Most farmers and hatchery managers cannot calculate individual inbreeding values for fish that they culture, because individual fish cannot be identified and because pedigrees are not recorded. However, this does not mean that inbreeding values cannot be determined for hatchery fish. Even though individual inbreeding values cannot be calculated, population averages can be and should be determined. Average inbreeding values should be determined for all hatchery populations every generation, and the acquisition and use of these data should be an integral part of hatchery management. The average inbreeding value is as important as average harvest weight or any other data that can be gathered.
The average inbreeding value of a population is determined from the population's effective breeding number (Ne). Effective breeding number is one of the most important bits of information that can be gathered about a hatchery population, and Ne should be determined every generation for all hatchery populations.
The first part of this chapter will define Ne, will describe what factors influence Ne, and will then show how Ne is calculated. Effective breeding number is traditionally calculated by counting the number of males and number of females that produce viable offspring. This direct approach will be the technique described in this chapter.
Recently, protocols have been developed to assess Ne indirectly by examining changes in gene frequency and by assessing something called “linkage disequilibrium,” which is the difference between the actual and expected co-occurrence of two alleles at two loci. In some cases, these indirect procedures may provide a more accurate estimate of Ne, because they account for differential reproduction and offspring survival, which can have a marked impact on Ne. However, these techniques are more expensive because they require detailed biochemical work and they also require highly trained scientists who are skilled at biochemical population genetics, requirements that are not available at most farms or hatcheries.
Consequently, the traditional approach of enumerating the males and females that spawn and that produce viable offspring will be the only technique that will be described in this chapter. This approach is easy and the math is relatively simple. Those who wish to learn about the indirect procedures will find papers that describe these techniques in Recommended Reading.
The second part of the chapter will show how Ne is used to determine the average inbreeding value of a hatchery population.
The techniques outlined in this chapter are fairly simple. Subsistence-level farmers do not need to know how to determine Ne and average inbreeding, but any farmer or hatchery manager who wants to increase yields and productivity or to manage populations that are being cultured in order to restock lakes and rivers should learn how to determine these values and should incorporate these protocols into yearly work plans.
Effective breeding number and the average inbreeding value are important population descriptors, because they help explain trends in yield, fecundity, and other important production phenotypes. These values also enable a farmer to predict if problems will occur as a result of inbreeding depression or the loss of genetic variance. Proper brood stock management cannot be accomplished without this information.
The information generated in this chapter will be used in Chapter 5 to show how Ne affects genetic drift-random changes in gene frequency-and in Chapters 7 and 8 which will describe techniques that can be used to manage Ne and inbreeding.
The Ne of a hatchery population is one of the most important bits of information about the population. Unfortunately, most hatchery managers do not know what Ne is, do not know how to determine Ne, and do not know how Ne influences inbreeding and brood stock management. If asked to describe the size of their hatchery population, most farmers or hatchery managers would produce a census or an approximate number. Those with good records would probably be able to give the number of male and female brood fish. This information is used to determine the number of fingerlings that can be produced, to calculate the amount of feed that must be ordered, or to estimate yield. As important as this information is, it does not describe the population genetically. In order to describe a population genetically, one must determine Ne.
Effective breeding number is one of the most important concepts in brood stock management, because it gives an indication about the genetic stability or genetic health of the population. This is because Ne is inversely related to inbreeding and to genetic drift. The relationship between Ne and inbreeding will be described later in this chapter. That with genetic drift will be described in Chapter 5.
If hatchery populations were infinitely large, an understanding of Ne would be unnecessary. However, hatchery populations are usually small and are often closed. A closed population is one where immigration (the introduction of fish from another population) is not allowed; consequently, fish from other populations are not allowed to mate with or hybridize with fish from a closed population. Hatchery managers often maintain closed populations for various reasons; chief among them is the desire to minimize health problems by preventing the introduction of diseases that often accompany acquired fish.
When working with a closed, finite population, the best way to describe it is not by total number of fish, but by Ne. Effective breeding number is determined by the number of male and female brood fish that produce viable offspring, the sex ratio of the brood fish that spawned, the variance of family size, and the mating system that is used.
In most situations where fish cannot be identified and where mating is random (fish are paired without regard to phenotypic value, or fish swim free in a pond and choose their own mates), Ne can be determined by using the following formula:
where: number of males and number of females are the number of male and female brood fish that produce viable offspring. If all offspring for a brood fish die, that male or female is not included when determining Ne.
If matings and offspring production cannot be monitored, Ne cannot be determined. If fish spawn in ponds and eggs are allowed to hatch in the ponds and if offspring are not harvested until they are mixed schools of fry or fingerlings, Ne will be difficult, if not impossible, to determine.
The formula shows that Ne is determined both by the number of male and female brood fish and by the sex ratio. For example, if 53 female brood fish produced eggs and if 25 males were used to fertilize those eggs and if all brood fish produced viable offspring, Ne is:
Ne = 67.95
This example illustrates a fundamental concept of brood stock management: The genetic size of the population (Ne) and the number of fish that produce offspring are not always the same; the genetic size is usually smaller. Seventy-eight brood fish produced offspring, but Ne was only 67.95. The reason Ne was smaller is because the sex ratio was skewed; in this example, the sex ratio was 2.12 females: 1 male (53 females:25 males).
Effective breeding number and the number of brood fish that produce offspring will be the same only when the sex ratio is 1:1. In the above example, there were 78 total brood fish, so a 1:1 sex ratio would have been 39 females:39 males. If these numbers were used to produce offspring, Ne would have been:
Ne = 78
Figure 17 shows the effect of number of males and females and of the sex ratio in determining Ne. The information presented in Figure 17 clearly shows that the best way to increase Ne is to increase the number of males and the number of females that spawn and produce viable offspring and to bring the sex ratio closer to a 1:1 ratio. Increasing one sex while keeping the other sex at a fixed number has little effect on increasing Ne after a certain point; this can be seen by examining the curves for 1, 2, 10, and 25 males in Figure 17.
Unequal reproductive success can affect Ne. If some fish produce thousands of viable offspring while others produce only dozens, Ne will be less than expected. Variance in family size is a major reason why Ne is often smaller than expected. Variance in family size effects Ne as follows: When there are single pair matings, variance in family size is equal for both sexes, and Ne becomes:
Where: NeUR is effective breeding number when there is unequal reproductive success and variance in offspring production is variance in family size.
Figure 17. Effective breeding numbers produced by various combinations of males and females.
From: Tave, D. 1993. Genetics for Fish Hatchery Managers, 2nd ed. Van Nostrand Reinhold, New York, New York, USA.
For example, if Ne in a population is 100, there are single pair matings (50 males and 50 females), and the variance in family size (variance in offspring production) is 5, Ne becomes:
NeUR = 57.14
The variance in family size reduced predicted Ne by 43%, and yielded an effective breeding population of just 28 males and 28 females.
When males mate with more than one female and/or females mate with more than one male, there is unequal reproductive success for the two sexes, and Ne becomes:
Family size often assumes what statisticians call a “Poisson distribution,” and when this occurs the mean and the variance are the same. If this occurs, mean family size can be used.
The Ne that has been calculated in the previous examples is that for a single generation. It is important to determine Ne for every generation, because the values are not independent. The average Ne for a series of generations is determined by calculating the harmonic mean:
where: Ne mean is the mean effective breeding number over t generations; and Ne1, Ne2, and Net are the effective breeding numbers in generations 1, 2, and t, respectively. The math needed to determine mean Ne is fairly simple and can be done “by hand”; however, inexpensive hand-held calculators are preprogrammed to determine inverses (the “1/x” key) so their use reduces this formula to simple arithmetic.
Because the mean is determined by the harmonic average, the generation with the smallest Ne will have a controlling influence on average Ne. This means that if a disease or other calamity reduces Ne, subsequent increases in Ne will have little effect on increasing the average.
For example, if the Ne's for five generations are 50, 10, 65, 85, and 100, the mean Ne is:
Ne mean = 31.82
In this example, Ne went below 50 only once, but mean Ne was 31.82 because Ne dropped to 10 for a single generation. Even though Ne was 85 and 100 in the final two generations, if just 16 males and 16 females had been spawned each of the five generations, mean Ne would have been slightly larger (Ne mean = 32).
This example illustrates one of the most important concepts of brood stock management: Ne must be managed each and every generation. The population's Ne must be considered to be one of its most important descriptors. If Ne declines to a low level for just a single generation, this will have a devastating impact on mean Ne, even if Ne is maintained at high levels before and after the single-generation reduction. When a harmonic mean is used to determine the mean, the generation with the lowest value has a disproportionate effect on the mean. The implications of this on the genetic health of a hatchery populations will be discussed in Chapter 7.
Once Ne has been determined, a simple formula can be used to calculate the average inbreeding value in the population:
where F is the average inbreeding value in the population. For example, if Ne = 100, the average inbreeding value for the fish in the population is:
F=0.005
This value is assigned to every offspring produced by those brood fish.
The formula shows that F is inversely related to Ne: a large Ne produces a small F; a small Ne produces a large F. Figure 18 shows how much inbreeding will be produced by various Ne's for a single generation. The relationship between F and Ne in this formula clearly demonstrates why it is important to calculate Ne and why management of Ne is of paramount importance in the management of hatchery populations.
Figure 18. The relationship between effective breeding number (Ne) and average inbreeding (F) in a population. This is the average inbreeding produced by a single generation of mating.
From: Tave, D. 1990. Effective breeding number and broodstock management: I. How to minimize inbreeding. Pages 27–38 in R.O. Smitherman and D. Tave, eds. Proceedings Auburn Symposium on Fisheries and Aquaculture. Alabama Agricultural Experiment Station, Auburn University, Alabama, USA.
The inbreeding that is calculated in this manner is genetically identical to that which was determined in Chapter 3; it is a measure of the increase in homozygosity that occurred as a result of the mating of relatives. The only difference is: when individual inbreeding values are determined, the values are customized for each individual, based on its family pedigree; when Ne is used to determine inbreeding, the value that is calculated is the population average. Consequently, some fish will be more inbred than the average while others will be less inbred, and there is no easy and inexpensive way to determine which fish are above the mean and which are below. But if pedigrees do not exist, this is the only way to produce an estimate of the inbreeding that has accumulated in the population.
The reason why F and Ne are inversely related is that chance encounters between relatives increase in small populations, and inbreeding is the mating of relatives. When mating is random, closed populations produce consanguineous matings by chance, and the probability of consanguineous matings is inversely related to Ne. For example, if a fish has 50 relatives in a population of 1,000,000, the odds of that fish mating with a relative (assuming there is a 50:50 sex ratio) is 25/500,000 or 0.00005 if mates are chosen at random without regard to pedigree. If a fish has 50 relatives in a population of 1,000 fish, the odds of mating with a relative jumps to 25/500 or 0.05, which is 1,000 times greater. Consequently, the probability of mating with a relative and of producing inbred offspring increases as Ne decreases.
Once inbreeding is produced in a small closed population as a result of a small Ne, the inbreeding itself lowers future Ne in a positive feedback cycle:
where: NeF is the effective breeding number in an inbred population, and F is the average inbreeding in the population.
The positive feedback illustrated in this formula means that reductions in Ne increase inbreeding, which in turn lowers Ne, which in turn increases inbreeding, etc. Because it is a positive feedback, the system takes a bad situation and continually makes it worse.
Farmers and hatchery managers should determine the Ne of a hatchery population every generation. A population's Ne is one of the most important pieces of information about that population because it describes its genetic size. An understanding of Ne and what it means is of major importance because Ne helps determine the genetic stability of the population in that it determines the average inbreeding in the population and influences genetic drift. Once Ne has been quantified, it can be used to determine the average inbreeding value in the population. The link between Ne and genetic drift will be discussed in Chapter 5.
The average inbreeding value in the population is inversely related to Ne, which means that inbreeding can become a problem in small, closed populations, which are typical of those found at most farms and fish culture stations. Once inbreeding occurs, the problem can quickly become worse as inbreeding and Ne are linked in a positive feedback loop. Consequently, small Ne's will produce levels of inbreeding which will cause growth rate, fecundity, and other traits to decline over succeeding generations.
It is important to understand what Ne is, how it is determined, and how it affects inbreeding so that it can be managed to prevent unwanted inbreeding from decreasing productivity and profits. Techniques that can be used to manage Ne and thus prevent inbreeding from reaching levels that cause problems will be discussed in Chapter 7, and recommended Ne's that should be maintained to prevent inbreeding from reaching levels that cause problems will be presented in Chapter 8.
