Genetic distance sampling: a novel sampling method for obtaining core collections using genetic distances with an application to cultivated lettuce

Abstract

This paper introduces a novel sampling method for obtaining core collections, entitled genetic distance sampling. The method incorporates information about distances between individual accessions into a random sampling procedure. A basic feature of the method is that automatically larger samples are obtained if accessions are further apart and smaller samples if accessions are closer together. Genetic distance sampling can be used in conjunction with predefined stratifications of the accessions. Sample sizes are determined automatically; they depend on the distances between accessions within strata. The method is applied to the collection of cultivated lettuce of the Centre for Genetic Resources, the Netherlands. In this paper, genetic distances between accessions are obtained using AFLP marker data. However, genetic distance sampling can be applied using any measure of genetic distance between accessions. Some properties of genetic distance sampling are discussed.

Appendix

In order to provide a mathematical description of the relationship between the average band frequencies of AFLP markers in samples obtained using genetic distance sampling (y) and the corresponding band frequencies in the entire collection (x) a third-order polynomial was used:

$$ y = f(x) = \alpha x^{3} + \beta x^{2} + \gamma x + \delta . $$

This function provides enough flexibility to describe the relationship between y and x. Since the band frequency of non-polymorphic AFLP markers remain unchanged under any sampling procedure, f(0) = 0, leading to δ = 0, and f(1) = 1, leading to γ = 1 − α − β. As a consequence,

$$ f(x) = a(x^{3} - x) + \beta (x^{2} - x) + x. $$

(1)

Since the simple matching coefficient treats the presence of bands in the same way as the absence of bands, it follows that f(1/2) = 1/2, leading to β = −3/2 α. As a consequence,

$$ f(x) = \alpha {\left( {x^{3} - \frac{3} {2}x^{2} + \frac{1} {2}x} \right)} + x. $$

(2)

In order to achieve that the function f(x) is non-decreasing, the value of α should be smaller or equal to 4. If α is positive (negative), the slope of f(x) is greater (smaller) than unity if x is either close to 0 or 1.

Using the least-squares criterion, the function g(x) = f(x) − x can be fitted to the data by simple linear regression with zero intercept. For the data used in this study expression (1) did not provide a significantly better fit to the data compared to expression (2). Therefore, only results using expression (2) have been presented. It would also be possible to use weighted linear regression with weights proportional to 1/x(1 − x). This would give more weight to points with x close to 0 or 1 in comparison to points with x close to 1/2. For the current data this leads to even larger estimates of α.