Number of people and number of generations affected by a single deleterious mutation.

A deleterious autosomal dominant or X-linked mutation persists for only a finite number of generations and affects only a finite number of people. The purpose of this paper is to determine the distributions of the time to extinction and of the total number of people who eventually carry the mutant gene. The theory of subcritical branching processes forms the framework for our investigation. Autosomal mutations can be treated using single type branching processes and X-linked mutations using two-type branching processes. Application is made to Huntington's chorea and Becker's muscular dystrophy.     

Calculation of the equilibrium distribution for a deleterious gene by the finite Fourier transform

In a population of constant size every deleterious gene eventually attains a stochastic equilibrium between mutation and selection. The individual probabilities of this equilibrium distribution can be computed by an application of the finite Fourier transform to an appropriate branching process formula. Specific numerical examples are discussed for the autosomal dominants, Huntington's chorea and chondrodystrophy, and for the X-linked recessive, Becker's muscular dystrophy.     

The frequency distribution of nucleotide variation in Drosophila simulans.

Patterns of codon bias in Drosophila suggest that silent mutations can be classified into two types: unpreferred (slightly deleterious) and preferred (slightly beneficial). Results of previous analyses of polymorphism and divergence in Drosophila simulans were interpreted as supporting a mutation-selection-drift model in which slightly deleterious, silent mutants make significantly greater contributions to polymorphism than to divergence. Frequencies of unpreferred polymorphisms were inferred to be lower than frequencies of other silent polymorphisms. Here, I analyzed additional D. simulans data to reevaluate the support for these ideas. I found that D. simulans has fixed more unpreferred than preferred mutations, suggesting that this lineage has not been at mutation-selection-drift equilibrium at silent sites. Frequencies of polarized unpreferred polymorphisms are not skewed toward rare alleles. However, frequencies of unpolarized unpreferred codons are lower in high-bias genes than in low-bias genes. This supports the idea that unpreferred codons are borderline deleterious mutations. Purifying selection on silent sites appears to be stronger at twofold-degenerate codons than at fourfold-degenerate codons. Finally, I found that X-linked polymorphisms occur at a higher average frequency than polymorphisms on chromosome arm 3R, even though an average X-linked site is significantly less likely to be polymorphic than an average site on 3R. This result supports a previous analysis of D. simulans indicating different population genetics of X-linked versus autosomal mutations.     

Development of the dendritic branching pattern of the Medial Giant Interneuron in the grasshopper embryo

The embryonic development of the grasshopper's Medial Giant Interneuron (MGI) was examined by injecting the cell with the fluorescent dye Lucifer Yellow at a series of stages in its growth. Particular attention was given to the way in which this neuron constructs its stereotyped dendritic branching pattern. The MGI's dendrites originate as secondary processes which sprout at characteristic points along the neurite after the primary growth cone has passed. These processes then arborize to form a miniature version of their adult branching pattern before the end of embryonic life. While growing, the dendritic branches are covered with a radiant profusion of filopodia; however, these filopodia are ephemeral structures and disappear once the cell matures. By contrast there is no significant reduction in either the number or the spatial extent of the actual dendrites at any embryonic stage. This implies that the stereotyped branching pattern of the mature MGI is primarily determined by a precise pattern of initial growth, and that secondary pruning of branches does not play an important role in shaping the final form of this cell. The coordinate ingrowth of the first cercal sensory axons was examined by cobalt filling the embryonic nerve, and the means by which these sensory axons make their initial contacts with the MGI's dendrites is herein discussed. The following paper considers the degree to which this sensory innervation regulates dendritic growth and branching.     

Positive and negative selection on mammalian Y chromosomes

Y chromosomes are genetically degenerate in most organisms studied. The loss of genes from Y chromosomes is thought to be due to the inefficiency of purifying selection in nonrecombining regions, which leads to the accumulation of deleterious mutations via the processes of hitchhiking, background selection, and Muller's ratchet. As the severity of these processes depends on the number of functional genes linked together on the nonrecombining Y, it is not clear whether these processes are still at work on the old, gene-poor mammalian Y chromosomes. If purifying selection is indeed less efficient in the Y-linked, compared to the X-linked genes, deleterious nonsynonymous substitutions are expected to accumulate faster on the Y chromosome. However, positive selection on Y-linked genes could also increase the rate of amino acid-changing substitutions. Thus, the previous reports of an elevated nonsynonymous substitution rate in Y-linked genes are still open to interpretation. Here, we report evidence for positive selection in two out of three studied mammalian Y-linked genes, suggesting that adaptive Darwinian evolution may be common on mammalian Y chromosomes. Taking positive selection into account, we demonstrate that purifying selection is less efficient in mammalian Y-linked genes compared to their X-linked homologues, suggesting that these genes continue to degenerate.     

Long Runs of Homozygosity Are Enriched for Deleterious Variation

Exome sequencing offers the potential to study the population-genomic variables that underlie patterns of deleterious variation. Runs of homozygosity (ROH) are long stretches of consecutive homozygous genotypes probably reflecting segments shared identically by descent as the result of processes such as consanguinity, population size reduction, and natural selection. The relationship between ROH and patterns of predicted deleterious variation can provide insight into the way in which these processes contribute to the maintenance of deleterious variants. Here, we use exome sequencing to examine ROH in relation to the distribution of deleterious variation in 27 individuals of varying levels of apparent inbreeding from 6 human populations. A significantly greater fraction of all genome-wide predicted damaging homozygotes fall in ROH than would be expected from the corresponding fraction of nondamaging homozygotes in ROH (p < 0.001). This pattern is strongest for long ROH (p < 0.05). ROH, and especially long ROH, harbor disproportionately more deleterious homozygotes than would be expected on the basis of the total ROH coverage of the genome and the genomic distribution of nondamaging homozygotes. The results accord with a hypothesis that recent inbreeding, which generates long ROH, enables rare deleterious variants to exist in homozygous form. Thus, just as inbreeding can elevate the occurrence of rare recessive diseases that represent homozygotes for strongly deleterious mutations, inbreeding magnifies the occurrence of mildly deleterious variants as well.     

Deleterious Mutations, Apparent Stabilizing Selection and the Maintenance of Quantitative Variation

Apparent stabilizing selection on a quantitative trait that is not causally connected to fitness can result from the pleiotropic effects of unconditionally deleterious mutations, because as N. Barton noted, "...individuals with extreme values of the trait will tend to carry more deleterious alleles...." We use a simple model to investigate the dependence of this apparent selection on the genomic deleterious mutation rate, U; the equilibrium distribution of K, the number of deleterious mutations per genome; and the parameters describing directional selection against deleterious mutations. Unlike previous analyses, we allow for epistatic selection against deleterious alleles. For various selection functions and realistic parameter values, the distribution of K, the distribution of breeding values for a pleiotropically affected trait, and the apparent stabilizing selection function are all nearly Gaussian. The additive genetic variance for the quantitative trait is kQa2, where k is the average number of deleterious mutations per genome, Q is the proportion of deleterious mutations that affect the trait, and a2 is the variance of pleiotropic effects for individual mutations that do affect the trait. In contrast, when the trait is measured in units of its additive standard deviation, the apparent fitness function is essentially independent of Q and a2; and beta, the intensity of selection, measured as the ratio of additive genetic variance to the "variance" of the fitness curve, is very close to s = U/k, the selection coefficient against individual deleterious mutations at equilibrium. Therefore, this model predicts appreciable apparent stabilizing selection if s exceeds about 0.03, which is consistent with various data. However, the model also predicts that beta must equal Vm/VG, the ratio of new additive variance for the trait introduced each generation by mutation to the standing additive variance. Most, although not all, estimates of this ratio imply apparent stabilizing selection weaker than generally observed. A qualitative argument suggests that even when direct selection is responsible for most of the selection observed on a character, it may be essentially irrelevant to the maintenance of variation for the character by mutation-selection balance. Simple experiments can indicate the fraction of observed stabilizing selection attributable to the pleiotropic effects of deleterious mutations.     

The role of background selection in shaping patterns of molecular evolution and variation: evidence from variability on the Drosophila X chromosome

In the putatively ancestral population of Drosophila melanogaster, the ratio of silent DNA sequence diversity for X-linked loci to that for autosomal loci is approximately one, instead of the expected "null" value of 3/4. One possible explanation is that background selection (the hitchhiking effect of deleterious mutations) is more effective on the autosomes than on the X chromosome, because of the lack of crossing over in male Drosophila. The expected effects of background selection on neutral variability at sites in the middle of an X chromosome or an autosomal arm were calculated for different models of chromosome organization and methods of approximation, using current estimates of the deleterious mutation rate and distributions of the fitness effects of deleterious mutations. The robustness of the results to different distributions of fitness effects, dominance coefficients, mutation rates, mapping functions, and chromosome size was investigated. The predicted ratio of X-linked to autosomal variability is relatively insensitive to these variables, except for the mutation rate and map length. Provided that the deleterious mutation rate per genome is sufficiently large, it seems likely that background selection can account for the observed X to autosome ratio of variability in the ancestral population of D. melanogaster. The fact that this ratio is much less than one in D. pseudoobscura is also consistent with the model's predictions, since this species has a high rate of crossing over. The results suggest that background selection may play a major role in shaping patterns of molecular evolution and variation.     

A model of apical bifurcation applicable to trees and other organisms

An elementary model to describe the branching process in tree-like structures is proposed. It is assumed that when the apex reaches a certain size, it divides into two apices. The relative sizes of the two apices depends upon a branching parameter λ, which is the critical parameter of the model. With this constraint, and with some supplementary assumptions about growth, a computer is used to generate branched structures, and these are analyzed using Strahler's method of ordering. Diverse tree-like structures can be obtained by simply varying the parameter λ, and these resemble natural structures when compared on the basis of Strahler's technique.     

Spatial Structure of the Branching Streamer Channels in a Corona Discharge

The dendritic structure of streamer channels in a corona discharge is described by using fractal theory. It is found that, for a needle-plane discharge, the fractal dimension of the plasma structure is D = 2.16 ± 0.05. The computed spatial distributions of the branching ratios are compared with the available experimental data. The influence of the branching processes on the distribution of chemically active radicals in streamer corona discharges is studied.     

THE INEVITABILITY OF UNCONDITIONALLY DELETERIOUS SUBSTITUTIONS DURING ADAPTATION

Studies on the genetics of adaptation from new mutations typically neglect the possibility that a deleterious mutation might fix. Nonetheless, here we show that, in many regimes, the first mutation to fix is most often deleterious, even when fitness is expected to increase in the long term. In particular, we prove that this phenomenon occurs under weak mutation for any house‐of‐cards model with an equilibrium distribution. We find that the same qualitative results hold under Fisher's geometric model. We also provide a simple intuition for the surprising prevalence of unconditionally deleterious substitutions during early adaptation. Importantly, the phenomenon we describe occurs on fitness landscapes without any local maxima and is therefore distinct from “valley crossing.” Our results imply that the common practice of ignoring deleterious substitutions leads to qualitatively incorrect predictions in many regimes. Our results also have implications for the substitution process at equilibrium and for the response to a sudden decrease in population size.     

Further characterization of the long-run population distribution of a deleterious gene

In a population of constant size, there is an equilibrium distribution for every deleterious autosomal dominant gene. This equilibrium represents the balance between selection and mutation. The purpose of this paper is to describe an approximate method of computing the equilibrium distribution and an exact method of computing its cumulants. If the surrounding population has experienced prolonged growth or decline, then an equilibrium does not develop. However, one can show that the variance of the number of carriers divided by the current population size does stabilize; this quantity is an increasing function of the growth rate.     

Afferent innervation shapes the dendritic branching pattern of the Medial Giant Interneuron in grasshopper embryos raised in culture

In the adult grasshopper the Medial Giant Interneuron (MGI) receives synaptic input from the peripheral sensory neurons of the cercus. We prevented this innervation in grasshopper embryos by cutting off one or both cerci at a stage when the first sensory axons are just beginning to reach the central nervous system (CNS), and the MGI has not yet formed its mature branching pattern. Following this operation the embryos were raised in vitro for 3–9 days, and the MGI injected with the fluorescent dye Lucifer Yellow to determine its morphology. The development of the deprived cells was then compared to that of the normal MGI (described in M. Shankland and C. S. Goodman, 1982, Develop. Biol., 92, 483–500) and of cultured, but unoperated, controls to ascertain whether these presynaptic axons influence the embryonic growth and branching of the MGI's dendrites. The results of these experiments show that dendrite formation is enhanced in regions of the neuropil containing sensory axon terminals and that the afferents exert their influence locally on restricted portions of the branching structure. The enhanced growth of innervated dendrites appears to occur at the expense of dendritic outgrowth elsewhere, suggesting that the growing dendrites may be competing for a limited supply of some cellular component necessary for continued growth. Thus, the MGI's final branching pattern is at least partially dictated by the spatial distribution of presynaptic axons within the embryonic nervous system.     

The population genetics of X-autosome synthetic lethals and steriles

Epistatic interactions are widespread, and many of these interactions involve combinations of alleles at different loci that are deleterious when present in the same individual. The average genetic environment of sex-linked genes differs from that of autosomal genes, suggesting that the population genetics of interacting X-linked and autosomal alleles may be complex. Using both analytical theory and computer simulations, we analyzed the evolutionary trajectories and mutation-selection balance conditions for X-autosome synthetic lethals and steriles. Allele frequencies follow a set of fundamental trajectories, and incompatible alleles are able to segregate at much higher frequencies than single-locus expectations. Equilibria exist, and they can involve fixation of either autosomal or X-linked alleles. The exact equilibrium depends on whether synthetic alleles are dominant or recessive and whether fitness effects are seen in males, females, or both sexes. When single-locus fitness effects and synthetic incompatibilities are both present, population dynamics depend on the dominance of alleles and historical contingency (i.e., whether X-linked or autosomal mutations occur first). Recessive synthetic lethality can result in high-frequency X-linked alleles, and dominant synthetic lethality can result in high-frequency autosomal alleles. Many X-autosome incompatibilities in natural populations may be cryptic, appearing to be single-locus effects because one locus is fixed. We also discuss the implications of these findings with respect to standing genetic variation and the origins of Haldane's rule.     

Continuously stable strategies as evolutionary branching points

Evolutionary branching points are a paradigmatic feature of adaptive dynamics, because they are potential starting points for adaptive diversification. The antithesis to evolutionary branching points are continuously stable strategies (CSS's), which are convergent stable and evolutionarily stable equilibrium points of the adaptive dynamics and hence are thought to represent endpoints of adaptive processes. However, this assessment is based on situations in which the invasion fitness function determining the adaptive dynamics have non-zero second derivatives at CSS. Here we show that the scope of evolutionary branching can increase if the invasion fitness function vanishes to higher than first order at CSS. Using classical models for frequency-dependent competition, we show that if the invasion fitness vanishes to higher orders, a CSS may be the starting point for evolutionary branching. Thus, when invasion fitness functions vanish to higher than first order at equilibrium points of the adaptive dynamics, evolutionary diversification can occur even after convergence to an evolutionarily stable strategy.     

Recombination and hitchhiking of deleterious alleles

When new advantageous alleles arise and spread within a population, deleterious alleles at neighboring loci can hitchhike alongside them and spread to fixation in areas of low recombination, introducing a fixed mutation load. We use branching processes and diffusion equations to calculate the probability that a deleterious allele hitchhikes and fixes alongside an advantageous mutant. As expected, the probability of fixation of a deleterious hitchhiker rises with the selective advantage of the sweeping allele and declines with the selective disadvantage of the deleterious hitchhiker. We then use computer simulations of a genome with an infinite number of loci to investigate the increase in load after an advantageous mutant is introduced. We show that the appearance of advantageous alleles on genetic backgrounds loaded with deleterious alleles has two potential effects: it can fix deleterious alleles, and it can facilitate the persistence of recombinant lineages that happen to occur. The latter is expected to reduce the signals of selection in the surrounding region. We consider these results in light of human genetic data to infer how likely it is that such deleterious hitchhikers have occurred in our recent evolutionary past.     

Branching and self-organization in marine modular colonial organisms: a model

Despite the universality of branching patterns in marine modular colonial organisms, there is neither a clear explanation about the growth of their branching forms nor an understanding of how these organisms conserve their shape during development. This study develops a model of branching and colony growth using parameters and variables related to actual modular structures (e.g., branches) in Caribbean gorgonian corals (Cnidaria). Gorgonians exhibiting treelike networks branch subapically, creating hierarchical mother-daughter relationships among branches. We modeled both the intrinsic subapical branching along with an ecological-physiological limit to growth or maximum number of mother branches (k). Shape is preserved by maintaining a constant ratio (c) between the total number of branches and the mother branches. The size frequency distribution of mother branches follows a scaling power law suggesting self-organized criticality. Differences in branching among species with the same k values are determined by r (branching rate) and c. Species with rr/2 or c>r>0). Ecological/physiological constraints limit growth without altering colony form or the interaction between r and c. The model described the branching dynamics giving the form to colonies and how colony growth declines over time without altering the branching pattern. This model provides a theoretical basis to study branching as a simple function of the number of branches independently of ordering- and bifurcation-based schemes.     

Mutation accumulation may be a minor force in shaping life history traits

Is senescence the adaptive result of tradeoffs between younger and older ages or the nonadaptive burden of deleterious mutations that act at older ages? To shed new light on this unresolved question we combine adaptive and nonadaptive processes in a single model. Our model uses Penna''s bit-strings to capture different age-specific mutational patterns. Each pattern represents a genotype and for each genotype we find the life history strategy that maximizes fitness. Genotypes compete with each other and are subject to selection and to new mutations over generations until equilibrium in gene-frequencies is reached. The mutation-selection equilibrium provides information about mutational load and the differential effects of mutations on a life history trait - the optimal age at maturity. We find that mutations accumulate only at ages with negligible impact on fitness and that mutation accumulation has very little effect on the optimal age at maturity. These results suggest that life histories are largely determined by adaptive processes. The non-adaptive process of mutation accumulation seems to be unimportant at evolutionarily relevant ages.     

Study on mechanical properties of polyethylene with chain branching in atomic scale by molecular dynamics simulation

The effects of length and content of chain branching on the mechanical properties of polyethylene (PE) in atomic scale were examined by molecular dynamics (MD) simulations. Methyl-, ethyl- and butyl-groups were adopted as branched chains to distribute along PE backbones. Plastic flow deformation was captured by providing a uniaxial tensile loading at a given strain rate, which shows the characteristic of rate dependence. Current results are in reasonable agreements with existing experimental data. The statistical results show that the longer length of chain branching induces lower equilibrium density and higher yield strength of branched PE. In addition, higher content of chain branching brings higher equilibrium density and lower yield strength of branched PE. It is assumed that the distribution of dihedral angles influences the deformation of PE definitely. The non-bond interactions contribute to the load-bearing capacity of PE largely. Branched PE shows big differences on mechanical behaviours comparing with the linear one. Chain branching distribution also greatly affects the performance of PE, which needs a further discussion.