J. Anim. Sci. 2004. 82:2329-2332
© 2004 American Society of Animal Science
ANIMAL GROWTH, PHYSIOLOGY, AND REPRODUCTION |
A dynamic model for ovulation rate reveals an effect of the estrogen receptor genotype on ovarian follicular development in the pig
T. K. Soboleva*,1,
A. B. Pleasants*,
B. T. T. M. van Rens
,
T. van der Lende and
A. J. Peterson*
* AgResearch Ruakura, Hamilton, New Zealand, and
and
Animal Breeding and Genetics Group, Wias, Wageningen University, Wageningen, The Netherlands
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Abstract
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A mathematical model that describes the recruitment and growth of ovarian follicles was fitted to data on ovulation rate and the measurements of plasma estradiol collected at times during the estrous cycle for individual gilts. The method of least squares was used to obtain estimates of the parameters of the mathematical model. The estimated model parameters were the maximum estradiol production for a follicle, development of each follicle after commitment, and a function describing the initial estradiol production of committed follicles. The estimated parameters for each pig were classified by estrogen receptor (ER) genotype (AA or BB) and analyzed using a multivariate analysis of variance. There were differences between genotypes (P < 0.05) for the parameter that described the initial distribution of individual follicles at recruitment. Gilts with ER genotype BB recruited follicles that varied more in size but had fewer very small follicles, indicating that the ER gene affects the relative estradiol secretion of the follicles at commitment. This analysis is an example of a general approach to genetic studies that uses a mathematical model of the physiology as a statistical basis for estimating gene action.
Key Words: Estrogen Receptor Genotype • Follicle Growth • Modeling • Ovulation Rate • Pig
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Introduction
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Genes influence physiological processes in plants and animals to express observable traits. These physiological processes are typically complex, including feedbacks, thresholds, and other nonlinear features and are also affected by environmental variation. Unless an accounting is taken of these physiological processes, any study of gene function may end in confusion owing to the extra variation in the phenotype induced by the nonlinear features.
In this study, we demonstrate the advantages of using a biologically based nonlinear model that describes physiological processes underlying a trait of interest to investigate the effect of a specific gene on this trait and its underlying processes. The advantage of this approach is that the parameters of the model can be directly associated with an aspect of the physiology. More specifically, we report on the use of a mathematical model that describes the commitment and growth of ovarian follicles, developed by Soboleva et al. (2000)
, to investigate the mode of action of the ER-
gene on ovulation rate in the pig. This gene, for which two alleles (A and B) have been described (Rothschild et al., 1991), has been reported to be associated with litter size in the pig (Rothschild et al., 1996; Short et al., 1997; Van Rens et al., 2002). It is hypothesized that this gene acts on some aspect of the recruitment and growth of follicles during the estrous cycle. The difficulty in studying the action of this gene is that follicular growth is a nonlinear process, in which different genes can affect different aspects of this growth. Such nonlinear interactions make direct observation of the gene effects through the phenotype (in this case estradiol measurements and number of ovulations) problematic.
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Materials and Methods
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Estradiol Data Collection
The plasma estradiol concentrations on which this study is based came from an experiment that had been designed to compare several physiological traits for gilts with different ER genotypes. The design of the experiment was described extensively by Van Rens et al. (2000). Briefly, 24 Meishan synthetic gilts with an ER genotype of AA (n = 9) or BB (n = 15) that were distributed across two batches delivered at an 8-wk interval by the Pig Improvement Company (Abingdon, U.K.) were used. The gilts were surgically fitted with a permanent jugular vein catheter between 4 and 11 d after they had shown their second estrus after arrival and were artificially inseminated twice (at an interval of 24 h) at their third estrus after arrival. Blood samples were collected at 4-h intervals from d 16 after second estrus until d 2 after second insemination. Thereafter, blood samples were collected at 12-h intervals until d 10 after the second insemination. All blood samples were centrifuged immediately after collection. The plasma was stored at –20°C until analysis of LH and estradiol as described by Van Rens et al. (2000). The gilts of both batches were treated similarly. Estrous detection was performed by the back pressure test in presence of a vasectomized adult boar twice daily (at 0800 and 1600) or, during the 4-h interval sampling period, three times daily (at 0800, 1600, and 2400).
On d 35 or 36 after the first insemination, the gilts were slaughtered to study reproductive variables (i.e., morphometry and weight of reproductive tract, embryos, and ovaries). The number of corpora lutea individually dissected from the ovaries reflects the ovulation rate of the gilt.
The Mathematical Model
A dynamic model describing the growth of ovarian follicles from recruitment to ovulation was developed by Soboleva et al. (2000). This model is based on the general biological assumptions that 1) the estradiol secretion rate of an individual follicle is a measure of the maturity of the follicle; 2) the maturation rate of each follicle at any time depends on the current maturity of the follicle and on the circulating concentrations of other hormones (e.g., FSH and LH); 3) all follicles inherit the same developmental program; 4) the amount of circulating estradiol at any time is the sum of the contributions made by each follicle; and 5) the time of estradiol feedback on gonadotropin concentrations are negligible compared with the time of follicle growth. It is also assumed that the more estradiol all follicles produce, the greater is the potential to induce the LH surge, leading to ovulation.
All these assumptions are accumulated by the set of differential equations for follicular development.
 | [1] |
Here xi is the amount of estradiol produced by the ith committed follicle at time t; index i changes from 1 to N, where N is the total number of follicles committed. The parameters of this model are as follows: k, which is a rate constant defining the time scale of the process; 
, which can be identified with the rate of estradiol production by each follicle; E, the maximum estradiol production for an individual follicle; and m, which governs the range of ovulations that occur and is viewed in the context of this model as a species-specific parameter. It controls the minimum possible number of ovulations, and fixing it on some particular level makes data fitting procedure more stable (e.g., it has assigned value m = 1 for the mono-ovulatory species and m = 8.5 for gilts).
When the total level of estradiol production
reaches some critical level, it triggers the LH surge. This surge leads to the ovulation of a number of follicles within a certain range of maturity reflected by individual estradiol secretion.
The initial conditions of Eq. [1]
define the estradiol output of each follicle at recruitment, which occurs between d 14 and 16 of the estrous cycle involving the synchronized emergence of some 40 to 50 follicles from the proliferating pool. Based on data presented by Grant et al. (1989), the distribution of estradiol production from follicles at commitment can be described by points on the three-parametric curve:
 | [2] |
Each point on the curve corresponds to estradiol production xi by individual ith follicle. The value of estradiol production of the biggest committed follicle is restricted by parameter s. Although no two follicles are at identical stages of development, there is a group of follicles with similar levels of maturity. Parameters r and Q describe the relative difference in estradiol production among committed follicles.
Statistical Methods
The parameters of this model (, E, s, r, Q) and the rate constant k were estimated by least squares for each of the 24 gilts by fitting the model to the time series of estradiol measurements made on each gilt. Adding an error term changed the solution of Eq. [1] with the initial conditions [2] into a nonlinear regression problem. Including the information on the number of ovulations as a constraint served to identify the parameters of the nonlinear regression based on the Eq. [1].
Because Eq. [1] cannot be integrated directly, a numerical integration routine was used. The residuals to be minimized were calculated as deviations from the numerical integration. The criteria for acceptable estimators were 1) minimum sum of squares of the residuals, 2) nonsignificant autocorrelation between the residuals, 3) residuals with a Gaussian frequency distribution, and 4) the difference in the estradiol output at ovulation between the smallest follicle ovulating and the largest follicle not ovulating should be much larger than the differences in estradiol output between the follicles that do ovulate (i.e., the atretic pool of follicles is clearly separated from the ovulating pool of follicles). The parameters of Eq. [1] estimated for each gilt were analyzed using a multivariate analysis of variance that classified each gilt by batch and ER genotype.
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Results
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The least squares means and standard errors for the parameters of Eq. [1] and [2] for the gilts classified by ER genotype are shown in Table 1. Although the batch number significantly affected the parameters (P < 0.015) and s (P < 0.01), all of the variation in parameter s (which reflects the estradiol output of the largest committed follicle) was associated with variation in parameter . The average estimate of the rate constant k was 1.1 ± 0.31. There was no significant effect of genotype on parameter E, the maximum estradiol output of a follicle at ovulation.
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Table 1. Least squares means and standard errors for the repeated measures analysis of the parameters of Eq. [1] and [2], the number of small follicles recruited, and the number of follicles ovulated classified by estrogen receptor genotype
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An increase in the number of follicles ovulated was associated with a decrease in the maximum estradiol output of a follicle at ovulation (b = –1.88 ± 0.489 follicles ovulated per unit of maximum estradiol output per follicle). This is consistent with the model assumption that the ovulating LH surge is triggered by the total amount of estradiol in the system.
The only parameter associated with differences in genotype was the parameter r (P < 0.05), which reflects the initial distribution of estradiol output of individual follicles at commitment. Parameter Q was marginally significant (P < 0.10), and this was due to the high correlation between parameters r and Q. These two parameters describe the estradiol output at commitment of individual follicles relative to one another. This means that, although the largest follicles recruited were of the same size for both genotypes, gilts with the genotype BB recruited follicles that varied more in size and had a lower number of small follicles. This result is shown in Figure 1, where the initial estradiol secretion is plotted for each follicle recruited. Note that because parameter s was not significantly different between genotypes, the same value of s (the average value) was used for both genotypes in Figure 1. This suggests that the ER gene affects the relative estradiol secretion of the follicles at commitment. This result was not affected by performing the analysis with as covariate, indicating that the genetic effect was independent of any effect in the measured initial conditions induced by variation in this parameter. That is, the genetic effect persisted after accounting for the fact that the first estimate of follicle estradiol production occurred after the follicles commenced growing.
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Figure 1. Initial estradiol production by the ith follicle at commitment in estrogen receptor genotype AA ( ) and BB (x). (Note parameter s, the value of estradiol production of the largest committed follicle, which is not different between genotypes, has been normalized to accentuate the significant differences in r (P < 0.05) and Q (P < 0.10); parameters that describe the relative initial differences in estradiol production among committed follicles).
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Although not reaching significance, more follicles at a greater level of estradiol production were ovulated in the BB gilts, and conversely, less follicles at a lower level were ovulated in these BB gilts; the reverse was observed for the AA gilts. Thus, in the BB genotype, 11 out of 15 (73%) sows ovulated with plateau estradiol estimates, whereas, in the AA genotype, only 4 sows out of 9 (44%) ovulated with the same estradiol profile.
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Discussion
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The early analysis of this experiment by van Rens et al. (2000) did not find any association of ER genotype with either length of the estrous cycle or the profile of estrogen production during the cycle. Their analysis was based on the direct measurements of estrogen profile and ovulation rate. Equation [1], however, describes these traits in terms of a nonlinear interaction among the follicles, with an outcome dependent on the parameters of the model. That is, the nonlinear interaction expressed by Eq. [1] will obscure any genetic effect on the phenotype (ovulation rate and estradiol production) if such a genetic effect acts through values of the parameters. In this case, the standard methods of estimating gene action that rely on linear methods would be insufficient.
The analysis of the nonlinear regression estimates of the parameters from Eq. [1] and [2] suggests that the ER genotype is associated with the relative estradiol output of individual follicles at commitment. The source of the difference in the relative estradiol output in relation to ER genotype was not determined, but it does not affect the number of follicles ovulated (in agreement with Van Rens et al., 2002). The model also provided an estimate of the estradiol production of the follicles at ovulation, and although it is not conclusive, it is suggestive that follicle estradiol levels at commitment influence follicle estradiol levels at ovulation. In turn, this may influence the fertilization rate and/or the developmental capacity of the embryos throughout pregnancy after successful fertilization. Van Rens et al. (2002) found in a larger group of gilts (including the gilts from this study) evidence for an association between ER genotype and fetal mortality. Furthermore, there was evidence that differences in fetal mortality between gilts with different ER genotypes were probably due to a difference in placental development. It may therefore be postulated that the observed difference in relative estradiol output of individual follicles at ovulation has carryover effects on the development of the embryonic membranes after fertilization of the ovulated oocytes. It has been suggested that the developmental potential of human oocytes is related to concentrations of follicular estrogens, especially to the ratio estrogen:androgen in the follicular fluid (Andersen, 1993; Xia and Younglai, 2000). It remains to be determined, however, whether the same holds for other species, including the pig.
The analysis of ovulation rate in gilts reported here is an example of a general approach to genetic studies, in which mathematical models of the physiology are used as a statistical approach to estimate gene action. By fitting the individual estradiol profiles of all gilts to a nonlinear regression based on the integration of Eq. [1] estimates of the parameters were obtained that are hypothesized to be "closer" to the gene effects. This hypothesis can then be tested using the estimated parameters for each animal as the dependent variable.
It is anticipated that better estimates of a QTL for a trait would be obtained if the methods for detecting QTL were based on the parameter estimates from a model of the underlying physiology, such as the model in the current study. This method also has the advantage of partitioning the variation in a phenotypic measurement among a number of parameters in the model. In this way, multiple genes acting on various aspects of the physiology can be studied, an option not possible using the direct phenotypic measurements. Of course, a successful exercise of this type requires that the correct model of the physiological process of interest is constructed, and this is an important limitation on the general applicability of this technique. Nonetheless, when this can be done, it is a powerful methodology.
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Implications
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Mathematical modeling of physiological processes underlying complex traits can be a powerful tool in research aimed at a better understanding of the effect of specific genes. It is anticipated that better estimates of a quantitative trait locus for a trait would be achieved when such modeling is applied. In this respect, development of correct models of physiological processes underlying economically important traits in livestock should be stimulated.
1 Correspondence: Private Bag 3123 (phone: +64 7 838 5916; fax: + 64 7 838 5117; e-mail: tanya.soboleva{at}agresearch.co.nz).
Received for publication December 9, 2003.
Accepted for publication May 4, 2004.
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Literature Cited
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Rothschild, M. F., R. G. Larson, C. Jacobson, and P. Pearson. 1991. Pvu II polymorphisms at the porcine oestrogen receptor locus (ESR). Anim. Genet. 22:448.[Medline]
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Van Rens, B. T. T. M., P. N. De Groot, and T. Van der Lende. 2002. The effect of estrogen receptor genotype on litter size and placental traits at term in F2 crossbred gilts. Theriogenology 57:1635–1649.[Medline]
Van Rens, B. T. T. M., W. Hazeleger, and T. Van der Lende. 2000. Periovulatory hormone profiles and components of litter size in gilts with different estrogen receptor (ESR) genotypes. Theriogenology 53:1375–1387.[Medline]
Xia, P., and E. V. Younglai. 2000. Relationship between steroid concentrations in ovarian follicular fluid and oocyte morphology in patients undergoing intracytoplasmic sperm injection (ICSI) treatment. J. Reprod. Fertil. 118:229–233.[Abstract]
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Abstract
Introduction
