Computationally efficient map construction in the presence of segregation distortion

Abstract

Key message

We present a novel estimator for map construction in the presence of segregation distortion which is highly computationally efficient. For multi-parental designs this estimator outperforms methods that do not account for segregation distortion, at no extra computational cost.

Abstract

Inclusion of genetic markers exhibiting segregation distortion in a linkage map can result in biased estimates of genetic distance and distortion of map positions. Removal of distorted markers is hence a typical filtering criterion; however, this may result in exclusion of biologically interesting regions of the genome such as introgressions and translocations. Estimation of additional parameters characterizing the distortion is computationally slow, as it relies on estimation via the Expectation Maximization algorithm or a higher dimensional numerical optimisation. We propose a robust M-estimator (RM) capable of handling tens of thousands of distorted markers from a single linkage group. We show via simulation that for multi-parental designs the RM estimator can perform much better than uncorrected estimation, at no extra computational cost. We then apply the RM estimator to chromosome 2B in wheat in a multi-parent population segregating for the Sr36 introgression, a known transmission distorter. The resulting map contains over 700 markers, and is consistent with maps constructed from crosses which do not exhibit segregation distortion.

Appendix

Proof of Eq. 9

$$\begin{aligned} {\mathbb E} ( s_{xz} )&=\frac{p_{xaz}}{4 p_{.a.}} + \frac{p_{x h z}}{2 p_{.h.}} + \frac{p_{x b z}}{4 p_{.b.}} + \mathbb E ( \epsilon _{x z} )\\&= \frac{1}{4} \mathbb P_d (M_1 = x, M_3 = z \vert M_2 = a ) + \frac{1}{2}\mathbb P_d (M_1 = x, M_3 = z \vert M_2 = h )+ \frac{1}{4} \mathbb P_d (M_1 = x, M_3 = z \vert M_2 = b ) + {\mathbb E} (\epsilon _{x z} )\\&=\frac{1}{4} \mathbb P_u (M_1 = x, M_3 = z \vert M_2 = a ) + \frac{1}{2}\mathbb P_u (M_1 = x, M_3 = z \vert M_2 = h )+ \frac{1}{4} \mathbb P_u (M_1 = x, M_3 = z \vert M_2 = b )+{\mathbb E}(\epsilon _{xz})\\&= \mathbb P_u (M_1 = x, M_2 = a, M_3 = z ) + \mathbb P_u(M_1 = x, M_2 = h, M_3 = z )+ \mathbb P_u (M_1 = x, M_2 = b, M_3 = z )+{\mathbb E}(\epsilon _{x z})\\&= \mathbb P_u (M_1 = x, M_3 = z )+{\mathbb E} (\epsilon _{x z} ) \end{aligned}$$

Proof of Eq. 14

It is not true that

$$\begin{aligned} \mathbb P_{f,d} (M_1 = x, M_3 = z\vert M_2 \ne A) = \mathbb P_{f,u} (M_1 = x, M_3 = z\vert M_2 \ne A) \end{aligned}$$

However if we take $p_f = |F|^{-1}$ we can rewrite

$$\begin{aligned} \sum _{f \in F} | F |^{-1} \mathbb P_{f, d}( M_1 = x, M_3 = z \vert M_2 \ne A) \end{aligned}$$

as

$$\begin{aligned} | F |^{-1} \sum _{f \in F} \sum _{B \le y \le H} \mathbb P_{f, u}( M_1 = x, M_3 = z \vert M_2 = y)\mathbb P_{f, d}( M_2 = y \vert M_2 \ne A) \end{aligned}$$

We now make the approximation that $\mathbb P_{f, u} ( M_1 = x, M_3 = z \vert M_2 = y )$ has the same value for all funnels when $x$, $y$ and $z$ are distinct. Similarly, assume that there is a unique value across all funnels for $x = y, y \ne z$ and another value for $x \ne y, y = z$. Note that this holds exactly for $r_{12} = r_{23} = 0$ and $r_{12} = r_{23} = 0.5$, and it is always true that $\mathbb P_{f,u }( M_1 = x, M_3 = x \vert M_2 = x )$ is funnel-independent. If $x \ne A$ and $z \ne A$, we now have

$$\begin{aligned}&| F |^{-1}\left( \sum _{f \in F} \mathbb P_{f,u } ( M_1 = x, M_3 = z \vert M_2 = x )\mathbb P_{f, d} ( M_2 = x \vert M_2 \ne A)\right. \\&\quad + \sum _{f \in F} \mathop {\sum _{B \le y \le H}}_{y \ne x, y \ne z} \mathbb P_{f,u } ( M_1 = x, M_3 = z \vert M_2 = y )\mathbb P_{f, d}( M_2 = y \vert M_2 \ne A)\\&\quad + \left. \sum _{f \in F} \mathbb P_{f,u } ( M_1 = x, M_3 = z \vert M_2 = z )\mathbb P_{f, d} ( M_2 = z \vert M_2 \ne A )\right) \end{aligned}$$

From our assumption about funnel independence this becomes

$$\begin{aligned}&\mathbb P_{u} ( M_1 = x, M_3 = z \vert M_2 = x )c_x + \mathop {\sum _{y \ne A}}_{y \ne x, y \ne z} \mathbb P_u ( M_1 = x, M_3 = z \vert M_2 = y )c_y\\&\quad + \mathbb P_u( M_1 = x, M_3 = z \vert M_2 = z )c_z \end{aligned}$$

Here $\mathbb P_u$ refers to the approximate funnel-independent values. However $c_x$, $c_y$ and $c_z$ are all exactly $\frac{1}{7}$, so this is equal to

$$\begin{aligned} \frac{8}{7} \mathbb P_u\left( M_1 = x, M_2 \ne y, M_3 = z\right) \end{aligned}.$$

Similar arguments can be made for the cases $x = A, z = A$, etc. Substituting this approximation back into Eq. 13 gives the desired result. Note that as we did not use the fact that $M_2$ was between $M_1$ and $M_3$, these approximations are equally as valid if the marker order is $M_2, M_1, M_3$.

Simulation of error terms

As the expectation and distribution of $\epsilon _{xz}$ from Sect. 2.2 are analytically intractable, we characterized them through simulation. We considered the case of three dominant markers $M_1, M_2$ and $M_3$. The recombination fractions $r_{12}$ and $r_{23}$ took on values $0.05, 0.1, 0.2, 0.3, 0.4$ and $0.5$. The true genotypic probabilities $p_{.a.}$ and $p_{.h.}$ took on values $\frac{1}{10}, \frac{2}{10}, \ldots , \frac{8}{10}$, with the restriction that $p_{.a.} + p_{.h.} \le 0.9$. The last genotypic fraction $p_{.b.}$ had value $1 - p_{.a.} - p_{.h.}$. All eight possible combinations of dominant founders at the markers were considered. In total, 10,368 different sets of parameters were considered. For each set of parameters, 30,000 F2 populations of $300$ individuals were generated. For each population, the values $\epsilon _{00}, \epsilon _{01}, \epsilon _{10}$ and $\epsilon _{11}$ were calculated.

Figure 4 shows a histogram of the estimated values of ${\mathbb E} [\epsilon _{00} ]$ across all the scenarios considered. Note that Fig. 4 is not a histogram of a distribution, but it shows that the expectation was close to zero in every scenario considered. The behaviour of the other three expectations is similar. In considering whether it is reasonable to assume that $\epsilon _{xz} \simeq 0$, it is therefore sufficient to examine the variance of $\epsilon _{xz}$.

Table 1 Simulation parameters that produced the five largest values of $\mathrm{Var} (\epsilon _{01} )$

Table 1 lists the five scenarios for which the largest value of $\mathrm{Var} (\epsilon _{01} )$ was observed. These scenarios all involve extreme distortion. They also involve specific choices of dominant founders. For example, in the first scenario Eq. 11 implies that

$$\begin{aligned} \epsilon _{01} = \frac{1}{4} \left( \frac{\hat{p}_{001}}{\hat{p}_{.0.}} - \frac{\hat{p}_{001}}{p_{.0.}}\right) + \frac{3}{4}\left( \frac{\hat{p}_{011}}{\hat{p}_{.1.}} - \frac{\hat{p}_{011}}{p_{.1.}}\right) . \end{aligned}$$

In this case $p_{.1.} = 1 - p_{.a.}=0.2$, which is relatively small. So the difference in the second term can potentially be large, whereas the difference in the first term will tend to be small. Now consider the same scenario but with the dominant founder at $M_2$ being $a$. From Eq. 12, in this case

$$\begin{aligned} \epsilon _{01}&= \frac{3}{4} \left( \frac{\hat{p}_{001}}{\hat{p}_{.0.}} - \frac{\hat{p}_{001}}{p_{.0.}}\right) + \frac{1}{4}\left( \frac{\hat{p}_{011}}{\hat{p}_{.1.}} - \frac{\hat{p}_{011}}{p_{.1.}}\right) . \end{aligned}$$

The difference in the first term will tend to be small, and the difference in the second will be potentially large. However it is now multiplied by a factor of $\frac{1}{4}$ rather than $\frac{3}{4}$, and as a result $\mathrm{Var} (\epsilon _{01} )$ is expected to be smaller in this case.

When applying the approximation $\epsilon _{xz} \simeq 0$ we actually make four approximations simultaneously. The worst case for each individual approximation is not expected to be representative of the performance when actually applied to recombination fraction estimation. For example, in the first scenario in Table 1, the largest values of $|\epsilon _{00}|, |\epsilon _{10}|$ and $|\epsilon _{11}|$ observed across 30,000 populations were $0.0186, 0.0089$ and $0.11$. In the specific population that gave a value of $-0.297$ for $\epsilon _{01}$, the corresponding values of the other error terms were $0.014, 0.0062$ and $-0.022$. In general, we observed that scenarios with large values for one of the error terms have very small values for other terms. Hence for nearly all situations we expect the approximation to perform reasonably well.