Skip to main content
SpringerLink
Log in

Low-dimensional representation of genomic sequences

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

Numerous data analysis and data mining techniques require that data be embedded in a Euclidean space. When faced with symbolic datasets, particularly biological sequence data produced by high-throughput sequencing assays, conventional embedding approaches like binary and k-mer count vectors may be too high dimensional or coarse-grained to learn from the data effectively. Other representation techniques such as Multidimensional Scaling (MDS) and Node2Vec may be inadequate for large datasets as they require recomputing the full embedding from scratch when faced with new, unclassified data. To overcome these issues we amend the graph-theoretic notion of “metric dimension” to that of “multilateration.” Much like trilateration can be used to represent points in the Euclidean plane by their distances to three non-colinear points, multilateration allows us to represent any node in a graph by its distances to a subset of nodes. Unfortunately, the problem of determining a minimal subset and hence the lowest dimensional embedding is NP-complete for general graphs. However, by specializing to Hamming graphs, which are particularly well suited to representing biological sequences, we can readily generate low-dimensional embeddings to map sequences of arbitrary length to a real space. As proof-of-concept, we use MDS, Node2Vec, and multilateration-based embeddings to classify DNA 20-mers centered at intron–exon boundaries. Although these different techniques perform comparably, MDS and Node2Vec potentially suffer from scalability issues with increasing sequence length whereas multilateration provides an efficient means of mapping long genomic sequences.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Aguirre S, Maestre AM, Pagni S, Patel JR, Savage T, Gutman D, Maringer K, Bernal-Rubio D, Shabman RS, Simon V, Rodriguez-Madoz JR, Mulder LC, Barber GN, Fernandez-Sesma A (2012) DENV inhibits type I IFN production in infected cells by cleaving human STING. PLoS Pathog 8(10):e1002–934

    Google Scholar 

  • Asgari E, Mofrad MR (2015) Continuous distributed representation of biological sequences for deep proteomics and genomics. PloS One 10(11):e0141–287

    Google Scholar 

  • Baum LE, Petrie T (1966) Statistical inference for probabilistic functions of finite state Markov chains. Ann Math Stat 37(6):1554–1563

    MathSciNet  MATH  Google Scholar 

  • Bennett J, Lanning S et al (2007) The Netflix prize. In: Proceedings of KDD cup and workshop, New York, vol 2007, p 35

  • Berman P, DasGupta B, Kao MY (2005) Tight approximability results for test set problems in bioinformatics. J Comput Syst Sci 71(2):145–162

    MathSciNet  MATH  Google Scholar 

  • Blumenthal LM (1953) Theory and applications of distance geometry. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Bock JR, Gough DA (2001) Predicting protein–protein interactions from primary structure. Bioinformatics 17(5):455–460

    Google Scholar 

  • Breathnach R, Benoist C, O’hare K, Gannon F, Chambon P (1978) Ovalbumin gene: evidence for a leader sequence in mRNA and DNA sequences at the exon–intron boundaries. Proc Natl Acad Sci 75(10):4853–4857

    Google Scholar 

  • Cáceres J, Hernando C, Mora M, Pelayo IM, Puertas ML, Seara C, Wood DR (2007) On the metric dimension of cartesian products of graphs. SIAM J Discrete Math 21(2):423–441

    MathSciNet  MATH  Google Scholar 

  • Cai C, Han L, Ji ZL, Chen X, Chen YZ (2003a) SVM-Prot: web-based support vector machine software for functional classification of a protein from its primary sequence. Nucleic Acids Res 31(13):3692–3697

    Google Scholar 

  • Cai YD, Feng KY, Li YX, Chou KC (2003b) Support vector machine for predicting \(\alpha \)-turn types. Peptides 24(4):629–630

    Google Scholar 

  • Chaouche FA, Berrachedi A (2006) Automorphisms group of generalized Hamming graphs. Electron Notes Discrete Math 24:9–15

    MathSciNet  MATH  Google Scholar 

  • Chartrand G, Eroh L, Johnson MA, Oellermann OR (2000) Resolvability in graphs and the metric dimension of a graph. Discrete Appl Math 105(1):99–113

    MathSciNet  MATH  Google Scholar 

  • Chvátal V (1983) Mastermind. Combinatorica 3(3–4):325–329

    MathSciNet  MATH  Google Scholar 

  • Cook SA (1971) The complexity of theorem-proving procedures. In: Proceedings of the third annual ACM symposium on theory of computing. ACM, pp 151–158

  • Cortes C, Vapnik V (1995) Support-vector networks. Mach Learn 20(3):273–297

    MATH  Google Scholar 

  • Davis J, Goadrich M (2006) The relationship between precision-recall and roc curves. In: Proceedings of the 23rd international conference on machine learning. ACM, pp 233–240

  • Fix E, Hodges JL Jr (1951) Discriminatory analysis-nonparametric discrimination: consistency properties. Tech. rep, DTIC Document

  • Gary MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. WH Freeman and Company, New York

    Google Scholar 

  • Gehrke J, Ginsparg P, Kleinberg J (2003) Overview of the 2003 KDD cup. ACM SIGKDD Explor Newsl 5(2):149–151

    Google Scholar 

  • Grover A, Leskovec J (2016) node2vec: scalable feature learning for networks. In: Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 855–864

  • Hamming RW (1950) Error detecting and error correcting codes. Bell Labs Techn J 29(2):147–160

    MathSciNet  MATH  Google Scholar 

  • Harary F, Melter R (1976) On the metric dimension of a graph. Ars Comb 2:191–195

    MATH  Google Scholar 

  • Harrison MA (1963) The number of transitivity sets of Boolean functions. J Soc Ind Appl Math 11(3):806–828

    MathSciNet  MATH  Google Scholar 

  • Hauptmann M, Schmied R, Viehmann C (2012) Approximation complexity of metric dimension problem. J Discrete Algorithms 14:214–222

    MathSciNet  MATH  Google Scholar 

  • Hayes WS, Borodovsky M (1998) How to interpret an anonymous bacterial genome: machine learning approach to gene identification. Genome Res 8(11):1154–1171

    Google Scholar 

  • Hoff KJ, Tech M, Lingner T, Daniel R, Morgenstern B, Meinicke P (2008) Gene prediction in metagenomic fragments: a large scale machine learning approach. BMC Bioinf 9(1):217

    Google Scholar 

  • Jaakkola TS, Diekhans M, Haussler D (1999) Using the Fisher kernel method to detect remote protein homologies. ISMB 99:149–158

    Google Scholar 

  • Karp RM (1972) Reducibility among combinatorial problems. In: Complexity of computer computations. Springer, pp 85–103

  • Khuller S, Raghavachari B, Rosenfeld A (1996) Landmarks in graphs. Discrete Appl Math 70(3):217–229

    MathSciNet  MATH  Google Scholar 

  • Kratica J, Kovačević-Vujčić V, Čangalović M (2009) Computing the metric dimension of graphs by genetic algorithms. Comput Optim Appl 44(2):343–361

    MathSciNet  MATH  Google Scholar 

  • Krzanowski WJ (2000) Principles of multivariate analysis: a user’s perspective. OUP, Oxford

    MATH  Google Scholar 

  • Leslie CS, Eskin E, Noble WS (2002) The spectrum kernel: a string kernel for SVM protein classification. Pac Symp Biocomput 7:566–575

    Google Scholar 

  • Li J, Lim SP, Beer D, Patel V, Wen D, Tumanut C, Tully DC, Williams JA, Jiricek J, Priestle JP, Harris JL, Vasudevan SG (2005) Functional profiling of recombinant NS3 proteases from all four serotypes of dengue virus using tetrapeptide and octapeptide substrate libraries. J Biol Chem 280(31):28,766–28,774

    Google Scholar 

  • Lozupone C, Knight R (2005) UniFrac: a new phylogenetic method for comparing microbial communities. Appl Environ Microbiol 71(12):8228–8235

    Google Scholar 

  • Lozupone C, Lladser ME, Knights D, Stombaugh J, Knight R (2011) UniFrac: an effective distance metric for microbial community comparison. ISME J 5(2):169–172

    Google Scholar 

  • Maaten Lvd, Hinton G (2008) Visualizing data using t-SNE. J Mach Learn Res 9((Nov)):2579–2605

    MATH  Google Scholar 

  • Mikolov T, Chen K, Corrado G, Dean J (2013) Efficient estimation of word representations in vector space. ArXiv e-prints 1301.3781

  • Mikolov T, Sutskever I, Chen K, Corrado GS, Dean J (2013) Distributed representations of words and phrases and their compositionality. In: Advances in neural information processing systems, pp 3111–3119

  • Mladenović N, Kratica J, Kovačević-Vujčić V, Čangalović M (2012) Variable neighborhood search for metric dimension and minimal doubly resolving set problems. Eur J Oper Res 220(2):328–337

    MathSciNet  MATH  Google Scholar 

  • Ng P (2017) dna2vec: consistent vector representations of variable-length k-mers. ArXiv e-prints 1701.06279

  • Opsahl T (2011) Why Anchorage is not (that) important: binary ties and sample selection. http://toreopsahl.com/2011/08/12/why-anchorage-is-not-that-important-binary-tiesand-sample-selection. Accessed September 2013

  • Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer P, Weiss R, Dubourg V, Vanderplas J, Passos A, Cournapeau D, Brucher M, Perrot M, Duchesnay E (2011) Scikit-learn: machine learning in Python. J Mach Learn Res 12:2825–2830

    MathSciNet  MATH  Google Scholar 

  • Perozzi B, Al-Rfou R, Skiena S (2014) Deepwalk: online learning of social representations. In: Proceedings of the 20th ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 701–710

  • Reese MG, Kulp D, Tammana H, Haussler D (2000) Genie—gene finding in Drosophila melanogaster. Genome Res 10(4):529–538

    Google Scholar 

  • Sarda D, Chua GH, Li KB, Krishnan A (2005) pSLIP: SVM based protein subcellular localization prediction using multiple physicochemical properties. BMC Bioinform 6(1):152

    Google Scholar 

  • Sciabola S, Cao Q, Orozco M, Faustino I, Stanton RV (2012) Improved nucleic acid descriptors for siRNA efficacy prediction. Nucleic Acids Res 41(3):1383–1394

    Google Scholar 

  • Slater PJ (1975) Leaves of trees. Congressus Numerantium 14(549–559):37

    MATH  Google Scholar 

  • Stanke M, Steinkamp R, Waack S, Morgenstern B (2004) AUGUSTUS: a web server for gene finding in eukaryotes. Nucleic Acids Res 32(suppl 2):W309–W312

    Google Scholar 

  • Yang KK, Wu Z, Bedbrook CN, Arnold FH (2018) Learned protein embeddings for machine learning. Bioinformatics 34(15):2642–2648

    Google Scholar 

  • Yates A, Akanni W, Amode MR, Barrell D, Billis K, Carvalho-Silva D, Cummins C, Clapham P, Fitzgerald S, Gil L et al (2016) Ensembl 2016. Nucleic Acids Res 44(D1):D710–D716

    Google Scholar 

  • Yu CY, Chang TH, Liang JJ, Chiang RL, Lee YL, Liao CL, Lin YL (2012) Dengue virus targets the adaptor protein MITA to subvert host innate immunity. PLoS Pathog 8(6):e1002–780

    Google Scholar 

Download references

Acknowledgements

The authors thank the reviewers for their very insightful comments on the original version of this manuscript. This research was partially funded by the NSF IGERT Grant 1144807, and NSF IIS Grant 1836914. The authors acknowledge the BioFrontiers Computing Core at the University of Colorado–Boulder for providing High-Performance Computing resources (funded by National Institutes of Health 1S10OD012300), supported by BioFrontiers IT group.

Author information

Authors and Affiliations

  1. Department of Computer Science, The University of Colorado, Boulder, CO, 80309-0526, USA

    Richard C. Tillquist

  2. Department of Applied Mathematics, The University of Colorado, Boulder, CO, 80309-0526, USA

    Manuel E. Lladser

Authors
  1. Richard C. Tillquist
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuel E. Lladser
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel E. Lladser.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tillquist, R.C., Lladser, M.E. Low-dimensional representation of genomic sequences. J. Math. Biol. 79, 1–29 (2019). https://doi.org/10.1007/s00285-019-01348-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-019-01348-1

Keywords

Search

Navigation