Calculation of the number of discounted expressions

According to the definition given in the ‘Introduction’ section, discounted gene expressions of the genotypes of selected parents (boars or gilts) as a consequence of one mating can be described as the accumulation of parent genes in the progeny over generations and years at all production levels of a breeding or crossbreeding system. The number of genes expressing a trait in different parities of breeding females and their progeny and in different generations was discounted to the year of parent selection with a discount rate for the appropriate time interval. In pigs, a half-year period was applied as the time unit for the discounting based on the average length of the sow reproductive cycle. A flow diagram, called transition matrix P , was constructed to follow the movement of genes of selected parents through the pig populations over time. Its elements (p _ij ) give the probability that genes from the male and female parent sex and age class j are transmitted to the progeny sex and age class i during one time unit. Matrix P is a quadratic matrix that has a block structure containing different numbers of blocks depending on the breeding or crossbreeding system and on the mating type (artificial or natural mating). The general structure of matrix P is:

$$P{\equals}\left[ {\matrix{ {P_{{11}} } \hfill & {P_{{12}} } \hfill & {...} \hfill & {P_{{1n}} } \hfill \cr {P_{{21}} } \hfill & {P_{{22}} } \hfill & {...} \hfill & {P_{{2n}} } \hfill \cr \vdots \hfill & \vdots \hfill & \ddots \hfill & \vdots \hfill \cr {P_{{n1}} } \hfill & {P_{{n2}} } \hfill & {...} \hfill & {P_{{nn}} } \hfill \cr } } \right]$$

The blocks (submatrices) of matrix P represent the age classes of breeding males, females and slaughter animals in a purebred breeding system or these age classes in all links (with different breeds and crossbreeds) of a crossbreeding system. The submatrices that represent transmission of genes from sires to sires and dams to dams are of structure S ₁ (equation (1)). The submatrices representing transmission of genes from sires to dams and from dams to sires are of structure S ₂ (see equation (1)). The submatrices showing the transmission of genes from sires to boars that are used for natural mating in different links of a crossbreeding system and that do not produce further generations of sires and the submatrices for the transmission of genes from dams to crossbred females not reproducing themselves have the form S ₃ (equation (1)). The submatrices for the transmission of genes from sires and dams to slaughter animals of appropriate genotypes have the structure S ₄ with only one row (one age class, see equation (1)).

(1) $$\eqalignno{ S_{1} {\equals} & \left[ {\matrix{ x & x & x & \cdots & x & x \cr 1 & 0 & 0 & \cdots & 0 & 0 \cr 0 & 1 & 0 & \cdots & 0 & 0 \cr 0 & 0 & 1 & \cdots & 0 & 0 \cr 0 & 0 & 0 & \ddots & 0 & 0 \cr 0 & 0 & 0 & \cdots & 1 & 0 \cr } } \right]\,,\,\,S_{2} {\equals}\left[ {\matrix{ x & x & x & \cdots & x & x \cr 0 & 0 & 0 & \cdots & 0 & 0 \cr 0 & 0 & 0 & \cdots & 0 & 0 \cr 0 & 0 & 0 & \cdots & 0 & 0 \cr 0 & 0 & 0 & \ddots & 0 & 0 \cr 0 & 0 & 0 & \cdots & 0 & 0 \cr } } \right],\,\,\cr & S_{3} {\equals}\left[ {\matrix{ 0 & 0 & 0 & \cdots & 0 & 0 \cr 1 & 0 & 0 & \cdots & 0 & 0 \cr 0 & 1 & 0 & \cdots & 0 & 0 \cr 0 & 0 & 1 & \cdots & 0 & 0 \cr 0 & 0 & 0 & \ddots & 0 & 0 \cr 0 & 0 & 0 & \cdots & 1 & 0 \cr } } \right] \cr\cr & {\;S_{4} {\equals}\left[ {\matrix{ x \hfill & x \hfill & x \hfill & \cdots \hfill & x \hfill & x \hfill \cr } } \right]} $$

Each of the elements in the first row of submatrices S ₁ to S ₄ (designated x) is the proportion of genes in the sex–age class i at time t, which comes from sex–age class j at time t-1, where i refers to the row number of the matrix, j refers to the column number of the matrix and x assumes values between 0 and 0.5. Thus, the values of x depend on the contribution of each sex–age class of parents to the progeny gene pool. The sign x was used here in the submatrices S ₁ to S ₄ in light of generality, because which of the x values will have a zero or non-zero value depends on the parent selection strategy, the ages of females and males at the first mating and the number of animal sex–age classes in specific breeds or crossbreeds. The element 1 in the submatrices shows the aging of breeding animals. The concrete form of the transmission matrix P with concrete values of x for the evaluated three-way crossbreeding system, in which artificial insemination is assumed for all levels (in all links) of this system, is given in the Supplementary Material S1 as an example. The number of sex–age classes in the blocks of matrix P for breeding boars (ncS) was calculated as:

(2) $${\rm ncS}\,{\equals}\,1\,{\plus}\,{\mathop{\rm int}} \left( {{\rm lifeB}\,{\plus}\,0.5} \right)$$

where 1 indicates the first age class of sires, that is, young breeding boars before starting their use for mating, and the next number is the integer from the productive lifetime of breeding boars since their first mating to culling (lifeB), which is given as the number of sow reproductive cycles (i.e. the length of each age class is equal to the average length of the farrowing interval). The number of sex–age classes for dams of a breed or crossbreed (ncD) was calculated as:

(3) $${\rm ncD}\,{\equals}\,{\mathop{\rm int}} \left( {{{{\rm ageG1}} \over {{\rm fintS}}}} \right)\,{\plus}\,{\rm RR}$$

where the first term in the equation is the integer from the age of gilts of the appropriate breed or crossbreed at first farrowing (ageG1) expressed as the number of sow reproductive cycles (fintS is the average farrowing interval), and RR is the maximum number of reproductive cycles a sow can reach.

The non-zero elements in each row of matrix P must sum to 1. The sum of non-zero elements in the first row of submatrices for sires and dams must be 0.5 because sires and dams contribute to progeny with half of their genes. The values of the elements x in the first rows of submatrices for sires to sires and dams to sires can be established by program users as input parameters for gene flow. This will be the case when sires and dams for the production of the next generation of breeding boars are selected only from specific age classes of sires and dams in the particular breeding programme. Otherwise, these values are calculated in the same way as the elements x in the first rows of the remaining submatrices, namely from the age structure of the boar and sow populations for each breed and crossbreed, as described below.

The first element (p ₁₁ ) in the first row of the submatrices for sires to sires, sires to dams and sires to slaughter animals is always zero because boars are not used for mating when they are younger than 6 months. The value of the second element (p ₁₂ ) is half of the proportion of young boars in the boar population. Further values for the elements higher than two in these submatrices are assumed to be equal and are, therefore, calculated as:

(4) $$p_{{1(2{\plus}i)}} \,{\equals}\,\left( {{{0.5\,\hbox{-}\,p_{{12}} } \over {{\rm ncS}\,\hbox{-}\,2}}} \right)\quad for\quad i{\equals}1,\,\ldots\,\,,{\rm ncS}\hbox{-}2$$

where ncS has the same meaning as in equation (2). The values for the non-zero elements p _ij in the first row of the submatrices for dams to sires (for cases in which there is no selection of age classes for dams of sires), dams to dams and dams to slaughter animals are calculated as follows:

(5) $$\eqaligno{ P_{1( {\rm{acD}} - {\rm{RR}} +i)} = \left\{ {\matrix{\hskip-0pt{0.5\, \times \,pS_i \hskip 30pt\matrix{{ {\rm{\hskip 110pt dams\hbox{-}to\hbox{-}purebred\,\,animals}} }\hfill { {\rm{in\,\,purebreeding\hskip 15pt}} } \cr } {\rm{ }} } \cr {\matrix{\hskip-2pt{0.5\, \times \,pS_i } { \times \,{\displaystyle{{1\,-\, {\rm{pcross}} S_{\left( {i-1} \right)} } \over {\sum\limits_{i = 1}^{ {\rm{RR}} } {\left( {(1- {\rm{pcross}} S_{\left( {i-1} \right)} )\, \times \,pS_i } \right)} }}}} \cr } \quad\matrix{{ {\rm{dams\hbox{-}to\hbox{-}purebred \quad animals}} } { {\rm{in \quad crossbreeding}} } \cr } {\rm{ }} } \cr {\,\matrix{\hskip0pt{0.5\, \times \,pS_i } { \times {\displaystyle{{ {\rm{pcross}} S_{\left( {i-1} \right)} } \over {\sum\limits_{i = 1}^{ {\rm{RR}} } {\left( { {\rm{pcross}} S_{\left( {i-1} \right)} \, \times \,pS_i } \right)} }}}} \hfill \cr } \quad\matrix{{ \hskip 25pt{\rm{dams\hbox{-}to\hbox{-}crossbred \quad animals}} } { {\rm{in \quad crossbreeding\hskip 0pt}} }\hfill \cr } {\rm{ }} } \cr } } \right. \cr}$$

where pS _i is the proportion of sows in parity i (i=1,…RR), RR has the same meaning as in equation (3) and pcrossS _i is the fraction of sows crossed on each parity i (pcrossS ₀ is the fraction of crossed gilts). The term in the first line of equation (5) is valid in cases in which there are only purebred matings in the particular breed (e.g. all matings in the herds with the dam breed Czech Landrase, which is used in the sire position in the three-way crossbreeding system, are purebred matings). The terms in the second and third lines of equation (5) are valid in cases in which crossbred matings exist for sows of a particular breed. The values calculated according to the second line of equation (5) are the first elements of the first row of the submatrices for dams to purebred animals (sires, dams and slaughter animals), and those calculated according to the third line are the first elements of the first row of the submatrices for dams to crossbred animals (to crossbred dams and crossbred slaughter animals).

To calculate NDEs of a breeding animal selected for mating in the sex–age class n _sc , a vector m is defined with the same dimension as the dimension of matrix P . At time t=0 (time of selection of this breeding animal), all elements of vector m are zero, except the element n _sc , which has a value of 1. Then, the proportions of genes of the selected animal in each of the sex–age classes for all animal groups defined in matrix P at time t are calculated as:

(6) $$m^{t} \,{\equals}\,P^{t} \,\,{\times}\,m^{0} $$

Discounting the proportions of genes in each sex–age class of animals at time t to the time of parent selection with a discount rate u and summing these discounted proportions over the defined investment period inv, a vector msum is obtained with elements msum _j , which represents the NDEs of the selected animal in each sex–age class j of all animal groups defined in the appropriate breeding (crossbreeding) system:

(7) $$msum\,{\equals}\,\mathop{\sum}\limits_{t{\equals}1}^{{\rm inv}} {m^{t} \,\,{\times}\,\left( {1{\plus}u} \right)^{{{\minus}t}} } $$

Calculation of the absolute and relative economic weights for maternal and direct pig traits

According to Wolfová and Nitter (Reference Wolfová and Nitter2004), most traits can be considered to belong to one of the two groups in animal breeding. The first group occurs only once in the animal’s life (e.g. growth and carcass traits). The second group is realised by adult females several times in their life (e.g. fertility traits, but also survival rates of sows, determining the average length of the sow productive lifetime). Survival rates of sows are defined as the proportion of sows that survive to the end of the current reproductive cycle. Only some sows die or are culled during each reproductive cycle, and therefore, sow survival rate is realised (expressed) repeatedly during the lives of females. For simplicity, we designated the first group as direct traits and the second group as maternal traits. Some traits (e.g. growth of piglets until weaning, piglet survival) are affected by direct and maternal genetic effects and can be defined as a trait of the piglet (the direct component of a trait realised once by the offspring) as well as a trait of the sow (the maternal component realised several times during the female’s life). Many studies in the literature have estimated genetic parameters for both maternal and direct effects of piglet growth and piglet survival (Lundgren et al., Reference Lundgren, Canario, Grandinson, Lundeheim, Zumbach, Vangen and Rydhmer2010; Roehe et al., Reference Roehe, Shrestha, Mekkawy, Baxter, Knap, Smurthwaite, Jarvis, Lawrence and Edwards2010; Bouwman et al., Reference Bouwman, Bergsma, Duijvesteijn and Bijma2014). However, in some countries, genetic parameters for both effects of a trait are not available yet. For example, in the Czech Republic, piglet survival rates at birth and until weaning are treated as sow reproductive traits, whereas growth of piglets is treated as a piglet trait. Therefore, piglet survival rates were matched to maternal traits, and piglet growth was matched to direct traits in our study; however, we included two possibilities for the allocation of traits to both trait groups in our bioeconomic model and computer program. Each trait could be allocated to the maternal or direct trait group, or maternal and direct components of a trait could be defined. In the last case, separate genetic parameters had to be estimated for both trait components and inserted in the appropriate input parameter files of the program for calculation of the relative trait EWs.

For calculation of NDEs for the group of direct and maternal traits or trait components, we defined so-called realisation vectors hd and hm , respectively. A realisation vector describes the extent of the expression of a trait in the individual sex–age classes of animals, and therefore, both realisation vectors had the same dimensions as matrix P . Because MEVs of all traits (direct and maternal) were calculated per sow in the appropriate link of a breeding system in our bioeconomic model, both trait groups are realised only through sows. The realisation vector for direct traits contains non-zero elements only in columns that belong to the age class of slaughter animals (one age class for each involved breed and crossbreeds). These non-zero elements are equal to the fraction of sows of the appropriate breed and crossbreed from the total number of sows involved in the evaluated breeding (crossbreeding) system. In the realisation vector for maternal traits, the non-zero elements are in age classes of adult females of all involved breeds and crossbreeds according to age structure of sow herds in the appropriate links of the evaluated breeding system. The concrete forms of the vectors hd and hm for the three-way crossbreeding system are given in the Supplementary Material S2.

An additional step was calculation of the matrix hed with the elements hed _ij , where i values are rows of the matrix (each row belongs to a direct trait i) and j values are the matrix columns (their number is equal to the dimension of the vector hd ). The elements hed _ij were obtained by multiplying each jth elements of vector hd with the sum of the MEVs of the direct trait i expressed in purebred and crossbred progeny in all links of the crossbreeding system, taking into account the ratios of the number of sows in these links. Similarly, the elements hem _ij of the matrix hem for maternal trait i were calculated. The EWs for a direct and maternal trait i or for the direct and maternal components of trait i (EWd _i and EWm _i , respectively) were then calculated as:

(8) $$\eqalignno{ & {\rm EWd}_{i} {\equals}\,\mathop{\sum}\limits_{j{\equals}1}^{{\rm acm}} {{\rm hed}_{{ij}} \,\,{\times}\,{\rm msum}_{j} } \cr & {\rm EWm}_{i} {\equals}\,\mathop{\sum}\limits_{j{\equals}1}^{{\rm acm}} {{\rm hem}_{{ij}} \,\,{\times}\,{\rm msum}_{j} } $$

where acm is the dimension of matrix P , that is, the total number of sex–age classes of all animal groups involved in the specified breeding (crossbreeding) system, and msum _j are the elements of vector msum from equation (7). To compare the EWs of traits measured in different units, standardised EWs for direct (SEWd _i ) and maternal (SEWm _i ) traits i or trait components were calculated as follows:

(9) $$\eqalignno{ & {\rm SEWd}_{i} {\equals}\,{\rm EWd}_{i} {\times}{\rm gstdd}_{i} \cr & {\rm SEWm}_{i} {\equals}\,{\rm EWm}_{i} {\times}{\rm gstdm}_{i} $$

where gstdd _i and gstdm _i are the genetic standard deviations of the direct and maternal traits i or trait components, respectively. Because the standardised EWs have the same unit for all traits, they can be expressed as the percentage of the sum of all standardised EWs. Both positive and negative values can occur; therefore, it is useful to calculate the sum from the absolute values of the standardised EWs. The standardised EW of the direct (maternal) trait i or a direct (maternal) trait component expressed as the percentage of the sum of all standardised EWs for both direct and maternal traits or trait components (ewsum) are called relative EWs and are calculated as follows:

(10) $$\eqalignno{ & {\rm REWd}_{i} {\equals}\,100{\times}{{{\rm abs}({\rm SEWd}_{i} )} \over {{\rm ewsum}}} \cr & {\rm REWm}_{i} {\equals}\,100{\times}{{{\rm abs}({\rm SEWm}_{i} )} \over {{\rm ewsum}}} $$

where REWd _i and REWm _i are the relative EWs for direct and maternal trait i, respectively.

Economic weights for traits of breeds involved in the three-way crossbreeding system

The three-way crossbreeding system of the Czech National Programme for Pig Breeding with breeds Czech Large White (CLW), Czech Landrace (CL) and Pietrain (PN) was described in a previous paper by Krupa et al. (Reference Krupa, Krupová, Wolfová and Žáková2017). Therefore, only the parameters needed for the calculation of NDEs for the evaluated traits will be given here. The following traits were assumed to be direct traits: lifetime daily gain of finished animals, age of gilts at first mating, conception rate of gilts, survival rate of piglets in the nursery, survival rate of young animals after nursery, dressing percentage and lean meat content of finished animals and feed conversion of pigs in the nursery and in finishing. To the group of maternal traits, the following traits were assigned: number of piglets born alive, survival rate of piglets at birth and until weaning, conception rate of sows and productive lifetime of sows. For trait definitions, see Wolfová et al. (Reference Wolfová, Wolf, Krupová, Krupa and Žáková2017).

In the Czech National Programme for Pig Breeding, sires and dams to produce the next generation of breeding boars are selected from all age groups of boars and sows according to the estimated breeding values of animals. Therefore, the number of sex–age classes in matrix P was deduced from the age structure of boar and sow populations in each breed and crossbreed, which was calculated by Krupa et al. (Reference Krupa, Krupová, Wolfová and Žáková2017). The numbers of classes with further variables needed for the construction of matrix P and of realisation vectors hm and hd and for the calculation of NDEs for the groups of direct and maternal traits in each breed are given in Table 1. For slaughter animals, five age classes were defined, namely one class for each of the purebred slaughter animals of breeds CLW, CL and PN and one class for each of the crossbred slaughter animals of genotypes CLW×CL and (CLW×CL)×PN. Because artificial insemination is assumed for purebreeding as well as for crossbreeding, insemination doses of a boar can be used for mating in different links of the crossbreeding system. Therefore, only one group of boars in each breed was defined. Four groups of sows were involved: sows of breeds CLW, CL and PN and of the crossbreed CLW×CL. Thus, using the numbers of sire and dam age-classes given in Table 1, the total number of sex–age classes in the evaluated three-way crossbreeding system (i.e. the dimension acm of matrix P ) was 70. An investment period of 8 year was chosen because the relative NDEs for direct and maternal traits in pigs are expected to stabilise at about 8 years (Wolfová and Nitter, Reference Wolfová and Nitter2004).

Table 1

Parameters used for calculation of the number of discounted gene expressions (NDEs) for maternal and direct traits for specific breeds of pigs

Breeds: CLW=Czech Large White; CL=Czech Landrace; PN=Pietrain.

¹ These values represent the double values of variable p ₁₂ in equation (4) for the appropriate breed.

² These values divided by 100 represent the variables pdA, pdB, pdC and pdAB, respectively, which are used in the Supplementary Material S2.

To determine the importance of using the gene flow procedure for estimation of the EWs of pig traits in crossbreeding systems, the absolute approximate EWs of traits calculated by Krupa et al. (Reference Krupa, Krupová, Wolfová and Žáková2017) for the same three-way crossbreeding system were compared with EWs of traits calculated here. The approximate trait EWs were originally calculated per sow of each breed; therefore, they were first converted to be expressed per the same base as was used in this study, namely per sow in the whole crossbreeding system. Then, the relative approximate EWs were calculated by applying equation (10). Both the approximate absolute and approximate relative EWs were compared with the absolute EWs and relative EWs calculated here.