Breeding without breeding: minimum fingerprinting effort with respect to the effective population size

Abstract

We present a probabilistic model to minimize the fingerprinting effort associated with the implementation of the “breeding without breeding” scheme under partial pedigree reconstruction. Our approach is directed at achieving a declared target population’s minimum effective population size (N _e ), following the pedigree reconstruction and genotypic selection and is based on the graph theory algorithm. The primary advantage of the proposed method is to reduce the cost associated with fingerprinting before the implementation of the pedigree reconstruction for seed parent–offspring derived from breeding arboreta and production or natural populations. Stochastic simulation was conducted to test the method’s efficiency assuming a simple polygenic model and a single trait. Hypothetical population consisted of 30 parental trees that were paired at random (selfing excluded), resulting in 600 individuals (potential candidates for forwards selection). The male parentage was assumed initially unknown. The model was used to estimate the minimum genotyping sample size needed to reaching the prescribed N _e . Results were compared with the known pedigree data. The model was successful in revealing the true relationship pattern over the whole range of N _e . Two to three offspring entered genotyping to meet the N _e  = 2 while 41 to 43 were required to satisfy the N _e  = 14. Importantly, genetic gain was affected at the lower limits of the genotyping effort. Doubling the number of parents resulted in considerable reduction of the genotyping effort at higher N _e values.

Appendix

ad(iii):

Probabilities are estimated by the Monte Carlo method:

Algorithm M

Let the number of all outcomes be denoted as A and that of all successful outcomes as S.

(i _M ):

Set $A\leftarrow 0$, $S \leftarrow 0$.

(ii _M ):

$A \leftarrow A + 1$

Assign randomly male parents to offspring, so that i ^th offspring is sired by a male parent y with a probability p _i,y.

(iii _M ):

Find the biggest subset of offspring M with all parents different. {algorithm MinRedl2}

If $\vert M \vert \geqq N$, increase $S \leftarrow S + 1$.

(iv _M ):

If $A \geqq o_{\max}$ or

if P is outside the confidence interval logit(S, A, α), {we reject H ₀ : p = P} return result $\hat{p}= S/A$, else go to (ii _M ).

ad(iii _M ):

The problem of finding the largest subset of offspring with different parents was converted to the problem of finding a maximum matching in a general graph, a well-developed subject in graph theory (Agnarsson and Greenlaw 2008).

As a graph, we define a pair (V, E) containing a set of vertices V and edges E connecting pairs of vertices. In our settings, the vertices are declared as parents while the edges are respective offspring. A pairing M is a subset of E where no two edges share common vertex. Vertices sharing an edge with a vertex V will be denoted as its neighbors.

For finding maximum matchings in graphs, we utilized the fast approximation algorithm MinRed12 (MR) by Magun (1998).

(i _MR ):

$\vert M \vert \leftarrow 0$

(ii _MR ):

Remove all vertices without edges and all duplicate edges; consider vertex v with the lowest number of neighbors Δ _v .

Note: This vertex is always removed. Along with the vertex, potential neighbor is removed as well (this is determined by the number of neighbors):

(iii _MR ):

If Δ _v  = 1, remove v and its neighbor w,

if Δ _v  = 2, remove v and merge its neighbors w ₁ and w ₂, else remove w with the lowest number of neighbors Δ _w .

Note: While the first case (Δ _v  = 1) is obvious, the second case (Δ _v  = 2) is questionable (which one should be removed). As one vertex must always be removed, the two are simply merged. We are not interested at the exact form of such a pairing. It is sufficient to declare the number of paired vertices. If Δ _v  > 2, a heuristic approach is used (this is the only approximation step in the algorithm). Let us choose a particular neighbor w out of all potential neighbors v so that it has the fewest number of neighbors. This one is removed. The general idea is that vertices with the lowest number of neighbors interfere the least to the pairing among other vertices. Details are provided in Magun (1998).

(iv _MR ):

$\vert M \vert \leftarrow \vert M \vert + 1$ {Add edge vw to pairing}

If $\vert M \vert \geqq N$, finish.

If there are any more edges in the graph, go to (ii _MR ).

Note: We are testing a null hypothesis that $\hat{p} = P$ against the alternative that $\hat{p}\ne P$. If H ₀ is rejected, we can stop testing the current set. As a result, an outcome of this algorithm must always satisfy this testing, irrespective of its exact score value.

For hypothesis testing, we will use the logit interval, based on the approximation of binomial distribution of logit functions. Such a function is used in the next step with the exception that the exact binomial interval is used for X = 0, 1,Z − 1,Z. If $\hat{p}$ is inside this interval, we are not rejecting the hypothesis.

ad(iv _M ):

To test, whether the probability is estimated accurately, an interval logit(X, Z, α) is used in the algorithm, where X = number of successful tests, Z = number of all tests, and α is the probability of the type I error. The value c = − 0.5 is taken from Edwardes (1998).

Set

$$\begin{array}{rll} x&=&X-c,\quad z=Z-2c, \\ \varphi &=& \text{exp} \bigg( \Phi (1-\alpha / 2) \sqrt{\frac{z}{x(z-x)}}\bigg), \end{array}$$

where Φ is the cumulative distribution function for the normal distribution. Then

$$\begin{array}{rll} &&\text{logit}(X,Z,\alpha) \\&& = \left\{ \begin{array}{rl} \Big\langle 0,1-\sqrt[Z]{\alpha/2} \Big\rangle & \text{if } X=0 \\ \Big\langle 1-\sqrt[Z]{1-\alpha}, \displaystyle\frac{x}{x+(z-x)/\varphi} \Big\rangle & \text{if } X=1 \\ \Big\langle \displaystyle\frac{x}{x+(z-x)\varphi},\sqrt[Z]{1-\alpha} \Big\rangle & \text{if } X\!=\!Z\!-\!1 \\ \Big\langle \sqrt[Z]{\alpha/2},1 \Big\rangle & \text{if } X=Z \\ \Big\langle \displaystyle\frac{x}{x+(z-x)\varphi},\displaystyle\frac{x}{x+(z-x)/\varphi} \Big\rangle & \text{else.} \end{array} \right. \end{array} $$

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