To facilitate development of methods for mapping multiple interacting QTL using variance component models, it is necessary to significantly improve the computational efficiency. To do so, three key components need to be addressed: estimation of relationship between relatives (i.e. IBD matrix estimation), variance component estimation of QTL at a given genomic location (or combination of locations) and the genome scan for genomic locations (or combinations of locations) to evaluate (i.e. the global optimization algorithm). Recently, more efficient algorithms for variance component estimation [9] and optimization methods for QTL genome scans [6, 7] have been described. Here, we propose a new approach to describe IBD between individuals in the form of continuous marker bracket IBD functions. This improves memory usage and computational efficiency in the estimation of genome-wide IBD as well as facilitates implementation of existing and development of new and more efficient global optimization algorithm for detection of individual and multiple QTL.

We have shown that it is possible to estimate the IBD function for a marker bracket exactly by re-implementing existing IBD algorithms using Pong-Wong algorithm [10] as an example. This work illustrates that the IBD-function based approach generates the same IBD's and estimates of variance components as the original algorithm as well as a potential to improve both computational and memory usage. This approach should be applicable for most IBD estimation algorithms, although we have not shown this here, but will require an effort to be made for each particular IBD estimation algorithm in reformulating the computational algorithm and updating analysis software. This is the preferable strategy to obtain the optimal IBD functions and achieve maximal computational improvements by this approach.

In situations where re-implementation of the original algorithm is not achievable, but where IBD functions could be useful, we provide a general curve-fitting based algorithm to estimate these functions from IBD matrices provided by any existing algorithm. To illustrate the applicability of this approach, we have made an in-depth study of its properties. By using input from one IBD estimation method [3], we show that our approximation has a very small effect on the IBDs as well as on subsequently estimated QTL variance components. Moreover, a less detailed study with another IBD estimation method (Merlin; [4]) yielded similar results as the comparisons with LOKI (results not shown).

IBD probabilities can theoretically be calculated exactly, but in real datasets this will be computationally prohibitive. The methods used in practice thus aim to approximate the true IBD and depending on the assumptions made, methods will generally obtain different IBD matrices for a specific dataset. Here, we clearly show that differences in underlying assumptions in the algorithms may have a much greater impact on both the direct estimates if IBD's, as well as on variance component estimation, than from using IBD functions as approximations of a specific IBD estimation method.

By re-implementing an already fast deterministic IBD estimation algorithm by Pong-Wong et al [10] to conduct IBD-function estimation, it is possible to improve the computational efficiency in calculating IBD by a factor of 2 to 2.6 in most realistic situations. The algorithm used in the LOKI [3] software can simultaneously compute IBD estimates for any number of pre-defined locations in a marker bracket. Theoretically, our method to compute IBD functions from a limited set of IBD matrices does therefore not, decrease the computation-time of this method, but will decrease the memory requirement significantly. In practice, however, even though LOKI can theoretically compute any number of IBD matrices in a marker-bracket in a single run without significant additional computational cost, the limiting factor for performing a dense genome scan is the memory requirement to store a large number of IBD matrices across the genome. Thus, since in our work LOKI is not re-implemented to directly report the IBD functions for marker brackets, the IBD-function estimation approach will be useful to allow an increase in precision in the genome-scan, without an accompanying increase in memory requirement.

Our new approach facilitates the development of efficient algorithms for multi-dimensional genome scans for interacting QTL [15] by significantly reducing the hardware requirement for storing genome-wide IBD in RAM, which is a requirement for efficient analyses as storing and accessing large amount of data from disk would slow down the analyses significantly. This improved efficiency in the analysis makes analyses based on more advanced genetic modelling, including e.g. epistasis, more computationally tractable and thus of interest to a larger group of users.

The computational efficiency in QTL mapping in general can be improved considerably by replacing the commonly used grid search with a more efficient optimization algorithm [6, 7]. Using this approach, the number of locations (or combination of locations) tested for QTL decreases and the computational efficiency is improved regardless of the IBD- or variance component estimation algorithms used. An inherent property of non-exhaustive grid searches is that the locations where IBD are needed cannot be predicted before the analysis. Thus, IBD either have to be computed on a pre-defined dense grid with high computational efficiency and high requirement for data storage or computed serially with a resulting decrease in the computational efficiency. By using the CF-IBD algorithm, however, it is possible to retain the computational efficiency in calculating IBD matrices in parallel, while minimizing the memory requirement in the analyses.

An existing optimization algorithm that would be interesting to explore for single- and multi-dimensional QTL searches in the variance component framework would be DIRECT [16]. This algorithm has been shown to be efficient in multi-dimensional QTL detection [7], and there greatly improves the computational efficiency. In addition to this, the availability of continuous IBD functions also opens up new opportunities to develop novel optimization algorithms for detecting QTL that explicitly uses the fact that there exists an underlying continuous function describing the genetic relationships between individuals across the genome.