a simple, powerful and promising methodology to immunology and amino acid sequence studies.

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Notes

  1. Allele: an alternative form of a gene that occurs at a specified chromosomal position (locus) [22].

  2. ftp://ftp.ebi.ac.uk/pub/databases/imgt/mhc/hla/DRB_prot.fasta.

  3. We have found from a number of different experiments that “they do not cluster”. (Perhaps the geometric phenomenon here is in the higher dimensional scaled topology, i.e. the Betti numbers β i >0, for i>0 [4].)

  4. Both the data set and the 5-fold partition are available at http://www.cbs.dtu.dk/suppl/immunology/NetMHCII-2.0.php.

  5. Both the data set and the 5-part partition are available at http://www.cbs.dtu.dk/suppl/immunology/NetMHCIIpan-2.0.

  6. The data set was downloaded from http://www.immuneepitope.org/list_page.php?list_type=mhc&measured_response=&total_rows=64797&queryType=true, on May 23, 2012.

  7. The code is published in http://www.cbs.dtu.dk/cgi-bin/nph-sw_request?netMHCIIpan.

  8. Another way of measuring distance between clusters is the Hausdorff distance.

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Acknowledgements

The authors would like to thank Shuaicheng Li for pointing out to us that the portions of DRB alleles that contact with peptides can be obtain from the non-aligned DRB amino acid sequences by the use of two markers, “RFL” and “TVQ”. We thank Morten Nielsen for his criticism on over-fitting.

We thank Yiming Cheng for his suggestions on the computer code which were very helpful for speeding up the algorithm for evaluating K 3. He also discussed with us the influence on HLA–peptide binding prediction of using different representations of the alleles, and of adjusting the index β in the kernel according to the sequence length. Although the topics are not included in the paper, they have some potential for future work.

Also, we appreciate Felipe Cucker for reviewing our draft, making many improvements. We thank Santiago Laplagne for pointing out a bug in the codes for Table 2.

The work described in this paper is supported by GRF grant [Project No. 9041544] and [Project No. CityU 103210] and [Project No. 9380050].

Author information

Authors and Affiliations

  1. Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong

    Wen-Jun Shen & Hau-San Wong

  2. Microsoft Search Technology Center Asia, Beijing, China

    Quan-Wu Xiao

  3. Department of Statistical Science, Duke University, Durham, NC, USA

    Xin Guo

  4. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Stephen Smale

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  5. Stephen Smale
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Correspondence to Stephen Smale.

Additional information

Communicated by Teresa Krick.

Appendix: The BLOSUM62-2 Matrix

Appendix: The BLOSUM62-2 Matrix

We list the whole BLOSUM62-2 matrix in Table 8. Table 9 explains the amino acids denoted by the capital letters.

Table 8 The BLOSUM62-2 matrix
Full size table
Table 9 The list of the amino acids
Full size table

From the Introduction, we see that the matrix Q can be recovered from the BLOSUM62-2 once the marginal probability vector p is available. The latter vector is obtained by

$$p = \bigl(\mbox{[BLOSUM62-2]}\bigr)^{-1} v_1, $$

where \(v_{1} = (1,\ldots,1)\in\mathbb{R}^{20}\) is a vector with all its coordinate being 1. The matrix Q can be obtained precisely from http://www.ncbi.nlm.nih.gov/IEB/ToolBox/CPP_DOC/lxr/source/src/algo/blast/composition_adjustment/matrix_frequency_data.c#L391.

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Shen, WJ., Wong, HS., Xiao, QW. et al. Introduction to the Peptide Binding Problem of Computational Immunology: New Results. Found Comput Math 14, 951–984 (2014). https://doi.org/10.1007/s10208-013-9173-9

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