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CROP BREEDING & GENETICS

Pollen Dispersion within a Population, Nonrandom Mating Theory, and Number of Replications in Polycross Nurseries

Dep. of Agronomy, Purdue Univ., 915 West State St., West Lafayette, IN 47907-2054

^* Corresponding author (wnyquist{at}purdue.edu).

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

 
A breeder is uncertain how many replications to include in a polycross. Our objective was to develop a method to answer the question. Four isolated, space-planted nurseries on a 0.91-m grid of 29 rows by 35 plants were established of ryegrass (Lolium perenne L.), a wind-pollinated species, with rows in each nursery oriented differently with respect to the compass. The middle row was homozygous for the dominant fluorescent marker gene. Seeds from the seven middle plants in the seven rows closest to the middle row on both sides were observed for the frequency of the dominant gene. From these observations a two-parameter exponential pollen dispersal function was fitted for the mean of all eight directions combined. The variance among the cross-pollination frequencies q_jk was calculated under two pollination assumptions: (i) pollination confined within each replication and (ii) pollination across all replications. After bulking equal quantities of seed from all replications we have the variance ² of the mean cross frequencies _jk. The square root of this variance, , was used as a measure of the closeness to random mating. The required number of replications to achieve specific levels of closeness to random mating is given for 3 to 100 clones for Assumption 1. The measure of closeness to random mating is given for 3 to 100 clones and up to 30 replications for Assumption 2. When pollination is assumed to occur across replications, a 70% reduction in the variance of is obtained compared to pollination within replications. Hence, fewer replications are required in this case.

Abbreviations: DST, daylight savings time.

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

 
APOLYCROSS is commonly used in the breeding of cross-fertilizing, wind-pollinated, perennial species. A polycross is the interpollination by natural hybridization of a group of genotypes, generally selected, grown in isolation from other compatible genotypes in such a way as to promote random interpollination. An assumption in the use of the polycross is that each clone has an equal chance of pollinating, or being pollinated by, any of the others. This implies that clones must be flowering at the same time. To promote random mating a common procedure in a polycross nursery is to increase the number of contiguous replications, rerandomizing the clones in each replication. Two experimental designs, the Latin square and the randomized complete-block, are commonly used for the arrangement of clones in the polycross nursery (Fehr, 1987, p. 152–154). If the number of different genotypes or clones per replication is 10 or less, the Latin square is often used. If more than 10 clones per replication are to be interpollinated, a randomized complete-block design with adequate replication is normally preferred (Stuber, 1980). For this latter design the overall shape of the polycross nursery is as square as possible to maximize the approach to random pollination. Models for the movement of pollen are based on the dispersion of air-borne pathogenic spores, comprehensively reviewed by Gregory (1945). When the seed is harvested from the polycross nursery, equal quantities of seed from each replication are bulked for each clone to constitute the so-called polycross progeny. Under these conditions the expected underlying structure of the polycross is the modified diallel cross (reciprocal crosses with no selfed progenies). Although theoretical aspects of the diallel cross have been studied extensively (Cockerham, 1963), nonrandom mating has not been incorporated in any diallel analysis. When the genotypes or clones are selected or fixed, the differences among the polycross progenies provide a test for combining ability. Or, alternatively, when the clones are not selected, the among-polycross variance component provides an estimate of the additive variance, assuming all nonadditive genetic variation to be negligible, and the component can be adjusted for nonrandom mating.

By the use of marker genes, nonrandom pollination in polycrosses has been reported for perennial ryegrass (Lolium perenne L.) (Wit, 1952), maize (Zea mays L.) (Gutierrez and Sprague, 1959), smooth bromegrass (Bromus inermis Leyss) (Knowles, 1969), and orchardgrass (Dactylis glomerata L.) (Carlson, 1971). By evaluation of separate replicate entries of a 20-clone, 10-replicate polycross nursery of smooth bromegrass, Hittle (1954) found replicate entries for 12 of 20 clones significant for one or more forage traits studied, indicating nonrandom mating. However, using the same technique, Wassom and Kalton (1958), for orchardgrass, and Ives and Thomas (1959), for smooth bromegrass, reported random pollination. Wright (1962, 1965), Olesen and Olesen (1973), Olesen (1976), and Morgan (1988) developed systematic, balanced designs with neighborhood balance for the polycross when the number of clones is one less than a prime number. Each replication is long and narrow. The number of replications is equal to the number of clones. Although some authors have reported nonrandom pollination in a polycross, no one has attempted to estimate the required number of replications for a given level of nonrandom mating.

The objective of this study was to determine the extent of pollen dispersal about a single source or plant within a population by use of a dominant marker gene and thereby determine the required number of replications in a polycross nursery to achieve a certain level of nonrandom mating.

	MATERIALS AND METHODS

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

 
Nurseries
Four separate nurseries of perennial ryegrass, a wind-pollinated, self-incompatible species, were established on the Agronomy Farm at the University of California, Davis, in the growing season 1962–1963, each isolated from all other ryegrass. Each nursery was a 29 by 35 spaced-plant nursery on a 0.91-m (3-foot) grid. Rows 1 to 14 and another 14 rows, rows 16 to 29, each contained 35 spaced plants of an experimental strain derived from commercial ryegrass in which all dominant fluorescent individuals had been removed and, hence, the strain was breeding true for nonfluorescence. The contaminant row, row 15, contained 35 spaced plants of another experimental strain of perennial ryegrass derived from intercrossing 12 homozygous, true breeding clones for the dominant fluorescent gene at a single locus in a 30-replication polycross nursery (Nyquist, 1963; Okora et al., 1999). This gene causes the seedling roots grown on filter paper to produce a brilliant bluish-white fluorescence under ultraviolet light. The compound is an oxazole alkaloid, named annuloline, C₁₇H₁₀NO(OCH₃)₃ (Axelrod and Belzile, 1958) whose structure is known (Karimoto et al., 1964). Each of the four nurseries was planted with the direction of the rows oriented differently with respect to the compass. The contaminant row was oriented in a west to east direction in nursery A, northwest to southeast direction in nursery B, north to south in nursery C, and northeast to southwest in nursery D.

To determine the distance which pollen traveled in both directions from the contaminant row, seed was harvested from the seven middle plants in each of the seven rows closest to the contaminant row on both sides of that row, namely, rows 8 to 14 and 16 to 22. One hundred twenty-five or more seeds from each of the 49 (= 7 x 7) plants on each side of the contaminant row in each of the four nurseries were examined for fluorescence. A total of seven plants were missing in the four nurseries among the 392 (= 49 x 2 x 4) possible plants.

Heading date and mature natural plant height were recorded for each of the 35 fluorescent plants in the contaminant row and for each of the seven middle nonfluorescent plants in each of rows 1 to 14 and 16 to 29 in each of the four nurseries. These data were collected to consider adjusting the observed percent fluorescence if the fluorescent and nonfluorescent strains differed for heading date or natural plant height. The date of heading after 31 March was recorded for each plant when the tip of one or more spikes had begun to emerge above the flag leaf. Plant height was measured (in centimeters) from the soil surface to the tip of the tallest, naturally occurring spike.

The date of first anthesis after 31 March was recorded for fluorescent plants 1 to 14 and plants 22 to 35 in the contaminant row 15. Date of first anthesis of a plant was recorded when the first florets on a plant extruded their anthers, shedding pollen. In addition, date of first anthesis was recorded for the seven middle nonfluorescent plants in rows 1 to 3 and 27 to 29. These plants or rows were selected to avoid walking between plants to be harvested later for seed for fluorescence determination and to avoid possibly carrying pollen manually on one's trousers once anthesis had started to occur. These observations of first anthesis were recorded in nurseries B and C only.

Fluorescence Determination
Counts of fluorescent and nonfluorescent seedlings were conducted in 1965 in the State Seed Testing Laboratory located on the Purdue University campus in West Lafayette, IN, using standard germination conditions for ryegrass of 20°C for 16 h in darkness and 30°C for 8 h under lights. All tests were conducted in lots of 25 seeds from each plant, using Number 2434, 11 by 40 cm, double-folded, accordion-like filter paper (supplied by Carl Schleicher & Schull, Dassel/Kr. Einbeck, West Germany). A minimum of five lots of 25 seeds each was tested from each plant over time. For some plants with poorer germination one or more additional lots of 25 seeds were tested. The average number of seedlings observed for fluorescence was 118.1 per plant with a standard deviation of an individual of 13.0 with a range from 101 to 187.

Relation between Percent Fluorescence and Heading Date and Plant Height
To determine the influence of heading date on percent fluorescence, if any, a weighted analysis of covariance between percent fluorescence and the covariate heading date for one-way classification (Snedecor and Cochran, 1967, section 14.2) was conducted by use of the generalized linear models (GLM) procedure in SAS (SAS Institute, 2004). All rows of observed plants from the four nurseries were combined. The total number of observations in the covariance analysis was 384. The initial model was w_ijY_ij = µ + _i + ß_i(w_ijX_ij – X–..) + w_ijß_ij, where Y_ij = percent fluorescence for the jth plant in the ith row; _i = effect of the ith row, i = 1, ..., 56 (= 7 x 2 x 4); ß_i = effect of the ith regression coefficient, i = 1, ..., 56; X_ij = heading date of jth plant, j = 1, ..., n_i (= 7, most commonly), in the ith row, i = 1, ..., 56; _ij = random error effect of the ijth plant. The weight for each of the individual plants was the reciprocal of the binomial variance, w_ij=, where _ij is the proportion of fluorescent seedlings for the jth plant in the ith row, j = 1, ..., n_i, i = 1, ..., 56, and k_ij is the total number of seedlings observed for the jth plant in the ith row. Plants with no fluorescent individuals were assumed arbitrarily to have one-tenth of a fluorescent individual (_ij=0.1/k_ij) to avoid a variance of zero. The regression coefficients were tested for equality in the initial model (Steel et al., 1997, section 17.8) and found to be homogenous (F = 0.89, P = 0.6961). Hence, the final model had a single regression coefficient. The same procedure was applied to plant height with similar results.

Daily Pollination Cycle
The daily pollination cycle was studied on four different days, 18, 20, 22, and 25 May 1963, by attaching a Vaseline-coated microscope slide to each of four weather vanes located in the middle of each quadrant in nursery C only. The weather vanes were designed so that the Vaseline side of the slide was always facing into the wind and the slide was oriented with a 45° angle from the vertical position. The single slide on each weather vane was located 45 cm above the soil surface. A new set of slides was exposed every 30 min, starting at 0930 h Pacific daylight savings time (DST) and ending at 1700 h. For counting the number of pollen grains, pollen grains were stained with cotton blue dye and covered with a 22-mm square cover slip. Ten systematically arranged circular microscope fields, each with an area of 1.84 mm², were counted on each slide, using a microscope with a 10x ocular and a 10x objective. The mean number of pollen grains per single microscope field was plotted with respect to the end of the 30-min time periods.

Wind Direction and Speed
At a nearby weather station, wind direction was recorded every minute to the nearest one of the eight compass directions. Since most anthesis or pollen shedding occurred between 10 and 31 May and between 1100 and 1500 h Pacific DST each day, the mean wind direction was calculated in degrees for each of those 176 (= 22 d x 8 30-min periods d^–1) 30-min periods where 0° was equated to north, 45° to northeast, and so on. The distribution of the 176 mean directions with respect to the eight compass directions, north (337.5° to 22.5°), northeast (22.5° to 67.5°), ..., and northwest (292.5° to 337.5°), was also calculated. We also calculated the mean wind speed for each of the 16 15-min periods from 1100 to 1500 h Pacific DST from 10 to 31 May.

Pollen Dispersal Function
The dispersion of the fluorescent gene carried by pollen from the contamination row within each nursery for each of the eight compass directions was fitted to a two-parameter exponential function, _i=ae^bx_i, where _i is the predicted percent fluorescence in the ith row from the contamination row, i = 1, 2, ..., 7, parameter a is the value of the function when x_i = 0, b = shape parameter for the curve, and x_i is the distance from the contamination row in meters, using the nonlinear regression procedure NLIN in SAS. A weighted regression was performed where the ith weight w_i for the mean proportion of fluorescence for the number of plants, n_i, in the ith row was w_i=1/_i.²=. The weight was the reciprocal of the variance of the linear combination _i.=. The variance of _ij is the binomial variance defined in a previous section.

An average pollen dispersal function was obtained by a weighted fitting of the two-parameter exponential model to the mean fluorescent percentage averaged across the eight compass directions for each distance from the contaminant row. This average pollen dispersal function was then used for calculating the required number of replications as described below.

Nonrandom Mating Theory of the Polycross
To determine the number of replications in a polycross for a certain level of nonrandom mating, we examined the nonrandom mating theory for the polycross. The mating structure for an n-clone polycross can be defined by an n x n matrix for the ith replication. We define the conditional probability for the jth clone (row), acting as a female parent, in the ith replication, mating with the kth clone (column), acting as a male parent, as

[1]

where n is the number of clones in the polycross, and b is the number of replications or blocks in the polycross, and 0 q_jk⁽ⁱ⁾ 1. Note that pollen from clone k normally may come from any replication—not just the ith replication (i.e., pollination is normally not confined within each replication). When equal amounts of seed from each replication are bulked for each clone, we have the mean of the j x k matings averaged across b replications, namely,

[2]

where _jk=1 for j = 1, ..., n. Both q_jk⁽ⁱ⁾ and are elements in different n x n Q matrices with elements on the diagonal equal to zero and elements in each row sum to one. When random mating is achieved in the polycross, technically known as random mating with avoidance of self-fertilization (see Nyquist, 1990, section 6.4.1.2), we have

[3]

[4]

When nonrandom mating occurs, we can define the following parameters:

[5]

[6]

[7]

[8]

The first variance q² (Eq. [5]), the variance of q itself before bulking across b replications, is the result of nonrandom pollination per se. Pollination may be across all replications. The second variance ² (Eq. [6]), the variance of , is the combined result of nonrandom pollination and the mean of b values. The quantity _.k is a measure of the relative contribution of the kth clone as a male parent averaged across (n – 1) polycross progenies. Clones may differ in pollen production due to different numbers of influorescences and other variables. The breeder normally cannot control the relative contributions of the clones as male parents, but weighing the amount of seed produced by each clone and computing its proportional amount may provide a rough estimate of _.k, assuming a high correlation between weight of seed produced and number of pollen grains shed.

Assuming clones to be a random sample from a random-mating population in linkage equilibrium, we express the true variation among the means of polycross progenies and the variation among individuals within polycross progenies as a function of the covariances of relatives (C_F and C_H are covariances of full and half sibs, respectively) and the nonrandom mating parameters (see Appendix for derivation): The genetic variance component among-polycross progenies is

[9]

(see Eq. [A13] in Appendix where _q²=²). The genetic variance component within-polycross progenies is

[10]

(see Eq. [A18] in Appendix where _q²=²), where C_I is the genetic covariance of an individual with itself and is equal to _A² + _D² + _AA² + _AD² + _DD² + ...

Number of Replications in a Polycross
To estimate the number of replications to achieve the desired level of closeness to random mating, the following assumptions or conditions were made:

1. All clones were synchronous in flowering behavior with respect both to days and within days.
   
2. All clones produced the same number of pollen grains.
   
3. The pollen dispersal function was independent of the genotype of the pollen source.
   
4. The pollen dispersal function was independent of the direction from the pollen source, i.e., no asymmetrical pollen dispersion existed due to different wind directions.
   

Two different assumptions with respect to the extent of pollination were made:

1. A clone in any replication was only pollinated by the other clones in that replication.
   
2. Every clone in every replication was pollinated by the other clones in all replications.
   

Assumption 1. Pollination Confined within each Replication
To compute the required number of replications a program was written in SAS. First, we calculated the mean variance of q_jk for n clones. Each replication consisted of an r x c = n plant grid arrangement (r rows and c columns); its shape was as square as possible (e.g., for six clones we considered only a 2 x 3 = 6 arrangement; we did not consider a 1 x 6 = 6). We considered various combinations of the number of rows and columns through a total of 36 clones. With these restrictions all clone numbers were not considered. After 36 clones we considered only square arrangements through 100 clones. One clone or plant was randomly assigned to each of the r x c positions (11, 12, ..., 1c, 21, ..., 2c, ..., r1, ..., rc). For the corner position 11, we calculated the distance x in meters between the position 11 and each of the other n – 1 clonal positions. Then we calculated the predicted value of the pollen density function for each of the n – 1 clonal distances, using the average estimated two-parameter density function fitted to the mean fluorescent percentage averaged across the eight compass directions. Then we divided each pollen density by the sum of the n – 1 pollen densities to estimate the probability q_jk⁽ⁱ⁾ of clone j in position 11 being pollinated by each of the other clones, k = 1, 2, ..., n, k j, in the other positions in that same ith replication. The variance q_jk⁽ⁱ⁾ of was computed for clone j. This above procedure was repeated for each of the other n – 1 clonal positions in the ith replication. The mean variance of q_jk for the n clones for the ith replication was calculated. Since this variance for the ith replication was the same for every replication for this first assumption of pollination confined within each replication, we denoted the variance with a sub (1), _q(1)². This variance _q(1)² was based on only one replication.

Second, the required number of replications to approximate random mating to a specific degree was computed for the measure of closeness to random mating, the standard error of the mean, ₍₁₎==D₍₁₎. The departure D₍₁₎ or ₍₁₎ from zero for random mating was set equal to an arbitrary constant, say 0.01, and b' was the calculated number of replications. The mean =_jk(1)=, where b is the number of replications in the polycross nursery, comes about from the process of taking equal quantities of seed from each replication for each clone, bulking the seed, and mixing the seed to constitute the polycross progeny for the jth clone. The variance of _jk(1) is ₍₁₎²=. Solving this expression for the required number of replications, we obtain b_q(1)²/₍₁₎²=_q(1)²/D₍₁₎² .

Assumption 2. Pollination across All Replications
A second program, similar to the first one, was written in which pollination was not restricted within a single replication. This program used a different random permutation of clone placement for every replication and was repeated 1000 times for each of various combinations of different numbers of plant rows and columns per replication and different numbers of replication rows and replication columns. The overall nursery size was equal to the number of plant rows times the number of plant columns. The number of plant rows was equal to the number of rows per replication times the number of replication rows, and likewise the number of plant columns was equal to the corresponding product. The shape of the overall polycross nursery was as square as possible for the above conditions; the shape of the individual replication was not important. The ratio of the larger nursery dimension to the smaller nursery dimension in terms of number of plants was as close to one as possible. The number of replications was limited to 30 or less. Since pollination was not confined within a single replication in this case, but extended across all b replications, we denoted that variance of q itself before bulking with a sub (b), namely, _q(b)². The variance was computed for each of the various combinations. The variance of the mean was _(b)²=_q(b)²/b after bulking of the seed across the b replications.

	RESULTS AND DISCUSSION

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

 
Heading Date for Fluorescent and Nonfluorescent Plants
The mean heading dates were 22.82 ± 0.43 d after 31 March (n = 138) for all fluorescent plants and 24.73 ± 0.18 (n = 764) for the observed nonfluorescent plants. The difference –1.91 was highly significant (P < 0.0001). The estimated variances for heading date of the two populations, 21.67 for fluorescent and 26.40 for nonfluorescent, were homogeneous (P = 0.1504).

This difference in heading date between fluorescent and nonfluorescent plants implied a difference in flowering date between the fluorescent and nonfluorescent plants. Direct observation for flowering date on a subset of the above plants was 44.02 ± 0.29 (n = 56) for fluorescent plants and 45.05 ± 0.24 (n = 81) for nonfluorescent plants. The difference –1.03 was highly significant (P = 0.0064). A difference in flowering date between fluorescent and nonfluorescent plants could affect the observed pollen dispersal function. If nonfluorescent plants flowered much later than fluorescent plants, the tail of the pollen dispersal function would be shortened.

Because the two populations differed in heading time and flowering time, a weighted analysis of covariance between percent fluorescence and the covariate heading date was performed to determine if percent fluorescence was influenced by heading date. A marginally significant regression coefficient estimate b = –0.021 (P = 0.0549) was obtained in the analysis of covariance. The interpretation of this regression coefficient is that the observed percent fluorescence decreased by 0.021% for an increase of one heading day. In other words, later heading nonfluorescent plants, which were observed for fluorescent offspring, were more likely to be pollinated by nonfluorescent plants rather than the fluorescent contaminant plants because nonfluorescent plants headed later than fluorescent plants. If the nonfluorescent plants had headed at the same time as the fluorescent plants, the percent fluorescence would have been only 0.046% [= –0.021(22.82 – 25.02)] higher—less than 1/20th of 1%. This effect was considered trivial, so the raw unadjusted data for percent fluorescence were used in calculating the pollen dispersal functions.

Natural Plant Height for Fluorescent and Nonfluorescent Plants
The mean, fully developed, natural plant height was 76.20 ± 0.75 (n = 138) for all fluorescent plants and 79.63 ± 0.32 (n = 764) for the observed nonfluorescent plants. The difference –3.43 was highly significant (P < 0.0001). The estimated variances of the two populations, 63.59 for fluorescent and 79.65 for nonfluorescent, were homogeneous (P = 0.1009).

Again, since the fluorescent and nonfluoresent populations differed in natural plant height, this difference may affect the observed fluorescent percentage (i.e., pollen would not be carried as far from the contaminant row with a lower plant height). Hence, an analysis of covariance similar to that for heading date was performed and showed a nonsignificant regression coefficient 0.0041 (P = 0.5727) between percent fluorescence and the covariate plant height. Therefore no adjustment of the observed fluorescent percentage was required.

Daily Pollination Cycle
The daily pollination cycles for 18, 20, 22, and 25 May are shown in Fig. 1 . Pollen shedding began during the 30-min period ending at 1130 h Pacific DST on 18 May, 1200 h on 20 May and 22 May, and 1100 h on 25 May, depending primarily on temperature. Maximum pollen shedding occurred in the 30-min period ending at 1230 Pacific DST for all 4 d. It was more difficult to determine when pollen shedding ceased, but little new pollen was probably released after 1500 h.

View larger version (19K): [in this window] [in a new window]   Figure 1. Distribution of the mean number of pollen grains per microscope view (1.84 mm2) shed by the end of the half-hour periods during the day on four different days at Davis, CA, in 1963.

Relation between Percent Fluorescence and Distance
The observed percent fluorescence and the fit obtained by the two-parameter exponential function with respect to the distance from the contaminant plant row for the eight compass directions in the four nurseries are shown in Fig. 2 . The estimates of the two parameters for each of the eight compass directions are presented in Table 1. A weighted regression for a two-parameter exponential function satisfactorily fit each of the eight distributions. Time of pollen dispersion during the day, wind direction, and wind speed influenced the eight pollen dispersion functions, although no attempt was made to quantify these effects. The highest percentages of fluorescence were observed for the northeast direction. This agreed with the senior author's impression that the prevailing wind direction, particularly during the last hour or so of the pollination period, was from the southwest direction, but it disagreed somewhat with the wind direction records reported above. Although asymmetric pollen dispersion occurred, we desired to use the average pollen dispersal function, =17.5004e^–0.4800x (Table 2), for calculating the required number of replications.

View larger version (19K): [in this window] [in a new window]   Figure 2. Percentage fluorescence observed at different distances from the contamination row in each of four separate isolated nurseries for each compass direction, N (north), NE (northeast), E (east), SE (southeast), S (south), SW (southwest), W (west), and NW (northwest), in Davis, CA, in 1963. The curves represent the weighted fitting of the two-parameter exponential function, yi = aebxi, to the observed values.

One of the objectives of this study was to observe the dispersion of pollen about a single plant within a population to simulate the situation in a polycross nursery. A direct approach of observing the dispersion of pollen about a single fluorescent individual surrounded entirely by nonfluorescent individuals was considered unsatisfactory for several reasons. The percentage of fluorescence would have been extremely low even on plants most adjacent to the single fluorescent one. Very large samples sizes would have been required. In addition, the random fluorescent plant could have been unusually early or late compared to the nonfluorescent plants. This would have complicated interpretation of the results. Instead we created a design where the single fluorescent plant and each of the seven individual plants at different distances from the single fluorescent plant in a cardinal compass direction (north, east, south, or west) in the direct approach were expanded to full row of 35 plants. To minimize the border effects due to finiteness, only the middle seven plants in each of the seven rows nearest to the fluorescent row were observed for percent fluorescence. The assumption is made that the pollen dispersal functions observed here are unbiased estimates of the true pollen dispersal functions within a population.

Number of Replications in a Polycross
Assumption 1: Pollination Confined within Each Replication
The variance of q itself before bulking all of the replications and the number of replications for different numbers of plant rows and columns per replication on a 0.91-m grid for two values, 0.010 and 0.005, of the measure of closeness to random mating are given in Table 3. For the measure of closeness D₍₁₎=₍₁₎=, the standard error of the mean pollination frequency, the calculated number of replications decreases as the number of rows and/or columns increases, except for the category of only two rows. For only two rows per replication the required number of replications increases as the number of columns increases. The variance _q(1)² also increases as the number of columns increases in contrast to all other situations. The measure D₍₁₎ sets the standard deviation for the mean frequency of different crossed seeds equal to a sufficiently small value of 0.010 or 0.005. Comparing the calculated number of replications in the two columns for the two different values of the measure, the calculated number of replications is four times greater for a reduction of one half of the standard deviation of the mean pollination frequency.

(1)

(1)

Assumption 2: Pollination across All Replications
The variance _q(b)² resulting from pollination across all replications of clones in a 0.91-m grid nursery is given as the first of two elements in every cell in Table 4. For each row in Table 4 the variance decreases monotonically as the number of replications increases, except for a few cells in the first three rows. For these few cells the overall shape of the nursery showed the greatest departure from a square (i.e., the ratio of the larger nursery dimension to the smaller nursery dimension was equal to or greater than 2.0 for these cells). It is believed that the variance declines as the shape of the overall nursery approaches a square, because the variance of the interclonal distance is minimized for a square. For the assumption of pollination across all replications, the shape of the individual replication is unimportant. What is important is the squareness of the overall polycross nursery. In the use of the Latin square for the layout of a polycross, the shape of the replication is long and narrow. However, the overall nursery shape is square. In general, we retained the same number of rows and columns per replication in Table 4 as in Table 3 to obtain the lowest variance or ratio closest to one. However, occasionally a less square replication shape gave a lower variance or ratio closer to one. Therefore for some cells in Table 4 we adopted a less square replication shape. For example, for the cell for 10 replications (= 5 replication rows x 2 replication columns) of a 2 x 2 replication shape (4 clones), the overall nursery shape was (2 x 5) x (2 x 2) = 10 x 4, giving a ratio of 2.5 (= 10/4) and a variance _q(10)²=610.14x10^–6. Using the next less square replication shape of 1 x 4 the overall nursery shape was more square, giving a lower ratio of 1.6 and a lower variance of _q(10)²=463.35x10^–6.

The second element in every cell is a measure for closeness to random mating. It combines the variance due to pollination across all replications and the averaging of the cross frequencies across replications due to bulking. The measure declines as the number of replications increases and as the number of rows and/or columns increases, but does not decline monotonically as the number of clones increases.

Example 2.
We continue Example 1 of 16 clones. Because the variance of the cross frequency _q(b)² is dependent on the number of replications themselves, we can better answer the inverse question: what is the value of the measure D_(b), given the number of replications? The variance of the cross frequency _q(b)² for pollination across four replications from Table 4 is equal to 173.34 x 10^–6, and the variance of the mean cross frequency averaged across four replications is (173.34 x 10^–6)/4 = 43.34 x 10^–6. The value for the measure of the deviation from random mating is D₍₄₎=₍₄₎==6.58x10^–3=0.007, as given for the second element in the cell in Table 4. The value is less than 0.012 in Example 1 because ₍₄₎=6.58x10^–3<₍₁₎=11.69x10^–3 in Example 1.

The variance _q(b)² from this more extensive pattern of pollination across all replications decreases with an increase in the number of replications for all replication sizes (Table 4). To assess the relative gain from this more extensive pattern of pollination, we compared the overall mean of 216 [= (10 – 1) x 24] variances _(b)² based on b > 1 replication for pollination across all replications and the mean from bulking across b replications to the corresponding mean of 216 variances ₍₁₎² based on pollination within each replication only and the mean from bulking across b replications. The mean variance from pollination across all replications was only 30% of the mean variance resulting from pollination confined within each replication followed by bulking seed.

Many factors influence the number of replications in a polycross to achieve a desired level of nonrandom mating. We have not studied all of the factors in detail either singly or in combination. One of the major factors is the nature of the pollen dispersal function which, in turn, is influenced by many factors. Pollen is acted on by a vertical force, gravity, by a horizontal force, wind velocity, and by turbulence of the air (Gregory, 1945). Rising masses of heated air tend to carry pollen upward and thus increase the dispersal distance. Any factors such as higher wind velocities and higher buoyancy of pollen, which would extend the tail of the pollen dispersal function, would permit a smaller number of replications. Asymmetry with respect to wind direction or pollen dispersion would influence the number of replications, but we assumed a symmetrical distribution. The use of a marker gene to estimate the pollen dispersal function within a population integrated the combined effects of many of these factors. Other factors which influence the number of replications include the spacing of the grid, the size and shape of the replication and/or polycross nursery. Concerning the grid spacing, we used only one grid spacing of 0.91 m which is believed to be approximately equal to that used for most forage grasses. It is believed that a smaller grid distance would require fewer replications, but the functional relation has not been examined. The grid distance needs to be sufficiently great to avoid erroneous mixing of seed at harvest time. As the size or number of clones per replication increases, the required number of replications decreases. Since the minimum variance _q² from pollination per se occurs for a square, each nursery should approach a square as closely as possible. The shape of the individual replication is not important. A square nursery will minimize the number of replications for a given level of nonrandom mating. Concerning the extent of pollination we considered two cases: (i) an unrealistic assumption of pollination confined within each replication and (ii) pollination across all replications. The first one was considered for simplicity; it made the pollination pattern and the number of replications independent. The second one was more realistic and additionally was found to reduce the variance by 70%. Hence, fewer replications were required.

Pollination distances may become quite great for large replication sizes (large clone numbers) and many replications. Hence, some inaccuracy or lack of fit may exist, using our pollen dispersal function for these situations. We predicted far beyond the range of the original distance values.

The authors largely ignore the problem of different flowering times which generally exist. It is desirable to select clones which flower about the same time.

The polycross is most commonly used for selected clones (considered fixed effects in analysis of variance), but occasionally clones are a random sample from a population and estimates of variance components are desired (Aastveit and Aastveit, 1990). The bias in the among-polycross progeny variance component due to nonrandom mating compared to random mating can be calculated for specific cases. For example, from Table 4 for 100 clones and four replications (n = 100, b = 4), ₍₄₎²=_q(4)²/b=(25.69x10^–6)/4=6.42x10^–6. Then the ratio of C. for nonrandom mating to C. for random mating is equal to 1.000808. We assumed _A²=100 and _D²=24, so hence C_F=_A²+_D²=56 and C_H=_A²=25. The variance _.k of was assumed to be equal to zero. The upward bias of 0.08% is trivial, and can be ignored in this case.

	APPENDIX

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

 
Genetic Composition of the Variance Components among and within Polycross Progenies in Terms of Covariances of Relatives under Nonrandom Mating
Chorush (1970) derived the among and within genetic variance components in terms of expectations of effects for additive, dominance, and different types of epistasis. A more general approach is the derivation of those components in terms of covariances of relatives. Cockerham (1963) (see Nyquist, 1990, Chapters 9 and 10) expressed the variance components of many complex mating designs in terms of covariances of relatives. He wrote the model twice, once with no primes on the subscripts for the first relative and again with primes on the subscripts for the second relative, and then defined the different kinds of relatives in terms of the subscripts as follows:

j = j', k = k', full sibs (FS); j = k', k = j', reciprocal full sibs (RFS);

j = j', k k', maternal half sibs (MHS); j j', k = k', paternal half sibs (PHS); and

(j = k', k j'; j k', k = j') reciprocal half sibs (RHS).

Variance Component among Polycross Progenies
The basic approach is as follows. First, consider the covariance between two random individuals within the jth polycross progeny, and then, second, subtract the covariance of a random individual within the jth polycross progeny and another random individual in another random polycross progeny j'(j' j). The first covariance is a function of full sibs and maternal half sibs, and the second covariance is a function of reciprocal full sibs, reciprocal half sibs, and paternal half sibs. The desired covariance is the average of the differences summed over j.

To simplify notation in this Appendix we omitted the bar

in _jk, so q_jk=_jk=. Let us consider the first covariance C_(1)j. Two random individuals are full sibs with probability 1–q_jk² and are maternal half sibs with probability 1–q_jk² . Hence, the first covariance for polycross progeny j is

[A1]

The second covariance C_(2)j for polycross progeny j is the covariance between a random individual in polycross progeny j and a random individual in polycross j'(j' j). To be specific let j = 1 and create an n x n diagram for each of the other polycross progenies j' = 2, 3, ..., n, identifying the kind of relative with respect to each random individual in polycross progeny 1, and weighting each kind of relative by its frequency q_1kq_j'k for j' = 2, 3, ..., n. For j = 1 and j' = 2, we write the expression for the second covariance C₍₂₎₁₂ for a random individual from polycross progeny 1 and a random individual from polycross progeny 2, namely,

[A2]

Or, in general, for j and j'(j' j) we have

[A3]

We now desire to calculate the average covariance between a random individual in the jth polycross progeny and a random individual from any of the other (n – 1) polycross progenies. Thus we have

[A4]

We now reduce the covariance C_(1)j for two random individuals from polycross progeny j by the second covariance C_(2)j, or by the extent to which a random individual in polycross progeny j is related to a random individual from the other (n – 1) polycross progenies. So we have the desired covariance C_j for the jth polycross progeny

[(A5)]

Next we must average this adjusted covariance over j to obtain the desired covariance C. which equals the variance component for the variation among polycross progeny means. Thus, by using Eq. [A1] and [A4], Eq. [A5] becomes

[A6]

We now note that the last term in Eq. [A6] is

[A7]

Note that q_.k=q_ik. Eq. [A6] becomes

[A8]

Equation [A8] may be reparameterized, using

[A9]

[A10]

[A11]

and substituting the above in Eq. [A8], we obtain another form for Eq. [A8]

[A12]

Equation [A12] can be simplified if we assume C_FS = C_RFS = C_F and C_MHS = C_RHS = C_PHS = C_H. This first assumption (C_FS = C_RFS) also implies that . Then with some algebraic manipulation, we have

[A13]

Making the assumptions of disomic inheritance, noninbred parents, and no linkage, we have for t loci:

[A14]

[A15]

Variance Component within Polycross Progeny
Two approaches for the derivation of the variance component within polycross progenies are presented. In the first approach, any random offspring from the j x k cross in the jth polycross progeny is a random individual from a random-mating population because parent j and k are two random unrelated individuals from a random-mating population. Hence, the genetic covariance of such a random offspring with itself C_I is equal to the total genetic variance of a random-mating population. From this total covariance we need to subtract the average covariance C_(1)j of two random offspring from polycross progeny j or the extent to which two random offspring within polycross progeny j are related. Note that this covariance C_(1)j is unadjusted for the average covariance between polycross progeny j and the other (n – 1) j' polycross progenies. Hence, the within covariance or within variance component for the jth polycross progeny is

[(A16)]

Equation [A16] must be averaged over all n polycross progenies to obtain the desired within covariance C_w or within variance component. Thus, from Eq. [A16] and the first part of Eq. [A6]

[A17]

This is the general expression for the variance component for variation within polycross progenies in terms of covariances of relatives. We can re-express C_w in terms of the nonrandom mating parameter to obtain

[A18]

The second approach uses the idea that the within covariance is equal to the total covariance minus the among covariance C.

[A19]

It is evident from the above considerations that the covariance of an individual with itself C_I must be reduced by the average covariance of a random individual from a random polycross progeny and a random individual from the other (n – 1) polycross progenies which is equal to C₍₂₎. Hence,

[A20]

Substituting Eq. [A20] and C₍₁₎ – C₍₂₎ from Eq. [A6] in Eq. [A19], we obtain

[A21]

which agrees with Eq. [A17].

	ACKNOWLEDGMENTS

 
We extend our appreciation to Abdel-Moneium Elahmadi, who collected some of the heading and flowering data for this paper, and to Ira Chorush, who laid the foundation for nonrandom mating in a polycross.

	NOTES

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

 
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Received for publication June 15, 2006.

	REFERENCES

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION APPENDIX REFERENCES

 

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