The Journal of Heredity 2002:93(3)
© 2002 The American Genetic Association 93:214-217
Brief Communication |
Optimal Positioning of Markers to Control Genetic Background in Marker-Assisted Backcrossing
From the Station de Génétique Végétale, INRA/UPS/INAPG, Ferme du Moulon, 91190 Gif sur Yvette, France.
Address correspondence to Bertrand Servin at the address above or e-mail: servin{at}moulon.inra.fr.
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Abstract |
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 Top Abstract IntroductionMethodsResults and DiscussionAppendix: Analytical Derivation...Computing {int}0d(x, d, L)dxComputing {int}dL/2 PMM(x, d,...Obtaining {Pi}References |
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Molecular markers are commonly used in backcross breeding programs in plants. As genetic maps contain more and more markers, it is of interest to determine which markers are to be used for selection. Here we describe how one can compute an optimal positioning of markers resulting in a maximization of the expected proportion of recipient genome. This criterion allows us to take selection into account and to produce relevant results regarding the final efficiency of background selection in backcross programs.
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Introduction |
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TopAbstractIntroductionMethodsResults and DiscussionAppendix: Analytical Derivation...Computing {int}0d(x, d, L)dxComputing {int}dL/2 PMM(x, d,...Obtaining {Pi}References |
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Molecular markers have proven very useful in improving backcross breeding schemes. Particularly, markers allow us to estimate the genomic composition of individuals, and selection on markers can speed up the recipient genome recovery on noncarrier chromosomes (background selection). Several studies have shown that few markers (typically 2–4 markers/Morgan) are necessary to control genetic background in marker-assisted backcrossing (Hospital et al. 1992; Visscher et al. 1996). Yet more than 2–4 markers/Morgan are generally available. If we assume that all markers on the genetic map have the same technical benefits (codominance, polymorphism between parents), the choice of the markers to use for background selection has to be made according to their positions on the genetic map.
Few studies have evaluated the optimal positioning of markers to improve background selection efficiency. Hospital et al. (1992) showed the impact of marker positions on background selection efficiency based on simulation studies, and determined roughly the optimal positioning of two markers on a chromosome of 100 cM. Visscher (1996) computed the optimal positioning of markers, defined as the positions for which markers best explained the variation in genomic composition of the chromosomes. In Visscher (1996), the proportion of variance explained by markers is derived analytically, based on previous calculations from Hill (1993), under the assumption of no background selection on markers. The idea is that markers that explain most of the variation prior to selection would be the most efficient to select for. However, this ignores the effects of selection on markers over successive generations. Note that it is widely acknowledged in population and quantitative genetics that such effects of selection are barely amenable analytically.
We suggest here a different approach to determine the optimal positioning of markers, taking the effects of selection into account. The aim of a backcross selection program is that any locus but the gene introgressed from the donor line eventually returns to a homozygous recipient type. Even without background selection on markers, this is just a matter of time (i.e., of the number of backcross generations). The aim of selection on markers is to go faster toward fixation than without selection on markers. However, it is known that selection on the markers themselves is very efficient, with 2–4 markers/Morgan. And obviously, once they are fixed, markers become useless for selection. Hence what is really important is not whether markers will be fixed or not, but how much of the genome outside the markers will be fixed for recipient type by the time the markers are fixed.
Based on these considerations, we propose to define optimal marker positions as the positions that maximize the genomewide proportion of loci that are fixed for homozygous recipient type once the markers are fixed for homozygous recipient type (i.e., selection on markers has been successful). This is evaluated from the expected probability that any locus on the genome is of recipient type, given that all markers are of recipient type.
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Methods |
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TopAbstractIntroductionMethodsResults and DiscussionAppendix: Analytical Derivation...Computing {int}0d(x, d, L)dxComputing {int}dL/2 PMM(x, d,...Obtaining {Pi}References |
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Throughout the article, we will assume that recombination takes place without interference and will use Haldane's mapping function to compute genetic distances from recombination rates. Since only one chromosome of each pair is segregating in backcrossing, the analytical derivations and numerical applications are related to the segregating chromosome throughout the article. The criterion computed (
) is the expected proportion of recipient genome, given that all markers are of recipient type, which obviously can be addressed on a per chromosome basis as
 | ((1)) |
thus depends on the number and the positioning of markers, and on the backcross generation at which all markers are of homozygous recipient type (i.e., the last generation of background selection). For a given number of markers m, the positioning of markers on a chromosome is described by a single parameter d (see Figure 1), the distance between the first telomere (T1) and the first marker (M1); d is also the distance between the last marker (Mm) and the second telomere (T2). For m > 2, the other markers (M2 to Mm-1) are equally spaced in the segment [M1,Mm], as was also done by Visscher (1996), who used the same parameter d.
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Hence the chromosome is composed of two segments delimited by a telomere and a marker (herein called TM segments), of size d, and (m - 1) segments delimited by two successive markers (herein called MM segments), of size (L-2d)/(m-1).
The closed form of for two markers at generation BC1 can be obtained by analytical derivations (see appendix):
 | ((2)) |
(1 - e-2d) and rM1M2 = (1 - e-2(L-2d). To find optimal marker positions, must then be maximized for d 
(0,
).
Computing for more markers or for more advanced backcross generations is hardly amenable analytically. We then computed P(X|M) using MDM, a program designed for the numerical computation of expected genotype frequencies at multiple loci (Servin et al. 2002). From the results of MDM, was approximated by summing P(X|M) for discrete values of x, equally spaced along the chromosome, with a step of 0.1 cM. A smaller step was tried but did not produce significantly more accurate results. We derived on the chromosome for different numbers m of markers and for different positionings of the markers.
The closer a locus at position x is to a marker, the higher is P(X|M). When d increases, the size of the MM segments decreases and P(X|M) at any locus on MM segments increases. Conversely, when d increases, the size of TM segments increases, and P(X|M) decreases at any locus on TM segments. As is a linear combination of these probabilities, it presents a maximum (noted *) for an optimal value of d (noted d*), giving the optimal positioning of the markers.
Figure 2 illustrates variations in the genomic composition as a function of d for a chromosome of 100 cM controlled by two markers of fully recipient genotype at generation BC3. As explained above, the proportion of recipient genome on MM segments (dotted lines) increases with d and the proportion of recipient genome on TM segments (scattered lines) decreases as d increases. Finally, presents a maximum of coordinates (d*, *) indicated by the dot in Figure 2. Qualitatively, similar results are obtained with any number of markers and at any backcross generation.
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Results and Discussion |
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TopAbstractIntroductionMethodsResults and DiscussionAppendix: Analytical Derivation...Computing {int}0d(x, d, L)dxComputing {int}dL/2 PMM(x, d,...Obtaining {Pi}References |
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Optimal Positioning
Table 1 shows the optimal positioning (d*) of two to four markers on a 100 cM chromosome in backcross generations BC1–BC3, as well as the corresponding
*. The theoretical proportion of recipient genome without selection on markers is also recalled as a comparison (
). Finally, Table 1 recalls the optimal positioning of Visscher (1996), expressed as corresponding d values.
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It is seen from Table 1 that optimal d* values slightly increase with the backcross generations. This can be interpreted as follows. In BC1, the optimal length of MM segments, (L - 2d*)/(m - 1), is much larger than the length of TM segments (d*), because segments flanked by two selected markers are better controlled than segments flanked by only one marker. But, as meioses accumulate in advanced backcross generations, the apparent recombination rate between markers increases, and MM segments tend to be not better controlled than TM segments. The optimal size of MM segments relative to TM segments then need to be reduced compared to its value in BC1.
The variation of d* is more important between generation BC1 and BC2 than between more advanced generations (e.g., BC2 and BC3) as seen in Table 1. Indeed, as only one generation of recombination has taken place in BC1, it is very likely that the MM segments are introgressed as a whole and are fully recipient, because the probability that double recombination events occurred between the markers is very low, except for very large MM segments. In later backcross generations, loss of control on MM segments is due not only to (rare) double recombinations between the markers, but also to (more frequent) single recombinations between the markers occurring twice in different generations. Thus the apparent recombination rate between markers increases faster between generation BC1 and BC2 than between two other backcross generations.
Suboptimal Positioning
Even with dense genetic maps, it can be hard to find markers exactly positioned at optimal positions d*. In this case, it is of interest to know the impact on genome content of using markers not exactly placed in the optimal positions described above (suboptimal positioning of markers). The last two columns of Table 1 show the positions (d*- and d*+) defining an interval in which * - 1% 
*.
Using more markers on a chromosome leads to better control of the return to the recipient genome because the regions controlled by the markers overlap. Thus better control of the genomic background can be achieved either by using more markers, that can be suboptimally placed, or by using fewer markers, optimally placed. For example, in generation BC2, using four suboptimally placed markers leads to an expected of 97.6%, which can be obtained with three well-placed markers (* = 97.6%).
For a given number of markers, the impact of suboptimal positioning of markers is less important when the backcross generation is more advanced. Indeed, even when no selection on markers is performed, the recurrent genome content increases, due to backcrossing. Thus for a given number of markers, the same value of can be reached either by optimally placing markers and performing fewer backcross generations or by suboptimally placing the markers and performing more backcross generations. For example, a chromosome of fully recipient genotype at three markers will present an expected proportion of recipient genome of = * = 97.6% at generation BC2 if markers are optimally placed. If markers are suboptimally placed, the same return to the recipient genome will be obtained at generation BC3 ( 97.3%).
Optimization Criterion
We found that optimal positions of two markers on a chromosome of 100 cM are about 20 cM from the telomeres (from 18.6 cM in BC1 to 22.9 cM in BC3 as recalled from Table 1). These results slightly differ from those in Visscher (1996), where the optimal marker positions are around 28 cM from the telomeres in the same conditions. Generally the positioning described in Visscher (1996) is farther from the telomeres than ours. The main difference is that optimal d* values here are given conditional on the success of selection, whereas the values given by Visscher (1996) are obtained assuming no selection on markers. This could explain the difference between our respective results. Conversely, our results for two markers fit well those of Hospital et al. (1992), obtained by simulations that take fully into account selection on markers. In fact, they found that the optimal positioning of two markers on a 100 cM chromosome was roughly at 20 cM from the telomeres. This argues for a better relevance of our optimization criterion to predict marker positions that maximize the response to selection (i.e., the return to recipient parent genome) compared to the one used by Visscher (1996), because takes selection into account.
The optimal positioning given in Visschcher (1996) is based on the linear prediction of the proportion of recipient genome in a population composed of individuals presenting every possible genotype at markers. However, the relationship between the proportion of recipient genome and the possible genotypes at markers is linear only in BC1. Indeed, using such a linear predictor in BC1 leads to results very close to the ones obtained using our estimate . For more advanced backcross generations, the relationship is no longer linear, and a linear predictor is not a good estimate of the proportion of recipient genome.
Efficiency of Background Selection
The proportion of recipient genome (*) obtained for the optimal positioning is high compared to the theoretical values when no selection on markers is performed (), as shown in Table 1. For example, a noncarrier chromosome presenting three optimally placed markers that are of recipient type in BC3 will have 99.2% of recipient genome. Without selection on markers, the same return to the recipient genome would be obtained only in BC6. Thus when all markers are of recipient type, it is expected that most of the genome is of recipient type. This confirms previous studies by showing that few markers can efficiently control large chromosomal regions (Hospital et al. 1992; Visscher et al. 1996).
Although the criterion we used to infer optimal positioning is based on the success of selection on markers, our study does not allow us to predict the efficiency of selection on markers, but previous studies have shown that it is very efficient. For example, Hospital et al. (1992) considered background selection on two markers per chromosome for 10 noncarrier chromosomes of 100 cM. They showed that, selecting down to a proportion of 10% individuals at each generation, homozygous recipient genotypes at all markers can be obtained as early as BC3. In the case of fewer noncarrier chromosomes and/or higher selection intensity, background selection may succeed in only one or two generations.
Our method could be extended to background selection on carrier chromosomes, but the optimal positioning will then depend on the position of the target gene (or of markers controlling it) and on the positions of markers used to reduce the linkage drag around the target gene.
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Appendix: Analytical Derivation of for Two Markers in Generation BC1
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TopAbstractIntroductionMethodsResults and DiscussionAppendix: Analytical Derivation...Computing {int}0d(x, d, L)dxComputing {int}dL/2 PMM(x, d,...Obtaining {Pi}References |
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We consider a chromosome controlled by two markers (M1 and M2) positioned as explained in Figure 1. We denote rM1M2 the recombination rate between M1 and M2, and rTM the recombination rate between T1 and M1. As the distance between T2 and Mm is also d, rTM is also the recombination rate between T2 and M2. We assume that recombination rates are related to genetic distances by Haldane's mapping function, and thus TM1M2 =
(1 - e-2(L-2d)) and rTM = (1 - e-2d). We also consider an unmarked locus, noted X, placed at position x on the chromosome.
As recalled from Equation (1) in the method section,
 | ((A.1)) |
M) is the probability to have the three loci X, M1, and M2 of homozygous recipient genotype, and P(M) is the probability to have both markers M1 and M2 of homozygous recipient genotype.
As markers are placed symmetrically to the center of the chromosome, and as only P(X M) is a function of x, Equation (A.1) can be rewritten as
 | ((A.2)) |
(d, L) = 2/LP(M) and P(M) = (1 - rM1M2).
P(X M) has to be divided into two parts to compute Equation (A.2), depending on the relative positions of X and M1:
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Let r1 denote the recombination rate between X and M1, and r2 the recombination rate between X and M2. Using Haldane's mapping function we have
 | (6) |
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Computing  0d(x, d, L)dx
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TopAbstractIntroductionMethodsResults and DiscussionAppendix: Analytical Derivation...Computing {int}0d(x, d, L)dxComputing {int}dL/2 PMM(x, d,...Obtaining {Pi}References |
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As X is on the TM segment, PTM(x, d, L) =
(1 - r2 - rM1M2r1). In this case, r1 = (1 - e-2(d-x)). Developing r1 and r2 as functions of x, we obtain
 | ((A.3)) |
Integrating Equation (A.3) gives
 | (A.4) |
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Computing dL/2 PMM(x, d, L)dx
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TopAbstractIntroductionMethodsResults and DiscussionAppendix: Analytical Derivation...Computing {int}0d(x, d, L)dxComputing {int}dL/2 PMM(x, d,...Obtaining {Pi}References |
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As X is on the MM segment, PTM(x, d, L) =
(1 - rM1M2 - r1r2). In this case, r1 = (1 - e-2(x-d)). Developing r1 and r2 as functions of x, we obtain
 | ((A.5)) |
Integrating Equation (A.5) gives
 | ((A.6)) |
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Obtaining
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TopAbstractIntroductionMethodsResults and DiscussionAppendix: Analytical Derivation...Computing {int}0d(x, d, L)dxComputing {int}dL/2 PMM(x, d,...Obtaining {Pi}References |
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Finally, from Equation (A.2),
 | ((A.7)) |
 | ((A.8)) |
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Acknowledgments |
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The authors wish to thank P. M. Visscher and one anonymous referee for their helpful comments.
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Footnotes |
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Corresponding Editor: Leif Andersson
Received April 23, 2001
Accepted December 31, 2002
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References |
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TopAbstractIntroductionMethodsResults and DiscussionAppendix: Analytical Derivation...Computing {int}0d(x, d, L)dxComputing {int}dL/2 PMM(x, d,...Obtaining {Pi}References |
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Hill WG, 1993. Variation in genetic composition in backcrossing programs. J Hered 84:212–213.
Hospital F, Chevalet C, and Mulsant P, 1992. Using markers in gene introgression breeding programs. Genetics 132:1199–1210.[Abstract]
Servin B, Dillmann C, Decoux G, and Hospital F (2002). MDM: a program to compute fully informative genotype frequencies in complex breeding schemes. J Hered 93:227–228
Visscher PM, 1996. Proportion of the variation in genomic composition in backcrossing programs explained by genetic markers. J Hered 87:136–138.
Visscher PM, Haley CS, and Thompson R, 1996. Marker assisted introgression in backcross breeding programs. Genetics 144:1923–1932.[Abstract]
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