We consider a large well-mixed wild B. rapa population in which there is yearly repeated input of C genomes as a result of cross-fertilization by pollen from a nearby B. napus crop. B. rapa and F1 hybrids are strictly self-incompatible (Hansen et al., 2003). In addition to 10 A-chromosome pairs, F1 hybrids entering the population have nine unpaired C-chromosomes. We used the following procedure.
(1) Plants are assumed to carry between zero and nine C-chromosomes, and their frequencies are denoted as f0, f1, f2…f9; the few plants having more than nine C-chromosomes are included in the last category.
We assumed that all chromosomes are unpaired and inherited independently of one another. In reality, when two plants with many C-chromosomes mate there is of course a chance that their offspring will have one or more matching pairs of C-chromosomes. In that case each gamete will contain one copy of that chromosome rather than zero (25% probability), one (50%) or two (25%) copies as the binomial distribution dictates. This slight change in the distribution of C-chromosomes over gametes again has little effect on our results.
(3) As an alternative to the binomial distribution we multiply the frequencies of gametes having i C-chromosomes by their relative fitness (Table 1), as in Lu & Kato (2001), where fertilization fitness = Fi = (observed frequency of class i/theoretical frequency of class i) and relative fitness = Ri = Fi /F0. The relative fitness of gametes with zero C-chromosomes is thus always 1.
There are initially Pi gametes (0 < i < 9) with i C-chromosomes before selection. After biased transmission the new frequency Pi′ of surviving gametes with i C-chromosomes is P
This equation shows that it does not matter whether we use fertilization fitness or relative fitness, because if Fi /F0 is substituted for Ri, the term F0 is in both the numerator and denominator and thus cancels out. The equation also shows that how many copies of a certain class are passed on to the next generation depends on the frequency of other classes in the population. If a population consists of nearly 100% plants with zero C-chromosomes, then , and the contribution to the next generation of a rare class with i chromosomes equals its relative fitness Ri. This is a severe restriction.
To illustrate this point and to clarify the calculation, first consider a cross between a parent with zero and another parent with, for example, four C-chromosomes. With unbiased transmission the offspring resulting from such a cross would differ in the number of C-chromosomes they possess, but the average would be two, that is, 50% transmission of C-chromosomes. Secondly, the same cross with biased transmission is outlined in Table 2. The initial frequency of gametes with zero to four C-chromosomes formed is calculated using the binomial equation (see paragraph (2)). Then, after biased transmission, the expected number of C-chromosomes in the surviving gametes is 0.747 instead of 2: 18.6% transmission instead of 50%. Selection against C-chromosomes is thus much stronger than expected when one compares the observed vs expected frequencies in Table 1. The strong effect is a result of the fact that gametes with zero chromosomes occur at a fairly high frequency in this biparental cross and have the highest relative fitness. We return to this issue in the Discussion. It would also be possible to select in the embryo instead of the gametic phase, but we did not pursue this option.
Table 2. Example illustrating how using relative fitness affects the number of C-chromosomes passed on to the next generation in a cross between a parent with zero and a parent with four C-chromosomes
|No. of C-chromosomes (i)||Initial Pi in gametes||Relative fitness (Ri)1||After biased transmission||New Pi||C-chromosomes passed on|
| || || ||∑PiRi = 0.1195|| ||Total = 0.747|
(4) New hybrids, all containing nine C-chromosomes, are then added to the population each year in the seed stage. This is because estimates of hybridization rates in the literature are typically based on seeds and young plants, that is, after selection among gametes and embryos has taken place.
(5) Plants with zero C-chromosomes produce more offspring than all others. Plants carrying one to nine C-chromosomes have fitness 1 – s, whereas pure B. rapa plants have fitness 1. Thus s is the selection coefficient (0 < s < 1) at the level of the adult plant. A high positive value of s means strong selection against plants carrying C-chromosomes.
(6) After normalization all frequencies add up to 1, and the cycle is repeated for 10 or 50 generations.
In a recent review, Devos et al. (2009) reported that between 0 and 69% of all seeds produced by B. rapa plants near an oilseed rape field were hybrid. Simard et al. (2006) reported a similar range, but noted that the hybridization rate rapidly declined with increasing density of B. rapa plants and distance to the crop. Warwick et al. (2003) reported 13.6% hybrid seeds near oilseed rape crops. In a mixed experimental population, Halfhill et al. (2002) reported between 0.7 and 16.8% hybrid seeds. Of course, under conditions where species distributions only partly overlap there is less opportunity for hybridization. Landbo et al. (1996) found no hybrids in seeds produced by autumn-flowering B. rapa and an average hybridization rate of 3%. We chose hybridization rates of 10% and 1% for our model.
F1 hybrids typically grow well (Hauser et al., 1998a; Halfhill et al., 2005), but their percentage seed set and pollen viability are lower than those of B. rapa (Mikkelsen et al., 1996; Ammitzbøll et al., 2005; Allainguillaume et al., 2006). For instance, Hauser et al. (1998b) documented that F1 hybrids produced 7.5 times more seed pods than B. rapa under field conditions, but because of reduced seed set this resulted in only a threefold fitness advantage of these hybrids over B. rapa. In a competition experiment, Sutherland et al. (2006) demonstrated that F1 hybrids produced c. 80% of the seed output of B. rapa. Allainguillaume et al. (2006) estimated that F1 hybrids yielded 46.9% of the seed output of B. rapa. Average pollen viability in F1 hybrids may vary between 54 and 84.5% (Warwick et al., 2003; Leflon et al., 2006). F1 hybrids thus generally do quite well.
Contrary to the strong performance of hybrids, BC1 plants generally produce very low seed numbers per pod and do worse than B. rapa, although Hauser et al. (1998b) noted that some BC1 plants were as fit as B. rapa and that survival after planting was high for all types. Rose et al. (2009) reported lower seed yield of some backcross lines but none of the differences were significant. Rose et al. (2009) regressed seed yield of hybrids under competitive conditions against the number of B. rapa markers in the genome and found a significant positive correlation. This could suggest that the number of C-chromosomes present is negatively correlated with seed yield, but regression data are currently not available.
Taking all data into account, Andersson & de Vicente (2010) concluded that overall fitness costs of carrying C-chromosomes may be low. This is certainly true for the F1 but debatable for the BC1, for which there are fewer data. The fit plants may be the ones that have already lost their C-chromosomes. Our model covers a range of selection regimes against hybrids: slightly reduced fitness of hybrids (s = 0.1; i.e. 90% of the fitness of pure B. rapa), strongly reduced fitness (s = 0.9) and intermediate cases (s = 0.3, 0.5, 0.7).