A Quantitative Model for Nonrandom Generalized Transduction, Applied to the Phage P22–SALMONELLA TYPHIMURIUM System

A mathematical model for nonrandom generalized transduction is proposed and analyzed. The model takes into account the finite number of transducing particle classes for any given marker. The equations for estimation of the distance between markers from contransduction frequency data are derived and standard errors of the estimates are given. The obtained relationships depend significantly on the number of classes of transducing fragments. The model was applied to estimate the number of transducing fragment classes for a given marker in transduction with phage P22 of Salmonella typhimurium. It was found that the literature data on frequencies of contransduction in crosses with mutual substitution of selective and nonselective markers can be rationalized most accurately by assuming that the mean number of classes is equal to 2. An improved method for analysis of cotransduction data is proposed on the basis of our model and the results of calculation. The method relies on solving a set of algebraic equations for cotransduction frequencies of markers located within one phage length. The method allows a relatively precise determination of distances between markers, positions of transducing particle ends and deletion or insertion lengths. The approach is applied to the trp-cysB-pyrF and aroC-hisT-purF-dhuA regions of the Salmonella typhimurium chromosome.     

A corrected Haldane’s map function to calculate genetic distances from recombination data

The estimation of genetic distances from recombination data has no direct relationship due to the fact that multiple crossovers do not generate recombinant gametes that can be recognized in the progeny. The Haldane’s map function is the most widely used mathematical formulation able to relate the observed recombination frequency with the actual number of crossovers. Here I show that the model in which the Haldane’s correction is based on is not correct, and I present a modified map function that takes into consideration the actual number of recombinant gametes produced in cells in which different number of crossovers have occurred. My correction generates shorter genetic distances than the Haldane’s one.     

A Mathematical Model of Interference for Use in Constructing Linkage Maps from Tetrad Data

In determining genetic map distances it is necessary to infer crossover frequencies from the ratios of recombinant and parental progeny. To do this accurately, in intervals where multiple crossovers may occur, a mathematical model of chiasma interference must be assumed when mapping in organisms displaying such interference. In Saccharomyces cerevisiae the model most frequently used is that of R.W. Barratt. An alternative to this model is presented. This new model is implemented using a microcomputer and standard numerical methods. It is demonstrated to fit ranked tetrad data from Saccharomyces more closely than the Barratt model and thus generates more accurate estimates of map distances when used with two-point data. A computer program implementing the model has been developed for use in calculating map distances from tetrad data in Saccharomyces.     

Reduced mismatch repair of heteroduplexes reveals "non"-interfering crossing over in wild-type Saccharomyces cerevisiae

Using small palindromes to monitor meiotic double-strand-break-repair (DSBr) events, we demonstrate that two distinct classes of crossovers occur during meiosis in wild-type yeast. We found that crossovers accompanying 5:3 segregation of a palindrome show no conventional (i.e., positive) interference, while crossovers with 6:2 or normal 4:4 segregation for the same palindrome, in the same cross, do manifest interference. Our observations support the concept of a "non"-interference class and an interference class of meiotic double-strand-break-repair events, each with its own rules for mismatch repair of heteroduplexes. We further show that deletion of MSH4 reduces crossover tetrads with 6:2 or normal 4:4 segregation more than it does those with 5:3 segregation, consistent with Msh4p specifically promoting formation of crossovers in the interference class. Additionally, we present evidence that an ndj1 mutation causes a shift of noncrossovers to crossovers specifically within the "non"-interference class of DSBr events. We use these and other data in support of a model in which meiotic recombination occurs in two phases-one specializing in homolog pairing, the other in disjunction-and each producing both noncrossovers and crossovers.     

Serial-Section Analysis of Clustering within Anthers of Maize Microsporocytes with Specific Crossovers

The distribution within anthers of maize plants of microsporocytes of differing crossover class was studied by application of the serial-sectioning technique to plants heterozygous for a paracentric inversion. In such material, cells with four-strand double crossovers within the inversion are distinguishable from cells with other classes of crossovers within the inverted region and from cells with no crossovers within the inversion. Evidence was found for coarse clustering of cells with four-strand double crossovers within the inversion and for coarse clustering of cells of other crossover classes (with at least one crossover within the inversion). The local frequencies of these two major categories of crossover cells appeared to vary independently of each other, with regional increases in each occurring at the expense of noncrossover cells. There is also some suggestion that cells with four-strand doubles within the inversion may be clustered independently of other double-crossover classes within the inversion, but these other classes are not directly scorable.     

Clustering of specific crossovers in maize microsporocytes

Analysis of systematic scan data from microsporocyte smear preparations suggests that cells with a crossover in a specific region tend to be geographically clustered to some extent within anthers. Crossover classes observed include single crossovers within a heterozygous inversion, three-strand double crossovers within and proximal to the inversion, and four-strand double crossovers within the inversion. Evidence for clustering within scans of cells of the first two classes is reported, but results are inconclusive for the third class. No evidence for within-scan correlation of frequencies of any two of the crossover classes was found. It is inferred that a large part of the variability in crossover frequency observed is probably due to factors which alter crossover interference. Implications related to the frequency and distribution of crossover sites are discussed.     

MOUSE-III: Learning rules of conformational analysis from X-ray data

MOUSE-III is learning program that finds rules of conformational analysis from raw cristallographie data. The program perceives molecular features, finds conformational classes in the data and then learns rules that link features to classes. The rules that MOUSE learns are capable of correctly assigning conformations to ring systems that were not used for training with greater than 95% accuracy, when MOUSE was presented with sufficient data. The rules also show a compression of as much as 99% when compared to the raw data. This is accomplished through abstraction and generalization. The algorithm is presented along with a carefully worked example. An example of a learned rule is also presented and analyzed. Some conclusions about the scope and limitations of the learning process are presented.     

CPROP: a rule-based program for constructing genetic maps.

Gene mapping assigns chromosomal coordinates to genetic loci based on analysis of fragmentary ordering and metric data. In assembling genetic maps, geneticists use rules of inference to derive new facts about order and distance between loci from experimentally derived conclusions about order and distance. They construct comprehensive maps by merging related sets of data and resolving conflicts between them. In this article we describe software that formalizes and automates some of these rules of inference to yield a useful map construction utility called CPROP.     

Meiosis V: Matric and path coefficient solutions of tri- and quadrivalents

Possible classes of spatial ordering between chromosomal components of multivalents and their motion with respect to each other are deduced on the basis of the propositions: (A) that the proximity of centromeres, or similar reference centers, to each other determines which chromosomes are to migrate to opposite poles, irrespective of homology and (B) that segregation from multivalents proceeds as a Markov process consisting of at least three steps taking place during or prior to metaphase I. Segregation matrices constructed from data on the (relative) poles to which chromosomes in a quadrivalent may proceed, taken together with proposition (A), permit deducing possible classes of spatial ordering between elements within multivalents early in meiotic prophase. Transient and ergodic states of centromere ordering are compared to resonance interaction between electrons within single molecules. Outcomes deduced by the method here correspond with many of those proposed in the literature, but some do not. Some of the conclusions are as follows. For trivalents, any one of the three centromeres, or similar reference points, may locate geometrically between the other two early in meiotic prophase and coorient with them both. In some types of quadrivalents the four reference points (centromeres, for example) first separate into cooriented pairs (= presegregation) each of which then segregates independently, or not, of the other pair. In some segregation modes of other quadrivalent types, three of the four centromeres evidently behave as a trivalent, coorienting and segregating independently of the fourth element. It is argued that structures known to be present during later states of meiosis (e.g. chiasmata) cannot, alone, account for observed segregation ratios.     

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.     

Variable selection and pattern recognition with gene expression data generated by the microarray technology

Lack of adequate statistical methods for the analysis of microarray data remains the most critical deterrent to uncovering the true potential of these promising techniques in basic and translational biological studies. The popular practice of drawing important biological conclusions from just one replicate (slide) should be discouraged. In this paper, we discuss some modern trends in statistical analysis of microarray data with a special focus on statistical classification (pattern recognition) and variable selection. In addressing these issues we consider the utility of some distances between random vectors and their nonparametric estimates obtained from gene expression data. Performance of the proposed distances is tested by computer simulations and analysis of gene expression data on two different types of human leukemia. In experimental settings, the error rate is estimated by cross-validation, while a control sample is generated in computer simulation experiments aimed at testing the proposed gene selection procedures and associated classification rules.     

A parametric copula model for analysis of familial binary data

Modeling the joint distribution of a binary trait (disease) within families is a tedious challenge, owing to the lack of a general statistical model with desirable properties such as the multivariate Gaussian model for a quantitative trait. Models have been proposed that either assume the existence of an underlying liability variable, the reality of which cannot be checked, or provide estimates of aggregation parameters that are dependent on the ordering of family members and on family size. We describe how a class of copula models for the analysis of exchangeable categorical data can be incorporated into a familial framework. In this class of models, the joint distribution of binary outcomes is characterized by a function of the given marginals. This function, referred to as a "copula," depends on an aggregation parameter that is weakly dependent on the marginal distributions. We propose to decompose a nuclear family into two sets of equicorrelated data (parents and offspring), each of which is characterized by an aggregation parameter (alphaFM and alphaSS, respectively). The marginal probabilities are modeled through a logistic representation. The advantage of this model is that it provides estimates of the aggregation parameters that are independent of family size and does not require any arbitrary ordering of sibs. It can be incorporated easily into segregation or combined segregation-linkage analysis and does not require extensive computer time. As an illustration, we applied this model to a combined segregation-linkage analysis of levels of plasma angiotensin I-converting enzyme (ACE) dichotomized into two classes according to the median. The conclusions of this analysis were very similar to those we had reported in an earlier familial analysis of quantitative ACE levels.     

Genotyping new BXD recombinant inbred mouse strains and comparison of BXD and consensus maps

Nine additional BXD recombinant inbred (RI) strains have been developed from the F₂ cross of C57BL/6J and DBA/2J mouse strains. A tenth line stopped breeding in the F₁₂ generation. F₂₀ generation breeding pairs from the nine surviving strains and an F₁₂ pair from the extinct line were genotyped at 319 genetic markers (primarily microsatellites) spanning most of the genome. Where typing data were lacking, the established set of 26 BXD strains also were genotyped at these same loci. The availability of these additional nine strains enhances the value of the BXD RI set for analysis of complex phenotypic traits. The proportion of loci still segregating at the F₂₀ generation was found to closely approximate expectation, suggesting that selection favoring the retention of heterozygosity is not a strong factor. However, the number of crossovers between adjacent markers was frequently less than predicted from consensus map distances. A significant deficiency of recombinants was observed on Chrs 3, 4, 14, and X. On Chr 14, the estimated cumulative BXD map distance between the most proximal and distal markers was only 30.2 cM, compared with a distance of 60.0 cM in the consensus map. On the X Chr, the estimated and predicted cumulative distances were 38.8 and 69.5 cM, respectively. Over all chromosomes, the BXD RI map is 14.5% shorter than predicted from the consensus map. It is suggested that distances in some of the consensus maps are inflated. Alternatively, recombinant genotypes could be selected against during inbreeding owing to allelic interactions affecting fitness. The latter interpretation implies that relatively strong intrachromosomal epistasis is common. Received: 2 October 1998 / Accepted: 15 December 1998     

Expression and functional analysis of AtMUS81 in Arabidopsis meiosis reveals a role in the second pathway of crossing-over

Meiotic crossovers/chiasmata, that are required to ensure chromosome disjunction, arise via the class I interference-dependent pathway or via the class II interference-free pathway. The proportions of these two classes vary considerably between different organisms. In Arabidopsis, about 85% of chiasmata are eliminated in Atmsh4 mutants, denoting that these are class I events. In budding and fission yeasts Msh4-independent crossovers arise largely or entirely via a Mus81-dependent pathway. To investigate the origins of the 15% residual (AtMSH4-independent) chiasmata in Arabidopsis we conducted a cytological and molecular analysis of AtMUS81 meiotic expression and function. Although AtMUS81 functions in somatic DNA repair and recombination, it is more highly expressed in reproductive tissues. The protein is abundantly present in early prophase I meiocytes, where it co-localizes, in a double-strand break-dependent manner, with the recombination protein AtRAD51. Despite this, an Atmus81 mutant shows normal growth and has no obvious defects in reproductive development that would indicate meiotic impairment. A cytological analysis confirmed that meiosis was apparently normal in this mutant and its mean chiasma frequency was similar to that of wild-type plants. However, an Atmsh4 / Atmus81 double mutant revealed a significantly reduced mean chiasma frequency (0.85 per cell), compared with an Atmsh4 single mutant (1.25 per cell), from which we conclude that AtMUS81 accounts for some, but not all, of the 15% AtMSH4-independent residual crossovers. It is possible that other genes are responsible for these residual chiasmata. Alternatively the AtMUS81 pathway coexists with an alternative parallel pathway that can perform the same functions.     

The Mus81/Mms4 endonuclease acts independently of double-Holliday junction resolution to promote a distinct subset of crossovers during meiosis in budding yeast

Current models for meiotic recombination require that crossovers derive from the resolution of a double-Holliday junction (dHJ) intermediate. In prokaryotes, enzymes responsible for HJ resolution are well characterized but the identification of a eukaryotic nuclear HJ resolvase has been elusive. Indirect evidence suggests that MUS81 from humans and fission yeast encodes a HJ resolvase. We provide three lines of evidence that Mus81/Mms4 is not the major meiotic HJ resolvase in S. cerevisiae: (1) MUS81/MMS4 is required to form only a distinct subset of crossovers; (2) rather than accumulating, dHJ intermediates are reduced in an mms4 mutant; and (3) expression of a bacterial HJ resolvase has no suppressive effect on mus81 meiotic phenotypes. Our analysis also reveals the existence of two distinct classes of crossovers in budding yeast. Class I is dependent upon MSH4/MSH5 and exhibits crossover interference, while class II is dependent upon MUS81/MMS4 and exhibits no interference. mms4 specifically reduces crossing over on small chromosomes, which are known to undergo less interference. The correlation between recombination rate and degree of interference to chromosome size may therefore be achieved by modulating the balance between class I/class II crossovers.     

Nondisjunction of chromosome 15: origin and recombination.

Thirty-two cases of uniparental disomy (UPD), ascertained from Prader-Willi syndrome patients (N = 27) and Angelman syndrome patients (N = 5), are used to investigate the pattern of recombination associated with nondisjunction of chromosome 15. In addition, the meiotic stage of nondisjunction is inferred by using markers mapping near the centromere. Two basic approaches to the analysis of recombination are utilized. Standard methods of centromere mapping are employed to determine the level of recombination in specific pairwise intervals along the chromosome. This method shows a significant reduction in recombination for two of five intervals examined. Second, the observed frequency of each recombinant class (i.e., zero, one, two, three, or more observable crossovers) is compared with expected values. This is useful for testing whether the reduction in recombination can be attributed solely to a proportion of cases with no recombination at all (because of asynapsis), with the remaining groups showing normal recombination (or even excess recombination), or whether recombination is uniformly reduced. Analysis of maternal UPD(15) data shows a slight reduction in the multiple-recombinant classes, with a corresponding increase in both the zero- and one-recombinant classes over expected values. The majority, more than 82%, of the extra chromosomes in maternal UPD(15) cases are due to meiotic I nondisjunction events. In contrast, most paternal UPD(15) cases so far examined appear to have a postzygotic origin of the extra paternal chromosome.     

Genetic Mapping in Escherichia coli K-12 by Radiation-Induced Crossing-Over

Conventional methods for chromosomal mapping in Escherichia coli are (i) interruption of matings to obtain minimum marker entry times, (ii) linkage analysis of recombinants, and (iii) cotransduction. Method (i) has a resolution of about 0.5 min (5 x 10(4) nucleotides) and is not useful for distances less than about 1 min; methods (ii) and (iii) are capable of better resolution but are generally not very reproducible and no general theory is available for translating crossing-over and cotransduction frequencies into physical chromosomal distances. We found that when merozygotes are irradiated (X rays or ultraviolet light) soon after marker transfer, high linkage values (0.8 to 1.0) between nearby marker pairs decrease with radiation dose to 0.5. Our results are quantitatively consistent with the idea that radiations induce crossing-over lesions proportional to dose, and the number of such lesions between two markers is proportional to the physical separation of the markers in the range that can also be measured by interruption of mating (0.5 to 4.0 min). Additivity relations among markers are also satisfied. We used this technique to measure the distances (0.1 to 1.0 min) between several pairs of closely linked markers.     

Gene ordering in partitive clustering using microarray expressions

A central step in the analysis of gene expression data is the identification of groups of genes that exhibit similar expression patterns. Clustering and ordering the genes using gene expression data into homogeneous groups was shown to be useful in functional annotation, tissue classification, regulatory motif identification, and other applications. Although there is a rich literature on gene ordering in hierarchical clustering framework for gene expression analysis, there is no work addressing and evaluating the importance of gene ordering in partitive clustering framework, to the best knowledge of the authors. Outside the framework of hierarchical clustering, different gene ordering algorithms are applied on the whole data set, and the domain of partitive clustering is still unexplored with gene ordering approaches. A new hybrid method is proposed for ordering genes in each of the clusters obtained from partitive clustering solution, using microarray gene expressions.Two existing algorithms for optimally ordering cities in travelling salesman problem (TSP), namely, FRAG_GALK and Concorde, are hybridized individually with self organizing MAP to show the importance of gene ordering in partitive clustering framework. We validated our hybrid approach using yeast and fibroblast data and showed that our approach improves the result quality of partitive clustering solution, by identifying subclusters within big clusters, grouping functionally correlated genes within clusters, minimization of summation of gene expression distances, and the maximization of biological gene ordering using MIPS categorization. Moreover, the new hybrid approach, finds comparable or sometimes superior biological gene order in less computation time than those obtained by optimal leaf ordering in hierarchical clustering solution.     

Crossover interference in humans

Crossing-over between homologous chromosomes facilitates proper disjunction of chromosomes during meiosis I. In many organisms, gene functions that are essential to crossing-over also facilitate the intimate chromosome pairing called "synapsis." Many organisms--including budding yeast, humans, zebrafish, Drosophila, and Arabidopsis--regulate the distribution of crossovers, so that, most of the time, each chromosome bundle gets at least one crossover while the mean number of crossovers per chromosome remains modest. This regulation is obtained through crossover interference. Recent evidence suggests that the organisms that use recombination functions to achieve synapsis have two classes of crossovers, only one of which is subject to interference. We statistically test this two-pathway hypothesis in the CEPH data and find evidence to support the two-pathway hypothesis in humans.     

Extensive interallelic polymorphisms drive meiotic recombination into a crossover pathway

Recombinants isolated from most meiotic intragenic recombination experiments in maize, but not in yeast, are borne principally on crossover chromosomes. This excess of crossovers is not explained readily by the canonical double-strand break repair model of recombination, proposed to account for a large body of yeast data, which predicts that crossovers (COs) and noncrossovers (NCOs) should be recovered equally. An attempt has been made here to identify general rules governing the recovery of the CO and NCO classes of intragenic recombinants in maize. Recombination was analyzed in bz heterozygotes between a variety of mutations derived from the same or different progenitor alleles. The mutations include point mutations, transposon insertions, and transposon excision footprints. Consequently, the differences between the bz heteroalleles ranged from just two nucleotides to many nucleotides, indels, and insertions. In this article, allelic pairs differing at only two positions are referred to as dimorphic to distinguish them from polymorphic pairs, which differ at multiple positions. The present study has revealed the following effects at these bz heteroalleles: (1) recombination between polymorphic heteroalleles produces mostly CO chromosomes; (2) recombination between dimorphic heteroalleles produces both CO and NCO chromosomes, in ratios apparently dependent on the nature of the heteroalleles; and (3) in dimorphic heterozygotes, the two NCO classes are recovered in approximately equal numbers when the two mutations are point mutations but not when one or both mutations are insertions. These observations are discussed in light of a recent version of the double-strand break repair model of recombination that postulates separate pathways for the formation of CO and NCO products.