Thursday, 16 January 2014

Managing risk in animal breeding schemes

Animal breeding is not exact science in the sense that normally one cannot exactly predict the outcome, or even select the "ingredients". Each gamete (an egg cell or a sperm) is different, and their combination and further cellular divisions all include an effect of randomness. So each breeding scheme has risks. This post will address some of those risks and how to minimize their impact.


Breeding always requires some inbreeding. This is because we want to increase the genes from one or few excellent animals, so we use them for males/females of several generations. Consider horse racing and show jumping: it's common to list the famous parents, siblings, half-sibs and grandparents of any horse to prove its value.

The change of inbreeding can be calculated as
ΔF= 1 / 2Ne
where Ne is the effective population size. If the pnumber of parents of different sexes isn't equal, then we estimate

ΔF ≈ (1 / 8Nm) + (1 / 8Nf)

Inbreeding works in two ways: inbreeding increases variance between lines/populations, but decreases variance among a line/population. Remember that inbreeding depression, the negative effect of inbreeding on genetic diversity, can be negated by breeding two animals of completely different lines.

Genomic selection versus progeny testing

Both agenomic selection schemes (GS) and progeny testing schemes (PT) have their own risks. Professors Alban Bucket  and Jarmo Juga from the University of Helsinki have studied the risks in bovines. They state that in GS schemes the rate of inbreeding is slightly higher than in PT schemes, but reciprocally the genetic response is much higher in GS than in PT. The choice becomes a matter of balancing the risks. How high of an inbreeding level do we accept to get strong genetic response? 

Bucket and Juga state that if the amount of sires is not increased, the risk is comparable between GS and PT schemes. The risk in GS can be further minimized by increasing the amount of MOET (multiple ovulation, embryo transfer) and the number of genotyped females.This increases the genetic diversity and allows effective Mendelian variance. However, increasing the amount of AI bulls in a GS scheme increases the risks of inbreeding.

Preserving genetic diversity

As has been stated earlier, selection and inbreeding impact genetic diversity in two ways: the variance between lines increases, while the variance within lines decreases. If a line equals a breed, the impact can be very strong.One example can be found from the study by Uimari and Tapio, who studied how the effective population size has changed over generations in two pig breeds. During 50 generations, selection has decreased the effective population size from 600 to a mere 50. The decrease is simply due to breeding selection.

The impact of selection to the Ne of two pig breeds.
(c) Uimari and Tapio

Maintaining genetic diversity should be duly considered in every breeding scheme. By genotyping a large amount of animals it is possible to ensure diversity by pairing unrelated animals. By genotyping one can also ensure that rare alleles stay in the population, and that there is enough heterozygozity. These two go often hand in hand: rare alleles are found most often in heterozygotes than in homozygotes. By genotyping one can also preserve traits of specific interest and genomically control the level of inbreeding.

FAO, The Food and Agriculture Organization, has created a simple chart about preserving genetic diversity. The chart is part of their publication considering The State of the Worlds ANGR for Food and Agriculture (ANGR = Animal genetic resources). It shows that the actions required are rather simple. Because really - 
all it takes is the courage to look beyond monetary gain and efficiency.

Monday, 13 January 2014

Calculating breeding values

Basics of animal breeding have been covered earlier in this blog. We've discussed the very basics of animal breeding as well as the  Mathematics of animal breeding . Optimization of animal breeding schemes has also been briefly considered. Today we take a closer look at calculating the breeding value using statistical concepts and information from various sources.

Estimated Breeding Value

When calculating EBV (estimated breeding value) for an animal, we usually want to combine information from various sources. We have results from the animal itself, but also from its relatives. However, EBV is always for one trait only.

The formula for an EBV is

 = b1x1 + b2x2 + ... bnxn

where  is the EBV, b1 is the regression coefficient for trait 1 and x1 is the result for trait 1. Seems simple, doesn't it? Now all we need to do is calculate the b values. The key to the b values is to remember where dealing with several bits of information at once, and every "bit" is actually an equation

 Â = (y1 - μ) = a1 + e1

where  is the EBV, y1 is the animal's own result in trait 1, μ is the population mean result in trait 1, a1 is the additive genetic effects contributing to the trait and e denotes the environmental factors contributing to the trait. Simply put: an EBV consists of genetics and environmental factors.

So, the b's must fulfill equations for traits 1 ... n at the same time. Instead of a group of equations we use matrices to calculate the b's. Only then can we continue to calculating the actual selection index. The matrix notation for calculating b's is
b = P-1G

Here P and G are matrices. P includes the variances and covariances between phenotypic results. The marking P-1 simply means that the transpose of the matrix P is used in the calculation. G is another matrix, which links the information sources to true breeding values. In the G matrix Ai denotes the true breeding value of animal i. Info1 and Info2 below are different information sources, for example animals 1 and 2.

You might remember that we never know the actual breeding value A, so we always work with the estimated BV, denoted as Â. However, with enough information and correct calculations it is assumed that A = Â. The trick in matrix G is to consider the genetic relationships between the information sources, here animals 1 and 2. If Info 1 is the animal itself, so that the source for info 1 = i, then Cov(Info 1, Ai) = Var(Ai). More generally,

Cov(Info x, Ay) = a(x,y) * Var(A)


Var(A) = h2 * s.d. (P)

where a(x,y) is the coefficient of genetic relationship between animals x and y. If x and y are full siblings, their coefficient of genetic relationship is 0,5, and Cov(Info x, Ay) = 0,5 Var(A). If they're half-sibs, it's Cov(Info x, Ay) = 0,25 Var(A) and so on. s.d. (P) is the standard deviation of the trait P, or the trait for which we are calculating the breeding value for. Standard deviation of P is the square root of the variance of P.

Selection index and economic breeding value

So now we can calculate the EBV for one trait. What if we want to combine several traits into one number? Then we need a selection index. It works like EBV, but combines information from several sources and several traits into one.

A selection index can either be optimal or common. The difference is in the coefficients: optimal coefficients minimize the variance between true breeding values and estimated breeding values. In a common selection index the coefficient b is said to be "any b0", but in the optimal index b = P-1Cv. The optimal index considers covariances, and breeding accuracies impact the b coefficients.

Here we can see a new matrix, C. C is used if the measured traits are not the same as the traits to be improved. For example, we might measure weight and thickness of back fat, but we want to improve weights and percentage of lean meat. Now we need the matrix C, which relates to the other matrices as shown in the picture below.

If we want to include money to the calculations, we get a total breeding value. Money is used in breeding values to give economical weights to each trait. This weight is entirely decided by animal breeders, and based on what they think is most important. Economical weight isn't linked to genetics or phenotype in any way. It is simply a way to put the traits into some order of importance.Often economical values is derived from actual profits or costs regarding the trait in question. The weight is currency per 1 unit of increase/decrease in the trait, for example euros per +- 1 kg of meat or dollars per +- 1 weaned piglet. The economical value can be used to ompare the costs and profits between different breeding schemes.

(c) Wikipedia Commons
For example:
We have two schemes for pig breeding. One scheme gives us - 0,5 piglets per sow, but 10 kg more meat since the surviving piglets are heavier. The other scheme gives + 0,7 piglets but -6 kg meat.  Let us assume that 1 kg of meat is +10 euros and 1 piglet is 15 euros.

Now the first scheme yields (-0,5 * 15) + (10*10) = 92,5 euros, and the second scheme (0,7 * 15) + (-6 * 10) = -49,5 euros. With these exaggerated numbers it is easy to see which scheme would be more profitable for the producer.

Economic weight can also be used when restricting a selection index. We may want to improve one trait, but leave another trait untouched. In that case the economic value of the trait, which is not allowed to change, is set to 0,

Total breeding value

Using indices and economic breeding values we can calculate a total breeding value for an animal. The formulas are

H = v'g
I = b'x where b = P-1Gv

 H = total breeding value, estimated using the index I
v = economical weights of traits
g = breeding values of traits
I = selection index
b = regression coefficients for traits
x = vector of observations [result1    result2    resultn]
P = covariances and variances between observations
G = covariances and variances between traits to be improved and measured traits.

Additional information and sources

Mrode, R. A. Linear Models for the Prediction of Animal Breeding Values, 2nd edition. CABI Publishing, USA. ISBN-13: 978-085199-000-2

Cameron, N. D. Selection Indices and Prediction of Genetic Merit in Animal Breeding. CAB International, USA. ISBN-13: 978-085199-169-6

GenUp-software for playing with genetics: