Singular effect of linkage on long-term genetic gain in Fisher's infinitesimal model.

During the founding of the field of quantitative genetics, Fisher formulated in 1918 his "infinitesimal model" that provided a novel mathematical framework to describe the Mendelian transmission of quantitative traits. If the infinitely many genes in that model are assumed to segregate independently during reproduction, corresponding to having no linkage, directional selection asymptotically leads to a constant genetic gain at each generation. In reality, genes are subject to strong linkage because they lie on chromosomes and thus segregate in a correlated way. Various approximations have been used in the past to study that more realistic case of the infinitesimal model with the expectation that the asymptotic gain per generation is modestly decreased. To treat this system even in the strong linkage limit, we take the genes to lie on continuous chromosomes. Surprisingly, the consequences of genetic linkage are in fact rather singular, changing the nature of the long-term gain per generation: the asymptotic gain vanishes rather than being simply decreased. Nevertheless, the per-generation gain tends to zero sufficiently slowly for the total gain, accumulated over generations, to be unbounded.

Biologically meaningful regulatory logic enhances the convergence rate in Boolean networks and bushiness of their state transition graph.

Boolean models of gene regulatory networks (GRNs) have gained widespread traction as they can easily recapitulate cellular phenotypes via their attractor states. Their overall dynamics are embodied in a state transition graph (STG). Indeed, two Boolean networks (BNs) with the same network structure and attractors can have drastically different STGs depending on the type of Boolean functions (BFs) employed. Our objective here is to systematically delineate the effects of different classes of BFs on the structural features of the STG of reconstructed Boolean GRNs while keeping network structure and biological attractors fixed, and explore the characteristics of BFs that drive those features. Using $10$ reconstructed Boolean GRNs, we generate ensembles that differ in BFs and compute from their STGs the dynamics' rate of contraction or 'bushiness' and rate of 'convergence', quantified with measures inspired from cellular automata (CA) that are based on the garden-of-Eden (GoE) states. We find that biologically meaningful BFs lead to higher STG 'bushiness' and 'convergence' than random ones. Obtaining such 'global' measures gets computationally expensive with larger network sizes, stressing the need for feasible proxies. So we adapt Wuensche's $Z$-parameter in CA to BFs in BNs and provide four natural variants, which, along with the average sensitivity of BFs computed at the network level, comprise our descriptors of local dynamics and we find some of them to be good proxies for bushiness. Finally, we provide an excellent proxy for the 'convergence' based on computing transient lengths originating at random states rather than GoE states.

Preponderance of generalized chain functions in reconstructed Boolean models of biological networks.

Boolean networks (BNs) have been extensively used to model gene regulatory networks (GRNs). The dynamics of BNs depend on the network architecture and regulatory logic rules (Boolean functions (BFs)) associated with nodes. Nested canalyzing functions (NCFs) have been shown to be enriched among the BFs in the large-scale studies of reconstructed Boolean models. The central question we address here is whether that enrichment is due to certain sub-types of NCFs. We build on one sub-type of NCFs, the chain functions (or chain-0 functions) proposed by Gat-Viks and Shamir. First, we propose two other sub-types of NCFs, namely, the class of chain-1 functions and generalized chain functions, the union of the chain-0 and chain-1 types. Next, we find that the fraction of NCFs that are chain-0 (also holds for chain-1) functions decreases exponentially with the number of inputs. We provide analytical treatment for this and other observations on BFs. Then, by analyzing three different datasets of reconstructed Boolean models we find that generalized chain functions are significantly enriched within the NCFs. Lastly we illustrate that upon imposing the constraints of generalized chain functions on three different GRNs we are able to obtain biologically viable Boolean models.

Best practices for the execution, analysis, and data storage of plant single-cell/nucleus transcriptomics.

Single-cell and single-nucleus RNA-sequencing technologies capture the expression of plant genes at an unprecedented resolution. Therefore, these technologies are gaining traction in plant molecular and developmental biology for elucidating the transcriptional changes across cell types in a specific tissue or organ, upon treatments, in response to biotic and abiotic stresses, or between genotypes. Despite the rapidly accelerating use of these technologies, collective and standardized experimental and analytical procedures to support the acquisition of high-quality data sets are still missing. In this commentary, we discuss common challenges associated with the use of single-cell transcriptomics in plants and propose general guidelines to improve reproducibility, quality, comparability, and interpretation and to make the data readily available to the community in this fast-developing field of research.

Leveraging developmental landscapes for model selection in Boolean gene regulatory networks.

Boolean models are a well-established framework to model developmental gene regulatory networks (DGRNs) for acquisition of cellular identities. During the reconstruction of Boolean DGRNs, even if the network structure is given, there is generally a large number of combinations of Boolean functions that will reproduce the different cell fates (biological attractors). Here we leverage the developmental landscape to enable model selection on such ensembles using the relative stability of the attractors. First we show that previously proposed measures of relative stability are strongly correlated and we stress the usefulness of the one that captures best the cell state transitions via the mean first passage time (MFPT) as it also allows the construction of a cellular lineage tree. A property of great computational importance is the insensitivity of the different stability measures to changes in noise intensities. That allows us to use stochastic approaches to estimate the MFPT and thereby scale up the computations to large networks. Given this methodology, we revisit different Boolean models of Arabidopsis thaliana root development, showing that a most recent one does not respect the biologically expected hierarchy of cell states based on relative stabilities. We therefore developed an iterative greedy algorithm that searches for models which satisfy the expected hierarchy of cell states and found that its application to the root development model yields many models that meet this expectation. Our methodology thus provides new tools that can enable reconstruction of more realistic and accurate Boolean models of DGRNs.

SeSAM: software for automatic construction of order-robust linkage maps.

BACKGROUND: Genotyping and sequencing technologies produce increasingly large numbers of genetic markers with potentially high rates of missing or erroneous data. Therefore, the construction of linkage maps is more and more complex. Moreover, the size of segregating populations remains constrained by cost issues and is less and less commensurate with the numbers of SNPs available. Thus, guaranteeing a statistically robust marker order requires that maps include only a carefully selected subset of SNPs. RESULTS: In this context, the SeSAM software allows automatic genetic map construction using seriation and placement approaches, to produce (1) a high-robustness framework map which includes as many markers as possible while keeping the order robustness beyond a given statistical threshold, and (2) a high-density total map including the framework plus almost all polymorphic markers. During this process, care is taken to limit the impact of genotyping errors and of missing data on mapping quality. SeSAM can be used with a wide range of biparental populations including from outcrossing species for which phases are inferred on-the-fly by maximum-likelihood during map elongation. The package also includes functions to simulate data sets, convert data formats, detect putative genotyping errors, visualize data and map quality (including graphical genotypes), and merge several maps into a consensus. SeSAM is also suitable for interactive map construction, by providing lower-level functions for 2-point and multipoint EM analyses. The software is implemented in a R package including functions in C++. CONCLUSIONS: SeSAM is a fully automatic linkage mapping software designed to (1) produce a framework map as robust as desired by optimizing the selection of a subset of markers, and (2) produce a high-density map including almost all polymorphic markers. The software can be used with a wide range of biparental mapping populations including cases from outcrossing. SeSAM is freely available under a GNU GPL v3 license and works on Linux, Windows, and macOS platforms. It can be downloaded together with its user-manual and quick-start tutorial from ForgeMIA (SeSAM project) at https://forgemia.inra.fr/gqe-acep/sesam/-/releases.

Relative importance of composition structures and biologically meaningful logics in bipartite Boolean models of gene regulation.

Boolean networks have been widely used to model gene networks. However, such models are coarse-grained to an extent that they abstract away molecular specificities of gene regulation. Alternatively, bipartite Boolean network models of gene regulation explicitly distinguish genes from transcription factors (TFs). In such bipartite models, multiple TFs may simultaneously contribute to gene regulation by forming heteromeric complexes, thus giving rise to composition structures. Since bipartite Boolean models are relatively recent, an empirical investigation of their biological plausibility is lacking. Here, we estimate the prevalence of composition structures arising through heteromeric complexes. Moreover, we present an additional mechanism where composition structures may arise as a result of multiple TFs binding to cis-regulatory regions and provide empirical support for this mechanism. Next, we compare the restriction in BFs imposed by composition structures and by biologically meaningful properties. We find that though composition structures can severely restrict the number of Boolean functions (BFs) driving a gene, the two types of minimally complex BFs, namely nested canalyzing functions (NCFs) and read-once functions (RoFs), are comparatively more restrictive. Finally, we find that composition structures are highly enriched in real networks, but this enrichment most likely comes from NCFs and RoFs.

A preference for link operator functions can drive Boolean biological networks towards critical dynamics.

Boolean modelling is a powerful framework to understand the operating principles of biological networks. The regulatory logic between biological entities in these networks is expressed as Boolean functions (BFs). There exist various types of BFs (and thus regulatory logic rules) which are meaningful in the biological context. In this contribution, we explore one such type, known as link operator functions (LOFs). We theoretically enumerate these functions and show that, among all BFs and even within the biologically consistent effective and unate functions (EUFs), the LOFs form a tiny subset. We then find that the AND-NOT LOFs are particularly abundant in reconstructed biological Boolean networks. By leveraging these facts, namely, the tiny representation of LOFs in the space of EUFs and their presence in the biological dataset, we show that the space of acceptable models can be shrunk by applying steady-state constraints to BFs, followed by the choice of LOFs which satisfy those constraints. Finally, we demonstrate that among a wide range of BFs, the LOFs drive biological network dynamics towards criticality.

Quantitative modelling of fine-scale variations in the Arabidopsis thaliana crossover landscape.

In, essentially, all species where meiotic crossovers (COs) have been studied, they occur preferentially in open chromatin, typically near gene promoters and to a lesser extent, at the end of genes. Here, in the case of Arabidopsis thaliana, we unveil further trends arising when one considers contextual information, namely summarised epigenetic status, gene or intergenic region size, and degree of divergence between homologs. For instance, we find that intergenic recombination rate is reduced if those regions are less than 1.5 kb in size. Furthermore, we propose that the presence of single nucleotide polymorphisms enhances the rate of CO formation compared to when homologous sequences are identical, in agreement with previous works comparing rates in adjacent homozygous and heterozygous blocks. Lastly, by integrating these different effects, we produce a quantitative and predictive model of the recombination landscape that reproduces much of the experimental variation.

Minimum complexity drives regulatory logic in Boolean models of living systems.

The properties of random Boolean networks have been investigated extensively as models of regulation in biological systems. However, the Boolean functions (BFs) specifying the associated logical update rules should not be expected to be random. In this contribution, we focus on biologically meaningful types of BFs, and perform a systematic study of their preponderance in a compilation of 2,687 functions extracted from published models. A surprising feature is that most of these BFs have odd "bias", that is they produce "on" outputs for a total number of input combinations that is odd. Upon further analysis, we are able to explain this observation, along with the enrichment of read-once functions (RoFs) and its nested canalyzing functions (NCFs) subset, in terms of 2 complexity measures: Boolean complexity based on string lengths in formal logic, which is yet unexplored in biological contexts, and the so-called average sensitivity. RoFs minimize Boolean complexity and all such functions have odd bias. Furthermore, NCFs minimize not only the Boolean complexity but also the average sensitivity. These results reveal the importance of minimum complexity in the regulatory logic of biological networks.

Enhancing backcross programs through increased recombination.

BACKGROUND: Introgression of a quantitative trait locus (QTL) by successive backcrosses is used to improve elite lines (recurrent parent) by introducing alleles from exotic material (donor parent). In the absence of selection, the proportion of the donor genome decreases by half at each generation. However, since selection is for the donor allele at the QTL, elimination of the donor genome around that QTL will be much slower than in the rest of the genome (i.e. linkage drag). Using markers to monitor the genome around the QTL and in the genetic background can accelerate the return to the recurrent parent genome. Successful introgression of a locus depends partly on the occurrence of crossovers at favorable positions. However, the number of crossovers per generation is limited and their distribution along the genome is heterogeneous. Recently, techniques have been developed to modify these two recombination parameters. RESULTS: In this paper, we assess, by simulations in the context of Brassicaceae, the effect of increased recombination on the efficiency of introgression programs by studying the decrease in linkage drag and the recovery of the recurrent genome. The simulated selection schemes begin by two generations of foreground selection and continue with one or more generations of background selection. Our results show that, when the QTL is in a region that initially lacked crossovers, an increase in recombination rate can decrease linkage drag by nearly ten-fold after the foreground selection and improves the return to the recurrent parent. However, if the QTL is in a region that is already rich in crossovers, an increase in recombination rate is detrimental. CONCLUSIONS: Depending on the recombination rate in the region targeted for introgression, increasing it can be beneficial or detrimental. Thus, the simulations analysed in this paper help us understand how an increase in recombination rate can be beneficial. They also highlight the best methods that can be used to increase recombination rate, depending on the situation.

Plant roots sense soil compaction through restricted ethylene diffusion.

Soil compaction represents a major challenge for modern agriculture. Compaction is intuitively thought to reduce root growth by limiting the ability of roots to penetrate harder soils. We report that root growth in compacted soil is instead actively suppressed by the volatile hormone ethylene. We found that mutant Arabidopsis and rice roots that were insensitive to ethylene penetrated compacted soil more effectively than did wild-type roots. Our results indicate that soil compaction lowers gas diffusion through a reduction in air-filled pores, thereby causing ethylene to accumulate in root tissues and trigger hormone responses that restrict growth. We propose that ethylene acts as an early warning signal for roots to avoid compacted soils, which would be relevant to research into the breeding of crops resilient to soil compaction.

CNVmap: A Method and Software To Detect and Map Copy Number Variants from Segregation Data.

Single nucleotide polymorphisms (SNPs) are used widely for detecting quantitative trait loci, or for searching for causal variants of diseases. Nevertheless, structural variations such as copy-number variants (CNVs) represent a large part of natural genetic diversity, and contribute significantly to trait variation. Numerous methods and softwares based on different technologies (amplicons, CGH, tiling, or SNP arrays, or sequencing) have already been developed to detect CNVs, but they bypass a wealth of information such as genotyping data from segregating populations, produced, e.g., for QTL mapping. Here, we propose an original method to both detect and genetically map CNVs using mapping panels. Specifically, we exploit the apparent heterozygous state of duplicated loci: peaks in appropriately defined genome-wide allelic profiles provide highly specific signatures that identify the nature and position of the CNVs. Our original method and software can detect and map automatically up to 33 different predefined types of CNVs based on segregation data only. We validate this approach on simulated and experimental biparental mapping panels in two maize populations and one wheat population. Most of the events found correspond to having just one extra copy in one of the parental lines, but the corresponding allelic value can be that of either parent. We also find cases with two or more additional copies, especially in wheat, where these copies locate to homeologues. More generally, our computational tool can be used to give additional value, at no cost, to many datasets produced over the past decade from genetic mapping panels.

Probabilities of Multilocus Genotypes in SIB Recombinant Inbred Lines.

Recombinant Inbred Lines (RILs) are obtained through successive generations of inbreeding. In 1931 Haldane and Waddington published a landmark paper where they provided the probabilities of achieving any combination of alleles in 2-way RILs for 2 and 3 loci. In the case of sibling RILs where sisters and brothers are crossed at each generation, there has been no progress in treating 4 or more loci, a limitation we overcome here without much increase in complexity. In the general situation of L loci, the task is to determine 2 L probabilities, but we find that it is necessary to first calculate the 4 L "identical by descent" (IBD) probabilities that a RIL inherits at each locus its DNA from one of the four originating chromosomes. We show that these 4 L probabilities satisfy a system of linear equations that follow from self-consistency. In the absence of genetic interference-crossovers arising independently-the associated matrix can be written explicitly in terms of the recombination rates between the different loci. We provide the matrices for L up to 4 and also include a computer program to automatically generate the matrices for higher values of L. Furthermore, our framework can be generalized to recombination rates that are different in female and male meiosis which allows us to show that the Haldane and Waddington 2-locus formula is valid in that more subtle case if the meiotic recombination rate is taken as the average rate across female and male. Once the 4 L IBD probabilities are determined, the 2 L probabilities of RIL genotypes are obtained via summations of these quantities. In fine, our computer program allows to determine the probabilities of all the multilocus genotypes produced in such sibling-based RILs for L<=10, a huge leap beyond the L = 3 restriction of Haldane and Waddington.

Assessing by Modeling the Consequences of Increased Recombination in Recurrent Selection of Oryza sativa and Brassica rapa.

Meiotic recombination generates genetic diversity but in most species the number of crossovers per meiosis is limited. Previous modeling studies showed that increasing recombination can enhance response to selection. However, such studies did not assume a specific method of modifying recombination. Our objective was to test whether two methods used to increase recombination in plants could increase genetic gain in a population undergoing recurrent selection such as in genomic selection programs. The first method, in Oryza sativa, used a mutant of anti-crossover genes, increasing global recombination without affecting the recombination landscape shape. The second one used the ploidy level of a cross between Brassica rapa and Brassica napus, increasing recombination especially in pericentromeric regions. Our modeling framework used these recombination landscapes and sampled quantitative trait loci positions from the actual gene distributions. We simulated selection programs with initially a cross between two inbred lines, for two species. Increased recombination enhanced the response to selection. The amount of enhancement in the cumulative gain largely depended on the species and the number of quantitative trait loci (2, 10, 20, 50, 200 or 1000 per chromosome). Genetic gains were increased up to 30% after 20 generations. Furthermore, increasing recombination in cold regions was the most effective: the gain was larger by 25% with the first method and 34% with the second one in B. rapa, and 12% compared to 16% in O. sativa In summary, increased recombination enhances the genetic gain in long-term selection programs, with visible effects after four to five generations.

Recombination suppression in heterozygotes for a pericentric inversion induces the interchromosomal effect on crossovers in Arabidopsis.

During meiosis, recombination ensures allelic exchanges through crossovers (COs) between the homologous chromosomes. Advances in our understanding of the rules of COs have come from studies of mutations including structural chromosomal rearrangements that, when heterozygous, are known to impair COs in various organisms. In this work, we investigate the effect of a large heterozygous pericentric inversion on male and female recombination in Arabidopsis. The inversion was discovered in the Atmcc1 mutant background and was characterized through genetic and next-generation sequencing analysis. Reciprocal backcross populations, each consisting of over 400 individuals, obtained from the mutant and the wild type, both crossed with Landsberg erecta, were analyzed genome-wide by 143 single-nucleotide polymorphisms. The negative impact of inversion became evident in terms of CO loss in the rearranged chromosome in both male and female meiosis. No single-CO event was detected within the inversion, consistent with a post-meiotic selection operating against unbalanced gametes. Cytological analysis of chiasmata in F1 plants confirmed that COs were reduced in male meiosis in the chromosome with inversion. Crossover suppression on the rearranged chromosome is associated with a significant increase of COs in the other chromosomes, thereby maintaining unchanged the number of COs per cell. The CO pattern observed in our study is consistent with the interchromosomal (IC) effect as first described in Drosophila. In contrast to male meiosis, in female meiosis no IC effect is visible. This may be related to the greater strength of interference that constrains the CO number in excess of the minimum value imposed by CO assurance in Arabidopsis female meiosis.

Role of Cis, Trans, and Inbreeding Effects on Meiotic Recombination in Saccharomyces cerevisiae.

Meiotic recombination is a major driver of genome evolution by creating new genetic combinations. To probe the factors driving variability of meiotic recombination, we used a high-throughput method to measure recombination rates in hybrids between SK1 and a total of 26 Saccharomyces cerevisiae strains from different geographic origins and habitats. Fourteen intervals were monitored for each strain, covering chromosomes VI and XI entirely, and part of chromosome I. We found an average number of crossovers per chromosome ranging between 1.0 and 9.5 across strains ("domesticated" or not), which is higher than the average between 0.5 and 1.5 found in most organisms. In the different intervals analyzed, recombination showed up to ninefold variation across strains but global recombination landscapes along chromosomes varied less. We also built an incomplete diallel experiment to measure recombination rates in one region of chromosome XI in 10 different crosses involving five parental strains. Our overall results indicate that recombination rate is increasingly positively correlated with sequence similarity between homologs (i) in DNA double-strand-break-rich regions within intervals, (ii) in entire intervals, and (iii) at the whole genome scale. Therefore, these correlations cannot be explained by cis effects only. We also estimated that cis and trans effects explained 38 and 17%, respectively, of the variance of recombination rate. In addition, by using a quantitative genetics analysis, we identified an inbreeding effect that reduces recombination rate in homozygous genotypes, while other interaction effects (specific combining ability) or additive effects (general combining ability) are found to be weak. Finally, we measured significant crossover interference in some strains, and interference intensity was positively correlated with crossover number.

Feedback between environment and traits under selection in a seasonal environment: consequences for experimental evolution.

Batch cultures are frequently used in experimental evolution to study the dynamics of adaptation. Although they are generally considered to simply drive a growth rate increase, other fitness components can also be selected for. Indeed, recurrent batches form a seasonal environment where different phases repeat periodically and different traits can be under selection in the different seasons. Moreover, the system being closed, organisms may have a strong impact on the environment. Thus, the study of adaptation should take into account the environment and eco-evolutionary feedbacks. Using data from an experimental evolution on yeast Saccharomyces cerevisiae, we developed a mathematical model to understand which traits are under selection, and what is the impact of the environment for selection in a batch culture. We showed that two kinds of traits are under selection in seasonal environments: life-history traits, related to growth and mortality, but also transition traits, related to the ability to react to environmental changes. The impact of environmental conditions can be summarized by the length of the different seasons which weight selection on each trait: the longer a season is, the higher the selection on associated traits. Since phenotypes drive season length, eco-evolutionary feedbacks emerge. Our results show how evolution in successive batches can affect season lengths and strength of selection on different traits.

High-throughput measurement of recombination rates and genetic interference in Saccharomyces cerevisiae.

Allelic recombination owing to meiotic crossovers is a major driver of genome evolution, as well as a key player for the selection of high-performing genotypes in economically important species. Therefore, we developed a high-throughput and low-cost method to measure recombination rates and crossover patterning (including interference) in large populations of the budding yeast Saccharomyces cerevisiae. Recombination and interference were analysed by flow cytometry, which allows time-consuming steps such as tetrad microdissection or spore growth to be avoided. Moreover, our method can also be used to compare recombination in wild-type vs. mutant individuals or in different environmental conditions, even if the changes in recombination rates are small. Furthermore, meiotic mutants often present recombination and/or pairing defects affecting spore viability but our method does not involve growth steps and thus avoids filtering out non-viable spores.

Haldane, Waddington and recombinant inbred lines: extension of their work to any number of genes.

In the early 1930s, J. B. S. Haldane and C. H. Waddington collaborated on the consequences of genetic linkage and inbreeding. One elegant mathematical genetics problem solved by them concerns recombinant inbred lines (RILs) produced via repeated self or brother-sister mating. In this classic contribution, Haldane and Waddington derived an analytical formula for the probabilities of 2-locus and 3-locus RIL genotypes. Specifically, the Haldane-Waddington formula gives the recombination rate R in such lines as a simple function of the per generation recombination rate r. Interestingly, for more than 80 years, an extension of this result to four or more loci remained elusive. In 2015, we generalized the Haldane-Waddington self-mating result to any number of loci. Our solution used self-consistent equations of the multi-locus probabilities 'for an infinite number of generations' and solved these by simple algebraic operations. In practice, our approach provides a quantum leap in the systems that can be handled: the cases of up to six loci can be solved by hand while a computer program implementing our mathematical formalism tackles up to 20 loci on standard desktop computers.