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Fertility assessment through heterospermic insemination of flow-sorted sperm in cattle¹

A. F. Flint^*, P. L. Chapman and G. E. Seidel, Jr.^2,*

^* Department of Biomedical Sciences and and Department of Statistics, Animal Reproduction and Biotechnology Laboratory, Colorado State University, Fort Collins 80523

² Correspondence:
ARBL Bldg., Foothills Campus (phone: 970-491-5287; fax: 970-491-3557; E-mail:
gseidel{at}colostate.edu).

	Abstract

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The ability to assess fertility of bovine sperm accurately and rapidly would be very useful for research and applications to the cattle industry. Sperm motility and other in vitro tests of sperm normality are only partially correlated with fertility, and lengthy breeding trials are expensive and time consuming. Heterospermic insemination by mixing sperm from more than one male provides an in vivo method to assess relative fertility among bulls that can be economical and rapid. Sperm that had been flow-sorted and cryopreserved from four groups of four bulls were inseminated in all combinations of three bulls within groups into nonsuperovulated heifers or superovulated heifers. Embryos were collected nonsurgically between d 13.5 and 20 following estrus and evaluated for paternity by genotyping. Following determination of paternity, a heterospermic index was created for each bull using a maximum likelihood function. These indices ranged from 0.22 ± 0.15 to 2.43 ± 0.43 (mean = 1.00, with a higher value indicative of greater fertility). In all four groups, either the high- or low-fertility bull was identified (P < 0.05) using a total of 25 to 36 genotypable embryos from nonsuperovulated heifers. The heterospermic rankings of bulls were similar for single and superovulated heifers for one group of bulls, but dissimilar for a second group. Heterospermic insemination followed by genotyping of embryos proved to be efficacious for rapidly ranking fertility of flow-sorted sperm from bulls when females were not superovulated, but results were less clear when females were superovulated.

Key Words: Cattle • Fertility • Genotypes • Insemination • Spermatozoa • X-Y Separation

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion Implications Appendix Literature Cited

 
Accurate sexing of sperm by flow cytometry-cell sorting (Seidel and Johnson, 1999) has created a need to maximize its application. One method to increase utilization of sexed sperm is to decrease sperm numbers per insemination. Unfortunately, when sperm numbers are lowered, the number of males that achieve acceptable pregnancy rates also decreases (Saacke, 1998), so it becomes important to determine fertility differences among males. Heterospermic insemination with mixtures of sperm from different males is a powerful test for male fertility (Beatty et al., 1969; Martin and Dziuk, 1977; Dziuk, 1995). This procedure controls variation in environmental factors, technicians, and sperm numbers (Berger, 1995), and is not affected by male-female interactions (Berger and Dally, 2001). Thus, when males differ slightly in fertility using homospermic insemination, heterospermic insemination can be used to exaggerate the differences. Heterospermic insemination has been estimated to be up to 170 times more efficient in assessing the fertility of bulls compared with homospermic procedures (Beatty et al., 1969).

Larger differences in fertility occur between bulls when relatively low numbers of sperm per inseminate are used than when high sperm numbers are used (Den Daas et al., 1998). For this reason, acceptable fertility rates can be obtained with traditional doses of sperm used in AI and natural breeding, even with lower-fertility bulls. However, with sexed sperm, it is necessary to lower the number of sperm inseminated to maximize application and decrease the cost per inseminate. Heterospermic insemination may be an invaluable tool for identifying those bulls that are unsuitable for sexed sperm programs. For this reason, the following experiment was done with three objectives: 1) to determine whether heterospermic insemination could be used to determine relative fertilities of sexed sperm from different bulls, 2) to estimate the number of 15-d embryos needed to discriminate differences in sire fertility, and 3) to evaluate if superovulation could be an appropriate model to reduce insemination numbers. The ultimate objective of the experiments was to develop a rapid, inexpensive, in vivo fertility test for bulls. We also developed a maximum likelihood method of calculating heterospermic indices.

	Materials and Methods

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Semen Collection and Processing
Ejaculates were collected from 15 bulls representing three breeds (Jersey, Holstein, and Angus) using an artificial vagina. Ejaculates containing less than 50% progressively motile sperm or more than 25% morphologically abnormal sperm were discarded. Semen was handled according to Certified Semen Services (Columbia, MO) recommendations. All sperm used in this study were incubated for 60 min at 34°C with Hoechst 33342 (ICN, Costa Mesa, CA) to stain the nucleus (Schenk et al., 1999). Sperm were then sorted by flow cytometry at 22 ± 2°C using an SX MoFlo flow cytometer (Cytomation, Inc., Fort Collins, CO) (Schenk et al., 1999). The study used X-chromosome-enriched, Y-chromosome-enriched, or bulk-sorted sperm. In each case, dead sperm were discarded. Bulk-sorted sperm were subjected to all steps of the sorting process, but they were sorted based only on viability, not on sex chromosome content.

Sperm were packaged in 0.25-mL polyvinyl chloride straws (IMV, Minneapolis, MN) in a 20% egg yolk Tris base extender (Schenk et al., 1999) at 20 x 10⁶ sperm/mL. The straws were frozen on racks in static liquid nitrogen vapor (Schenk et al., 1999) and stored in liquid nitrogen. Post-thaw progressive motility was determined subjectively (Schenk et al., 1999), and samples with less than 35% progressive motility post thaw were discarded.

Field Trials
Initially, most bulls were of unproven fertility. Therefore, before and during the heterospermic study, heifers were inseminated homospermically with flow-sorted sperm from individual bulls used in field trials to obtain limited fertility data. These field trial inseminations were done at the Rouse Beef Cattle Research Center (Saratoga, WY) with the Angus bulls and Angus heifers, and at Teague Diversified facilities (Wiggins, CO) with the Jersey bulls and Holstein heifers.

Insemination of Heifers
Semen from the 15 bulls was used to make four sets of four bulls. One bull was evaluated in two different sets. Set one was four Jersey bulls, set two comprised two Angus and two Holstein bulls, and sets three and four each comprised four Holstein bulls. The four groups of bulls were tested contemporaneously, not sequentially.

For the heterospermic inseminations, sperm from each set of bulls were mixed and inseminated in all possible combinations of three bulls (ABC, ABD, ACD, and BCD). The 0.25-mL straws containing sorted sperm were thawed nearly simultaneously in a 37°C water bath for 30 s to 1 min. Straws were blotted dry when removed from the water bath. Sperm from each bull were expelled into an individual prelabeled 5-mL tube. Some straws were loaded with a column of extender without sperm to wet the cotton plug, followed by extender containing sperm. In those cases, care was taken to remove any excess extender not containing sperm (in column-loaded straws) so it did not mix with the portion containing sperm (Schenk et al., 1999).

Equal numbers of motile sperm post thaw were used to give sperm from each bull the same opportunity to fertilize the oocyte; 200,000 motile sperm post thaw for each of the three bulls were chosen to achieve approximately a 50% embryo collection rate (Robl and Dziuk, 1988). The volume of extended sperm required for each bull was calculated for individual freeze codes. The volume calculated for each bull was then transferred into a separate 5-mL tube for each heifer. With a 1-mL tuberculin syringe, about 10 µL of Tris extender was drawn into a new 0.25-mL straw, followed by an air space about 4 mm in length, the mixed sperm, an additional 4-mm air space, and finally, additional extender to fill the straw, moving the inseminate exactly to the middle. The air spaces were added so as not to dilute the inseminate with the extender so that the volume could be measured accurately. Straws were loaded into 0.25-mL embryo transfer guns (IMV), and the guns were placed in sterile, side-opening sheaths (IMV), and then kept clean and warm (25 to 30°C) until insemination. All sperm handling after thawing was done at room temperature, approximately 20 to 22°C.

Heifers, 14 to 22 mo of age and primarily of Angus or Angus crossed with Charolais or Simmental breeding, were maintained in dry lots at University facilities. They were fed 8 to 10 kg of hay daily and were observed for standing estrus early each morning and late each afternoon. Over a 16-mo period, they were induced into estrus with an i.m. injection of 25 mg of PGF2 (Lutalyse, i.m.) 7 to 15 d following standing estrus or a previous treatment of PGF2. Heifers that were not superovulated were inseminated 12 or 24 h following standing estrus. To compensate for the low numbers of sperm used, each heifer was inseminated with one-half of the inseminate into each horn of the uterus by passing the embryo transfer gun past the palpable bifurcation of the uterus and depressing the plunger to a mark on the plunger for the first horn, followed by the remainder of the inseminate into the opposite horn. During insemination, the horns of the uterus were manipulated as little as possible to reduce trauma. To keep the time between thawing and insemination to 30 min or less, heifers were generally bred in groups of three or four. If more than three heifers were bred at one insemination time, new straws of sperm were thawed for each group of heifers. The number of inseminations totaled 364.

Superovulation of Heifers
For superovulation, heifers from the same population as the previous group were injected with Folltropin-V, a commercially available FSH (Vetrepharm, Ontario, Canada), prepared at 10 mg/mL, and frozen at -20°C until use. After thawing at 37°C to minimize damage to the hormone, vials were stored at 2 to 8°C for up to 10 d. Before the start of superovulation, ovaries of each heifer were palpated per rectum to verify the presence of a corpus luteum. Heifers were injected with the following quantities of 10 mg/mL of Folltropin at half-day intervals: 5, 5, 4, 4, 2, 2, 2, and 2 mL i.m. In addition, each heifer received 25 mg and 12 mg of PGF2 i.m. (Lutalyse), respectively, with the 6th and 7th FSH injections.

Heifers were superovulated in groups of three to six. Only bull sets 3 and 4 were evaluated using superovulation. The number of superovulated heifers totaled 75. Superovulated heifers were inseminated similarly to nonsuperovulated heifers with the following modifications. Due to poor collection efficiency initially, the dose of sperm for each breeding was increased from 200,000 to 400,000 motile sperm per bull per insemination, and each heifer was inseminated both 12 and 24 h following detection of standing estrus or first detection of an activated Kamar (Kamar Co., Steamboat Springs, CO).

Collection of Embryos
Embryos were collected between 13.5 and 20.5 d (mostly 14 to 17 d) following standing estrus. A Foley catheter, size 14 to 24 depending on cervix size, was passed through the cervix and into the uterus (Elsden et al., 1976). Embryos were collected in modified Dulbecco’s PBS (m-PBS; Elsden and Seidel, 1995) with 0.1% BSA. The uterus was filled with 50 to 200 mL of collection medium that was then collected. The filling and emptying process was repeated until the embryo was expelled or approximately 0.5 to 1 L of collection medium was dispensed. The collected medium was filtered through a 100-µm mesh filter. Collected material was observed under a dissecting microscope. A new filter was used for each heifer to avoid contamination with DNA from a previous embryo. The tubing was also flushed thoroughly between heifers to limit tissue contamination.

When embryonic tissue was found, it was removed using a Pasteur pipette and rinsed in m-PBS without BSA. Some embryos were torn when multiple embryos were collected from a donor. The pieces that could not be associated to individual embryos were discarded so as not to use information from the same embryo twice. Before being stored, each embryo was washed three times in BSA-free m-PBS. Each embryo was then bisected so there was backup tissue in case the genotyping laboratory was unable to obtain genotyping results. Care was taken not to reuse pipettes or dishes between embryos to limit biopsy contamination.

The embryo biopsies were placed directly into 1.5-mL microcentrifuge tubes. Each vial was labeled with an individual sample number to correlate to the number of the heifer, date of insemination and collection, and bulls used in the inseminate. Each of the labeled tubes was placed upright in a standard -20°C freezer until sufficient numbers of embryos were assembled for shipment to the Celera AgGen Laboratory (Davis, CA) for genotyping.

Evaluation of Paternity
Embryonic tissues, as well as blood or sperm samples from the 15 bulls, were shipped to Celera AgGen on dry ice by overnight courier. Celera AgGen proprietary procedures were used to evaluate DNA for paternity using a StockMarks Paternity PCR Typing Kit (Celera AgGen). The kit contained reagents for a mixed 11-plex in a single PCR reaction. The 11 proprietary primers were from the kit (StockMarks for Cattle Bovine Genotyping Kit) (Applied Biosystems Inc., Foster City, CA). To each 1 µL of DNA, the following were added: 3.0 µL of 10x StockMarks buffer, 4.0 µL of 1.25 mM for each deoxynucleotide triphosphate, 5.5 µL of primer mix, 0.5 µL of AmpliTaq Gold polymerase (Applied Biosystems, Inc.), and 1.0 µL of distilled water. The sample was then amplified using PCR with the following program: the reactions were heated for 15 min at 95°C followed by 31 cycles of denaturation (94°C, 45 s), annealing (61°C, 45 s), and extension (72°C, 60 s), then 60 min at 72°C, and a final step of 2 h at 25°C. After holding at 4°C until gel separation, 0.4 µL of the diluted sample was added to 2.0 µL of a loading mix containing deionized formamide, dye, and GS500 internal size standard 29:5:6 (Perkin-Elmer, Boston, MA). The DNA-loading mix samples were denatured at 95°C for 5 min and placed on ice immediately following the denaturing step. Using a 0.2-mm needle and a Hamilton multichannel syringe, the entire volume of the DNA-loading mix was loaded onto an ABI prism 377 gel (Applied Biosystems, Inc.). The gel was then run with 70 W of constant power up to 1,500 V for several hours, with the running time based on the movement of the dye in the gel.

A unique pattern of polymorphisms is produced from DNA from blood, sperm, and biopsies of embryos. A list of the three possible sires for each embryo was provided with each biopsy. After PCR, one of the three sire polymorphism profiles usually could be matched to the embryo polymorphism profile, thus determining the true sire.

Statistical Evaluation
Information on genotypes of embryos was analyzed using a competitive index developed with maximum likelihood procedures (see Appendix for the maximum likelihood analysis procedure) with S-plus programming so that the average index for a group of bulls was 1.00. The competitive indices were compared among single-ovulating heifers, superovulated heifers, and field matings. Other statistical comparisons were done with Student’s t-test, or by ² analyses, with the Fisher-Yates correction.

	Results

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Single ovulation collections of 13.5- to 20.5-d elongating embryos yielded 162 embryos from the 337 heifers from which embryo collection was attempted (48%). An additional 25 heifers were not collected because they had estrous cycles shorter than 15 d (n = 11), a uterine infection (n = 3), an impassable cervix (n = 7), or other reproductive problems (n = 4). There was no difference (P > 0.10) in collection rates of embryos from heifers that had been inseminated 12 h after estrus (50%) vs. those inseminated 24 h after estrus (47%). Differences in group fertility were noted. Bull groups 1, 2, and 4 were not different from each other in the percentage of embryos collected per heifer (P > 0.05; Table 1), but all had superior embryo collection to group 3 (P < 0.05).

Two groups of bulls were evaluated using only nonsuperovulated heifers. In group 1, bull J005 had the poorest fertility as calculated from heterospermic indices (P < 0.05). Bulls J002, J003, and J004 were not different from each other (P > 0.05, Table 3). In group 2, bull AN001 by far sired the most embryos. Fertilities of the three remaining bulls were not significantly different from each other, but were different from the fertility of bull AN001 (P < 0.05, Table 3).

View this table: [in this window] [in a new window]   Table 3. Heterospermic indices calculated from maximum likelihood analyses using results from genotyping embryos (from Tables 1 and 2)

Most of the field trial data using the bulls in the heterospermic trials turned out to be only marginally informative, primarily due to limited numbers of breedings per bull. Published information is available for bulls J002, J003, J004, J005, and AN004 (Seidel et al., 1999). There were no consistent differences in fertility among bulls J002, J003, J004, and J005 with 25 to 30 heifers bred per bull within each of three trials. In one of the trials, bull J002 had higher fertility than bull J004; however, in a subsequent trial, bulls J004 and J002 did not differ from each other. In another field trial involving 172 inseminations, bulls AN001 and AN004 had very similar fertility, but semen from one bull was shipped by air for 6 h before sorting and cryopreservation, so a direct comparison cannot be made (Seidel et al., 1999).

In 8 of 24 superovulation collections there was a significant heterospermic effect (embryos not sired in a 1:1:1 ratio, P < 0.05) as determined by chi-square (Table 4); approximately one of 24 collections would be expected to show a heterospermic effect by chance at P < 0.05.

	Discussion

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When heterospermic insemination is used, the ratio of offspring sired by each male is considered a measure of their relative fertility (Robl and Dziuk, 1988). Furthermore, potential fertilizing ability from heterospermic insemination was directly correlated to homospermic conception rate (Beatty et al., 1969; Dziuk, 1995). Thus, the ranking achieved using heterospermic insemination can be correlated with other measures of fertility and seminal quality such as 16-week non-return rate and spermatozoal assays for quality (Hammitt, 1985; Parrish and Foote, 1985; Saacke et al., 1980). The rankings for heterospermic insemination are highly correlated to homospermic nonreturn rates (Beatty et al., 1969; Martin and Dziuk, 1977; Stahlberg et al., 2000), giving an accurate rating of fertility. Heterospermic insemination is an accurate predictor of relative fertility in cattle (Beatty et al., 1969; Stewart et al., 1974), rabbits (Beatty, 1960), swine (Martin and Dziuk, 1977; Stahlberg et al., 2000), mice (Robl and Dziuk, 1988), and chickens (Martin and Dziuk, 1977). In cattle, the use of polymorphic DNA markers (Stahlberg et al., 2000) combined with 15-d embryo collection makes heterospermic evaluation of fertility more rapid than waiting for offspring to be born.

Interpretations of some previous experiments on heterospermic insemination (e.g., Beatty et al., 1969) are difficult because methods and assumptions for calculating heterospermic indices were not presented. In the current study, a maximum likelihood procedure was developed, which is described in detail in the Appendix. This procedure is sufficiently general that it can be used to calculate indices for heterospermic mixtures of sperm from two, three, or four males. Furthermore, a procedure was developed for estimating standard errors (see Appendix) that greatly increases the usefulness of this statistical approach. An S-plus computer program for the procedures described in the Appendix is available from the authors.

The current experiment was designed to develop an in vivo method to test the fertility of a group of bulls. In the experiment with nonsuperovulated heifers, the average number of genotypable embryos was seven per bull set, whereas the average number of genotypable embryos in the superovulation experiment was 16 per bull set. This led to heterospermic indices that were unable to distinguish the middle hierarchy in most cases. Whereas more embryos were obtained per set, many were not genotypable, probably in most cases because the commercial laboratory used did not optimize procedures for the small amount of tissue provided from embryos. With optimized procedures, nearly all embryos should be genotypable. When greater numbers of embryos were collected with superovulation, it was possible to discriminate between the fertility of more bulls within a group.

Superovulation would be a more efficient and rapid way to test bulls using heterospermic insemination. In the current study, superovulation rankings did not differ significantly from single ovulation rankings, but there are insufficient data to be sure that superovulation is an appropriate model. One concern with superovulation is that the superovulated uterine environment may not be the same as the single ovulation uterine environment. Another concern is that the embryos within a superovulated donor may not be completely independent experimental units. Superovulation may or may not be an appropriate model for the nonsuperovulated animal. However, using superovulated animals to test a group of males for use with embryo transfer and superovulation may be a better test of fertility than homospermic nonreturn rates.

Hammitt (1985) showed that 11 of 16 (69%) heterospermic inseminations resulted in a significant difference from 1:1 ratio of offspring from each male (P < 0.05). In the current experiment using superovulated donors, 33% (8/24) of the collections resulted in a significant heterospermic effect within a donor cow (P < 0.05; Table 4). The current experiment supports previous conclusions (Beatty et al., 1969; Hammitt, 1985) that there is a heterospermic effect in cattle, including when spermatozoa are challenged before insemination.

Heterospermic insemination is a powerful tool to determine the highest- and lowest-fertility bulls in a group. When fertility is challenged by lowering sperm numbers, sorting via flow cytometry, or cryopreservation, specific bulls are considerably more fertile than others. Stewart et al. (1974) observed that when fresh sperm were used, that there was no difference in the fertility among four bulls (Beatty et al., 1969). However, when frozen-thawed sperm were used, considerable differences appeared. Although fresh sperm were not used in the current study, the same concept could apply. Systems that challenge fertility tend to induce greater bull differences in fertility. Thus, a means to rank bulls based on these differences would be of great value in determining the best bulls to use with sorted sperm.

Two considerations should be evaluated when using heterospermic insemination to rank fertility. First, the use of three bulls simultaneously may result in more robust fertility discrimination and decreases the number of inseminations needed when four or more bulls are to be tested. However, the statistical model is very complicated. It would be worthwhile to evaluate two random bulls using heterospermic insemination. In this case, a majority of the time, one bull would sire greater than 70% of the embryos (offspring), whereas the other bull would sire less than 30% of the embryos or offspring (Beatty et al., 1969). Fertility differences between males are best observed when about 50% of the embryos are fertilized (Robl and Dziuk, 1988). The current experiment approached this value with a 48% embryo collection rate with nonsuperovulated donors. If one assumes a 70:30 ratio when embryos are collected from half of the donors, it is possible to approximate the number of donors and embryos needed to observe differences between males using a ² analysis corrected for continuity. The minimum number of embryos needed to distinguish 30:70 from 50:50 ratios of embryos between males at P < 0.05 is approximately 17 embryos from 34 donors for sperm of 2 bulls used in combination with each other. If superovulation were employed, 17 embryos might be recovered from approximately 5 donors at critical sperm numbers. Note that unless a reference sire of known fertility is used, heterospermic procedures only provide information on relative fertility among bulls tested.

A second, recently tested approach is to sort sperm into X and Y sperm populations (Seidel et al., 2003). Sperm sorted for X-chromosomes from bull A would be mixed with sperm sorted for Y-chromosomes from Bull B for one-half of the inseminations, and the converse for the other one-half. The sperm would be mixed pre-freeze or post-thaw and inseminated. The most fertile bull would simply be assessed by sexing the embryos or by sexing the fetuses by ultrasound; no other genotyping would be needed. This does assume no interaction between treatments and sorting procedures. Note that this approach also can be used to test treatments within individual bulls.

Accurate information on fertility of frozen semen from bulls is so difficult and costly to obtain that it is rarely available, even for the hundreds of bulls in major bull studs. For some bulls, nonreturn rates to AI and estimated relative conception rates determined from Dairy Herd Improvement records are available, but these data contain many inaccuracies. The heterospermic procedures proposed here also are costly but may be justified in some circumstances. Costs include feeding heifers, insemination and collection of embryos, genotyping embryos, and paying personnel. In a well-organized system, determining relative fertility of bulls probably would cost a few thousand dollars per bull evaluated.

	Implications

Top Abstract Introduction Materials and Methods Results Discussion Implications Appendix Literature Cited

 
The methods of in vitro sperm testing currently used in the artificial insemination industry are not highly correlated with male fertility, and testing fertility via traditional artificial insemination is expensive and time consuming. Heterospermic insemination plus genotyping of embryos is a quick and accurate method of assessing the relative fertilizing ability of sperm of individual males within a group of bulls. This procedure utilizes the same environment for fertility testing where results will be applied, while eliminating variation in environmental factors, technicians, sperm numbers, and male–female interaction. Minimal statistical procedures would need to be implemented if two males were competing, and fewer than 40 single-ovulating females would need to be used to gather fertility data for each of the combinations. The general procedure of heterospermic insemination could be implemented into many forms and could be applied in many different areas of reproductive research, as well as used to determine the fertility of males whose semen has been processed by various techniques.

	Appendix

Top Abstract Introduction Materials and Methods Results Discussion Implications Appendix Literature Cited

 
Mathematical methods for "heterospermic index" estimates and their standard errors.

I. Writing of the likelihood equations.

Assume that t trials compare s sires. Because not all sires are used in each trial, let:

where i = 1, ..., t, and j = 1, ..., s.

The results for fertilized embryos are given by n_ij = the number of embryos fertilized by sire j in trial i. We assume the following mathematical model for fertilized embryos:

1. Individual embryos are independent of one another.
   
2. Given that an embryo from trial i is fertilized, the probability that it was fertilized by sire j is given by the probability that an embryo that was fertilized by sire j is proportional to that sire’s "heterospermic index," _j, divided by the sum of the heterospermic index of all sires in that trial. So that all probabilities are positive, we assume that all _j are positive.
   
3. The vector of heterospermic index, = (₁, ₂, . . ., _s)^T, is only unique up to a constant, so we assume that the average heterospermic index of all sires is one. This restriction implies that the sum of probabilities for sires used in each trial equals one.
   

Given the above assumptions, the likelihood function of the vector given the observed data, is the product of the multinomial likelihoods in logarithm (base e) for the trials:

II. Iterative solution of the likelihood equations.

By the "Lagrange multiplier theorem" from advanced calculus, maxima and minima of the log-likelihood may be found where the constraint is satisfied, and the partial derivatives with respect to the _j are zero. The constant, , is called the "Lagrange multiplier." Taking the derivative of the coefficient of and using the fact that the derivative of log(y) is 1/y, and the chain rule, we compute the derivatives of the log-likelihood function.

We see no closed form solution to the above system of equations, although several numerical methods are available. For ease of programming and numerical stability, we choose an iterative numerical solution that exploits the fact that the left terms depends more strongly on the value of _j, than does the denominator of the middle summation.

Therefore, an iterative solution proceeds as follows:

1. Take initial values = 1 and ⁰ = 1.
   
2. From these compute
   
3. Update
   

   
   

4. Update
   

   
   

5. Update the value of by solving equation A.4 for , for each j, then averaging the results overj:
   

   
   

6. Update _j, t_j, and , using formulas 2, 3, and 4, but at each step, use the most recent version of _j, t_j, and .
   
7. Iterate until convergence.
   

In practice, the values were found to converge in only a few iterations, but 20 iterations were used to avoid the complication of programming tests of convergence. The iterative procedure was carried out using the statistical package S-Plus-6 for Windows (Insightful Corp., Seattle, WA). The converged to zero with balanced examples in our application.

III. Bootstrap standard errors for the _j estimates.

Standard errors for estimates were computed using the "bootstrap" method described by Efron and Tibshirani (1993). The idea of the bootstrap method is to simulate data sets randomly, as if the true values of the parameters were equal to their estimates. From each simulated data set, new _j estimates computed using the same method used to obtain the original parameter estimates. Standard errors for parameter estimates are then computed as the sample standard deviations of the estimates computed from the simulated data sets. Efron and Tibshirani (1993) note that 25 simulated data sets are usually sufficient, and that rarely are more than 200 needed to obtain an adequate standard error. We used 200. The simulation took less than 10 s for each data set. For some of the resampled data sets some sires had zero offspring. For those situations, convergence was found to be more stable when the was set to zero initially. The bootstrap applied to this problem could be considered to be in the class called "parametric bootstrap," because once the parameter estimate are fixed, the distribution of the simulated data sets follows the multinomial distribution (a parametric distribution). The standard errors computed by the bootstrap method should be considered very approximate for small samples, for the same reason that standard errors for binomial proportions computed by the usual method should be considered very approximate for small samples.

	Footnotes

 
¹ We thank personnel at XY, Inc., who did the semen collection and sperm preparation, and Z. Brink, who assisted with many aspects of managing the cattle. XY, Inc., Fort Collins, CO, funded the project.

Received for publication October 22, 2002. Accepted for publication February 27, 2003.

	Literature Cited

Top Abstract Introduction Materials and Methods Results Discussion Implications Appendix Literature Cited

 


Beatty, R. A. 1960. Fertility of mixed semen from different rabbits. J. Reprod. Fertil. 1:52–60.

Beatty, R. A., G. H. Bennett, J. G. Hall, J. L. Hancock, and D. L. Stewart. 1969. An experiment with heterospermic insemination in cattle. J. Reprod. Fertil. 19:491–502.[Medline]

Berger, T. 1995. Proportion of males with lower fertility spermatozoa estimated from heterospermic insemination. Theriogenology 43:769–775.

Berger, T., and M. Dally. 2001. Do sire-dam interactions contribute significantly to fertility comparisons in heterospermic insemination trials? Theriogenology 56:535–543.[Medline]

Braundmeier, A. G., and D. J. Miller. 2001. The search is on: Finding accurate molecular markers of male fertility. J. Dairy Sci. 84:1915–1925.[Abstract]

Den Daas, J. H., G. G. De Jong, L. M. T. E. Lansbergen, and A. M. Van Wagtendonk-De Leeuw. 1998. The relationship between the numbers of spermatozoa inseminated and the reproductive efficiency of individual dairy bulls. J. Dairy Sci. 81:1714–1723.[Abstract]

Dziuk, P. J. 1995. Factors that influence the proportion of offspring sired by a male following heterospermic insemination. Anim. Reprod. Sci. 43:65–88.

Efron, B., and R. Tibshirani. 1993. An Introduction to the Bootstrap. Chapman and Hall, New York.

Elsden, R. P., J. F. Hasler, and G. E. Seidel Jr. 1976. Nonsurgical recovery of bovine eggs. Theriogenology 6:523–532.[Medline]

Elsden R. P., and G. E. Seidel Jr. 1995. Procedures for Recovery, Bisection, Freezing, and Transfection of Bovine Embryos. Colorado State Univ., Animal Reproduction and Biotechnology Laboratory, Fort Collins.

Hammitt, D. G. 1985. The relationship between in vivo heterospermic fertility and in vitro tests of seminal quality. Ph.D. Diss., Iowa State Univ., Ames.

Martin, P. A., and P. J. Dziuk. 1977. Assessment of relative fertility of males (cockerels and boars) by competitive mating. J. Reprod. Fertil. 49:323–329.[Abstract]

Parrish, J. J., and R. H. Foote. 1985. Fertility differences among male rabbits determined by heterospermic insemination of fluorochrome-labeled spermatozoa. Biol. Reprod. 33:940–949.[Abstract]

Robl, J. M., and P. J. Dziuk. 1988. Comparison of heterospermic and homospermic inseminations as measures of male fertility. J. Exp. Zool. 245:97–101.[Medline]

Saacke, R. G. 1998. AI fertility: Are we getting the job done? Pages 6–13 in Proc. 17th Natl. Assoc. Anim. Breeders Tech. Conf. on AI and Reprod. Milwaukee, WI.

Saacke, R. G., W. E. Vinson, M. L. O’Connor, J. E. Chandler, J. Mullins, R. P. Amann, C. E. Marshall, R. A. Wallace, W. N. Vincel, and H. C. Kelgren. 1980. The relationship of semen quality and fertility: A heterospermic study. Pages 71–78 in Proc. 8th Natl. Assoc. Anim. Breeders Tech. Conf. on AI and Reprod. Milwaukee, WI.

Schenk, J. L., T. K. Suh, D. G. Cran, and G. E. Seidel Jr. 1999. Cryopreservation of flow-sorted bovine spermatozoa. Theriogenology 52:1375–1391.[Medline]

Seidel, G. E., Jr., Z. Brink, and J. L. Schenk. 2003. Use of heterospermic insemination with fetal sex as the genetic marker to study fertility of sexed sperm. Theriogenology 59:515.

Seidel, G. E., Jr., and L. A. Johnson. 1999. Sexing mammalian sperm—Overview. Theriogenology 52:1267–1272.

Seidel G. E., Jr., J. L. Schenk, L. A. Herickhoff, S. P. Doyle, Z. Brink, R. D. Green, and D. G. Cran. 1999. Insemination of heifers with sexed sperm. Theriogenology 52:1407–1420.[Medline]

Stahlberg, R., B. Harlizius, K. F. Weitze, and D. Waberski. 2000. Identification of embryo paternity using polymorphic DNA markers to assess fertilizing capacity of spermatozoa after heterospermic insemination in boars. Theriogenology 53:1365–1373.[Medline]

Stewart, D. L., R. L. Spooner, G. H. Bennett, R. A. Beatty, and J. L. Hancock. 1974. A second experiment with heterospermic insemination in cattle. J. Reprod. Fertil. 36:107–116.[Medline]

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