A Novel Method for Estimating Linkage Maps

Abstract

The goal of linkage mapping is to find the true order of loci from a chromosome. Since the number of possible orders is large even for a modest number of loci, the problem of finding the optimal solution is known as a NP-hard problem or traveling salesman problem (TSP). Although a number of algorithms are available, many either are low in the accuracy of recovering the true order of loci or require tremendous amounts of computational resources, thus making them difficult to use for reconstructing a large-scale map. We developed in this article a novel method called unidirectional growth (UG) to help solve this problem. The UG algorithm sequentially constructs the linkage map on the basis of novel results about additive distance. It not only is fast but also has a very high accuracy in recovering the true order of loci according to our simulation studies. Since the UG method requires n À 1 cycles to estimate the ordering of n loci, it is particularly useful for estimating linkage maps consisting of hundreds or even thousands of linked codominant loci on a chromosome.A LTHOUGH more and more genomes are being sequenced, high-quality linkage maps for many organisms remain useful. For a majority of organisms, obtaining appropriate linkage maps will be a necessary step for understanding their genetic architecture. With the advance of technology as well as increasing demand of high-density linkage maps, the number of loci simultaneously examined in experiments is steadily growing, which presents a considerable computational challenge for estimating the underlying linkage map since the number of possible orders of loci increases rapidly with the number of loci (Olson and Boehnke 1990;Mester et al. 2003). The construction of linkage maps has been recognized as a special case of the traveling salesman problem (TSP) (Liu 1998). The TSP is a classical non-deterministic polynomial time (NP)-complete problem (Wilson 1988;Olson and Boehnke 1990;Falk 1992) that has attracted the attention of mathematicians and computer scientists. Currently, there are two approaches to tackle the problem. One is to find the answer by performing exhaustive searches and the other is to find approximations. Even for just 10 loci on a chromosome, there are 1,814,400 possible orders. Thus it is extremely time consuming (if not entirely impossible) to exhaustively search all possible orders when the number of loci on a chromosome is .30 (Mester et al.

Evaluation of algorithms used to order markers on genetic maps

“…UG (Tan and Fu, 2006): based on the R matrix, the distance between all loci was calculated by d ij ¼ r ij þ (2/n ij ) P k r ik r jk , for every k, with r ij 4r ik , r ij 4r jk , and n ij individuals. The value T ij ¼ 2d ij À( P kai d ik þ P kaj d jk ) was calculated for every ioj.…”

“…The ordering of markers within linkage groups is considered a special case of the classical traveling salesman problem (Doerge, 1996;Liu, 1998;Mester et al, 2003;Tan and Fu, 2006). Basically, the problem consists in choosing the best order among m!/2 possible orders (m ¼ number of markers).…”

“…Therefore, approximate solutions that allow large-scale genetic mapping have been proposed (Liu, 1998). Several algorithms to obtain approximate solutions for ordering markers have been proposed, including simulated annealing (Thompson, 1984;Weeks and Lange, 1987), stepwise likelihood (Lathrop et al, 1984), branch and bound (Lathrop et al, 1985), try and ripple , seriation (SER) (Buetow and Chakravarti, 1987), rapid chain delineation (RCD) (Doerge, 1996), genetic and evolutionary algorithm (Mester et al, 2003), recombination counting and ordering (RECORD) (Van Os et al, 2005) and unidirectional growth (UG) (Tan and Fu, 2006). Some of them have been implemented into friendly user softwares, such as Linkage (Lathrop et al, 1984), MAPMAKER/EXP , JoinMap (Stam, 1993), Emap from the program suite QTL Cartographer (Basten et al, 2003) and OneMap (Margarido et al, 2007).…”

R Package ASMap: Efficient Genetic Linkage Map Construction and Diagnosis

“…Computational methods that exploited this knowledge were quickly adopted in construction algorithms including the evolution-strategy algorithm (Mester, Ronin, Hu, Peng, Nevo, and Korol 2003a;Mester, Ronin, Minkov, Nevo, and Korol 2003b) implemented in Multipoint and the RECORD algorithm (Van Os, Stam, Visser, and Van Eck 2005) available as command line software and later implemented in software packages IciMapping (Meng, Li, Zhang, and Wang 2015) and onemap (Margarido and Mollinari 2015). Other construction algorithm variants involving efficient solutions of the traveling salesman problem included the unidirectional growth (UG) algorithm (Tan and Fu 2006), the AntMap algorithm (Iwata and Ninomiya 2006), the MSTmap algorithm (Wu et al 2008) and Lep-MAP (Rastas, Paulin, Hanski, Lehtonen, and Auvinen 2013). Unfortunately these more recent algorithms have only been available as downloadable low-level source codes that require compilation before use.…”

“…In a more recent addition to the R contributed package list, onemap (Margarido and Mollinari 2015) contains a collection of linkage map construction tools specialized for a restricted set of inbred and outbred populations. The package implements well known marker ordering algorithms including SE-RIATION (Buetow and Chakravarti 1987), RECORD (Van Os et al 2005), RCD (Doerge 1996) and UG (Tan and Fu 2006). Unfortunately these algorithms have been implemented in native R code and lack the computational expediency of the low-level source code equivalents.…”