Maturing asci of Neurospora crassa from wild-type x histone H1 - GF P (inserted at his-3 ). Histone H1, being a chromosomal protein, allows the GFP-tagged nuclei (two per spore at this stage) to fluoresce in four of the eight ascospores; the remaining four ascospores carry the untagged nuclei from the wild-type parent. Almost all asci show the first-division segregation of hH1 - GF P because of its close proximity to the centromere of linkage group I (photo courtesy of Namboori B. Raju, Stanford University, USA). 

Contexts in source publication

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... 2 − 2 c + c 2 2 c − c 2 p 1 = p 4 = ; p 2 = p 3 = 4 4 p i = P ( n i ) = ∀ i = 1 , 2 , 3 , 4 1 1 1 ≤ p 1 ≤ ; 0 ≤ p 2 ≤ . 4 2 4 Restrictions on the parameters (see Catchpole and Morgan 20 ) ensure that the recombination probability is less than or equal to 1 (see Sec. 5.3). The likelihood 2 formulation in Eq. (4) is distinct from that in Zhao et al. 16 because no crossover process is postulated in Eq. (4). There are at least two ways to collect data either as random spores or as tetrads in Fig. 1. The MLE for random spores when two markers are scored is now described to allow comparisons with tetrad data. The following theorem is given without proof in Tewari et al. ...

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... standard errors have not been available before for the fungal genetic community. In Fig. 4, we see that the estimators r mle · td and r heu of the tetrad model are uniformly more efficient than r mle of the single spore model. Although this was intuitively expected, it was not quite obvious. Although identifying a tetrad type involves extracting more information (hence, an intuitively smaller variance) than for a single spore, we had different probability distributions for them, and a check was necessary to confirm this intuition. In the next section, we expand on the single spore model to account for multiple markers. It would appear that the amount of information in tetrads is usually at least a factor of 4 greater than that in single random spores. If obtaining tetrads is not more than four times the work of random spores, then tetrads are worth obtaining versus random spore data. This is intuitive because each tetrad in Fig. 1 allows us to observe potentially up to four distinct recombination events. The only situation where the relative efficiency approaches 1 is as the recombination fraction approaches ...

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... almost the beginning of genetics, an important goal has been to create maps of entire chromosomes. These maps fall into two classes, genetic and physical maps. 1 , 2 The former are constructed from information on how genes are transmitted from parent to offspring, whereas the latter are constructed by having an experimental approach designed to distinguish DNA fragments and to order these fragments. Computationally feasible maximum likelihood solutions for these respective problems were developed by Lander and Green 3 for a genetic map with many markers, and by Bhandarkar et al. 4 and Kececioglu et al. 5 for a physical map composed of many DNA fragments. The focus here is on constructing genetic maps with many markers. With l markers or genetic loci, each with two or more alternate types of a gene (i.e. alleles), the number of possible types of offspring is 2 l and hence the computational complexity of a likelihood-based approach to estimating a genetic map would appear to scale as O (25 l ). 6 At first glance, the computational complexity of the segregation of l markers to build a genetic map seems daunting. For a special case, Lander and Green 3 were able to develop a maximum likelihood procedure for ordering genetic markers that had a linear computational complexity in the number of markers. This is essential because geneticists have just completed the International HapMap 7 to hunt down most disease-causing genes, with thousands of markers scattered through the genome, and the single nucleotide polymorphisms (SNPs) from the Affymetrix chips in complex trait analysis are derived in part from the HapMap SNPs. Several model systems now possess genetic maps with thousands of markers, and their transmission to offspring can be observed simultaneously. 8 In essence, the simultaneous use of mapping data together with variation in a complex trait, such as many human diseases, provides a triangulation on genes that may influence a complex trait controlled by two or more genes. 9 The ideal data obtainable, from a geneticist’s perspective, is one in which he/she can observe the gametes of a parent directly (as opposed to the offspring) to understand the transmission of genes or genetic markers. Then, the transmission of genes in one parent does not mask how genes are transmitted in the other parent. In fungi, such as Neurospora crassa , the gametic products can be typed directly ( Fig. 1). The strings of spores in Fig. 1 are the gametes from a single cross. It is possible to engi- neer other organisms, such as the model plant system Arabidopsis thaliana , for tetrad analysis. 10 , 11 The question is, for this ideal kind of genetic data (in the best of all possible experimental worlds), can we construct a genetic map with many markers? Given the importance of tetrads to understanding how genes are transmitted together or separately in the hundreds of fungal laboratories employing these kinds of genetic analyses, one would think there would be a clear statistical methodology for the analysis of such multilocus data and the planning of such experiments. The problem examined here is very old and difficult, and is solved here for a case relevant to the fungal kingdom and organisms lucky enough to be engineered to have tetrads. We extend the work of Zhao and Speed to the case of many markers. Tetrad data are shown to be the best available ones for constructing genetic maps. In a previous paper, 13 we have demonstrated how the likelihood function involv- ing hundreds of genetic markers can be computed with a computational complexity linear in l . Here, we show that the inference tools based on the likelihood function for a genetic map, in fact, produce the correct map. Resulting genetic maps are independently verified against the sequence of the Neurospora crassa genome, 14 with the exception of linkage groups I, II, and V. Inferred recombination distances between markers and their standard errors are reported for the first time for this model system. In eukaryotic organisms, the vast majority of genes are found on the chromosomes in the cell’s nucleus. Many eukaryotic species are classified as either diploid, carrying two nearly identical pairs of nuclear chromosomes (i.e. two nuclear genomes) in each cell; or haploid, with only one chromosome set per cell. Most fungi and algae are haploids; whereas many other eukaryotes, including animals and flowering plants, are diploids. However, it is worth noting that diploid organisms produce haploid reproductive cells (such as eggs and sperm in animals); conversely, some haploid organisms, such as fungi, produce specialized diploid cells during the sexual phase of their life cycle. The letter n is used to designate the number of distinct chromosomes in one nuclear genome, so the haploid condition is designated as n (that is, 1 × n ) and the diploid state as 2 n (that is, 2 × n ). The symbol n is called the haploid chromosome number. In many familiar eukaryotes, recombination principally takes place in meiosis. In eukaryotes, the sexual cycle requires the production of specialized haploid cells (for example, egg and sperm) called gametes. This is achieved by DNA replication prior to meiosis in a diploid meiocyte, followed by two successive cell divisions, resulting in a tetrad of four haploid products or gametes. The two divisions of the nucleus that produce the tetrad of haploid gametes are called meiosis. Because of DNA replication prior to meiosis and pairing of homologs, each chromosomal type is represented in four copies called chromatids (or, in other words, two pairs of sister chromatids). The two pairs of sister chromatids align, constituting a bivalent (i.e. a group of four chromatids). It is at this stage that crossing over is thought to take place. For any particular bivalent, there can be one to several crossovers. A crossover can be represented by a double-stranded break, in which ends of chromatids reanneal with the wrong chromatid ends, as shown in Fig. 2. The crossovers can occur at any position along the chromatids, and the positions are different in different gametes. Furthermore, crossovers are usually only observed between nonsister chromatids (represented by the different colors in Fig. 2). If we designate the sister chromatids from one parent as 1 and 2, and from the other parent as 3 and 4, crossovers can be seen between 1 and 3, 1 and 4, 2 and 3, and 2 and 4, as shown in the different crossovers in Fig. 2. It is plausible that crossovers are equally likely over these pairs, being random events. This assumption is sometimes referred to as no chromatid interference (NCI). The process underlying meiotic recombination shuffles heterozygous allele pairs and deals them out in different combinations into the products of meiosis (such as the gametes of plants and animals). Being precise, meiotic recombination is defined as the production of haploid products of meiosis with genotypes differing from both haploid genotypes that originally combined to form the diploid parental meiocyte. The product of meiosis so generated is called a recombinant, and the process generating the recombinant is hypothesized to be a crossing over. Thus, crossing over is essentially a breaking-and-rejoining process between homologous DNA double helices in meiosis. There are two different mechanisms of meiotic recombination: independent assortment of heterozygous genes on different ...

Context 4

... almost the beginning of genetics, an important goal has been to create maps of entire chromosomes. These maps fall into two classes, genetic and physical maps. 1 , 2 The former are constructed from information on how genes are transmitted from parent to offspring, whereas the latter are constructed by having an experimental approach designed to distinguish DNA fragments and to order these fragments. Computationally feasible maximum likelihood solutions for these respective problems were developed by Lander and Green 3 for a genetic map with many markers, and by Bhandarkar et al. 4 and Kececioglu et al. 5 for a physical map composed of many DNA fragments. The focus here is on constructing genetic maps with many markers. With l markers or genetic loci, each with two or more alternate types of a gene (i.e. alleles), the number of possible types of offspring is 2 l and hence the computational complexity of a likelihood-based approach to estimating a genetic map would appear to scale as O (25 l ). 6 At first glance, the computational complexity of the segregation of l markers to build a genetic map seems daunting. For a special case, Lander and Green 3 were able to develop a maximum likelihood procedure for ordering genetic markers that had a linear computational complexity in the number of markers. This is essential because geneticists have just completed the International HapMap 7 to hunt down most disease-causing genes, with thousands of markers scattered through the genome, and the single nucleotide polymorphisms (SNPs) from the Affymetrix chips in complex trait analysis are derived in part from the HapMap SNPs. Several model systems now possess genetic maps with thousands of markers, and their transmission to offspring can be observed simultaneously. 8 In essence, the simultaneous use of mapping data together with variation in a complex trait, such as many human diseases, provides a triangulation on genes that may influence a complex trait controlled by two or more genes. 9 The ideal data obtainable, from a geneticist’s perspective, is one in which he/she can observe the gametes of a parent directly (as opposed to the offspring) to understand the transmission of genes or genetic markers. Then, the transmission of genes in one parent does not mask how genes are transmitted in the other parent. In fungi, such as Neurospora crassa , the gametic products can be typed directly ( Fig. 1). The strings of spores in Fig. 1 are the gametes from a single cross. It is possible to engi- neer other organisms, such as the model plant system Arabidopsis thaliana , for tetrad analysis. 10 , 11 The question is, for this ideal kind of genetic data (in the best of all possible experimental worlds), can we construct a genetic map with many markers? Given the importance of tetrads to understanding how genes are transmitted together or separately in the hundreds of fungal laboratories employing these kinds of genetic analyses, one would think there would be a clear statistical methodology for the analysis of such multilocus data and the planning of such experiments. The problem examined here is very old and difficult, and is solved here for a case relevant to the fungal kingdom and organisms lucky enough to be engineered to have tetrads. We extend the work of Zhao and Speed to the case of many markers. Tetrad data are shown to be the best available ones for constructing genetic maps. In a previous paper, 13 we have demonstrated how the likelihood function involv- ing hundreds of genetic markers can be computed with a computational complexity linear in l . Here, we show that the inference tools based on the likelihood function for a genetic map, in fact, produce the correct map. Resulting genetic maps are independently verified against the sequence of the Neurospora crassa genome, 14 with the exception of linkage groups I, II, and V. Inferred recombination distances between markers and their standard errors are reported for the first time for this model system. In eukaryotic organisms, the vast majority of genes are found on the chromosomes in the cell’s nucleus. Many eukaryotic species are classified as either diploid, carrying two nearly identical pairs of nuclear chromosomes (i.e. two nuclear genomes) in each cell; or haploid, with only one chromosome set per cell. Most fungi and algae are haploids; whereas many other eukaryotes, including animals and flowering plants, are diploids. However, it is worth noting that diploid organisms produce haploid reproductive cells (such as eggs and sperm in animals); conversely, some haploid organisms, such as fungi, produce specialized diploid cells during the sexual phase of their life cycle. The letter n is used to designate the number of distinct chromosomes in one nuclear genome, so the haploid condition is designated as n (that is, 1 × n ) and the diploid state as 2 n (that is, 2 × n ). The symbol n is called the haploid chromosome number. In many familiar eukaryotes, recombination principally takes place in meiosis. In eukaryotes, the sexual cycle requires the production of specialized haploid cells (for example, egg and sperm) called gametes. This is achieved by DNA replication prior to meiosis in a diploid meiocyte, followed by two successive cell divisions, resulting in a tetrad of four haploid products or gametes. The two divisions of the nucleus that produce the tetrad of haploid gametes are called meiosis. Because of DNA replication prior to meiosis and pairing of homologs, each chromosomal type is represented in four copies called chromatids (or, in other words, two pairs of sister chromatids). The two pairs of sister chromatids align, constituting a bivalent (i.e. a group of four chromatids). It is at this stage that crossing over is thought to take place. For any particular bivalent, there can be one to several crossovers. A crossover can be represented by a double-stranded break, in which ends of chromatids reanneal with the wrong chromatid ends, as shown in Fig. 2. The crossovers can occur at any position along the chromatids, and the positions are different in different gametes. Furthermore, crossovers are usually only observed between nonsister chromatids (represented by the different colors in Fig. 2). If we designate the sister chromatids from one parent as 1 and 2, and from the other parent as 3 and 4, crossovers can be seen between 1 and 3, 1 and 4, 2 and 3, and 2 and 4, as shown in the different crossovers in Fig. 2. It is plausible that crossovers are equally likely over these pairs, being random events. This assumption is sometimes referred to as no chromatid interference (NCI). The process underlying meiotic recombination shuffles heterozygous allele pairs and deals them out in different combinations into the products of meiosis (such as the gametes of plants and animals). Being precise, meiotic recombination is defined as the production of haploid products of meiosis with genotypes differing from both haploid genotypes that originally combined to form the diploid parental meiocyte. The product of meiosis so generated is called a recombinant, and the process generating the recombinant is hypothesized to be a crossing over. Thus, crossing over is essentially a breaking-and-rejoining process between homologous DNA double helices in meiosis. There are two different mechanisms of meiotic recombination: independent assortment of ...

Context 5

... demonstrated how the likelihood function involv- ing hundreds of genetic markers can be computed with a computational complexity linear in l . Here, we show that the inference tools based on the likelihood function for a genetic map, in fact, produce the correct map. Resulting genetic maps are independently verified against the sequence of the Neurospora crassa genome, 14 with the exception of linkage groups I, II, and V. Inferred recombination distances between markers and their standard errors are reported for the first time for this model system. In eukaryotic organisms, the vast majority of genes are found on the chromosomes in the cell’s nucleus. Many eukaryotic species are classified as either diploid, carrying two nearly identical pairs of nuclear chromosomes (i.e. two nuclear genomes) in each cell; or haploid, with only one chromosome set per cell. Most fungi and algae are haploids; whereas many other eukaryotes, including animals and flowering plants, are diploids. However, it is worth noting that diploid organisms produce haploid reproductive cells (such as eggs and sperm in animals); conversely, some haploid organisms, such as fungi, produce specialized diploid cells during the sexual phase of their life cycle. The letter n is used to designate the number of distinct chromosomes in one nuclear genome, so the haploid condition is designated as n (that is, 1 × n ) and the diploid state as 2 n (that is, 2 × n ). The symbol n is called the haploid chromosome number. In many familiar eukaryotes, recombination principally takes place in meiosis. In eukaryotes, the sexual cycle requires the production of specialized haploid cells (for example, egg and sperm) called gametes. This is achieved by DNA replication prior to meiosis in a diploid meiocyte, followed by two successive cell divisions, resulting in a tetrad of four haploid products or gametes. The two divisions of the nucleus that produce the tetrad of haploid gametes are called meiosis. Because of DNA replication prior to meiosis and pairing of homologs, each chromosomal type is represented in four copies called chromatids (or, in other words, two pairs of sister chromatids). The two pairs of sister chromatids align, constituting a bivalent (i.e. a group of four chromatids). It is at this stage that crossing over is thought to take place. For any particular bivalent, there can be one to several crossovers. A crossover can be represented by a double-stranded break, in which ends of chromatids reanneal with the wrong chromatid ends, as shown in Fig. 2. The crossovers can occur at any position along the chromatids, and the positions are different in different gametes. Furthermore, crossovers are usually only observed between nonsister chromatids (represented by the different colors in Fig. 2). If we designate the sister chromatids from one parent as 1 and 2, and from the other parent as 3 and 4, crossovers can be seen between 1 and 3, 1 and 4, 2 and 3, and 2 and 4, as shown in the different crossovers in Fig. 2. It is plausible that crossovers are equally likely over these pairs, being random events. This assumption is sometimes referred to as no chromatid interference (NCI). The process underlying meiotic recombination shuffles heterozygous allele pairs and deals them out in different combinations into the products of meiosis (such as the gametes of plants and animals). Being precise, meiotic recombination is defined as the production of haploid products of meiosis with genotypes differing from both haploid genotypes that originally combined to form the diploid parental meiocyte. The product of meiosis so generated is called a recombinant, and the process generating the recombinant is hypothesized to be a crossing over. Thus, crossing over is essentially a breaking-and-rejoining process between homologous DNA double helices in meiosis. There are two different mechanisms of meiotic recombination: independent assortment of heterozygous genes on different chromosomes, and crossing over between heterozygous genes on the same chromosome. Since we deal with genes on the same chromosome in this paper, we consider modeling only the crossing over. Figure 2 shows the way it works. In the figure, one parental genotype ( ab ) carries all of the mutant alleles, and the other parent ( AB ) contains all of the normal or wild-type alleles. A typical meiotic product (in a single crossover or SCO) aB , using the above definition of recombination, is clearly a recombinant, as it is genotypically different from either of the haploid parents ab and AB . In the figure, we see that different double crossovers (DCO) lead to different allelic combinations in recombinants. The tetrads are classified as parental ditype (PD), denoting no recombinants; tetrad type (T), denoting equal numbers of recombinants and parental gametes; and nonparental ditype (NPD), denoting all recombinants. These are observable categories by typing gametes (i.e. PD, T, and NPD) in asci, such as in Fig. 1. The modeling approach here is quite distinct from that of Lander and Green. Here, a detailed model of recombination involves exchanges between four chromatids during prophase I to generate all possible bivalent configurations in a given interval. The Lander and Green approach is simply based on counting recombination events in a given interval with the phase known. As a consequence, in their approach no chromosomal interference is assumed, while the likelihood function developed here does permit and is sensitive to chromosomal interference. Much work has been carried out on the genetic mapping problem, but most approaches posit some underlying crossover process. For example, Zhao and Speed 12 , 15 , 16 developed a model in which the crossover process is a stationary renewal process. 17 From this modeling framework, they can write down a likelihood specification that leads to the chi-square model as a special case. 18 With this likelihood formulation, they are able to test the performance of the model for several model systems 16 in a limited way. The limitation of their work is that they assume mathematically tractable processes to describe the crossover process in order to derive analytically tractable likelihood functions for a genetic map. As an example, in Foss et al. ’s 18 formulation of the chi-square model, the recombinational interme- diates ( C ) are assumed to be uniformily distributed along a chromosome (i.e. no chromosomal interference), while their resolution is assumed to follow a particular pattern. While the model and hence the likelihood has a parameter m to measure interference, the measure itself is quite abstract and hard to interpret. For example, in their model, they state that a nonexchange ( C ) is required to occur m times after each crossover resolution ( C x ) followed by a crossover resolution ( C x ). Our modeling approach below is distinct by not invoking a particular crossover process to formulate a likelihood function. Zhao and Speed 17 have calculated very general expressions for the probability of multilocus recombinants, but the limitation of their work is that there is no prescription on how to compute these probabilities except when the assumed process (for example, the chi-square model) allows sums over all possible exchanges, reducing them to explicit closed-form expressions (see Theorems 1 and 2 in Zhao et al. 16 ). Even in this circumstance, they do not present an analysis for more than 10 markers considered simultaneously. As a consequence, it is not clear how their method of likelihood maximization (simplex method) would scale to hundreds of markers without encountering a computational bottleneck. While their modeling and limited likelihood analysis have been illuminating with regard to recombination, they have invoked modeling assumptions about the crossover process that are unnecessary and difficult to verify. In their approach, the crossover process is used to connect the observed gametes to the bivalent configurations. In our modeling approach, we begin with the bivalent configurations, sidestepping the specification of a crossover process. Here, we do not make explicit assumptions regarding the underlying crossover process, as in earlier work. We develop a probabilistic framework which works directly with all possible bivalent configurations in distinct intervals along the chromosome, using the no-chromatid-interference (NCI) model. In this model, each possible chromatid exchange between nonsister chromatids in a given interval is equally likely. The term “bivalent configuration” refers to how chromatids are joined with their nonsister chromatids along the chromosome. In Fig. 2, all possible bivalent configurations in a tetrad for two markers are depicted. In the following sections, the model is laid out and its connection to the recombination fraction is detailed. For the reader’s convenience, the following glossary of mathematical terms and symbols is ...