Transitions and transversions in evolutionary descent: An approach to understanding

Summary In this paper I lay a quantitative theoretical groundwork for understanding the proportions of the possible types of base substitutions observed between 12 genes sharing a common ancestor and isolated from extant species. The experimentally observed types of base substitution between two sequenced genes do not give a direct measure of the types of base substitutions that occur during evolutionary descent. However, by use of a statistical assemblage of these observations, we can recover, without the assumption of parsimony, the conditional base substitution probabilities that determine this descent. Three methods—direct count, regression, and informational entropy maximization—are described by which these probabilities can be estimated from experimental data. The methods are complementary in that each is most useful for somewhat different types of experimental data. These methods are used to study the ratio of transversions to transitions during gene divergence. Though this ratio is not constant during divergence, it does approach a stable limiting value that in principle can vary from zero, corresponding to 100% transition differences, to infinity, corresponding to 0% transition differences. In practice the limiting ratio tends to hover around a value of two, which is expected on a random basis. However, base substitution pathways that are very nonrandom also may lead to a limiting ratio of exactly two, so that such a value is not diagnostic for random pathways. The limiting ratio can be directly calculated from a knowledge of the twelve conditional probabilities for each type of base substitution, or from a knowledge of the equilibrium base composition of the DNAs compared. An expression is given for this calculation. Fifteen years ago Jean Derancourt, Andrew Lebor and Emile Zuckerkandl (1967), analyzing the amino acid sequence of globin chains coded by nuclear genes, made the original observation that the proportion of transition differences decreases with increasing evolutionary time. Recently Brown et al. (1982) and Brown and Simpson (1982) have reported a decrease in the observed proportion of transition differences in mitochondrial DNA with increasing evolutionary divergence. The conditions that must be satisfied for this type of behavior to occur at stable base composition and with stable base substitution probabilities are defined. Multiple substitutionsper se do not lead to a decrease in transition differences with increasing evolutionary divergence.     

Exact and Asymptotic Power Determination For Dose‐Response Studies With Ordinal Response

In this article, we describe a conditional score test for detecting a monotone dose‐response relationship with ordinal response data. We consider three different versions of this test: asymptotic, conditional exact, and mid‐P conditional score test. Exact and asymptotic power formulae based on these tests will be studied. Asymptotic sample size formulae based on the asymptotic conditional score test will be derived. The proposed formulae are applied to a vaccination study and a developmental toxicity study for illustrative purposes. Actual significance level and exact power properties of these tests are compared in a small empirical study. The mid‐P conditional score test is observed to be the most powerful test with actual significance level close to the pre‐specified nominal level.     

Reproductive skew can provide a net advantage in both conditional and unconditional social interactions

We revisit a model for the evolution of costly social behaviour in the presence of reproductive skew. The model population is structured into groups, and reproductive skew is captured by assuming individuals adopt one of two social roles (dominant/subordinate). Unlike previous work, we adopt an ultimate perspective by tracking a mutant allele over the entire course of an invasion. Our main analysis applies the theory of branching processes, but a parallel analysis using the inclusive-fitness approach is also provided. Our first two results are modifications of known inequalities describing selective advantages for behaviours expressed conditional upon social status. We find that altruistic subordinate individuals are favoured more readily than previously thought; spiteful dominant individuals, however, are favoured less readily. Secondly, we identify the condition under which unconditional altruism (performed by both dominant and subordinate) will be adaptive. Our third main result shows that increasing the strength of selection can also change the range of parameters over which costly social behaviours are favoured. We find that stronger selection makes it relatively easier for subordinate altruism to emerge, but more difficult for dominant spite and unconditional altruism to occur. We discuss the possible implications of our results for human social evolution.     

Factors affecting the errors in the estimation of evolutionary distances between sequences

Phylogenetic methods that use matrices of pairwise distances between sequences (e.g., neighbor joining) will only give accurate results when the initial estimates of the pairwise distances are accurate. For many different models of sequence evolution, analytical formulae are known that give estimates of the distance between two sequences as a function of the observed numbers of substitutions of various classes. These are often of a form that we call "log transform formulae". Errors in these distance estimates become larger as the time t since divergence of the two sequences increases. For long times, the log transform formulae can sometimes give divergent distance estimates when applied to finite sequences. We show that these errors become significant when t approximately 1/2 |lambda(max)|(-1) logN, where lambda(max) is the eigenvalue of the substitution rate matrix with the largest absolute value and N is the sequence length. Various likelihood-based methods have been proposed to estimate the values of parameters in rate matrices. If rate matrix parameters are known with reasonable accuracy, it is possible to use the maximum likelihood method to estimate evolutionary distances while keeping the rate parameters fixed. We show that errors in distances estimated in this way only become significant when t approximately 1/2 |lambda(1)|(-1) logN, where lambda(1) is the eigenvalue of the substitution rate matrix with the smallest nonzero absolute value. The accuracy of likelihood-based distance estimates is therefore much higher than those based on log transform formulae, particularly in cases where there is a large range of timescales involved in the rate matrix (e.g., when the ratio of transition to transversion rates is large). We discuss several practical ways of estimating the rate matrix parameters before distance calculation and hence of increasing the accuracy of distance estimates.     

Duality,ancestral and diffusion processes in models with selection

The ancestral selection graph in population genetics was introduced by Krone and Neuhauser [Krone, S.M., Neuhauser, C., 1997. Ancestral process with selection. Theor. Popul. Biol. 51, 210–237] as an analogue of the coalescent genealogy of a sample of genes from a neutrally evolving population. The number of particles in this graph, followed backwards in time, is a birth and death process with quadratic death and linear birth rates. In this paper an explicit form of the probability distribution of the number of particles is obtained by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura [Kimura, M., 1955. Stochastic process and distribution of gene frequencies under natural selection. Cold Spring Harbor Symposia on Quantitative Biology 20, 33–53]. It is shown that the process of fixation of the allele in the diffusion model corresponds to convergence of the ancestral process to its stationary measure. The time to fixation of the allele conditional on fixation is studied in terms of the ancestral process.     

Random genetic drift and gamete frequency

An analytic expression of conditional expectation of transient gamete frequency, given that one of the two loci remains polymorphic, is obtained in terms of the diffusion process by calculating the moments of the distribution. Using this expression, a model where linkage disequilibrium is introduced by a single mutation is considered. The conditional expectation of the gamete frequency given that the locus with the mutant allele remains polymorphic is presented. The behavior is significantly different from the monotonic decrease observed in the deterministic model without random genetic drift.     

How to calculate the non-synonymous to synonymous rate ratio of protein-coding genes under the Fisher–Wright mutation–selection framework

First principles of population genetics are used to obtain formulae relating the non-synonymous to synonymous substitution rate ratio to the selection coefficients acting at codon sites in protein-coding genes. Two theoretical cases are discussed and two examples from real data (a chloroplast gene and a virus polymerase) are given. The formulae give much insight into the dynamics of non-synonymous substitutions and may inform the development of methods to detect adaptive evolution.     

Balancer-Cre transgenic mouse germ cells direct the incomplete resolution of a tri-loxP-targeted Cyp1a1 allele, producing a conditional knockout allele

To generate conditional alleles, genes are commonly engineered to contain recognition sites for bacteriophage recombinases, such as Cre recombinase. When such motifs (lox sites) flank essential gene sequences, and provided that Cre recombinase is expressed, Cre recombinase will excise the flanked sequence-creating a conditional knockout allele. Targeted conditional alleles contain a minimum of three lox sites. It would be desirable to have Cre recombinase perform partial resolution (i.e., recombination some of the time between only the two lox sites flanking the marker gene). Here we report use of the commercially available Balancer2-Cre transgenic mouse line to carry out this function from a tri-loxP-site-containing cytochrome p450 1A1 (Cyp1a1) targeted allele. Such incomplete resolution of this complex locus occurred progressively with age in germ cells of male mice; the conditional Cyp1a1 gene was recovered in offspring from mice containing the targeted Cyp1a1 allele and the Cre recombinase transgene. Removal of the marker gene resulted in a conditional Cyp1a1 allele whose expression was indistinguishable from that of the wild-type allele.     

Association studies of QTL for multi-allele markers by mixed models

In this paper, we extend association study methods of both Fan et al. [Hum Hered 2002;53:130-145], in which a quantitative trait locus (QTL) and a multi-allele marker are considered for trio families, and Fan and Xiong [Biostatistics 2003, in press], in which a QTL and a bi-allelic marker are considered for nuclear families. The objective is to build mixed models for association study between a QTL and a multi-allelic marker for nuclear families with any number of offspring. Two types of nuclear family data are considered: the first is genetic data of offspring from at least one heterozygous parents, and the second is genetic data of offspring of nuclear family. (1) For the data of offspring from at least one heterozygous parents, we assume that at least one parent is heterozygous at the marker locus, and we may infer clearly the transmission of parental marker alleles to the offspring. We show that it can be used in association study in the presence of linkage. The theoretical basis is the difference between the conditional mean of trait value given an allele is transmitted and the conditional mean of trait value given the allele is not transmitted from a heterozygous parent. To build valid models, we calculate the variance covariance structure of trait values of offspring. Besides, the reduction of the number of parameters is discussed under an assumption of tight linkage between the trait locus and the marker. (2) For the data of offspring of nuclear family, we show that it can be used in general association study. In this case, the theoretical basis is the difference between the conditional mean of trait values given an allele is transmitted from a parent and the population mean. Then, we calculate variance-covariance structure of trait values of offspring. (3) Based on the theoretical analysis, mixed models are built for each type of the data, and related test statistics are proposed for association study. By power calculation and comparison, we show that, in some instances, the proposed test statistics have higher power than that by collapsing alleles to be new ones. The proposed models are used to analyze chromosomes 4 and chromosome 16 data of the Oxford asthma data, Genetic Analysis Workshop 12.     

Early Warnings of Regime Shift When the Ecosystem Structure Is Unknown

Abrupt changes in dynamics of an ecosystem can sometimes be detected using monitoring data. Using nonparametric methods that assume minimal knowledge of the underlying structure, we compute separate estimates of the drift (deterministic) and diffusion (stochastic) components of a general dynamical process, as well as an indicator of the conditional variance. Theory and simulations show that nonparametric conditional variance rises prior to critical transition. Nonparametric diffusion rises also, in cases where the true diffusion function involves a critical transition (sometimes called a noise-induced transition). Thus it is possible to discriminate noise-induced transitions from other kinds of critical transitions by comparing time series for the conditional variance and the diffusion function. Monte Carlo analysis shows that the indicators generally increase prior to the transition, but uncertainties of the indicators become large as the ecosystem approaches the transition point.     

The effect of a barrier to gene flow on patterns of geographic variation.

Explicit formulae are given for the effects of a barrier to gene flow on random fluctuations in allele frequency; these formulae can also be seen as generating functions for the distribution of coalescence times. The formulae are derived using a continuous diffusion approximation, which is accurate over all but very small spatial scales. The continuous approximation is confirmed by comparison with the exact solution to the stepping stone model. In both one and two spatial dimensions, the variance of fluctuations in allele frequencies increases near the barrier; when the barrier is very strong, the variance doubles. However, the effect on fluctuations close to the barrier is much greater when the population is spread over two spatial dimensions than when it occupies a linear, one-dimensional habitat: barriers of strength comparable with the dispersal range (B approximately equal to sigma) can have an appreciable effect in two dimensions, whereas only barriers with strength comparable with the characteristic scale (B approximately equal to L=sigma/sqrt{2mu}) are significant in one dimension (mu is the rate of mutation or long-range dispersal). Thus, in a two-dimensional population, barriers to gene flow can be detected through their effect on the spatial pattern of genetic marker alleles.     

Proportions of different habitat types are critical to the fate of a resistance allele

We describe a simple deterministic theoretical framework for analysing the gene frequency evolution of two alternative alleles at a single genetic locus in a habitat comprising two environments in which the genotypes have different relative fitnesses. We illustrate this for adaptation of pest insects, where one allele (resistance to toxins expressed in transgenic crops) is favoured in one environment (transgenic plants) and the other allele (susceptibility to toxins) is favoured in the other environment (‘refuges’ of non-transgenic plants). The evolution of allele frequencies depends on selection pressure because of relative sizes of the environments and relative fitnesses of the genotypes in each environment. We demonstrate that there are critical threshold proportions for habitat division that determine equilibrium allele frequencies. The stability of the system depends on relationships between the relative genotype fitnesses. In some cases, the division of the habitat in exactly the threshold proportions removes selection pressure and maintains polymorphism at all allele frequencies.     

An analytical formula to estimate confidence interval of QTL location with a saturated genetic map as a function of experimental design

Analytical formulae are derived for the confidence interval for location of a quantitative trait locus (QTL) using a saturated genetic map, as a function of the experimental design, the QTL allele substitution effect, and the number of individuals genotyped and phenotyped. The formulae are derived assuming evenly spaced recombination events, rather than the actual unevenly spaced distribution. The formulae are useful for determining desired sample size when designing a wide variety of QTL mapping experiments, and for evaluating a priori the potential of a given mapping population for defining the location of a QTL. The formulae do not take into account the finite number of recombination events in a given sample.     

Freeze substitution Hi-C,a convenient and cost-effective method for capturing the natural 3D chromatin conformation from frozen samples

Chromatin interactions functionally affect genome architecture and gene regulation, but to date, only fresh samples must be used in High-through chromosome conformation capture(Hi-C) to keep natural chromatin conformation intact. This requirement has impeded the advancement of 3 D genome research by limiting sample collection and storage options for researchers and severely limiting the number of samples that can be processed in a short time. Here, we develop a freeze substitution Hi-C(FS-Hi-C) technique that overcomes the need for fresh samples. FS-Hi-C can be used with samples stored in liquid nitrogen(LN2):the water in a vitreous form in the sample cells is replaced with ethanol via automated freeze substitution.After confirming that the FS step preserves the natural chromosome conformation during sample thawing,we tested the performance of FS-Hi-C with Drosophila melanogaster and Gossypium hirsutum. Beyond allowing the use of frozen samples and confirming that FS-Hi-C delivers robust data for generating contact heat maps and delineating A/B compartments and topologically associating domains, we found that FS-HiC outperforms the in situ Hi-C in terms of library quality, reproducibility, and valid interactions. Thus, FS-HiC will probably extend the application of 3 D genome structure analysis to the vast number of experimental contexts in biological and medical research for which Hi-C methods have been unfeasible to date.     

遗传负荷理论的一种普适框架

遗传负荷表示种群由于遗传变异能力的存在而在平均适宜度上的损失,定量讨论各种遗传负荷,对研究现实发生水平上的物种进化具有重要意义,以往的遗传负荷理论从种群平衡出发,探讨现实发生水平上的物种进化,可是,进化是种群平衡的位移;这便构成了理论与现实之间的矛盾,为拓展以往的遗传负荷理论,给出了一个描述各种遗传负荷的普适理论框架,利用这个理论框架既能探讨平衡种群的遗传负荷,又能模写非平衡种群的遗传负荷及其变化,从而克服了以往的遗传负荷理论不能描述非平衡种群和不时与生物进化现实相悖的不足之处,为研究物种的进化提供了一种可靠的模拟方法。     

Generation of novel conditional and hypomorphic alleles of the Smad2 gene

Smad2 is an intracellular mediator of the transforming growth factor beta signaling (TGFbeta) pathway. It has been previously shown that, in the mouse, ablation of functional Smad2 results in embryonic lethality due to gastrulation defects. To circumvent the early lethality and study the spatially and temporally specific functions of Smad2, we utilized the Cre-loxP system to generate a Smad2 conditional allele. Here we show that a conditional allele, Smad2(flox), was generated. In this allele, exons 9 and 10 are flanked by loxP sites and the gene is functionally wildtype. Cre-mediated recombination results in a deletion allele which phenocopies our previously reported Smad2(DeltaC) null mutation. To generate this conditional allele, we first made a targeted mutation which introduced a floxed neo cassette into intron 10. This allele (Smad2(3loxP)) functions hypomorphically when placed opposite a null allele, and unlike the other published Smad2 hypomorphic allele, can be maintained in the homozygous state.     

State-dependent life-history equations

Matrix population models provide a natural tool to analyse state-dependent life-history strategies. Reproductive value and the intrinsic rate of natural increase under a strategy, and the optimal life-history strategy can all be easily characterised using projection matrices. The resultant formulae, however, are not directly comparable with the corresponding formulae for age structured populations such as Lotka's equations and Fisher's formula for reproductive value. This is because formulae involving projection matrices lose track of what happens to an individual over its lifetime and are only concerned with expected numbers of descendants one time step in the future. In contrast the usual age-dependent formulae explicitly followed a single individual through from birth to death.In this paper I show how the state-dependent formulae can be rewritten to be directly comparable with the standard age-structured formulae. Although the formulae are intuitively obvious the decomposition into current and future reproductive success differs from that previously given and is, I suggest, a more natural definition. The derivation of appropriate equations for optimal life-histories relies on results from dynamic programming theory; and is much more general and easier than previous derivations.The value of rewriting projection matrix results in terms of the lifetime of an individual organism is illustrated by an example in which the optimal plastic response to an environment is derived.     

On the distribution of allele frequencies in a diffusion model

An expression is derived and values tabulated for the expected allele frequencies and their variances, arranged in decreasing order in a population, from the finite and infinite alleles diffusion model in Watterson (1976). The neutral model and also a model with heterozygote selection are considered. Some observed ABO blood group allele frequencies are compared with the tabulated expected frequencies in the neutral three allele model. This extends the results of Watterson and Guess (1977) who tabulate the expected value of the most common allele. One test of neutrality previously advocated is to consider the distribution of F, the population homozygosity, conditional on G, the product of allele frequencies. However it is shown here that for a large number of alleles, F and G are asymptotically independent, the test would not be a good one in this case. A limit theorem is derived for the distribution of allele frequencies in the neutral model when the mutation rate is large. In this case F is shown to be asymptotically normal. An inequality is derived for the probability that the oldest allele in a population is amongst the r most frequent types. An inequality is also found for the probability that a sample will only contain representatives of the r most frequent allele types in the population.     

Sojourn times and substitution rate at overdominant and linked neutral loci

The sojourn times until fixation of an overdominant allele were investigated based on the diffusion equation. Furthermore, the rate of accumulation of mutations, or the substitution rate, was predicted from the mean extinction time of a common overdominant allele. The substitution rate calculated theoretically agreed well with that determined by computer simulation. Overdominant selection enhances the polymorphism at linked loci, while its effect on the sojourn times and the substitution rate at linked loci has not been studied yet. To solve these problems, a model that assumed two linked loci, each with infinite alleles, was examined by computer simulation. A decrease in the recombination rate between two loci markedly changed the distribution of sojourn times of a neutral allele. Although overdominant selection obviously increased the sojourn times and the polymorphism at a linked locus, the rate of nucleotide substitution at the neutral locus was not influenced significantly even if complete linkage was assumed. These results suggest that, in regions containing overdominant genes, linked neutral loci will exhibit elevated levels of polymorphism, but their rate of molecular evolution remains that predicted by neutral theory.     

Generation of a conditional null allele of jumonji

The jumonji (jmj) gene plays important roles in multiple organ development in mouse, including cardiovascular development. Since JMJ is expressed widely during mouse development, it is essential that conditional knockout approaches be employed to ablate JMJ in a tissue-specific manner to identify the cell lineage specific roles of JMJ. In this report, we describe the establishment of a jmj conditional null allele in mice by generating a loxP-flanked (floxed) jmj allele, which allows the in vivo ablation of jmj via Cre recombinase-mediated deletion. Gene targeting was used to introduce loxP sites flanking exon 3 of the jmj allele to mouse embryonic stem cells. Our results indicate that the jmj floxed allele converts to a null allele in a heart-specific manner when embryos homozygous for the floxed jmj allele and carrying the alpha-myosin heavy chain promoter-Cre transgene were analyzed by Southern and Northern blot analyses. Therefore, this mouse line harboring the conditional jmj null allele will provide a valuable tool for deciphering the tissue and cell lineage specific roles of JMJ.