## Abstract

The generation of special crosses between different inbred lines such as recombinant inbred strains (RIS) and intermated recombinant inbred populations (IRIP) is being used to improve the power of QTL detection techniques, in particular fine mapping. These approaches acknowledge the fact that recombination of linked loci increases with every generation, caused by the accumulation of crossovers appearing between the loci at each meiosis. This leads to an expansion of the map distance between the loci. While the amount of the map expansion of RIS and IRIP is known for infinite inbred generations, it is not known for finite numbers of generations. This gap was closed here. Since the recursive evaluation of the map expansion factors turned out to be complex, a useful approximation was derived.

THERE are several ways to improve knowledge of how quantitative trait loci (QTL) affect phenotypes (Darvasi 1998; Shalom and Darvasi 2002). It has been proven that recombinant inbred strains (RIS) and intermated recombinant inbred populations (IRIP), which are a combination of advanced intercross lines (AIL) and RIS, as well as AIL itself, are appropriate to fine map QTL (see Brockmann and Bevova 2002 and Winkler*et al*. 2003 for references). The common advantage of RIS, AIL, and IRIP is that linkage between nearby loci can be broken, thereby increasing recombination. The starting point for RIS, AIL, and IRIP usually consists of two inbred lines with extremely differing phenotypes or of divergently selected lines. Therefore, we consider only such loci that carry different alleles in the lines.

RIS: Recombinant inbred strains (Darvasi 1998) are derived from the systematic brother-sister mating of randomly selected pairs of the initial F

_{2}intercross. We denote the population of generation*i*with RIS(*i*), where*i*is the number of inbred generations;*i.e*.,*i*= 0 means the F_{2}intercross. In plant breeding, RIS are also generated by systematic self-fertilization.AIL: Advanced intercross lines (Darvasi and Soller 1995; Liu

*et al*. 1996) are generated from an initial intercross between different parental strains and a certain number*j*(mostly four or more) of subsequent random matings within every following generation. We denote the generated population with AIL(*j*) = F._{j}IRIP: Williams

*et al.*(2001) suggested a design whereby an AIL(*j*) is first produced, followed by an RIS(*i*), which further improves effectiveness. Lee*et al.*(2002) and Sharopova*et al.*(2002) applied this design and called the generated population an intermated recombinant inbred population. The design is abbreviated as IRIP(*j*,*i*).

Except for mutations and gene conversions, all of these advanced crosses cover only genetic material from two basic lines and therefore could be the result of an apparent F_{2} intercross and were analyzed with appropriate software, *e.g*., to estimate map lengths. It turned out (*cf*. Williams*et al*. 2001), that the map length increased if a new RIS or AIL generation was created and analyzed. Hence, map distances estimated with the advanced design are larger than those of the initial intercross. This property is called map expansion.

The map expansion of advanced designs was investigated by Haldane and Waddington (1931) (RIS), Liu*et al*. (1996) (AIL), and Winkler*et al.* (2003) (IRIP). Particularly, they obtained *x*_{RIS(∞)} = 4*x*, *x*_{AIL(}_{j}_{)} = *jx*/2, and *x*_{IRIP(}_{j}_{,∞)} = (*j*/2 + 3)*x* for small map distances *x*, where *x*_{*} is the map distance of the advanced design, *. The authors gave more or less the impression that the map expansion factor depends on the map distance *x*. This point is clarified here by showing that the map expansion is proportional to the initial map distance *x*; *i.e*., *x*_{*} = *ax*. In this way, map expansions evaluated for small map distances are valid for all map distances by appropriate scaling.

Practically, only finite numbers of generations may be produced. Thus the results of the described breeding experiments can be compared with theoretical expectations for AIL, but not for RIS and IRIP. Therefore it is desirable to extend theory to finite generation experiments also for RIS and IRIP. Such knowledge not only would allow us to compare theory and practice and to verify eventual differences, but also would allow us to plan the number of inbred generations necessary to achieve a certain degree of convergence to the asymptotic possessions. This is of economical interest since the creation of each generation costs money and time. For these reasons, exact recursive formulas were derived for *x*_{RIS(}_{i}_{)} and *x*_{IRIP(}_{j}_{,}_{i}_{)} for finite inbred generation numbers *i*. Since the recursive formulas turned out to be tedious, approximation formulas were added, giving a useful tool to determine the map expansions for each number of generations.

## METHODS AND RESULTS

Consider two loci on a chromosome. Let *x* be the map distance and θ the recombination fraction for a single meiosis and *x*_{*} and θ_{*} that for the advanced design, *. Let *x*_{*} = *g*(*x*) be the relation of the genetic scales. Since, by construction, map distances are additive, *g*(*x*_{1} + *x*_{2}) = *g*(*x*_{1}) + *g*(*x*_{2}) holds for two adjacent intervals; *i.e*., *g*(*x*) is a linear function. Furthermore, *x* = 0 is equivalent to *x*_{*} = 0; *i.e*., if there is no chiasma in each meiosis, there is none in the accumulated meiosis and vice versa. Therefore, *x*_{*} = *ax* is valid for all *x*.

From generation to generation, the map function θ = θ(*x*) is assumed. Let θ_{*} = θ_{*}(*x*_{*}) be the map function describing the process over several generations as it would have resulted from one apparent meiosis. Consider θ_{*} = θ_{*}[θ{*x*(*x*_{*})}] with *x*(*x*_{*}) = *x*_{*}/*a*. Then holds, *i.e*., There is one situation where the relations between *x*, *x*_{*}, θ, and θ_{*} are obvious. If one of these values is zero, all the others are also zero. For small map distances, map functions have slope one, regularly; *i.e*., Therefore, results and the relation between the genetic scales is 1

Application of this result to RIS and IRIP demands the determination of θ_{RIS(}_{i}_{)} and θ_{IRIP(}_{j}_{,}* _{i)}*. The asymptotic theory for RIS has been solved by Haldane and Waddington (1931) in the chapter on brother-sister mating, autosomal genes. We applied their equation (3.1) to evaluate the recombination fraction for each generation recursively. Since we found it more convenient to treat probabilities of mating types, instead of their numbers, we substituted

*C*,

*D*, … by

*C*/(no. mating types of

*C*),

*D*/(no. mating types of

*D*), … , respectively. So we obtained a recursion scheme (Table 1) to evaluate the probabilities

*C*

_{i}_{+1},

*D*

_{i}_{+1}, … of mating types AABB × AABB, AAbb × AAbb, … of generation

*i*+ 1 from the probabilities

*C*,

_{i}*D*, … of the mating types of the preceding generation. Note that there appeared to be discrepancies between Table 1 and Equation 3.1 of Haldane and Waddington. The reason is that their Equation 3.1 contains several misprints, for example, mating type

_{i}*Q*is missing in the expressions for types

*G*,

*H*, and

*I*, and type

*T*can be generated from itself.

For generation *i* the recombination fraction can be determined by adding the fractions of recombinant gametes of all mating types. From a mating type, from none up to four gametes may be recombinant. Therefore, the recombination fraction θ^{*}_{i} after *i* generations of inbreeding is 2This recombination fraction depends on the initial distribution of mating types. Brother-sister mating is started with the offspring of an F_{1} × F_{1} cross, *i.e*., all matings have type AABB × aabb, in the case of RIS and with the individuals of an AIL(*j*) in the case of IRIP. The recombination fractions of the initial cross are θ and θ_{AIL(}_{j}_{)}, respectively. Denoting the actual recombination fraction with , and are the genotype frequencies in the initial generation. The probability of the mating type AABB × AABB is therefore *C*_{0} = *P*(AABB) × *P*(AABB) × no. of types = (1 − θ_{0})^{4}/8. Analogously we obtained .

Application of the recursion scheme (Table 1) yielded , , , , … , … For θ → 0 and θ_{0} → 0, θ^{*}_{i} can be expressed by θ^{*}_{i} ∼ θ_{0} + α_{i}θ, where α* _{i}* is the constant coefficient of those polynomials of θ that are multipliers of (1 − 2θ

_{0})θ. Particularly, α

_{0}= 0, α

_{1}= 0.5 α

_{2}= 1, α

_{3}= 1.375, … , α

_{20}= 2.95578, … , and α

_{∞}= 3 holds. Following Equation 1, the relation between the genetic scales is 3Substituting the initial recombination fractions θ

_{0}= θ for RIS(

*i*) and θ

_{0}= θ

_{AIL(}

_{j}_{)}for IRIP(

*j*,

*i*) yielded 4Since the evaluation of α

*was tedious, an approximation α̃*

_{i}*≈ α*

_{i}*would be useful. With the software package CADEMO, we fitted several growth functions to the values of α*

_{i}_{2}, α

_{3}, up to α

_{20}. The four-parameter tanh-growth function performed best. Regarding the asymptotic condition α̃

_{∞}= 3, we fitted the function again and obtained the approximation 5For the initial and for the first generation the correct values α̃

_{0}= 0 and α̃

_{1}= 0.5 were assumed. The maximum absolute deviation between the true values and the approximations appeared to be <0.005. For RIS(

*i*), the true and the approximated map expansion factors are presented in Figure 1. The resolution is not sufficient to make a distinction between them. The theoretical and the approximated map expansion factors of IRIP (

*j*,

*i*) can be obtained from the RIS(

*i*) factors by adding

*j*/2 − 1. Therefore, the approximation has the same quality for these designs as for RIS(

*i*). Note that the map expansion factors of Equation 4 are independent of the map distance

*x*and of the genetic interference acting in each meiosis.

## DISCUSSION

The first result of the study clarified the discussion whether the map expansion factors of advanced designs depend on the lengths of the genetic interval: The factor is constant; *i.e*., the map expansion is proportional to the map distance *x*. Therefore, the observation of Winkler*et al.* (2003) that estimated map distances depended on the density of the marker map cannot be explained by a differing map expansion. Indeed, the estimate of the distance between two markers should not remarkably change, if another marker among them is additionally taken into account. So the reason for the observed phenomenon is probably the biased estimation, where the bias is disproportionally decreasing with growing marker density.

The evaluation of the map expansion factors for finite generations allows us to determine the generation number that is necessary to achieve a certain percentage of the asymptotic map expansion. With RIS, for example, 50% of the asymptotic map expansion factor is obtained already in the second generation of inbreeding, 90% is obtained in the tenth generation, 95% in the thirteenth generation, and 99% in the twenty-first generation (*cf*. Figure 1). These results encourage us to refrain from the generation of >21 rounds of inbreeding.

The approach applied here to brother-sister mating is also useful for other types of inbreeding. For example, the map expansion factors turned out to be (2^{i}^{+1} − 1)/2* ^{i} *for RIS(

*i*) and

*j*/2 + (2

*− 1)/2*

^{i}*for IRIP(*

^{i}*j*,

*i*), if inbreeding was realized by selfing.

If the aim of an advanced inbreeding experiment is not map expansion, but rather the generation of a certain degree of homozygozity or haplotype frequencies as in Perin*et al.* (2002), the recursive scheme (Table 1) is also useful to obtain theoretical expectations.

Williams*et al*. (2001) produced a RIS (32) and found an expansion of the genetic scale of factor 3.6, rather near the theoretical value of 3.997. Our hypothesis is, however, that the underestimation (which is in agreement with the findings of Winkler *et al.* mentioned above) was caused by disregarding genetic interference. We will show in a subsequent article that the apparent genetic interference of the accumulated generations differs from the genetic interference acting in each meiosis and that commonly used map functions lead to reduced map distance estimates in the advanced designs.

## Footnotes

Communicating editor: J. B. Walsh

- Received November 1, 2004.
- Accepted February 10, 2005.

- Genetics Society of America