Precision of Recombination Frequency Estimates After Random Intermating With Finite Population Sizes

Abstract

Random intermating of F₂ populations has been suggested for obtaining precise estimates of recombination frequencies between tightly linked loci. In a simulation study, sampling effects due to small population sizes in the intermating generations were found to abolish the advantages of random intermating that were reported in previous theoretical studies considering an infinite population size. We propose a mating scheme for intermating with planned crosses that yields more precise estimates than those under random intermating.

MARKER applications such as marker-assisted backcrossing, marker-assisted selection, and map-based cloning require linkage maps with precise estimates of the recombination frequency r between tightly linked loci. The amount of information per individual

(1)

(Mather 1936; Allard 1956), where n_m is the size of the mapping population and the expected variance of the recombination frequency estimate, is a statistic to compare alternative types of mapping populations with respect to the precision of recombination frequency estimates. To obtain a high mapping precision for tightly linked loci, t times intermated F₂ mapping populations ( populations) were suggested (Darvasi and Soller 1995) and developed in Arabidopsis (Liu et al. 1996) and maize (Lee et al. 2002). Liu et al. (1996) derived i_p for populations and found that i_p for their population was greater than that for an F₂ population if r < 0.131.

In their derivations, Liu et al. (1996) assumed random mating and infinite population sizes n_i in the intermating generations. However, Falke et al. (2006) hypothesized that for finite n_i sampling effects might overrule the increase in precision of estimates due to intermating. Martin and Hospital (2006) investigated estimation of recombination frequencies in recombinant inbred lines and found that maximum-likelihood estimates of r are biased if the relationship R = g(r) between r and the frequency R of recombinant gametes in the mapping population is nonlinear. The bias is determined by the size n_m of the mapping population. For intermated populations, g is nonlinear and, hence, maximum-likelihood estimates of r from intermated populations are biased. Knowledge about the relative extent of (a) the reduction in i_p due to finite sizes of intermating populations and (b) the bias of recombination frequency estimates due to finite sizes of mapping populations is important to assess the actual advantage of intermated populations over F₂ base populations for linkage mapping. However, no results are available.

Our objectives were to (1) investigate with computer simulations the extent of the bias of maximum-likelihood estimates of r depending on the finite size n_m of the mapping population assuming random mating with population size n_i = ∞ in the intermating generations, (2) investigate with computer simulations the effect of finite population sizes n_i in the intermating generations on the amount of information per individual i_p in the mapping population, and (3) propose a mating scheme for intermating with planned crosses that results in the same i_p values as random intermating with infinite population size.

Bias:

For intermated mapping populations, the relation g between the recombination frequency r and the frequency of recombinant gametes R is

(2)

(Darvasi and Soller 1995). Because g is nonlinear, the maximum-likelihood estimator (cf. Bailey 1961) of r is biased. Martin and Hospital (2006) employed a Taylor series expansion to derive a bias correction for arbitrary nonlinear g. Equation 18 of their derivations needs a correction. For g = f ⁻¹ it should read

(3)

With this modification, the general form of the bias correction according to Martin and Hospital (2006) is

(4)

where g′ and g″ are the first and second derivatives of g with respect to r. The bias-corrected estimator is then

(5)

For mapping populations, it can be calculated by using

(6)

and

(7)

We conducted simulations with Plabsoft (Maurer et al. 2008) to investigate the extent of the bias of and in and mapping populations of size n_m = 50, 100, 500, 100, and 5000, employing large population sizes n_i = 25,000 in the intermating generations. For each n_m we simulated 50,000 mapping populations in which and were estimated for locus pairs with map distances . From the 50,000 simulated mapping experiments, the bias of and was estimated as and .

For large population sizes (n_m ≥ 500) and small recombination frequencies (r < 0.1), the bias of was <10⁻⁴ in the and <3 × 10⁻⁴ in the mapping populations (Figure 1). However, for small populations (n_m = 50, 100) and r = 0.1 the bias amounted to 10⁻³ and 4 × 10⁻³ in the and mapping populations, respectively. Its absolute value was reduced efficiently by the bias correction. For example, for n_m = 50 and r = 0.05 the bias of in the was 3.6 × 10⁻⁴ and that of was −1.2 × 10⁻⁴. In the mapping population the bias of was 10⁻³ and that of was −7 × 10⁻⁵. For n_m = 50 and recombination frequencies >0.1, the bias of was considerable, reaching its maximum value of ∼0.04 in the interval 0.2 < r < 0.3. For recombination frequencies r > 0.25 the bias correction resulted in a serious overcorrection (Figure 1).

Figure 1.—

Estimates of the bias of the maximum-likelihood estimator (left) and the bias of the bias-corrected estimator (right) for (top) and (bottom) mapping populations depending on the recombination frequency r. In the intermating generations, populations sizes n_i = 25,000 were used. Sizes of the mapping populations were n_m = 50, 100, 500, 1000, and 5000.

The goal of using intermated mapping populations is to increase the precision of recombination frequency estimates for tightly linked loci. Therefore, the properties of an estimator must be favorable for small values of r. For these, biasedness is not a serious problem of the maximum-likelihood estimator . Nevertheless, the bias correction of Martin and Hospital (2006) with the modification presented in Equation 3 provides a means to reduce the bias to negligibly small values.

Amount of information per individual:

The precision of alternative types of mapping populations can be compared by expressing their i_p value as a proportion of the i_p value of an F₂ individual (Mather 1936):

(8)

For F₂ individuals Mather (1936) derived

(9)

and for individuals Liu et al. (1996) derived

(10)

The derivations of Liu et al. (1996) assume infinite population sizes n_i = ∞ in the intermating generations and, therefore, do not take into account an increase in the variance due to sampling effects caused by finite population sizes n_i.

Our investigations focus on the effect of finite population sizes n_i in the intermating generations and a finite population size n_m of an mapping population on the amount of information per individual i_p. The effect of finite n_i is accounted for by carrying out simulations with finite population sizes. The effect of finite n_m is accounted for by using a modified definition of the information content,

(11)

in which the variance is replaced by the mean squared error and, hence, the effect of the bias is considered.

We investigated the effect of finite population sizes n_i = 100, 200, 500 in the intermating generations on the amount of information i_p of individuals in the – mapping populations of size n_m = 100. For each type of mapping population and each n_i, we simulated 50,000 mapping populations in which was estimated for locus pairs with map distances . From the 50,000 simulated mapping experiments, MSE_r was estimated as , from which i_p and i_r (Equations 11 and 8) were determined.

For n_i = 100, the i_r values were <1 for all types of mapping populations, irrespective of the recombination frequencies r (Figure 2). For n_i = 200 and 500, the i_r values were >1 if the recombination frequencies were > ≈0.05 and ≈0.1, respectively. Even with n_i = 500, the i_r values were considerably smaller than the i_r values for infinite population sizes n_i = ∞ calculated with Equation 10 (Liu et al. 1996).

Figure 2.—

Relative amount of information per individual i_r in – mapping populations of size n_m = 100 depending on the recombination frequency r. Simulation results are given for population sizes n_i = 500, 200, and 100 in the intermating generations. For n_i = ∞ the theoretical values (Equation 10) are presented.

We conclude that the population sizes n_i of the intermating generations are the crucial factor for obtaining precise estimates of small r from populations. A substantial gain in precision compared to estimation of recombination frequencies from the F₂ base populations is achieved only if n_i ≥ 500 are employed.

Mating scheme with independent recombinations:

From the assumption of infinite population sizes in the intermating generations it follows that the individuals of a mapping population do not have common ancestors in the F₂ or intermating generations. Therefore, the recombination events in different individuals of the mapping population are stochastically independent. This stochastic independence is the key property of the model with infinite population sizes in the intermating generations, for which Liu et al. (1996) derived the information content per individual (Equation 10). For finite population sizes and random intermating, the above property of stochastic independence does not hold, because two individuals of the mapping populations can have a common ancestor with a probability larger than zero. This increases the standard error of the recombination frequency estimate and, hence, decreases the information content i_p per individual. A mapping population consisting of n_m individuals that have no common ancestors in F₂ or later generations, i.e., with stochastic independence of recombination events in different individuals, can be generated with the following planned crossing scheme and finite population sizes n_i.

For generating an mapping population of size n_m, an F₂ population of size 2^tn_m is generated. Then, 2^t−1n_m pairs of F₂ plants are crossed and from each cross one single plant is generated, resulting in an population of size 2^t−1n_m. The procedure is repeated in each subsequent generation, by producing exactly one progeny from the cross of two individuals of the parental population. Continuing the procedure for t generations results in a mapping population of size n_m.

Mapping populations generated with this “mating scheme with independent recombinations” have the same properties as mapping populations derived from large random-mating populations. In such populations, the amount of information i_p per individual is the same as in Equation 10. Hence, the mating scheme guarantees the maximum possible information content in the mapping population but reduces the efforts of employing large intermating populations.