Uncovering multiloci-ordering by algebraic property of Laplacian matrix and its Fiedler vector

MOTIVATION The loci-ordering, based on two-point recombination fractions for a pair of loci, is the most important step in constructing a reliable and fine genetic map. RESULTS Using the concept from complex graph theory, here we propose a Laplacian ordering approach which uncovers the loci-ordering of multiloci simultaneously. The algebraic property for a Fiedler vector of a Laplacian matrix, constructed from the recombination fraction of the loci-ordering for 26 loci of barley chromosome IV, 846 loci of Arabidopsis thaliana and 1903 loci of Malus domestica, together with the variable threshold uncovers their loci-orders. It offers an alternative yet robust approach for ordering multiloci. AVAILABILITY AND IMPLEMENTATION Source code program with data set is available as supplementary data and also in a software category of the website (http://biophysics.dgist.ac.kr) CONTACT crkim@pusan.ac.kr or iksoochang@dgist.ac.kr SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

[1]  Robert Tibshirani,et al.  An Introduction to the Bootstrap , 1994 .

[2]  Riccardo Velasco,et al.  Fast and Cost-Effective Genetic Mapping in Apple Using Next-Generation Sequencing , 2014, G3: Genes, Genomes, Genetics.

[3]  Bojan Mohar,et al.  Optimal linear labelings and eigenvalues of graphs , 1992, Discret. Appl. Math..

[4]  Simon de Givry,et al.  CarthaGene : multipopulation integrated genetic and radiation hybrid mapping , 2005 .

[5]  Bojan Mohar,et al.  Laplace eigenvalues of graphs - a survey , 1992, Discret. Math..

[6]  M. Nussbaum.  Spline Smoothing in Regression Models and Asymptotic Efficiency in $L_2$ , 1985 .

[7]  E. Lander,et al.  Construction of multilocus genetic linkage maps in humans. , 1987, Proceedings of the National Academy of Sciences of the United States of America.

[8]  S. Efroimovich.  Nonparametric Estimation of a Density of Unknown Smoothness , 1986 .

[9]  S. Goss,et al.  New method for mapping genes in human chromosomes , 1975, Nature.

[10]  K H Buetow,et al.  Multipoint gene mapping using seriation. II. Analysis of simulated and empirical data. , 1987, American journal of human genetics.

[11]  E A Thompson,et al.  Two-locus and three-locus gene identity by descent in pedigrees. , 1988, IMA journal of mathematics applied in medicine and biology.

[12]  K. Manly,et al.  Map Manager QTX, cross-platform software for genetic mapping , 2001, Mammalian Genome.

[13]  D E Weeks,et al.  Preliminary ranking procedures for multilocus ordering. , 1987, Genomics.

[14]  J. Ott.  Analysis of Human Genetic Linkage , 1985 .

[15]  C T Falk,et al.  A simple scheme for preliminary ordering of multiple loci: application to 45 CF families. , 1989, Progress in clinical and biological research.

[16]  J M Olson,et al.  Monte Carlo comparison of preliminary methods for ordering multiple genetic loci. , 1990, American journal of human genetics.

[17]  T. Ellis,et al.  NEIGHBOUR MAPPING AS A METHOD FOR ORDERING GENETIC MARKERS , 1997 .

[18]  M. Daly,et al.  Corrigendum to “MAPMAKER: An interactive computer package for constructing primary genetic linkage maps of experimental and natural populations” [Genomics 1 (1987) 174–181] , 2009 .

[19]  M. Daly,et al.  MAPMAKER: an interactive computer package for constructing primary genetic linkage maps of experimental and natural populations. , 1987, Genomics.

[20]  S. Knapp,et al.  Mapping quantitative trait loci using molecular marker linkage maps , 1990, Theoretical and Applied Genetics.

[21]  H E Stanley,et al.  Classes of small-world networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[22]  G. Getz,et al.  Coupled two-way clustering analysis of gene microarray data. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[23]  S. Judge.  Does the eye grow into focus? , 1990, Nature.

[24]  L. Amaral,et al.  The web of human sexual contacts , 2001, Nature.

[25]  D. Timothy Bishop,et al.  Construction of linkage maps with DNA markers for human chromosomes , 1985, Nature.

[26]  C. Stein,et al.  Estimation with Quadratic Loss , 1992 .

[27]  Jo L. Dicks,et al.  Computational approaches and software tools for genetic linkage map estimation in plants , 2009, Briefings Bioinform..

[28]  Stefano Lonardi,et al.  Efficient and Accurate Construction of Genetic Linkage Maps from the Minimum Spanning Tree of a Graph , 2008, PLoS genetics.

[29]  H. D. Simon,et al.  A spectral algorithm for envelope reduction of sparse matrices , 1993, Supercomputing '93. Proceedings.

[30]  J. Maccluer,et al.  Empirical power of three preliminary methods for ordering loci. , 1988, American journal of human genetics.

[31]  Jan H. J. Hoeijmakers,et al.  Genetic recombination: From competition to collaboration , 1998, Nature.

[32]  Huihui Li,et al.  Linkage Analysis and Map Construction in Genetic Populations of Clonal F1 and Double Cross , 2015, G3: Genes, Genomes, Genetics.

[33]  P. Bickel.  MINIMAX ESTIMATION OF THE MEAN OF A NORMAL DISTRIBUTION SUBJECT TO DOING WELL AT A POINT , 1983 .

[34]  R W Doerge,et al.  High-density haplotyping with microarray-based expression and single feature polymorphism markers in Arabidopsis. , 2006, Genome research.

[35]  T. Anderson,et al.  Genomic linkage map of the human blood fluke Schistosoma mansoni , 2009, Genome Biology.

[36]  Shlomo Havlin,et al.  Universal behavior of optimal paths in weighted networks with general disorder. , 2006, Physical review letters.

[37]  Roeland E. Voorrips,et al.  Software for the calculation of genetic linkage maps , 2001 .

[38]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[39]  E A Thompson,et al.  Crossover counts and likelihood in multipoint linkage analysis. , 1987, IMA journal of mathematics applied in medicine and biology.

[40]  M. Fiedler.  A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory , 1975 .

[41]  Seishi Ninomiya,et al.  AntMap: Constructing Genetic Linkage Maps Using an Ant Colony Optimization Algorithm , 2006 .

[42]  Richard G. F. Visser,et al.  RECORD: a novel method for ordering loci on a genetic linkage map , 2005, Theoretical and Applied Genetics.

[43]  Harry Eugene Stanley,et al.  Catastrophic cascade of failures in interdependent networks , 2009, Nature.

[44]  J. Ooijen,et al.  JoinMap® 4, Software for the calculation of genetic linkage maps in experimental populations , 2006 .

[45]  Minho Kang,et al.  A simple and exact Laplacian clustering of complex networking phenomena: Application to gene expression profiles , 2008, Proceedings of the National Academy of Sciences.

[46]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[47]  Alex Pothen,et al.  PARTITIONING SPARSE MATRICES WITH EIGENVECTORS OF GRAPHS* , 1990 .

[48]  C. Hackett,et al.  Effects of genotyping errors, missing values and segregation distortion in molecular marker data on the construction of linkage maps , 2003, Heredity.