Incorporating experimental mortality improves statistical inference of crossover patterning along meiotic chromosomes

Abstract

Classic recombination experiments designed to test genetic and environmental treatments do not directly measure chromosomal exchange, instead rates and distribution of crossing-over in ${\text{F}}_1$ meiocytes are inferred from genetic markers in ${\text{F}}_2$ adults. In Drosophila melanogaster females this procedure introduces substantial “missing data problems” because 75% of meiotic chromatids segregate to polar body nuclei and another 11% are transmitted to inviable ${\text{F}}_2$ zygotes which cannot be scored for recombination. To address these sources of uncertainty and bias, we extend the $Cx(Co{)}^m$ data-generating process by assuming: (i) programmed double-strand breaks occur as a Poisson point process, (ii) crossover maturation is a stationary renewal process, (iii) chromosome segregation is random one-half thinning of this process, (iv) fertilization by X- versus Y-bearing sperm is mendelian, and (v) egg-to-adult survival is binomially distributed with a ate parameter determined by ${\text{F}}_2$ marker alleles. To quantify experimental mortality, we performed egg counts for 6-point X chromosome testcrosses and marker-free X chromosome controls on identical genetic backgrounds under standard laboratory conditions. The 19,927 fly dataset revealed 44% ${\text{F}}_2$ experimental mortality, and likelihood ratio tests support a model where 36 of those 44% are due to sex-specific, marker-associated viability defects. Variability in X chromosome genetic map lengths with experimental mortality can be simulated, and we provide case-control 80% power curves to guide experimental design. We propose that differential mortality should be the de facto null hypothesis when comparing ${\text{F}}_2$ recombinant fractions, and we provide probabilistic models of the data-generating process to improve characterization of ${\text{F}}_1$ meiotic crossover patterning.

Introduction

Meiosis is the specialized cell division that produces haploid gametes from diploid germline cells in sexually reproducing eukaryotes. Meiotic recombination, exchange of genetic information between homologous chromosomes, occurs after the diploid genome has replicated but before the meiocyte undergoes two rounds of cell division to generate haploid meiotic products (Anderson 1925). Reciprocal exchange between homologs, termed crossing-over, promotes accurate segregation of chromosomes and, as a byproduct, generates new combinations of alleles in meiotic products (Bridges 1916). Therefore, rates and distribution of crossing-over along chromosomes have critical roles in both preventing aneuploidy and shaping genome-wide polymorphism in the presence of deleterious mutations (Charlesworth et al. 1993; Koehler et al. 1996).

There are three general approaches to estimating crossover rates (reviewed in Peñalba and Wolf 2020): (i) cytological analysis of meiocytes, (ii) from offspring genotypes in controlled matings or pedigrees, and (iii) with linkage disequilibrium in random samples from natural populations. The first method is direct observation of the physical manifestations of crossing-over while the third method involves model-based inference with extensive population genetic modeling. However, the second approach is somewhat ambiguous; some experimentalists treat observed offspring frequencies as direct recombination rate estimates, whereas mathematicians have developed probabilistic models to address incompleteness of the same data (cf. Haldane 1919; Crow 1990; Zhao and Speed 1996).

The fundamental challenge in designing recombination testcrosses is not ${\text{F}}_2$ sample size or sequencing coverage; but rather that meiotic events of interest occur in developing germaria of ${\text{F}}_1$ females and, therefore, are not directly observed (Fig. 1a,b). Furthermore, because each crossover event involves only two homologs in the four-strand bundle of prophase I, and only one strand will migrate to the oocyte pronucleus at telophase II (with the other three strands included in polar body nuclei), there is only a one-half chance of transmitting evidence of any given crossover to ${\text{F}}_2$ progeny (Mather 1935). For example, Fig. 1b illustrates a three-strand double exchange which has a 0.25 probability of transmitting the no-exchange strand, a 0.50 probability of transmitting a single exchange strand, and only a 0.25 probability of transmitting the strand carrying evidence of both crossovers.

Fig. 1.

a) D. melanogaster recombination testcross illustrated with color-coded karyotypes identifying strain-of-origin for acrocentric X, J-shaped Y, two metacentric autosomes, and the heterochromatic dot chromosomes. The meiosis of interest occurs in heterozygous ${\text{F}}_1$ female ovaries but recombination data are collected from ${\text{F}}_2$ adult phenotypic classes. b) The ${\text{F}}_1$ germarium in context of a single developing ovariole (with inset of enlarged, color-coded, linearized cartoon of the X chromosome prophase I four-strand bundle with 8 double strand breaks repaired as 5 gene conversions and a three-strand double exchange for reference). Note that programmed double-strand breaks, gene conversions, and crossovers occur during prophase I in the germarium long before chromosome segregation that follows metaphase I. c) The fate of single chromatids from the four-strand bundle. 75% of chromatids will segregate to polar body nuclei. Of the 25% of chromatids fated to oocyte pronucleus, one-half are fertilized by X-bearing sperm (pink) and the other half are fertilized by Y-bearing sperm (green). Of these zygotes only 56% will survive to adulthood, with 8% dying for unattributable reason (striped wedges) and 36% dying due to marker-associated viability effects (checked wedges). Thus, overall, there is only a 0.14 probability that any given chromatid from a four-strand bundle will be scored in ${\text{F}}_2$ adults for evidence of crossing-over in a recombination testcross (solid-colored pink and green fractions). Created in https://BioRender.com. Shiran, M. (2026) https://BioRender.com/0kx6oc0.

Incomplete transmission to the ${\text{F}}_2$ generation, however, is not the only difficulty in recombination testcrosses. Our experiments show, on average, only a 56% chance of ${\text{F}}_2$ zygotes surviving to adulthood to be scored for recombination (egg-to-adult viability varies based on fertilization by X- versus Y-bearing sperm and effect of genetic markers present in meiotic products). So, given a three-strand double exchange in the ${\text{F}}_1$ germarium (Fig. 1b), the double crossover detection probability is $0.56\times 0.25=0.14$, indicating a large “missing data problem” when scoring ${\text{F}}_2$ visible markers (Fig. 1c). While genome sequencing advances might seem to provide solutions (e.g. Miller et al. 2016), sequencing the ${\text{F}}_2$ individuals cannot address this issue, as even infinite sequencing coverage of infinite ${\text{F}}_2$ offspring in a marker-free cross has an upper limit of $0.92\times 0.25=0.23$ probability of detecting such a double crossover event (see Veller et al. 2022, for additional limitations of sequencing methods).

This missing data problem is best appreciated in human reproductive biology where 30% of pregnancies end in spontaneous miscarriage, approximately half of which are associated with aneuploidy stemming from either absence or misplacement of crossover events (Boué et al. 1975; Wilcox et al. 1988; Smits et al. 2020; Gruhn and Hoffmann 2022). Practically speaking, polar body biopsy is a long-standing standard practice to identify chromosomal aberrations in pre-implantation genetic testing (reviewed in Montag et al. 2004). More theoretically, Kong et al. (2004) first showed that if embryonic lethality is ignored there is an apparent increase of recombination associated with maternal age in human populations (see also Coop et al. 2008; Hussin et al. 2011; Bleazard et al. 2013; Campbell et al. 2015; Martin et al. 2015). However, when accounting for survivorship bias due to increased nondisjunction in aging mothers, the Kong et al. (2004) Icelandic dataset provides convincing evidence of natural selection on chromosome segregation in human meiosis.

To address these fundamental challenges in experimental testcrosses, we extend the $Cx(Co{)}^m$ model of the data-generating process as a stochastic function of the underlying meiotic events in ${\text{F}}_1$ females and then account for ${\text{F}}_2$ experimental mortality with sex-specific viability effects. Briefly, the underlying process-based model states that programmed double-strand breaks (C) occur as a homogeneous Poisson point process along the chromosome-level genetic map (i.e. a line segment on the real half-line with the centromere as the origin) (Haldane 1919). A proportion of such breaks repair as crossover events (Cx) in a stationary renewal process, with the intervening number of non-crossover gene conversion events (Co) events reflecting the strength of crossover interference (m) (Sturtevant 1913; Fisher 1947; Zhao and Speed 1996). Random one-half thinning of this process accounts for asymmetric female meiosis and the one-half chance of sampling evidence of any given reciprocal exchange event (Mather 1935). Fertilization of oocytes by X- versus Y-bearing sperm is mendelian and resulting zygote survival is modeled as a binomial with its rate parameter determined by ${\text{F}}_2$ phenotypic class.

The core assumptions in this process-based model were purely theoretical when established by Fisher (1947) and Owen (1949, 1950). They were later re-introduced as the “counting model” (Foss et al. 1993) on biological grounds and then further codified as the “gamma model” (McPeek and Speed 1995), the “Poisson-skip model” (Lange et al. 1997), and the “chi-square model” (Zhao et al. 1995b, which is the formulation we extend here). Historically, analysis of recombination testcrosses was solely aimed at building or refining genetic maps (e.g. Haldane 1919; Fisher 1947; Owen 1949; Stam 1979; Crow 1990; Speed 2005). More recently, the underlying models have been applied as mechanistic hypotheses for subcellular meiotic phenotypes to enable statistical inferences about effects of genetic mutations, karyotypic variation, heterochiasmy, and biological aging (e.g. Navarro et al. 1997; Zwick et al. 1999; Stahl et al. 2004; Petkov et al. 2007; Zhang et al. 2014, 2021; Koury 2023; Kapperud 2025).

Here, we demonstrate that both asymmetric female meiosis and experimental mortality introduce substantial uncertainty and bias when recombination inferences are based only on surviving ${\text{F}}_2$ adults. To account for these biases, we extend the $Cx(Co{)}^m$ model of Zhao et al. (1995b) to explicitly model zygosis and include sex-specific marker viability effects. The newly presented data is used to generate power curves for case-control experimental designs that can account for stochasticity in crossover patterning, chromosome segregation, fertilization, and experimental mortality. Furthermore, the Appendix section “Simulation of sampling distribution” outlines a procedure for using the data-generating process to allow rigorous null hypothesis statistical testing when comparing genetic or environmental treatments that may alter meiosis but also introduce segregation defects and experimental mortality.

Results

To investigate sources of experimental mortality in recombination studies, we present a new testcross dataset that includes absolute measures of ${\text{F}}_2$ egg-to-adult viability for the model genetic system D. melanogaster. We performed two testcrosses using either multiply-marked X chromosomes or a marker-free X chromosome controls on identical genetic backgrounds under normal laboratory conditions. Ten replicates of each cross, mated at a density of five females and five males, were serially transferred to generate ten 24-hour broods (Fig. 2). Immediately following tap transfer, eggs laid were recorded by digital photograph. Thus, when 14,190 ${\text{F}}_2$ eclosing adults were scored for visible marker recombination, per vial egg-to-adult survival rates could also be estimated from 19,927 previously imaged ${\text{F}}_2$ eggs.

Fig. 2.

Crossing scheme and experimental design for D. melanogaster multiply-marked X chromosome testcross (identical crossing scheme for marker-free controls not shown). Data structure follows the crossing design of ten replicate crosses serially transferred to create ten 24-h broods. The use of four unique isogenic genetic backgrounds (BDSC 6,326, DGRP 399, Canton-S, and Oregon-R) in P crosses prevents inbreeding effects from contributing to ${\text{F}}_2$ experimental mortality. Detailed marker stock construction can be found in the supplemental file S1 section V. Therefore, in a common garden design with the same environmental conditions and identical genetic backgrounds, all excess mortality in multiply-marked crosses versus marker-free controls can be attributed to the presence of genetic markers $y^1$, $s{c}^1$, $c{v}^1$, $v^1$, $f^1$, $ca{r}^1$, and $Dp(1;1)s{c}^{V1}$, $y^{+}$ highlighted in red script. Note $y^1$ is homozygous at its native locus and is not informative, but is required for the use of $Dp(1;1)s{c}^{V1}$, $y^{+}$ as a centromeric marker. Egg image created in BioRender. Koury, S. (2026) https://BioRender.com/5hxkje1.

In marker-free crosses, 7.9% of ${\text{F}}_2$ eggs failed to develop into ${\text{F}}_2$ adults. Assuming a 1:1 primary sex ratio, these failures were evenly distributed among males and females (7.0% vs. 8.7%, $F_{1,188}=0.56$, $p=0.46$) and showed no maternal age effects across 10 days ($b_{Y\cdot X}=-0.006$, $t_s(188)=-1.55$, $p=0.12$) (Supplementary Table S1). When X-linked visible markers $y^1$, $s{c}^1$, $c{v}^1$, $v^1$, $f^1$, $ca{r}^1$, and $Dp(1;1)s{c}^{V1}$, $y^{+}$ were segregating on the same genetic background, 44.3% of ${\text{F}}_2$ eggs failed to develop into ${\text{F}}_2$ adults and could not be scored for recombination. In contrast to marker-free crosses, multiply-marked cross failures were more prevalent in males than females (50.6% vs. 37.9%, $F_{1,128}=71.51$, $p=5.35\times {10}^{-14}$), again with no maternal age effect ($b_{Y\cdot X}=0.003$, $t_s(128)=1.24$, $p=0.22$) (Supplementary Table S2). Statistically significant replicate cross effects in the presence of markers, but not in marker-free crosses, suggests that markers both increase experimental mortality and also that mortality becomes more variable (Supplementary Table S1 and S2). Without a model of the underlying meiotic processes and explicit treatment of sex-specific marker viability, the degree of experimental mortality in this dataset is consistent with D. melanogaster X chromosome genetic map lengths anywhere between 50 and 100 centiMorgans, and thus a model-based approach is necessary to quantify uncertainty and bias in genetic map length estimates.

The ${\text{F}}_2$ mortality observed in multiply-marked crosses can be statistically associated with visible markers used to score recombination. To detect uneven recovery of markers in both sexes, we performed single-locus goodness-of-fit G-tests assessing which model best describes viability defects in this dataset. In order of increasing complexity these single-locus mendelian segregation models are:

• $H_0$ : no excess experimental mortality in multiply-marked crosses above and beyond the marker-free cross,

• $H_1$ : excess experimental mortality in multiply-marked crosses is random with respect to marker alleles,

• $H_2$ : excess experimental mortality in multiply-marked crosses is solely due to mutant alleles at marker loci, and

• $H_3$ : excess experimental mortality in multiply-marked crosses is due to viability effects associated with both mutant and wildtype alleles at marker loci.

The added viability effects in each hypothesis were estimated by the least squares solution to the five-equation systems given in supplemental file S1 section II “Single-locus analysis”. For all markers, the single-locus goodness-of-fit to $H_0$, $H_1$, and $H_2$ were rejected (Table 1, Supplementary Tables S3–S8). Thus, full viability models ($H_3$ estimating four parameters from the data with zero degrees of freedom) are necessary to describe experimental mortality in recombination testcrosses even with no experimental treatments. For all six markers, the best fit model estimated viability effects of both mutant and wildtype alleles that differ significantly between the sexes (Supplementary Table S3, Supplementary Fig. S2).

Table 1.

Above the line are single-locus goodness-of-fit for $H_0$, $H_1$, $H_2$ reporting the G statistic and denoting significance with *** to indicate $p<0.001$ for all G-tests.

Model	dfa	sc b	cv c	v d	f e	car f	$\mathit{D}\mathit{p}\mathbf{(}\mathbf{1}\mathbf{;}\mathbf{1}\mathbf{)}\mathit{s}{\mathit{c}}^{\mathit{V}\mathbf{1}}$ g
$H_0$	4	10,599***	10,965***	10,646***	10,501***	10,487***	10,484***
$H_1$	3	256***	622***	303***	159***	144***	141***
$H_2$	2	1,063***	641***	988***	1,289***	1,331***	1,342***
$H_3$	$v_{femalewildtype}$	0.77	0.82	0.77	0.74	0.74	0.74
	$v_{malewildtype}$	0.66	0.75	0.67	0.62	0.61	0.61
	$v_{femalemutant}$	0.59	0.54	0.58	0.62	0.62	0.62
	$v_{malemutant}$	0.43	0.34	0.41	0.47	0.48	0.48

Below the line are the zero degrees of freedom viability estimates from the full viability model $H_3$.

^asee supplemental file S1 section II “Single-locus analysis” for number and definition of viability parameters estimated from the data determining degrees of freedom

^b scute, X:396,060..397,497 all genomic locations as per Release 6 plus ISO1 MT, FB_2025_04

^c crossveinless, X:5,690,002..5,693,084

^d vermilion, X:10,923,972..10,925,631

^e forked, X:17,232,942..17,280,964

^f carnation, X:19,566,019..19,569,401

^gintrachromosomal duplication of roughly X:0..397,000 onto X chromosome right arm

However, the six marker loci reported in Table 1 do not segregate independently. Therefore, single-locus tests confound each other and distort viability estimates by a factor related to the genetic distances we seek to estimate. We address this problem by simultaneously estimating multi-locus marker viability effects, genetic map lengths, and crossover interference through maximizing the likelihood of observed ${\text{F}}_2$ phenotypic class counts under the extended $Cx(Co{)}^m$ model. Table 2 reports maximum likelihood estimates of the genetic map lengths for our full dataset (i.e. pooling all multiply-marked crosses and broods) under the multi-locus viability models corresponding to $H_0$, $H_1$, $H_2$, and $H_3$ (see Supplementary Table S9 and Supplementary Fig. S3 for multi-locus viability estimates). The estimated map lengths and strength of crossover interference are somewhat comparable across all four models, although assuming that all experimental mortality is solely due to mutant alleles ($H_2$) does lead to a slightly longer genetic map with stronger interference (Table 2).

Table 2.

Genetic map lengths estimated from multi-locus dataset for different deterministic map functions or stochastic process $Cx(Co)$^m using viability models $H_0$, $H_1$, $H_2$, and $H_3$.

Origin of Estimates	m a	Xb	sc-cv	cv-v	v-f	f-car	$\mathit{c}\mathit{a}\mathit{r}\mathbf{-}{\mathit{y}}^{\mathbf{+}}$ c	ℓd	${\mathit{\theta }}_{\mathit{i}}$ e	LR	df	p-value
Lindsley and Zimm (1992) Mapf		66.0	13.7	20.3	24.7	3.8	3.5g
Comeron et al. (2012) Maph		66.3	14.4	19.8	23.3	4.6	4.2i
Haldane’s Map Functionj		97.6	21.3	27.3	37.9	6.3	4.8
Kosambi’s Map Functionk		80.7	18.1	22.4	29.6	6.0	4.6
extended $Cx(Co)$m  $H_0$	4	76.5	17.3	21.0	27.5	6.1	4.6	33,136	6
extended $Cx(Co)$m  $H_1$	4	76.5	17.3	21.0	27.5	6.1	4.6	27,927	8	10,418	2	$<1.0\times {10}^{-325}$ l
extended $Cx(Co)$m  $H_2$	5	78.8	17.6	21.9	28.3	6.3	4.7	27,692	18	470	10	$<5.6\times {10}^{-95}$ m
extended $Cx(Co)$m  $H_3$	4	76.8	17.0	21.3	27.8	6.1	4.6	27,633	30	118	12	$<7.2\times {10}^{-20}$ n

Likelihood ratio tests can only be conducted on probabilistic model-based methods.

^aCrossover interference parameter m from $Cx(Co{)}^m$ model

^bFull X chromosome map consisting of 5 listed intervals

^c $y^{+}$ allele in intrachromosomal duplication $Dp(1;1)s{c}^{V1}$

^dLog-likelihood of the probabilistic model denoted as ℓ

^eThe number of parameters in the model denoted as ${\theta }_i$

^fLindsley and Zimm maps values included for comparison, these values are composites of multiple experiments

^gInferred from maximum map length without intrachromosomal duplication $Dp(1;1)s{c}^{V1}$ on X chromosome right arm

^hComeron map values included for comparison, these data do not come from classic recombination testcrosses

ⁱInferred from maximum map length without intrachromosomal duplication $Dp(1;1)s{c}^{V1}$ on X chromosome right arm

^jHaldane’s map function applied to raw recombinant fractions from recombination testcrosses presented here

^kKosambi’s map function applied to raw recombinant fractions from recombination testcrosses presented here

^lLikelihood ratio test of $H_1$ vs. $H_0$

^mLikelihood ratio test of $H_2$ vs. $H_1$

ⁿLikelihood ratio test of $H_3$ vs. $H_2$

Because the multi-locus versions of $H_0$, $H_1$, $H_2$, and $H_3$ can be strictly nested by constraining specified viability effects to unity, likelihood ratio tests can evaluate improved fit relative to increasing complexity of viability models (see Appendix section “Probability of observing recombination marker patterns”). Likelihood ratio tests revealed that the added parameters modeling experimental mortality were well-justified ($H_1$ vs. $H_0$: $LR=10418$, $df=2$, $p<1.0\times {10}^{-325}$), excess mortality was not random with respect to marker alleles ($H_2$ vs. $H_1$: $LR=470$, $df=10$, $p<5.6\times {10}^{-95}$), and both mutant and wildtype alleles at marker loci had sex-specific viability effects ($H_3$ vs. $H_2$: $LR=118$, $df=12$, $p<7.2\times {10}^{-20}$). Thus, similar to the single-locus tests, the multi-locus analysis strongly supports $H_3$ (excess experimental mortality in multiply-marked crosses is due to sex-specific viability effects associated with both mutant and wildtype alleles at marker loci) as the best viability model for the data-generating process.

To explore variability in genetic map lengths, we fit the extended $Cx(Co{)}^m$ model to data collected from single vials as the experimental unit. The observed variation was then partitioned between individual crosses and broods as fixed categorical effects using an ANOVA with Type III sum-of-squares (Supplementary Table S10). There was no significant effect of cross ($F_{6,54}=1.10,p=0.34$) or brood ($F_{9,54}=1.04$, $p=0.42$) on genetic map length. Finally, little improvement in statistical power can be achieved by pooling the data from multiple crosses or by broods (Supplementary Table S11 and S12). Figure 3 illustrates the 80% power curves in a case-control design with respect to the ${\text{F}}_1$ experimental units of analysis under the different data pooling strategies.

Fig. 3.

Case-control design 80% power curves for comparison of X chromosome genetic map lengths under different pooling schemes with respect to ${\text{F}}_1$ experimental units of analysis (data pooled by either vials, crosses, or broods). Step function is a product of replicates necessarily being a whole number $\ge 2$.

Discussion

The physical locations of meiotic crossing-over along the chromosome axis are of fundamental importance to a wide range of genetical research from cellular biology to evolutionary theory. Here, we show classic methods for characterizing the rate and distribution of crossing-over have a substantial missing data problem. In female-limited recombination of D. melanogaster, 75% of meiotic chromatids (and the accompanying evidence of meiotic events) are lost to polar body nuclei in ${\text{F}}_1$ females (Fig. 1c). Another 11% of chromatids are lost in ${\text{F}}_2$ zygotes that die before being scored for recombination (Fig. 1c). A further 5% of chromatids carry no evidence of recombination even though their four-strand bundle of origin may have had multiple exchanges (Fig. 1b). Therefore, expressing recombination rates as a fraction:

$$\frac{9\mathrm{\%}{\text{recombinant chromatids in viable F}}_2\text{progeny}}{14\mathrm{\%}{\text{of chromatids from the F}}_1\text{four-strand bundle}}$$ (1 )

approximates a 66 cM X chromosome but reveals little about the rate and distribution of actual meiotic events in prophase I of ${\text{F}}_1$ females.

While the missing data problem outlined above causes considerable uncertainty in estimates, perhaps more concerning is that both numerator and denominator in recombinant fractions are biased by viability effects of the visible markers scored for recombination. We show that experimental mortality (44%) is the major signal in recombination studies, and the majority of this death (36 of thos 44%) is due to sex-specific viability effects associated with both mutant and wildtype alleles. Thus, simple calculation of recombination rates from the observed ${\text{F}}_2$ adults can be strongly biased (Tables 1, 2, Tables S3–S9) and prohibits comparison of rates between experimental genotypes and/or treatments. To address this problem, we model the data-generating process by extending the $Cx(Co{)}^m$ model to a system of 129 equations defining the probability of observing an ${\text{F}}_2$ adult fly in any of 64 sex-specific phenotypic classes as well as the number of inviable ${\text{F}}_2$ zygotes that could not be scored for recombination (see Appendix section “Maximum likelihood solution to full system of equations”). This increased model complexity (24 additional parameters for our experiments) is well-justified by likelihood ratio tests and suggests that even more nuanced analyses of recombination datasets (e.g. McPeek and Speed 1995; Lange et al. 1997; Housworth and Stahl 2003) can be substantially improved by incorporating both measures and models of ${\text{F}}_2$ survival.

Experimental mortality in D. melanogaster genetic studies has long been recognized but is often overlooked in the design and analysis of recombination testcrosses (but see Bridges and Olbrycht 1926; Sturtevant and Beadle 1936; Redfield 1957; Hayman and Parsons 1962; Zwick et al. 1999; Koury 2023). As we note in the Introduction, embryonic lethality is a thoroughly-documented problem in human studies of recombination (Kong et al. 2004; Martin et al. 2015; Gruhn and Hoffmann 2022), and likely confounds analyses of ${\text{F}}_2$ data in all genetic model systems (Mittell and Morrissey 2024). A common approach to addressing this problem is assuming relative fitness of the wildtype is 1.0 and all viability defects are associated with mutant alleles (i.e. our ${\text{H}}_2$). However, we show here that maximum likelihood estimation using a mis-specified viability model like ${\text{H}}_2$ leads to biased inference of longer genetic maps with stronger crossover interference that fits the data significantly worse than the full viability model allowing for viability defects of wildtype alleles (Table 2). The present study builds on a published protocol for quantifying the missing data in recombination testcrosses Schneider et al. (2025), corrects biases that absolute mortality introduces to experimental observations, and can ultimately incorporate appropriate uncertainty to the maximum likelihood estimates through mechanistic simulation of sampling distributions (see Appendix section “Simulation of sampling distribution”).

The newly presented D. melanogaster recombination dataset, and analysis with the extended $Cx(Co{)}^m$ counting model, demonstrates that ${\text{F}}_2$ experimental mortality is overwhelmingly due to sex-specific viability effects associated with both mutant and wildtype alleles even under well-controlled, benign crossing conditions. Future recombination studies, especially those introducing strong genetic or environmental perturbations to meiosis, should directly measure experimental mortality as a likely source of substantial uncertainty and bias in any model system. Therefore, we propose the de facto null hypothesis in recombination studies be that changes in ${\text{F}}_2$ marker distribution results from experimental mortality and that analysis with probabilistic models of the data-generating process are essential for valid statistical inference about rates and distributions of meiotic events in ${\text{F}}_1$ females.

Materials and methods

Both multiply-marked and marker-free X chromosome crosses were conducted in a common garden design (i.e. same environmental conditions and identical genetic backgrounds). Multiply-marked X chromosome consisted of seven visible markers $y^1$, $s{c}^1$, $c{v}^1$, $v^1$, $f^1$, $ca{r}^1$, and $Dp(1;1)s{c}^{V1}$, $y^{+}$ in a stock reconstituted from Bloomington Drosophila Stock Center lines (BDSC 177, 1,515, and 4,914) (Grell 1978). This reconstituted multiply-marked chromosome was then background-replaced with autosomes of BDSC 6,326 using a balancer-translocation stock derived from BDSC 2,475 ($w^{*}$; $T(2;3)a{p}^{Xa}$/$In(2L)Cy$, $In(2R)Cy$; $TM3$, $S{b}^1$).

Centromeric marker $Dp(1;1)s{c}^{V1}$, $y^{+}$ required introgression of $y^1$ from BDSC 1,515 into Drosophila melanogaster Genetic Reference Panel (DGRP) line 399 (BDSC 25,192) by repeatedly backcrossing for >10 generations. Thus, all ${\text{F}}_1$ experimental females in multiply-marked testcrosses were whole autosome heterozygotes for BDSC 6,326 and DGRP 399, whole chromosome heterozygotes for the multiply-marked X chromosome with a homozygous $y^1$ at its native locus (Fig. 2). ${\text{F}}_1$ experimental females were crossed to whole genome heterozygotes for Oregon-R (BDSC 25,211) and Canton-S (BDSC 64,349) carrying visible markers $y^1$, $s{c}^1$, $c{v}^1$, $v^1$, $f^1$, and $ca{r}^1$ (Fig. 2). Marker-free crosses involved the same stocks to reproduce identical genetic backgrounds but with no X-linked markers (i.e. pure strains of Canton-S, Oregon-R, DGRP 399, and BDSC 6,326 with $w^{1118}$ allele replaced by $w^{+}$). See supplemental file S1 section V “Stock construction” for detailed stock construction notes and crossing diagrams.

Crosses were conducted in polystyrene vials with standard cornmeal Drosophila media containing 0.01 v/v McCormick blue food coloring to increase egg contrast in digital pictures. Additionally, 50 μL of live yeast paste (0.2 g/mL) with 0.02 v/v blue food coloring was added to induce egg laying behavior (Schneider et al. 2025). Vials were housed in a Percival incubator at ${21}^{\circ }$, 70% relative humidity, and 12:12 h light-dark cycle.

Virgin ${\text{F}}_1$ flies were collected over 72-hours, aged an additional 72 hours, then crossed at densities of five females to five males under light ${\text{CO}}_2$ anesthesia. After 24-hour recovery, mated flies were tap transferred onto fresh food establishing “Brood A”. Every subsequent 24-hours for 10 days, flies were tap transferred onto freshly prepared food to generate new broods (Broods B-J). Immediately after tap transferring flies, food surface was imaged for egg counting (Schneider et al. 2025). 72 h after imaging, a $15{\mathrm{cm}}^2$ piece of Kimwipe was inserted to food and wetted with 100 μL of 0.01M propionic acid 0.001M phosphoric acid.

Starting 10 d after egg laying and continuing until all viable adults had eclosed, sex-specific adult phenotypic classes were scored for recombination. ${\text{F}}_2$ adult flies from multiply-marked crosses in the later broods G, H, and I were collected into holding vials, but experienced substantial extra mortality (¿50%) before they could be scored. To prevent unbalancing the design from partial count data in select broods, adult phenotypic class data from all multiply-marked crosses in these three broods were excluded from analysis. Two independent digital egg counts were performed for all broods (including brood G, H, and I); because digital scoring consistently undercounts eggs per vial and individual observers differ in their undercounting, both egg counts were transformed using standard curves constructed from independent dataset of Schneider et al. (2025) (Supplementary Fig. S1). Number of ${\text{F}}_2$ eggs failing to develop into ${\text{F}}_2$ adults was calculated as the difference between the average of transformed egg counts and the number of ${\text{F}}_2$ adults scored for recombination.

Single-locus viability effects were tested using five ${\text{F}}_2$ adult classes (female wildtype, female mutant, male wildtype, male mutant, and lethal zygotes) independently for the six marker loci. The added viability effects in each hypothesis ($H_0$, $H_1$, $H_2$, and $H_3$) was estimated by the least squares solution to respective five-equation systems given in supplemental file S1 section V “Single-locus analysis”. Single-locus goodness-of-fit was assessed with G-tests using custom scripts in R (Sokal and Rohlf 1995; R Core Team 2025).

Multi-locus viability effects and genetic map lengths were estimated by maximizing the likelihood of 129 ${\text{F}}_2$ class counts ($2^6\text{ phenotypic classes}\times 2\text{sexes}+1\text{ for lethal zygotes}$) under the extended $Cx(Co{)}^m$ counting model for the full dataset (see Appendix section “Maximum likelihood solution to full system of equations”). Custom R scripts available on Github (Koury and Stevison 2026) used multinomial likelihood function to calculate the likelihood of observing the 129 ${\text{F}}_2$ classes counts for a set of model parameters under hypotheses $H_0$, $H_1$, $H_2$, and $H_3$ (see Data availability section). The iterative solution to maximizing this likelihood utilized the downhill simplex method with R package “dfoptim” where all viability estimates were constrained between 0 and 1 (Varadhan et al. 2023; R Core Team 2025).

To characterize the patterns of variability in the estimated genetic map lengths, we also fit the extended $Cx(Co{)}^m$ counting model to data with single vials, pooled broods, and pooled crosses as the experimental units. The observed variation was partitioned between crosses and/or broods as fixed effects with a Type III sum-of-squares ANOVA for each pooling scheme (Supplementary Table S10, S11, S12) (Fox and Weisberg 2019; R Core Team 2025). Case-control 80% power curves were determined by “Number of replications needed to detect a given ‘true’ difference between two means” method of Sokal and Rohlf (1995) using:

$$n\ge 2{\left(\frac{\sigma }{\delta }\right)}^2\{t_{\alpha [\nu ]}+t_{2(1-P[\nu ])}{\}}^2$$ (2 )

where n is the number of replications, σ is “true” standard deviation estimated from respective ANOVA tables, δ is minimal detectable difference, ν is degrees of freedom in the sample standard deviation, α is significance level, P is intended power of the test, with the values of $t_{\alpha [\nu ]}$ and $t_{2(1-P[\nu ]}$ coming from two tailed t-tables.

Appendix A: Extended $Cx(Co{)}^m$ model of crossover patterning

The counting model of crossing-over refers to a family of stochastic processes that share common assumptions in describing the patterning of genetic exchange along meiotic chromosomes (Fisher 1947; Owen 1949, 1950; Foss et al. 1993; McPeek and Speed 1995; Zhao et al. 1995b; Lange et al. 1997; Copenhaver et al. 2002). The chi-square $Cx(Co{)}^m$ formulation of the counting model proposes crossover precursors $(C)$ occur as a homogenous Poisson point process along the genetic map, with some crossover precursors maturing into crossover events $(Cx)$ in a stationary renewal process, and the remaining crossover precursors being designated as non-crossover events $(Co)$. All three events have straight-forward biological interpretations: crossover precursors C are programmed double-strand breaks, crossover Cx events are repair of double-strand breaks resulting in reciprocal genetic exchange between homologs, and non-crossover Co events are repair of double-strand breaks resulting in directional genetic exchange between homologs (i.e. non-crossover gene conversion events). Intuitively, the spacing of crossover Cx events with m number of intervening non-crossover Co events reflects the strength of crossover interference. Zhao et al. (1995b) provide a matrix algebra notation for this model and, where possible, we preserve their naming conventions.

Here, we extend the $Cx(Co{)}^m$ model to include both random and marker-associated, sex-specific viability effects. This extension requires explicit modeling of fertilization as well as expanding the system of equations to solve for different probabilities of observing reciprocal recombinant classes both within and between the sexes. We introduce the model and our extension by sequentially defining: (i) the probabilities of all possible crossovers in a single interval on the four-strand bundle during prophase I of the ${\text{F}}_1$ oocyte development, (ii) the probabilities of all possible crossover patterns in two adjacent intervals of the four-strand bundle in prophase I, (iii) the probabilities of any recombination pattern on a single strand after chromosome segregation in anaphase I and anaphase II, (iv) the probability of fertilization by X-bearing or Y-bearing sperm, and (v) the survival probability of the resulting zygote as a function of the ${\text{F}}_2$ genotype. This produces a probabilistic model of the data-generating process that can be represented by a system of equations, one for each observable ${\text{F}}_2$ phenotypic class plus one for lethal zygotes. Given a recombination dataset with an absolute measure of experimental mortality, the maximum likelihood solution to this system of equations can be used to estimate parameters in the underlying model(s) of meiosis.

Appendix B: Probability of crossover pattern on four-strand bundle

Let y be the rate of crossover precursor C formation along the real half-line. The chromosome level genetic map is defined as a line segment on the real half-line where this point process occurs, with the centromere serving as the origin. Assuming these events are Poisson distributed along the meiotic chromosome, the chance of s number of crossover precursors on the four-strand bundle is:

$$\frac{e^{-y}{y}^s}{s!}.$$ (B1 )

The fraction of precursor C events that mature to crossover Cx events is $1/p$ (where $p=m+1$). If programmed double-strand breaks occur randomly among the four strands in prophase I and there is no chromatid interference (Zhao et al. 1995a), then each strand has a one-half chance of being involved in a crossover event (Mather 1935).

However, in the renewal process that governs crossover maturation, the $1/p$ precursor C’s resolving as Cx crossover events are not randomly distributed along the four-strand bundle. This is because the $Cx(Co{)}^m$ model assumes a directional and deterministic pattern of a Cx followed by m number of Co’s and then a Cx followed by another m number of Co’s and so on (thus the nomenclature of $Cx(Co{)}^m$ counting model). Importantly, this renewal process is also assumed to be stationary, such that the first precursor C event has an equal chance of resolving as a crossover Cx event or any of the m number of sequential non-crossover Co events. This second assumption reintroduces a probabilistic element into the deterministic renewal process. Following Zhao et al. (1995b), the occurrence of k crossovers between two markers $l_1$ and $l_2$ (i.e. a single interval) can be computed as:

$$\frac{e^{-y}}{p}\sum _{i=1}^p\sum _{j=0}^{p-1}\frac{y^{pk-p+i+j}}{(pk-p+i+j)!}.$$ (B2 )

Where i and j are index variables in the double summation for the first interval, denoted by subscript one, which can conveniently be rewritten in matrix notation as:

$$\frac{1}{p}\mathbf{1}{\mathit{D}}_{{\mathit{k}}_{\mathbf{1}}}(y_1){\mathbf{1}}^{\mathbf{\prime }}.$$ (B3 )

Where $\mathbf{1}$ is a $1\times p$ vector, ${\mathbf{1}}^{\prime }$ is its transpose, and ${\mathit{D}}_{{\mathit{k}}_{\mathbf{1}}}(y_i)$ is the $p\times p$ matrix with $i,j^{\mathrm{th}}$ entry $e^{-y_1}{y}_1^{p{k}_1-p+i+j}$/$(p{k}_1-p+i+j)!$.

Within the first marker-defined interval, the stationarity of the renewal process means a random starting point in the $Cx(Co{)}^m$ sequence corresponds, in aggregate, to a random stopping point in this sequence irrespective of interval length. Nonetheless, the directionality of this process dictates that the realized stopping point of the $Cx(Co{)}^m$ sequence in the first interval for any given chromosome of a specific oocyte will fully determine the starting point in the adjacent second interval on that chromosome. Zhao et al. (1995b) provide a proof that the chance of $k_1$ crossovers between markers $l_1$ and $l_2$ with $k_2$ crossovers in the adjacent interval between markers $l_2$ and $l_3$ is:

$$e^{-y_2}\sum _{i=1}^p{p}_{k_1}^{p+1-i}\sum _{j=0}^{p-1}\frac{y_2^{p{k}_2-p+i+j}}{(p{k}_2-p+i+j)!}.$$ (B4 )

Rewriting equation (B4) in matrix notation, with subscripts 1 and 2 denoting first and second intervals, and the definition of $(p_{k_1}^1{p}_{k_1}^2{p}_{k_1}^3\cdots p_{k_1}^p)=\frac{1}{p}\mathbf{1}{\mathit{D}}_{{\mathit{k}}_{\mathbf{1}}}(y_1)$ yields:

$$(p_{k_1}^1{p}_{k_1}^2{p}_{k_1}^3\cdots p_{k_1}^p){\mathit{D}}_{{\mathit{k}}_{\mathbf{2}}}(y_2){\mathbf{1}}^{\mathbf{\prime }}.$$ (B5 )

Here, we see that termination of the $Cx(Co{)}^m$ sequence in interval 1 corresponds to the start of the renewal process in interval 2. Making further use of Zhao et al.’s (1995b) definition of $(p_{k_1}^1{p}_{k_1}^2{p}_{k_1}^3\cdots p_{k_1}^p)$ allows matrix algebra to account for all possible $Cx(Co{)}^m$ sequence starting and ending points both within an interval and across two adjacent intervals of any size as:

$$\frac{1}{p}\mathbf{1}{\mathit{D}}_{{\mathit{k}}_{\mathbf{1}}}(y_1){\mathit{D}}_{{\mathit{k}}_{\mathbf{2}}}(y_2){\mathbf{1}}^{\mathbf{\prime }}.$$ (B6 )

This approach can be extended to any number of adjacent intervals, or even non-adjacent intervals separated by a known genetic distance (e.g. a centromere or structural rearrangement).

Appendix C: Probability of recombination pattern on a single strand

Determining the probability of any crossover pattern on the four-strand bundle in prophase I is the primary goal of modeling meiotic recombination. Unfortunately, experimental observation of this crossover pattern requires labor-intensive cytological analysis. Therefore, to employ maximum likelihood methods for recombination testcross data we must define the probabilities of any recombination pattern on a single strand.

Recall that each crossover event involves only two homologs in the four-strand bundle of prophase I, and only one strand will migrate to the oocyte pronucleus at telophase II (with the other three strands included in polar body nuclei, see Fig. 1b in main text). Consequently, there is only a one-half chance of transmitting evidence of any given crossover to ${\text{F}}_2$ progeny (Mather 1935). This means the average number of crossover Cx events on each strand given s number of precursor C events is $s/2p$, which defines the total chromosome map length as $100y/2p$ centiMorgans (cM).

However, modeling the specific patterns of crossing-over that produced the $100y/2p$ cM genetic length, rather than the average, requires random one-half thinning of the stochastic point process defined in the first section. Therefore, the observation of no recombination $(N_z)$ in interval z is the result of either:

1. no precursor C events occurring in the interval z  $(s=0)$, denoted ${\mathit{D}}_{\mathbf{0}}(y_z)$,

2. some precursor C events $(s\ge 1)$, but none that mature into crossover Cx events,

3. some precursor C events $(s\ge 1)$ with some crossover Cx events, but none that involve the single strand migrating to the functional egg pole, or

4. some precursor C events $(s\ge 1)$ maturing to an even number of crossover Cx events that involve the same strand from the four-strand bundle.

In sum, all the ways of observing no recombination in interval z is given by Zhao et al.’s (1995b) Appendix Theorem 1:

$${\mathit{N}}_{\mathit{z}}={\mathit{D}}_{\mathbf{0}}(y_z)+\frac{1}{2}\sum _{s\ge 1}{\mathit{D}}_{\mathit{s}}(y_z).$$ (C1 )

In contrast, observation of recombination $(R_z)$ in interval z must necessarily come from $s\ge 1$ precursor C events and either an odd number of crossover Cx events or any even number of crossover Cx events that do not involve the same single strand. Therefore, using Mather’s (1935) proof and Zhao et al.’s (1995b) Appendix Theorem 1, we relate the observable recombination pattern with the underlying crossover pattern with:

$${\mathit{R}}_{\mathit{z}}=\frac{1}{2}\sum _{s\ge 1}{\mathit{D}}_{\mathit{s}}(y_z).$$ (C2 )

At this point our notation diverges from Zhao et al. (1995b) who coded recombination data as binary by interval and implicitly pooled reciprocal marker classes. Because we are interested in the inequivalence of reciprocal classes due to marker-associated viability effects, our recombination data are coded as binary by marker. Thus, a 6-point testcross with five intervals formerly had its “probability of recombination interval pattern” coded as $(i_1{i}_2{i}_3{i}_4{i}_5)$ but is here replaced with our “probability of the recombination marker pattern” $(l_1{l}_2{l}_3{l}_4{l}_5{l}_6)$, giving:

$$Pr(l_1{l}_2{l}_3{l}_4{l}_5{l}_6)=\frac{1}{2p}\mathbf{1}{\mathit{M}}_{\mathbf{1}}{\mathit{M}}_{\mathbf{2}}{\mathit{M}}_{\mathbf{3}}{\mathit{M}}_{\mathbf{4}}{\mathit{M}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }},$$ (C3 )

where ${\mathbf{M}}_{\mathbf{z}}\mathbf{=}{\mathbf{N}}_{\mathbf{z}}$ when $l_z=l_{z+1}$ and ${\mathbf{M}}_{\mathbf{z}}\mathbf{=}{\mathbf{R}}_{\mathbf{z}}$ when $l_z\ne l_{z+1}$. Please note that the denominator is $2p$ to reflect that our “recombination marker pattern” is the probability of observing only one of two reciprocal classes that result from the same “recombination interval pattern” notation.

Motivated by our experimental dataset from Drosophila melanogaster testcrosses, we assume an XY chromosomal sex-determination system where sex chromosome segregation is mendelian. This means there are twice as many ${\text{F}}_2$ phenotypic classes (now including sex denoted h) and the probability of the recombination marker pattern $(l_1{l}_2{l}_3{l}_4{l}_5{l}_{6}h)$ for a six-point testcross must be further reduced by a factor of two:

$$Pr(l_1{l}_2{l}_3{l}_4{l}_5{l}_{6}h)=\frac{1}{4p}\mathbf{1}{\mathit{M}}_{\mathbf{1}}{\mathit{M}}_{\mathbf{2}}{\mathit{M}}_{\mathbf{3}}{\mathit{M}}_{\mathbf{4}}{\mathit{M}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}.$$ (C4 )

Appendix D: Probability of observing recombination marker patterns

Our notation changes are necessary because sex-specificity of viability effects causes the probability of observing a given recombination marker pattern that is to no longer a simple one-half thinning of the crossover patterning process in ${\text{F}}_1$ female meiosis. Sex-specific, marker-associated viability effects make detection probabilities become a function of sex chromosome segregation in ${\text{F}}_1$  male meiosis as well as subsequent sex-specific ${\text{F}}_2$ survival rates. To explicitly incorporate both random and marker-associated experimental mortality into the extended $Cx(Co{)}^m$ model, we reduce the probability of observing a given sex-specific recombination marker pattern by the proportional viability of the marker-free cross as well as each respective marker allele in the multiply-marked cross assuming a multiplicative fitness function.

Comparison of marker-free controls and multiply-marked testcrosses demonstrates that 36 of the 44% experimental mortality is due to sex-specific, marker-associated viability effects. The remaining 8% of zygotic lethality is unattributable to sex, maternal age, or marker effects. The unattributable experimental mortality is: (1) assumed to be random, (2) estimated a priori from absolute mortality in marker-free crosses, (3) denoted $1-v_x$ with v representing absolute viability, and 4) treated as a constant during $Cx(Co{)}^m$ model fitting.

Both goodness-of-fit G-tests and likelihood ratio tests (single-locus and multi-locus datasets, respectively, in the accompanying paper) indicate there were marker-associated viability effects of both mutant and wildtype alleles at all loci, and that these effects differed between the sexes. Therefore, the proportional viability ranging from 0 to 1 are denoted $v_{alh}$ representing the viability effect of allele a, at locus l, in sex h, and the probability of observing recombination pattern $(l_1{l}_2{l}_3{l}_4{l}_5{l}_6)$ in ${\text{F}}_2$ sex h is reduced by:

$$v_{l_1{l}_2{l}_3{l}_4{l}_5{l}_{6}h}=v_x\times v_{a1h}\times v_{a2h}\times v_{a3h}\times v_{a4h}\times v_{a5h}\times v_{a6h}.$$ (D1 )

Using this notation, all four multilocus viability models in the accompanying paper ($H_0$, $H_1$, $H_2$, and $H_3$) can be expressed by constraining one or more v parameters in equation (D1) to unity, which also allows straight-forward likelihood ratio testing of these nested hypotheses.

As an example using the full viability model $H_3$, the probability of forming a yellow-bodied ${\text{F}}_2$ male zygote with evidence of a single exchange between vermilion and forked in our study was:

$$Pr(111000m)=\frac{1}{4p}\mathbf{1}{\mathit{N}}_{\mathbf{1}}{\mathit{N}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{3}}{\mathit{N}}_{\mathbf{4}}{\mathit{N}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}=\mathrm{0.044.}$$ (D2 )

However, the probability of observing a yellow-bodied ${\text{F}}_2$ male (i.e. surviving egg-to-adulthood development to be scored for recombination) was only:

$$Pr(111000m)=\frac{1}{4p}\mathbf{1}{\mathit{N}}_{\mathbf{1}}{\mathit{N}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{3}}{\mathit{N}}_{\mathbf{4}}{\mathit{N}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{111000m}=\mathrm{0.017.}$$ (D3 )

Including the example above, there are 128 unique, sex-specific phenotypic classes in a 6-point testcross. Additionally, the digital record of eggs laid per vial allows calculation of the number of inviable zygotes as the ${129}^{\mathrm{th}}$ class. Again taking the example of a yellow-bodied ${\text{F}}_2$ male with evidence of a single exchange between vermilion and forked, the probability of such a fly occurring in our experiment but not surviving to adulthood was:

$$\frac{1}{4p}\mathbf{1}{\mathit{N}}_{\mathbf{1}}{\mathit{N}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{3}}{\mathit{N}}_{\mathbf{4}}{\mathit{N}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}(1-v_{111000m})=\mathrm{0.027.}$$ (D4 )

Summing across all sex-specific phenotypic classes the total probability of observing an inviable zygote can be expressed as:

$$Pr(inviable)=\sum _a\sum _l\sum _h\frac{1}{4p}\mathbf{1}{\mathit{M}}_{\mathbf{1}}{\mathit{M}}_{\mathbf{2}}{\mathit{M}}_{\mathbf{3}}{\mathit{M}}_{\mathbf{4}}{\mathit{M}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}(1-v_{l_1{l}_2{l}_3{l}_4{l}_5{l}_{6}h}).$$ (D5 )

While any one phenotypic class’s contribution to zygotic lethality is small, together they constitute 44% of the total flies scored in the experiment and must be accounted for in any attempt to infer the rate and distribution of crossing-over in ${\text{F}}_1$ meiosis from the genetic markers of surviving ${\text{F}}_2$ adults.

Appendix E: Maximum likelihood solution to full system of equations

Definition of the probabilities of observing any sex-specific ${\text{F}}_2$ adult phenotypic marker pattern in a recombination testcross, as well as the probability of an egg in that testcross failing to develop into an observable adult, allows the use maximum likelihood methods to estimate underlying meiotic parameters of the extended $Cx(Co{)}^m$ model. Following Zhao et al. (1995b), we use the multinomial likelihood function:

$$\mathcal{L}(\theta |\text{count data})=\frac{\sum _{i=1}^{129}{x}_i!}{\prod _{i=1}^{129}{x}_i!}\prod _{i=1}^{129}Pr(l_1{l}_2{l}_3{l}_4{l}_5{l}_{6}h{)}_i,$$ (E1 )

where θ is the set of the extended $Cx(Co{)}^m$ model parameters, the count data is a vector of 129 observations, $x_i$ is the count in the $i^{th}$ phenotypic class, and the $Pr(l_1{l}_2{l}_3{l}_4{l}_5{l}_{6}h{)}_i$ is the probability of observing the $i^{th}$ phenotypic class. For reference, we provide the first 6 and last 7 equations in the system of 129 equations to solve for maximum likelihood estimates of the extended $Cx(Co{)}^m$ model for a 6-point testcross:

$$\{\begin{array}{c}Pr(000000f)=\frac{1}{4p}\mathbf{1}{\mathit{N}}_{\mathbf{1}}{\mathit{N}}_{\mathbf{2}}{\mathit{N}}_{\mathbf{3}}{\mathit{N}}_{\mathbf{4}}{\mathit{N}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{000000f}\\ Pr(111111f)=\frac{1}{4p}\mathbf{1}{\mathit{N}}_{\mathbf{1}}{\mathit{N}}_{\mathbf{2}}{\mathit{N}}_{\mathbf{3}}{\mathit{N}}_{\mathbf{4}}{\mathit{N}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{111111f}\\ Pr(100000f)=\frac{1}{4p}\mathbf{1}{\mathit{R}}_{\mathbf{1}}{\mathit{N}}_{\mathbf{2}}{\mathit{N}}_{\mathbf{3}}{\mathit{N}}_{\mathbf{4}}{\mathit{N}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{100000f}\\ Pr(011111f)=\frac{1}{4p}\mathbf{1}{\mathit{R}}_{\mathbf{1}}{\mathit{N}}_{\mathbf{2}}{\mathit{N}}_{\mathbf{3}}{\mathit{N}}_{\mathbf{4}}{\mathit{N}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{011111f}\\ Pr(110000f)=\frac{1}{4p}\mathbf{1}{\mathit{N}}_{\mathbf{1}}{\mathit{R}}_{\mathbf{2}}{\mathit{N}}_{\mathbf{3}}{\mathit{N}}_{\mathbf{4}}{\mathit{N}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{110000f}\\ Pr(001111f)=\frac{1}{4p}\mathbf{1}{\mathit{N}}_{\mathbf{1}}{\mathit{R}}_{\mathbf{2}}{\mathit{N}}_{\mathbf{3}}{\mathit{N}}_{\mathbf{4}}{\mathit{N}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{001111f}\\ \cdots \\ \cdots \\ \cdots \\ Pr(011010m)=\frac{1}{4p}\mathbf{1}{\mathit{R}}_{\mathbf{1}}{\mathit{N}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{3}}{\mathit{R}}_{\mathbf{4}}{\mathit{R}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{011010m}\\ Pr(100101m)=\frac{1}{4p}\mathbf{1}{\mathit{R}}_{\mathbf{1}}{\mathit{N}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{3}}{\mathit{R}}_{\mathbf{4}}{\mathit{R}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{100101m}\\ Pr(001010m)=\frac{1}{4p}\mathbf{1}{\mathit{N}}_{\mathbf{1}}{\mathit{R}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{3}}{\mathit{R}}_{\mathbf{4}}{\mathit{R}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{001010m}\\ Pr(110101m)=\frac{1}{4p}\mathbf{1}{\mathit{N}}_{\mathbf{1}}{\mathit{R}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{3}}{\mathit{R}}_{\mathbf{4}}{\mathit{R}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{110101m}\\ Pr(101010m)=\frac{1}{4p}\mathbf{1}{\mathit{R}}_{\mathbf{1}}{\mathit{R}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{3}}{\mathit{R}}_{\mathbf{4}}{\mathit{R}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{101010m}\\ Pr(010101m)=\frac{1}{4p}\mathbf{1}{\mathit{R}}_{\mathbf{1}}{\mathit{R}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{3}}{\mathit{R}}_{\mathbf{4}}{\mathit{R}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}{v}_{010101m}\\ Pr(inviable)=\sum _a\sum _l\sum _h\frac{1}{4p}\mathbf{1}{\mathit{M}}_{\mathbf{1}}{\mathit{M}}_{\mathbf{2}}{\mathit{M}}_{\mathbf{3}}{\mathit{M}}_{\mathbf{4}}{\mathit{M}}_{\mathbf{5}}{\mathbf{1}}^{\mathbf{\prime }}(1-v_{l_1{l}_2{l}_3{l}_4{l}_5{l}_{6}h}),\\ \end{array}$$

where all variables, vectors, matrices, and labels retain the definitions given in previous sections of this Appendix.

Appendix F: Simulation of sampling distributions

As stated above, the parameters of the data-generating process and the underlying model of meiosis can be estimated by maximum likelihood. The same data-generating process can be used to simulate the sampling distribution of any summary statistic or meiotic parameter of interest in that dataset. This is particularly useful for estimating confidence intervals as well as statistical comparisons between biological groups. The stated assumptions of the extended $Cx(Co{)}^m$ model provide a step-by-step guide for this process.

• First, the number and spacing of programmed double-strand breaks (C events) are simulated along the chromosome as a one-dimensional Poisson point process on the real half-line representing the genetic map. The rate of this process is given by $Cx(Co{)}^m$ estimates of y. Each C is assigned independently to one of the four strands with equal probability. This procedure explicitly assumes no genetic variation for programmed double-strand break placement.

• Second, $1/p$ of the C events mature into crossovers (Cx events) following the $Cx(Co{)}^m$ renewal process using the estimated m. This counting process is initiated at the centromere and has a randomly assigned starting point in the $Cx(Co{)}^m$ sequence to satisfy the stationarity assumption. Template strand for crossover repair is randomly assigned to one of two homologous chromosomes, conditional on the C event strand identity, with equal probability. This procedure explicitly assumes no chromatid interference and no sister chromatid exchange.

• Third, one of the four chromatids in the four-strand bundle is randomly selected with equal probability as the meiotic product segregating to the functional egg pole and ultimately being included in oocyte pronucleus. This procedure explicitly excludes nondisjunction and assumes no meiotic drive.

• Fourth, fertilization of the oocyte by X-bearing sperm versus Y-bearing sperm, and therefore the sex of the resulting zygote, is assigned with equal probabilities. This procedure assumes a chromosomal sex determination system with mendelian segregation of X and Y chromosomes in males.

• Fifth, zygotes are randomly classified as viable or inviable based on sex-specific, marker-associated viability effects estimated from the extended $Cx(Co{)}^m$ model. This requires each zygote be randomly assigned one of two equally likely reciprocal marker complements, and the probability of viable/inviable classification is dependent on the marker complement assignment.

• Finally, among the zygotes classified as viable conditional on reciprocal marker complement assignment, a second marker-independent round of viability selection was simulated. The probability of being reclassified as inviable was based on the empirical zygotic lethality rates estimated in marker-free control crosses in the accompanying paper.

This algorithm generates one data point classifiable as either one of the sex-specific phenotypic classes scored for recombination or an inviable zygote. Iteration of the process n times to match number of eggs laid in a given dataset can be solved for the maximum likelihood solution of parameter values in the extended $Cx(Co{)}^m$ model, giving a single simulated estimate of the parameters from the data-generating process. Repeating this process, for example 10,000 times, builds an empirical sampling distribution tailored to the experimental design of the dataset and allows construction of null hypothesis tests and confidence intervals.

Data availability

Strains are available upon request. Supplemental file S1 contains detailed strain construction notes as well as descriptions of all tables and figures in supplemental file S1. Raw count data, R code for all analyses, and further documentation of model structure and fitting procedures have been integrated into a GitHub repository with a static DOI (Koury and Stevison 2026) which can be found at https://github.com/StevisonLab/CxCoM-with-Viability.

Supplemental material available at GENETICS online.

Funding

This research was supported by the National Institute of General Medical Sciences of the National Institutes of Health under award number R35GM147501. The content is solely the responsibility of the authors and does not necessarily represent the official view of the National Institutes of Health (NIH).