Neutral two-locus multiple allele models with recombination

General formulae for the homozygosity and variance of linkage disequilibrium are derived for neutral, stationary, two-locus multiple allele models where there is a symmetric type of mutation at each locus. Particular cases examined are K allele models, the infinite alleles model, and the stepwise mutation model. The two-locus infinite allele model is examined at the molecular level and a joint probability generating function is found for the number of heterozygous sites at each locus in two randomly chosen gametes.     

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.     

Leg stiffness and joint stiffness while running to and jumping over an obstacle

During running, muscles of the lower limb act like a linear spring bouncing on the ground. When approaching an obstacle, the overall stiffness of this leg-spring system (k_leg) is modified during the two steps preceding the jump to enhance the movement of the center of mass of the body while leaping the obstacle. The aim of the present study is to understand how k_leg is modified during the running steps preceding the jump. Since k_leg depends on the joint torsional stiffness and on the leg geometry, we analyzed the changes in these two parameters in eight subjects approaching and leaping a 0.65 m-high barrier at 15 km h⁻¹. Ground reaction force (F) was measured during 5–6 steps preceding the obstacle using force platform and the lower limb movements were recorded by camera. From these data, the net muscular moment (M_j), the angular displacement (θ_j) and the lever arm of F were evaluated at the hip, knee and ankle. At the level of the hip, the M_j–θ_j relation shows that muscles are not acting like torsional springs. At the level of the knee and ankle, the M_j–θ_j relation shows that muscles are acting like torsional springs: as compared to steady-state running, the torsional stiffness k_j decreases from ~1/3 two contacts before the obstacle, and increases from ~2/3 during the last contact. These modifications in k_j reflect in changes in the magnitude of F but also to changes in the leg geometry, i.e. in the lever arms of F.     

On Nei and Roychoudhury''s Sampling Variances of Heterozygosity and Genetic Distance

Nei and Roychoudhury have developed an inequality relationship between the expected values of (j_x- g)² and (ĝ-g)² where j_x is a biased and ĝ is an unbiased estimate of population homozygosity g. Their inequality is somewhat weak and can be improved, as is demonstrated in this paper.     

Two-locus,fourth-order gene frequency moments: Implications for the variance of squared linkage disequilibrium and the variance of homozygosity

Identity coefficients are used to construct a sufficient set of equations to determine the fourth-order moments of gene frequencies for two linked loci. This allows the variance of the expected squared linkage disequilibrium to be found. It is shown that the coefficient of variation is generally greater than one and if the mutation rate is small, the standard deviation is more than four times the size of the mean. This demonstrates that squared linkage disequilibrium is a highly variable quantity. The variance of homozygosity for a gene which consists of two sites can also be obtained. Recombination between these sites increases the variance of homozygosity, suggesting that intragenic recombination significantly changes all the expected moments of gene frequencies if 4Nμ > 1.0 and r > μ (where N is the population size, μ is the mutation rate of the gene to neutral alleles, and r is the recombination rate between two sites within the gene).     

More on Nei and Roychoudhury''s Sampling Variances of Heterozygosity and Genetic Distance

The inequality relationship between the expected values of ( j_x-g)² and (ĝ-g) ², where j_x is a biased and ĝ is an unbiased estimate of population homozygosity g, were examined earlier by Nei and Roychoudhury (1974) and later by Mitra (1975). The improvement in the inequality still left much to be desired. In this paper a lower boundary of g has been obtained which may be regarded as ultimate for ensuring a smaller expected value of ( j_x-g)² than the corresponding value of ( ĝ-g)².     

Estimates of Inbreeding in a Natural Population: A Comparison of Sampling Properties

The average inbreeding coefficient f of a population can be estimated in several different ways based solely on the genotypic frequencies at a single locus. The means and variances of four different estimates have been compared. While the four estimates are equivalent when there are two alleles, the best estimates when there are three or more alleles are based upon total heterozygosity (see PDF) where x and y are the expected and observed number of heterozygotes) and the proportion of alleles that are homozygous (see PDF) where k = the number of alleles, a_ii = the number of A_iA_i homozygotes, and 2a_ij = the number of A_iA_j heterozygotes). Both are minimally biased estimates of f and have identical sampling variances when all alleles are equally frequent. However, when alleles have different frequencies, the choice between these two estimates depends on the gene frequencies and the true inbreeding coefficient of a population; f₂ is the best estimate when the true average inbreeding coefficient is suspected to be low or f = 0, while f₁ is best in populations with large average inbreeding coefficients. Approximate sampling variances of these two estimates are given for any f and any number of alleles with arbitrary gene frequencies; these approximations are accurate for samples as small as n = 100. The chi-square and maximum likelihood estimates of f are not as good for realistic sample sizes.     

The Relationship Between FST and the Frequency of the Most Frequent Allele

F_ST is frequently used as a summary of genetic differentiation among groups. It has been suggested that F_ST depends on the allele frequencies at a locus, as it exhibits a variety of peculiar properties related to genetic diversity: higher values for biallelic single-nucleotide polymorphisms (SNPs) than for multiallelic microsatellites, low values among high-diversity populations viewed as substantially distinct, and low values for populations that differ primarily in their profiles of rare alleles. A full mathematical understanding of the dependence of F_ST on allele frequencies, however, has been elusive. Here, we examine the relationship between F_ST and the frequency of the most frequent allele, demonstrating that the range of values that F_ST can take is restricted considerably by the allele-frequency distribution. For a two-population model, we derive strict bounds on F_ST as a function of the frequency M of the allele with highest mean frequency between the pair of populations. Using these bounds, we show that for a value of M chosen uniformly between 0 and 1 at a multiallelic locus whose number of alleles is left unspecified, the mean maximum F_ST is ∼0.3585. Further, F_ST is restricted to values much less than 1 when M is low or high, and the contribution to the maximum F_ST made by the most frequent allele is on average ∼0.4485. Using bounds on homozygosity that we have previously derived as functions of M, we describe strict bounds on F_ST in terms of the homozygosity of the total population, finding that the mean maximum F_ST given this homozygosity is 1 − ln 2 ≈ 0.3069. Our results provide a conceptual basis for understanding the dependence of F_ST on allele frequencies and genetic diversity and for interpreting the roles of these quantities in computations of F_ST from population-genetic data. Further, our analysis suggests that many unusual observations of F_ST, including the relatively low F_ST values in high-diversity human populations from Africa and the relatively low estimates of F_ST for microsatellites compared to SNPs, can be understood not as biological phenomena associated with different groups of populations or classes of markers but rather as consequences of the intrinsic mathematical dependence of F_ST on the properties of allele-frequency distributions.DIFFERENTIATION among groups is one of the fundamental subjects of the field of population genetics. Comparisons of the level of variation among subpopulations with the level of variation in the total population have been employed frequently in population-genetic theory, in statistical methods for data analysis, and in empirical studies of distributions of genetic variation. Wright’s (Wright 1951) fixation indices, and F_ST in particular, have been central to this effort.Wright’s F_ST was originally defined as the correlation between two randomly sampled gametes from the same subpopulation when the correlation of two randomly sampled gametes from the total population is set to zero. Several definitions of F_ST or F_ST-like quantities are now available, relying on a variety of different conceptual formulations but all measuring some aspect of population differentiation (e.g., Charlesworth 1998; Holsinger and Weir 2009). Many authors have claimed that one or another formulation of F_ST is affected by levels of genetic diversity or by allele frequencies, either because the range of F_ST is restricted by these quantities or because these quantities affect the degree to which F_ST reflects population differentiation (e.g., Charlesworth 1998; Nagylaki 1998; Hedrick 1999, 2005; Long and Kittles 2003; Jost 2008; Ryman and Leimar 2008; Long 2009; Meirmans and Hedrick 2011). For example, Nagylaki (1998) and Hedrick (1999) argued that measures of F_ST may be poor measures of genetic differentiation when the level of diversity is high. Charlesworth (1998) suggested that F_ST can be inflated when diversity is low, arguing that F_ST might not be appropriate for comparing loci with substantially different levels of variation. In a provocative recent article, Jost (2008) used the diversity dependence of forms of F_ST to question their utility as differentiation measures at all.One definition that is convenient for mathematical assessment of the relationship of an F_ST-like quantity and allele frequencies is the quantity labeled G_ST by Nei (1973), which for a given locus measures the difference between the heterozygosity of the total (pooled) population, h_T, and the mean heterozygosity across subpopulations, h_S, divided by the heterozygosity of the total population:GST=hT−hShT.(1)In terms of the homozygosity of the total population, H_T = 1 − h_T, and the mean homozygosity across subpopulations, H_S = 1 − h_S, we can writeGST=HS−HT1−HT.(2)The Wahlund (1928) principle guarantees that H_S ≥ H_T and, therefore, because H_S ≤ 1 and for a polymorphic locus with finitely many alleles, 0 < H_T < 1, G_ST lies in the interval [0,1].Using G_ST for their definition of F_ST, Hedrick (1999, 2005) and Long and Kittles (2003) pointed out that because h_T < 1, F_ST cannot exceed the mean homozygosity across subpopulations, H_S:F_ST = 1 ? h_S/h_T < 1 ? h_S = H_S.(3)Hedrick (2005) obtained this result by considering a set of K equal-sized subpopulations, in which each allele is private to a single subpopulation. In the limit as K → ∞, a stronger upper bound on F_ST as a function of H_S and K reduces to Equation 3 (see also Jin and Chakraborty 1995 and Long and Kittles 2003).While Hedrick (1999, 2005) and Long and Kittles (2003) have clarified the relationship between F_ST and the mean homozygosity H_S across subpopulations, their approaches do not easily illuminate the connection between F_ST and allele frequencies themselves. A formal understanding of the relationship between F_ST and allele frequencies would make it possible to more fully understand the behavior of F_ST in situations where markers of interest differ substantially in allele frequencies or levels of genetic diversity. Our recent work on the relationship between homozygosity and the frequency of the most frequent allele (Rosenberg and Jakobsson 2008; Reddy and Rosenberg 2012) provides a mathematical approach for formal investigation of bounds on population-genetic statistics in terms of allele frequencies. In this article, we therefore seek to thoroughly examine the dependence of F_ST on allele frequencies by investigating the upper bound on F_ST in terms of the frequency M of the most frequent allele across a pair of populations. We derive bounds on F_ST given the frequency of the most frequent allele and bounds on the frequency of the most frequent allele given F_ST. We consider loci with arbitrarily many alleles in a pair of subpopulations. Using theory for the bounds on homozygosity given the frequency of the most frequent allele, we obtain strict bounds on F_ST given the homozygosity of the total population. Our analysis clarifies the relationships among F_ST, allele frequencies, and homozygosity, providing explanations for peculiar observations of F_ST that can be attributed to allele-frequency dependence.     

Measures of Variation at DNA Repeat Loci under a General Stepwise Mutation Model

Polymorphisms at tandem repeat loci are caused by mutations with allele sizes occasionally altered by more than one repeat unit in both forward and backward directions. Such mutational changes may occur with asymmetric probabilities. Therefore, a one-step symmetric stepwise mutation model may not be appropriate for studying the population dynamics at all repeat loci. In this work, we evaluated the expectation and variance of the within-population variance of the allele size distribution in a finite population, and the expected homozygosity at a locus by the coalescence approach under a general stepwise mutation model, where mutational transitions of allele sizes can be arbitrary, including being asymmetric. Under the special cases of symmetric one-step, two-step, and multi-step geometric distributions of mutations, our general results reduce to the corresponding results obtained by earlier investigators. The general results indicate that in a finite population, which has reached a steady state under the (general stepwise) mutation and drift balance, the within-population variance of allele sizes has a simple expectation (i.e., proportional toNν, the product of the mutation rate,ν, and effective population size,N). However, its stochastic variance is a quadratic function of this composite parameter,Nν. Furthermore, this second-order variance does not decay with the number of alleles sampled from a population. Application of this theory to data on allele size distributions in unrelated Caucasians from the CEPH pedigree (obtained from the Genome Data Base) shows that the relationship of the variance and mean of within-population variance of allele sizes at tandem repeat loci, grouped by their chromosomal assignment, has a trend compatible with the theory. However, there is an indication that the second-order variance is generally underestimated. One reason for this departure might be that the CEPH sample may not represent a single homogeneous population that reached equilibrium at all tandem repeat loci.     

Identification of a plant introduction soybean line with genetic lesions affecting two distinct glycinin subunits and evaluation of impacts on protein content and composition

Unlike other oilseeds, soybean (Glycine max [L.] Merr) is also valuable due to its direct conversion into human food. One notable example is the cheese-like product tofu. The quality of tofu is improved when protein subunits derived from two glycinin genes, Gy1 and Gy4, are reduced or absent. Here we report the discovery that one exotic soybean plant introduction line, PI 605781 B, has not only a previously described loss-of-expression mutation affecting one glycinin gene (gy4), but also bears an extremely rare, potentially unique, frameshift mutation in the Glycinin1 gene (gy1-a). We analyzed glycinin gene expression via qRT-PCR with mRNA from developing seeds, which revealed that the novel allele dramatically reduced Gy1 mRNA accumulation. Similarly, both A₄A₅B₃ and A_1aB_1a protein subunits were absent or at undetectable levels, as determined by two-dimensional protein fractionation. Despite the reduction in glycinin content, overall seed protein levels were unaffected. The novel gy1-a allele was found to be unique to PI 605871B in a sampling of 247 diverse germplasm lines drawn from a variety of geographic origins.     

Sampling for Microsatellite-Based Population Genetic Studies: 25 to 30 Individuals per Population Is Enough to Accurately Estimate Allele Frequencies

One of the most common questions asked before starting a new population genetic study using microsatellite allele frequencies is “how many individuals do I need to sample from each population?” This question has previously been answered by addressing how many individuals are needed to detect all of the alleles present in a population (i.e. rarefaction based analyses). However, we argue that obtaining accurate allele frequencies and accurate estimates of diversity are much more important than detecting all of the alleles, given that very rare alleles (i.e. new mutations) are not very informative for assessing genetic diversity within a population or genetic structure among populations. Here we present a comparison of allele frequencies, expected heterozygosities and genetic distances between real and simulated populations by randomly subsampling 5–100 individuals from four empirical microsatellite genotype datasets (Formica lugubris, Sciurus vulgaris, Thalassarche melanophris, and Himantopus novaezelandia) to create 100 replicate datasets at each sample size. Despite differences in taxon (two birds, one mammal, one insect), population size, number of loci and polymorphism across loci, the degree of differences between simulated and empirical dataset allele frequencies, expected heterozygosities and pairwise F_ST values were almost identical among the four datasets at each sample size. Variability in allele frequency and expected heterozygosity among replicates decreased with increasing sample size, but these decreases were minimal above sample sizes of 25 to 30. Therefore, there appears to be little benefit in sampling more than 25 to 30 individuals per population for population genetic studies based on microsatellite allele frequencies.     

Genetic diversity of microsatellite loci in hierarchically structured populations

Microsatellite loci are widely used for investigating patterns of genetic variation within and among populations. Those patterns are in turn determined by population sizes, migration rates, and mutation rates. We provide exact expressions for the first two moments of the allele frequency distribution in a stochastic model appropriate for studying microsatellite evolution with migration, mutation, and drift under the assumption that the range of allele sizes is bounded. Using these results, we study the behavior of several measures related to Wright’s F_ST, including Slatkin’s R_ST. Our analytical approximations for F_ST and R_ST show that familiar relationships between N_em and F_ST or R_ST hold when the migration and mutation rates are small. Using the exact expressions for F_ST and R_ST, our numerical results show that, when the migration and mutation rates are large, these relationships no longer hold. Our numerical results also show that the diversity measures most closely related to F_ST depend on mutation rates, mutational models (stepwise versus two-phase), migration rates, and population sizes. Surprisingly, R_ST is relatively insensitive to the mutation rates and mutational models. The differing behaviors of R_ST and F_ST suggest that properties of the among-population distribution of allele frequencies may allow the roles of mutation and migration in producing patterns of diversity to be distinguished, a topic of continuing investigation.     

Polymorphism of the Serotonin System Genes in Some Finno-Ugric Populations

Polymorphic sites in the genes encoding monoamine oxidase A (MAO-A), serotonin transporter (hSERT) and 5-HT_2A receptor were typed in Khant and Komi ethnic groups with the purpose of revealing possible inerpopulation differences in genotype and allele frequencies. No statistically significant differences in the hSERT and 5-HT_2A gene frequencies were detected. At the same time, the populations examined had statistically significantly different MAO-A genotype and allele frequencies. These results obtained indicate the prevalence of the site gain alleles of theEcoRV and Fnu4HI RFLP loci at the MAO-A gene in Komis and the of the corresponding site loss alleles in Khants.     

Identification and mapping of five new genes in bacteriophage T7

Mutations that affect the single-stranded DNA-binding protein of bacteriophage T7 (gene 2.5) and four T7 proteins of unknown function (the gene 4.3, 4.5, 4.7 and 5.5 proteins) are described and mapped by three-factor crosses. An extensive search for mutants defective in the DNA-binding protein (M_r = 25,562) produced several strains in which this protein has an altered electrophoretic mobility but no strains that appear to lack it completely. The gene 2.5 mutation that was mapped produces a slightly short DNA-binding protein that appears functional by tests in vitro. It seems likely that a functional DNA-binding protein is needed for T7 growth but that conditional-lethal amber mutations in it are rare; the nucleotide sequence known to code for the gene 2.5 protein contains only 1 to 3 sites that would be expected to be readily mutable to conditional-lethal amber codons by N-methyl-N?nitro-N-nitrosoguanidine. The gene 4.3, 4.5 and 4.7 proteins (M_r ～ 8000 to 15,000) are eliminated by a deletion mutant that removes most of the DNA between genes 4 and 5. The gene 5.5 protein (M_r ～ 11,700) is made in relatively large amounts and is affected by two different mutations that were mapped between genes 5 and 6. One of these mutations appears to be an amber mutation that eliminates the protein entirely; the other decreases the electrophoretic mobility of the protein (an apparent increase in size). A larger protein (M_r ～ 18,000), found in small amounts and difficult to observe, is also affected by these two mutations; the relationship of this minor protein to the major gene 5.5 protein is not yet known. The genes 2 and 18 proteins have also been identified in patterns of protein synthesis during infection. The proteins specified by at least 34 different T7 genes have now been identified.     

Biomass Partitioning and Its Relationship with the Environmental Factors at the Alpine Steppe in Northern Tibet

Alpine steppe is considered to be the largest grassland type on the Tibetan Plateau. This grassland contributes to the global carbon cycle and is sensitive to climate changes. The allocation of biomass in an ecosystem affects plant growth and the overall functioning of the ecosystem. However, the mechanism by which plant biomass is allocated on the alpine steppe remains unclear. In this study, biomass allocation and its relationship to environmental factors on the alpine grassland were studied by a meta-analysis of 32 field sites across the alpine steppe of the northern Tibetan Plateau. We found that there is less above-ground biomass (M_A) and below-ground biomass (M_B) in the alpine steppe than there is in alpine meadows and temperate grasslands. By contrast, the root-to-shoot ratio (R:S) in the alpine steppe is higher than it is in alpine meadows and temperate grasslands. Although temperature maintained the biomass in the alpine steppe, precipitation was found to considerably influence M_A, M_B, and R:S, as shown by ordination space partitioning. After standardized major axis (SMA) analysis, we found that allocation of biomass on the alpine steppe is supported by the allometric biomass partitioning hypothesis rather than the isometric allocation hypothesis. Based on these results, we believe that M_A and M_B will decrease as a result of the increased aridity expected to occur in the future, which will reduce the landscape’s capacity for carbon storage.     

A Stochastic Four-State Model of Contingent Gating of Gap Junction Channels Containing Two “Fast” Gates Sensitive to Transjunctional Voltage

Connexins, a family of membrane proteins, form gap junction (GJ) channels that provide a direct pathway for electrical and metabolic signaling between cells. We developed a stochastic four-state model describing gating properties of homotypic and heterotypic GJ channels each composed of two hemichannels (connexons). GJ channel contain two “fast” gates (one per hemichannel) oriented opposite in respect to applied transjunctional voltage (V_j). The model uses a formal scheme of peace-linear aggregate and accounts for voltage distribution inside the pore of the channel depending on the state, unitary conductances and gating properties of each hemichannel. We assume that each hemichannel can be in the open state with conductance γ_h,o and in the residual state with conductance γ_h,res, and that both γ_h,o and γ_h,res rectifies. Gates can exhibit the same or different gating polarities. Gating of each hemichannel is determined by the fraction of V_j that falls across the hemichannel, and takes into account contingent gating when gating of one hemichannel depends on the state of apposed hemichannel. At the single-channel level, the model revealed the relationship between unitary conductances of hemichannels and GJ channels and how this relationship is affected by γ_h,o and γ_h,res rectification. Simulation of junctions containing up to several thousands of homotypic or heterotypic GJs has been used to reproduce experimentally measured macroscopic junctional current and V_j-dependent gating of GJs formed from different connexin isoforms. V_j-gating was simulated by imitating several frequently used experimental protocols: 1), consecutive V_j steps rising in amplitude, 2), slowly rising V_j ramps, and 3), series of V_j steps of high frequency. The model was used to predict V_j-gating of heterotypic GJs from characteristics of corresponding homotypic channels. The model allowed us to identify the parameters of V_j-gates under which small changes in the difference of holding potentials between cells forming heterotypic junctions effectively modulates cell-to-cell signaling from bidirectional to unidirectional. The proposed model can also be used to simulate gating properties of unapposed hemichannels.     

FCGR2C genotyping by pyrosequencing reveals linkage disequilibrium with FCGR3A V158F and FCGR2A H131R polymorphisms in a Caucasian population

The FCGR3A-V158F and FCGR2A-H131R polymorphisms are associated with clinical responses to therapeutic mAbs and with immune thrombocytopenic purpura (ITP). The FCGR2C-ORF/STOP polymorphism, controlling FcγRIIC expression on natural killer cells and therefore FcγRIIC-mediated antibody dependent cell-mediated cytotoxicity, is also associated with ITP. Using a new pyrosequencing assay to determine this polymorphism in a control population, we observed the expected allele frequencies (ORF:12.6%) and percentages of individuals with a single copy (10.0%) or 3 copies (12.1%) of FCGR2C, or with at least one FCGR2C-ORF allele (20.1%). No association of FCGR2C copy number variations with the FCGR3A-V158F or FCGR2A-H131R genotype was detected. More importantly, our results demonstrate a strong and a weaker linkage disequilibrium associating the FCGR2C-ORF allele with the FCGR3A-158V and the FCGR2A-131H allele, respectively.     

Genetic hazards of ionizing radiations: cytogenetic extrapolations from mouse to man

The frequencies of X-ray induced asymmetrical interchanges (dicentrics) and acentric fragments (deletions) at several doses were measured in the circullating leukocytes of six species. The leukocytes of the species used had similar DNA contents but different chromosome and chromosome arm numbers. The data for dicentrics were fitted separately for each species by regression analysis to the model Y_j = b_jD + c_jD². All species gave a good fit to this model. As expected, when the dicentric data for all species were pooled and fitted to this model a poor fit was obtained. However, if a term for arm number was included, so that the model Y_j = (N_j?1) (bD+cD²) was fitted, a significant amount of the variation among species could be accounted for. At each dose there was an approximately linear relationship between the yield of dicentrics and the arm number. Man, with an effective arm number of 81, had twice as many dicentrics as the mouse, with an effective arm number of. These results strongly suggest that the chromosome arm number of a species influences the yield of asymmetrica interchanges. The chromosome arm number did not appear to influence the yield of deletions, and the yields induced in the mouse and man at easch dose were equal.These results show that man is twice as sensitive as the mouse to the induction of translocations, whereas the two species are equally sensitive to the indcution of deletions and, in all probability, to the production of mutations.     

Calculation of the Raman frequencies as a function of pressure in the solid phases II and III (III’) of benzene

We calculate here the Raman frequencies of the lattice modes A(A _g ), B(B _2g ) and C(B _1g B _3g ) as a function of pressure at room temperature for the solid phases (II, III and III’) of benzene. This calculation is performed using volume data through the mode Grüneisen parameter. It is found that our calculated frequencies of those lattice modes increase with increasing pressure, as expected. Calculated frequencies are in good agreement with the measurements of the three lattice modes for the solid phases studied in benzene.     

The Inbreeding Effective Population Number and the Expected Homozygosity for an X-Linked Locus

Assuming random mating and discrete nonoverlapping generations, the inbreeding effective population number, (see PDF), is calculated for an X-linked locus. For large populations, the result agrees with the variance effective population number. As an application, the maintenance of genetic variability by the joint action of mutation and random drift is investigated. It is shown that, if every allele mutates at rate u to new types, then the probabilities of identity in state (and hence the expected homozygosity of females) converge to the approximate value (see PDF) at the approximate asymptotic rate (see PDF).