Classifications and comparisons of multilocus recombination distributions

Predicting recombination frequency from map distance

Map distance is one of the key measures in genetics and indicates the expected number of crossovers between two loci. Map distance is estimated from the observed recombination frequency using mapping functions, the most widely used of those, Haldane and Kosambi, being developed at the time when the number of markers was low and unobserved crossovers had a substantial effect on the recombination fractions. In contemporary high-density marker data, the probability of multiple crossovers between adjacent loci is negligible and different mapping functions yield the same result, that is, the recombination frequency between adjacent loci is equal to the map distance in Morgans. However, high-density linkage maps contain an interpretation problem: the map distance over a long interval is additive and its association with recombination frequency is not defined. Here, we demonstrate with high-density linkage maps from humans and stickleback fishes that the inverses of Haldane's and Kosambi's mapping functions systematically underpredict recombination frequencies from map distance. To remedy this, we formulate a piecewise function that yields more accurate predictions of recombination frequency from map distance. Our results demonstrate that the association between map distance and recombination frequency is context-dependent and without a universal solution.

Strategies and considerations for implementing genomic selection to improve traits with additive and non-additive genetic architectures in sugarcane breeding

Simulations highlight the potential of genomic selection to substantially increase genetic gain for complex traits in sugarcane. The success rate depends on the trait genetic architecture and the implementation strategy. Genomic selection (GS) has the potential to increase the rate of genetic gain in sugarcane beyond the levels achieved by conventional phenotypic selection (PS). To assess different implementation strategies, we simulated two different GS-based breeding strategies and compared genetic gain and genetic variance over five breeding cycles to standard PS. GS scheme 1 followed similar routines like conventional PS but included three rapid recurrent genomic selection (RRGS) steps. GS scheme 2 also included three RRGS steps but did not include a progeny assessment stage and therefore differed more fundamentally from PS. Under an additive trait model, both simulated GS schemes achieved annual genetic gains of 2.6-2.7% which were 1.9 times higher compared to standard phenotypic selection (1.4%). For a complex non-additive trait model, the expected annual rates of genetic gain were lower for all breeding schemes; however, the rates for the GS schemes (1.5-1.6%) were still greater than PS (1.1%). Investigating cost-benefit ratios with regard to numbers of genotyped clones showed that substantial benefits could be achieved when only 1500 clones were genotyped per 10-year breeding cycle for the additive genetic model. Our results show that under a complex non-additive genetic model, the success rate of GS depends on the implementation strategy, the number of genotyped clones and the stage of the breeding program, likely reflecting how changes in QTL allele frequencies change additive genetic variance and therefore the efficiency of selection. These results are encouraging and motivate further work to facilitate the adoption of GS in sugarcane breeding.

isqg: A Binary Framework for in Silico Quantitative Genetics

The dna is the fundamental basis of genetic information, just as bits are for computers. Whenever computers are used to represent genetic data, the computational encoding must be efficient to allow the representation of processes driving the inheritance and variability. This is especially important across simulations in view of the increasing complexity and dimensions brought by genomics. This paper introduces a new binary representation of genetic information. Algorithms as bitwise operations that mimic the inheritance of a wide range of polymorphisms are also presented. Different kinds and mixtures of polymorphisms are discussed and exemplified. Proposed algorithms and data structures were implemented in C++ programming language and is available to end users in the R package “isqg” which is available at the R repository (cran). Supplementary data are available online.

A natural class of multilocus recombination processes and related measures of crossover interference

Various classifications and representations of multilocus recombination structures are delimited. A class of recombination distributions called the count–location chiasma process is parametrized by a distribution of the number of crossover events and for such crossover events by a conditional distribution of crossover locations. A number of properties and examples of this recombination structure are developed connecting orderings among the recombination mapping functions and the nature of interference.

Principles of polymorphism and epistasis for multilocus systems

The nature of stable equilibrium configurations is described for general nonepistatic and generalized symmetric viability regimes in multilocus systems under conditions of tight and loose linkage. The influence of epistasis and symmetry can be better understood in terms of these standards. A dichotomy in the nature of stable polymorphisms emerges. More recombination, bisexuality, and multideme interactions facilitate the establishment of central type polymorphisms.

Analysis of central equilibrium configurations for certain multi-locus systems in subdivided populations

The multi-locus systems expressing non-epistatic and generalized symmetric selection lend themselves to the study of the stability of certain central polymorphic equilibria. These equilibria persist when any form of migration connects demes which share a common equilibrium. The analysis of the stability of the equilibrium in the global system is tractable, thus supplementing known protection results for two alleles at one locus with stability conditions on an internal equilibrium involving an arbitrary number of loci, each with an arbitrary number of alleles. Two of the principal findings are that stability of central Hardy–Weinberg type equilibria increase with ‘more’ migration and ‘more’ recombination. As a corollary, local stability in each deme implies stability in a system with migration superimposed; but instability in each deme when isolated does not imply instability when migration is superimposed.

Patterns of recombination and MLH1 foci density along mouse chromosomes: modeling effects of interference and obligate chiasma

Crossover interference in meiosis is often modeled via stationary renewal processes. Here we consider a new model to incorporate the known biological feature of "obligate chiasma" whereby in most organisms each bivalent almost always has at least one crossover. The initial crossover is modeled as uniformly distributed along the chromosome, and starting from its position, subsequent crossovers are placed with forward and backward stationary renewal processes using a chi-square distribution of intercrossover distances. We used our model as well as the standard chi-square model to simulate the patterns of crossover densities along bivalents or chromatids for those having zero, one, two, or three or more crossovers; indeed, such patterns depend on the number of crossovers. With both models, simulated patterns compare very well to those found experimentally in mice, both for MLH1 foci on bivalents and for crossovers on genetic maps. However, our model provides a better fit to experimental data as compared to the standard chi-square model, particularly regarding the distribution of numbers of crossovers per chromosome. Finally, our model predicts an enhancement of the recombination rate near the extremities, which, however, explains only a part of the pattern observed in mouse.

The length of the intact donor chromosome segment around a target gene in marker-assisted backcrossing

Recurrent backcrossing is an established procedure to transfer target genes from a donor into the genetic background of a recipient genotype. By assessing the parental origin of alleles at markers flanking the target locus one can select individuals with a short intact donor chromosome segment around the target gene and thus reduce the linkage drag. We investigated the probability distribution of the length of the intact donor chromosome segment around the target gene in recurrent backcrossing with selection for heterozygosity at the target locus and homozygosity for the recurrent parent allele at flanking markers for a diploid species. Assuming no interference in crossover formation, we derived the cumulative density function, probability density function, expected value, and variance of the length of the intact chromosome segment for the following cases: (1) backcross generations prior to detection of a recombinant individual between the target gene and the flanking marker; (2) the backcross generation in which for the first time a recombinant individual is detected, which is selected for further backcrossing; and (3) subsequent backcross generations after selection of a recombinant. Examples are given of how these results can be applied to investigate the efficiency of marker-assisted backcrossing for reducing the length of the intact donor chromosome segment around the target gene under various situations relevant in breeding and genetic research.

The relationship between count-location and stationary renewal models for the chiasma process

It is often convenient to define models for the process of chiasma formation at meiosis as stationary renewal models. However, count-location models are also useful, particularly to capture the biological requirement of at least one chiasma per chromosome. The Sturt model and truncated Poisson model are both count-location models with this feature. We show that the truncated Poisson model can also be expressed as a stationary renewal model, while the Sturt model cannot. More generally, we show that there is only one family of count-location models for the chiasma process that can also be expressed as stationary renewal models. The models in this family can exhibit either positive or negative interference.

Characterization of human crossover interference

We present an analysis of crossover interference over the entire human genome, on the basis of genotype data from more than 8,000 polymorphisms in eight CEPH families. Overwhelming evidence was found for strong positive crossover interference, with average strength lying between the levels of interference implied by the Kosambi and Carter-Falconer map functions. Five mathematical models of interference were evaluated: the gamma model and four versions of the count-location model. The gamma model fit the data far better than did any of the other four models. Analysis of intercrossover distances was greatly superior to the analysis of crossover counts, in both demonstrating interference and distinguishing between the five models. In contrast to earlier suggestions, interference was found to continue uninterrupted across the centromeres. No convincing differences in the levels of interference were found between the sexes or among chromosomes; however, we did detect possible individual variation in interference among the eight mothers. Finally, we present an equation that provides the probability of the occurrence of a double crossover between two nonrecombinant, informative polymorphisms.

A chromosome-based method to infer IBD scores for missing and ambiguous markers

We propose a probability model to impute missing identical-by-descent (IBD) vectors for linkage analysis, when adjacent marker loci are typed and interference is estimable. A chromosome-based IBD distribution, conditioned on available marker data, is computed using a fast algorithm to estimate the joint probability of genes IBD at several equally spaced linked loci. Weighted IBD vectors are then used in various test statistics for linkage analysis. As an example, we analyzed the 18 affected sib pairs in the GAW9 Problem 1 data set using Risch's lod-score test.

Robustness of the no-interference model for ordering genetic markers

Under the assumption of no chromatid interference, we derive constraints on the probabilities of the different recombination patterns among m + 1 genetic loci. An application of these constraints is a proof that the ordering of the loci that maximizes the likelihood under the assumption of no interference is, in fact, a consistent estimator of the true order even when there is interference.

The generalized multiplicative model for viability selection at multiple loci

Selection due to differential viability is studied in an n-locus two-allele model using a set indexation that allows the simplicity of the one-locus two-allele model to be carried to multi-locus models. The existence condition is analyzed for polymorphic equilibria with linkage equilibrium: Robbins' equilibria. The local stability condition is given for the Robbins' equilibria on the boundaries in the generalized non-epistatic selection regimes of Karlin and Liberman (1979). These generalized non-epistatic regimes include the additive selection model, the multiplicative selection model and the multiplicative interaction model, and their symmetric versions cover all the symmetric viability models.

Epistasis in the multiple locus symmetric viability model

The n-locus two-allele symmetric viability model is considered in terms of the parameters measuring the additive epistasis in fitness. The dynamics is analysed using a simple linear transformation of the gametic frequencies, and then the recurrence equations depend on the epistatic parameters and Geiringer's recombination distribution only. The model exhibits an equilibrium, the central equilibrium, where the 2n gametes are equally frequent. The transformation simplifies the stability analysis of the central point, and provides the stability conditions in terms of the existence conditions of other equilibria. For total negative epistasis (all epistatic parameters are negative) the central point is stable for all recombination distributions. For free recombination either a central point (segregating one, two, ... or n loci) or the n-locus fixation states are stable. For no recombination and some epistatic parameters positive the central point is unstable and several boundary equilibria may be locally stable. The sign structure of the additive epistasis is therefore an important determinant of the dynamics of the n-locus symmetric viability model. The non-symmetric multiple locus models previously analysed are dynamically related, and they all have an epistatic sign structure that resembles that of the multiplicative viability model. A non-symmetric model with total negative epistasis which share dynamical properties with the similar symmetric model is suggested.

Multilocus linkage analysis in humans: detection of linkage and estimation of recombination

Multilocus linkage analysis is investigated from the viewpoint of the efficiency of recombination estimates under different strategies for detecting linkage and determining gene order within a linkage group. We consider the appropriateness of assuming no interference with data available in human genetic studies. Examples are given to show the significance of multilocus analysis in humans. A computer program package, LINKAGE, for multilocus linkage analysis is described.

On chiasma formation point processes having the count location property

The count-location (C-L) chiasma formation schemes introduced by Karlin and Liberman (1979b) encompass a broad class of map functions involving positive, negative or no chiasma interference. The C-L schemes do not explicitly assume a specific mechanism of crossover formation, but rather a statistical property of the process. If viewed as a stochastic point process along the chromosome, it is shown that a crossing over mechanism having the C-L property is actually a rescaled mixture of Poisson processes. Surprisingly it turns out that these C-L point processes involve negative interference throughout the entire genome.

Preferential mating in symmetric multilocus systems

A class of multilocus models that incorporate both preferential mating and viability selection is studied. Symmetry in alleles is supposed, resulting in the phenotypes being dependent only on the location of heterozygous loci. Otherwise, an arbitrary number of loci, number of alleles per locus, and arbitrary recombination schemes, viability parameters, and preferential mating pattern are allowed. The conditions for stability of a central polymorphism, c(*), are indicated and interpreted. Mating and viability parameters enter as one combined quantity for each phenotypic class, which represents a generalized fitness. The effect on stability of c(*) of increasing the number of alleles per locus and the number of loci requires the formulation to be set in terms of frequency-dependent preference parameters.

The effect of migration and recombination on the equilibrium structure of populations subject to a common symmetric selection regime

The effect of migration and recombination on the equilibrium structure of populations subject to a common symmetric selection regime in all habitats is studied. Attention is restricted to a class of symmetric polymorphic equilibria which have been studied in two-deme systems by Bazykin (1972) and Karlin & McGregor (1972) for one locus and by Christiansen & Feldman (1975) for two loci. With increased migration and recombination the heterozygosity increases unless it is already at the maximum level. Although the populaton system as a whole is always at linkage equilibrium, the magnitude of linkage disequilibrium in the individual demes may either increase or decrease with more migration and recombination. In general, the less the migration and the less the recombination between interacting loci, the greater the possibilities of polymorphic equilibria.