Table 1 shows the equations of the three models, the estimations of their parameters, and their overall fit to the F₂ data from the 359 female quail. It also lists the estimations of the same parameters for the two F₀ lines (DD and LTI) used to produce the F₁ generation, parent of the F₂. Observed and predicted egg laying curves of the F₂ population are in Figure 1 (Model 1 for egg laying rate) and Figure 2 (Models 2 and 3 for 3-week egg production).

Table 2 lists the parameters of the curves for which the detection of QTL was performed, and gives their biological meaning. It also shows the corresponding statistics (number, mean, and range) from the set of the individual curves adjusted to the data. Five, 7 and 51 curves could not be obtained by nonlinear regression under Models 1, 2 and 3, respectively.

Table 3 shows the location of the chromosome-wide significant QTL, their position on the chromosome, the maximum F value obtained at this position, their genetic effects, the reduction of the residual variance obtained by fitting a QTL at this location, and the corresponding genome-wide significance.

Out of the 8 QTL found to be chromosome-wide significant (p_c<0.05) on 5 of the 12 autosomes scanned in this study, one was genome-wide suggestive (p_g <0.20) and none was genome-wide significant (p_g <0.05). One chromosome-wide significant QTL for the parameter t₂ related to the ascending phase of the egg laying curve in Model 2 was found on CJA13. Another QTL was close to chromosome-wide significance on CJA06 for the parameter y_P which describes the plateaued phase of the curve in the same model. Chromosome-wide significant QTL for the parameters associated with persistency in Models 1 and 3 were found on CJA10 and CJA14 for b, and on CJA05 and CJA10 for k₂. QTL were also found on CJA10 for the two parameters, c and d, of Model 1 which are related to sexual maturity.

In most cases, the genetic effect of the QTL was dominant, but the QTL was strictly additive in two instances, for y_P and t₂. The reduction of the residual variance due to fitting a QTL was small and did not exceed 2.2%.

For all three populations, the F₂ and the two F₀ lines, the three models provided a good overall fit (R²) to the egg laying data as they accounted for around 90% of the total variation. Similar figures have been obtained on quail with Model 1 17, but it seems that Model 3 underestimated the peak production of the F₂ (Figure 2). Moreover, it was more difficult to adjust this model to individual curves as indicated by the high proportion (14%) of instances for which the numerical estimation of the parameters did not converge to an acceptable set of values. Of course, the coefficient of determination R² of individual curves fitted for each F₂ quail was variable and often lower (in the range 99–58%) than for the global curve. Correspondingly, the variation of the biological parameters of the individual curves was much larger because they were estimated from 19 "observations" or less (only one value by bird and 3-week period). Because of the individual between-bird variations in the actual shape of the laying curves, it might be difficult to achieve more uniform goodness of fit for the within-bird curve parameters than that obtained in the present study, and as observed also with chicken 67. This means that the precision on the phenotypes (the estimated parameters of the curve) used for the QTL analysis varied between birds, possibly making the detection of QTL more difficult to achieve than for traits measured directly.

The majority of the chromosome-wide significant QTL corresponding to different parts of the laying curve obtained for the three models showed some dominance, which could be expected since there is consistent heterosis for egg production in poultry 18. The significant genome-wide additive QTL for total egg number found in the previous analysis of the same set of quail 12 was mapped on CJA06 at the same position as the additive QTL found for y_p (the number of eggs laid in 3 weeks at the time of maximum egg laying) in the present study. It is unlikely this convergence was coincidental, and it is an indication that the QTL which were only detected at the chromosome-wide level in the present work might truly correspond to regions with genes with some effect on the shape of the laying curve. Also, it is interesting to notice that the QTL analysis of the curves obtained under Model 1(for egg laying rate) and Model 3 (for 3-week egg number) led to the detection of a QTL for the decreasing phase of the curve (related to persistency) on CJA10. To our knowledge, there is no published report on similar QTL analyses of curves fitted to repeated observations of the same phenotype. Yet, this approach might be a good alternative to splitting the egg production into consecutive periods and searching for QTL for each period 11.

Two lines, LTI and DD, with different origins 1920 and selected for early egg production 13 or for high duration of tonic immobility 14 were crossed reciprocally (12 single-pair matings) to produce the F₁ generation. Ten males and 30 egg-laying females were selected at random across the F₁ families to produce the F₂ generation. Each F₁ sire was mated to three full sisters from another F₁ family to obtain 434 F₂ female chicks (39 to 45 birds per sire family and 12 to 19 quail per full-sib family), in three consecutive hatches. The 57–week egg laying test analyzed in this work was completed by 359 birds.

The F₂ quail were given successively standard starter (2901 kcal ME/kg, 11.5% moisture, 7% ash, 27% total protein, 8% fat and 4% crude fiber) and commercial layer (2709 kcal ME/kg, 11.5% moisture, 12% ash, 20% total protein, 4% fat and 4% crude fiber) diets. At 5 weeks of age they were assigned at random to individual cages of a 4-tier battery maintained at 22°C, in which they remained until the end of the experiment with free access to feed and drinking water.

Egg production was recorded daily and individually from 5 weeks of age onwards. Global egg laying curves for all the F₂ (and F₀) quail and individual curves fitted to the 19 consecutive 3-week egg production of each F₂ quail were obtained under three different nonlinear theoretical models 467 of egg laying. The parameters of the models with a biological meaning related to egg production were chosen as traits for which QTL analyses were carried out. For each model, the phenotypes were the estimates of the parameters of the individual laying curves.

Model 1 was: elr(t) = a exp(-bt)/(1+exp(-c(t-d))), where t is the number of weeks in test, elr is the egg laying rate (%), a is a scale parameter, b is the rate of decrease of egg laying, 1/c is an indicator of the variance in sexual maturity, and d is the mean number of weeks in test until sexual maturity 4. The traits for which QTL were sought under this model were b, c, and d. QTL analysis was not carried out on a because this parameter was highly correlated to b (r = 0.89) and has no biological interpretation.

Model 2 was: y(t) = (y_P/t₂)t - r (y_P/t₂) Ln [(exp(t/r) + exp(t₂/r))/(1+exp(t₂/r))] + r b₄ Ln [(exp(t/r) + exp((t₂+P)/r))/(1+ exp((t₂+P)/r))], where t is the number of weeks in test, y is the number of eggs laid over the last 3 weeks, r is the duration of transition and was taken equal to 0.3 as in 6, y_P is the level of constant production (that is, the 3-week production at the egg laying plateau), t₂ is the time at transition from rapid increase to constant egg production (that is, the number of weeks in test until the egg laying plateau), b₄ is the rate of decline in production (that is, the weekly rate of change in egg laying after the plateau) and P is the persistency of constant production (that is, the duration of the plateau, in weeks) 6. The traits for which QTL were sought under this model were y_P, t₂, and b₄. QTL analysis was not carried out on P because it was highly correlated to y_P (r = 0.71).

Model 3 was: y(t) = [21k₁ (1 - exp(-t))/(1+exp(-t))] - [21(k₁-k₂) (1 - exp(-t))/(1+exp(-(t-c₂)))], where t is the number of weeks in test, y is the number of eggs laid over the last 3 weeks, k₁ is the proportion of maximum production for the increasing phase of the curve, k₂ is the proportion of maximum production for the decreasing phase of the curve, and c₂ is the point of inflexion from the upper level of the increasing phase to the lower level of the decreasing phase of the curve (that is, a measure of persistency of egg production) 7. The traits for which QTL were sought under this model were k₁, k₂, and c₂.

The estimations of the parameters were obtained using the NLIN procedure of the SAS software 21. Bounds were imposed on estimates of b₄ (≤ 0), k₁ (≤ 1) and k₂ (≥ 0) to insure the coherence between these estimates and the biological meaning of the corresponding parameters of the curve. Under Models 1 and 2, individual egg laying curves could not be estimated for 5 and 7 quail, respectively, because convergence was not reached after 400 iterations, despite trying multiple starting values and using the Marquardt method. This problem was more severe under Model 3, however, for which 51 individual curves could not be obtained.

Genotyping was carried out at Gifu University, and all quail were typed for the 72 microsatellites listed in the Gifu augmented panel 15. The genotypic data were validated and stored in the MAPGENA database. All 24 F₀, 40 F₁ and 434 F₂ quail in the present experiment belonged to the resource family set up to establish the first microsatellite quail map 16 built from 72 loci resolved into 13 linkage groups, and to carry out a previous QTL study of the same population 12. Since then, additional work 222324 made it possible to assign all 13 linkage groups (total map distance of 576 cM with a 10 cM average spacing between loci) to chromosomes. Consequently, the unassigned linkage groups Q03, Q04, Q08, Q09, Q10 and Q11 of the first map have been renamed respectively, CJA03, CJA13, CJA09, CJA04, CJA20 and CJA10 in the present work. The sex chromosome was not included in the present study.

Standard QTL detection was performed using the line cross method 25 implemented in the QTL Express software 26, and it took into account a hatch effect (3 classes). Five thousand permutations 27 were carried out to set significance levels (p_c) for the most likely chromosome-wide QTL. Genome-wide significance (p_g) for a QTL detected on a given chromosome was obtained from p_c by: p_g = 1-(1-p_c)^1/r, where r was the ratio of the length of this chromosome over the total chromosome length spanned by the present study. Chromosome-wide significance was set up at p_c = 0.05. Genome-wide significant and suggestive thresholds were set up respectively at p_g = 0.05 and 0.20.