Using the Pareto principle in genome-wide breeding value estimation

Genome-wide breeding value (GWEBV) estimation methods can be classified based on the prior distribution assumptions of marker effects. Genome-wide BLUP methods assume a normal prior distribution for all markers with a constant variance, and are computationally fast. In Bayesian methods, more flexible prior distributions of SNP effects are applied that allow for very large SNP effects although most are small or even zero, but these prior distributions are often also computationally demanding as they rely on Monte Carlo Markov chain sampling. In this study, we adopted the Pareto principle to weight available marker loci, i.e., we consider that x% of the loci explain (100 - x)% of the total genetic variance. Assuming this principle, it is also possible to define the variances of the prior distribution of the ''big'' and ''small'' SNP. The relatively few large SNP explain a large proportion of the genetic variance and the majority of the SNP show small effects and explain a minor proportion of the genetic variance. We name this method MixP, where the prior distribution is a mixture of two normal distributions, i.e. one with a big variance and one with a small variance. Simulation results, using a real Norwegian Red cattle pedigree, show that MixP is at least as accurate as the other methods in all studied cases. This method also reduces the hyper-parameters of the prior distribution from 2 (proportion and variance of SNP with big effects) to 1 (proportion of SNP with big effects), assuming the overall genetic variance is known. The mixture of normal distribution prior made it possible to solve the equations iteratively, which greatly reduced computation loads by two orders of magnitude. In the era of marker density reaching million(s) and whole-genome sequence data, MixP provides a computationally feasible Bayesian method of analysis.     

Random Phenotypic Variation of Yeast (Saccharomyces cerevisiae) Single-Gene Knockouts Fits a Double Pareto-Lognormal Distribution

Background

Distributed robustness is thought to influence the buffering of random phenotypic variation through the scale-free topology of gene regulatory, metabolic, and protein-protein interaction networks. If this hypothesis is true, then the phenotypic response to the perturbation of particular nodes in such a network should be proportional to the number of links those nodes make with neighboring nodes. This suggests a probability distribution approximating an inverse power-law of random phenotypic variation. Zero phenotypic variation, however, is impossible, because random molecular and cellular processes are essential to normal development. Consequently, a more realistic distribution should have a y-intercept close to zero in the lower tail, a mode greater than zero, and a long (fat) upper tail. The double Pareto-lognormal (DPLN) distribution is an ideal candidate distribution. It consists of a mixture of a lognormal body and upper and lower power-law tails.

Objective and Methods

If our assumptions are true, the DPLN distribution should provide a better fit to random phenotypic variation in a large series of single-gene knockout lines than other skewed or symmetrical distributions. We fit a large published data set of single-gene knockout lines in Saccharomyces cerevisiae to seven different probability distributions: DPLN, right Pareto-lognormal (RPLN), left Pareto-lognormal (LPLN), normal, lognormal, exponential, and Pareto. The best model was judged by the Akaike Information Criterion (AIC).

Results

Phenotypic variation among gene knockouts in S. cerevisiae fits a double Pareto-lognormal (DPLN) distribution better than any of the alternative distributions, including the right Pareto-lognormal and lognormal distributions.

Conclusions and Significance

A DPLN distribution is consistent with the hypothesis that developmental stability is mediated, in part, by distributed robustness, the resilience of gene regulatory, metabolic, and protein-protein interaction networks. Alternatively, multiplicative cell growth, and the mixing of lognormal distributions having different variances, may generate a DPLN distribution.     

A simple proof that certain quantities are independent of the geographical structure of population

This paper gives a proof that certain quantities are independent of the geographical structure of a population. The quantities are: (1) the fixation probability of a mutant; (2) the sum of the quantity x(1 ? x), where x is the mutant frequency, while the mutant is segregating; and (3) the quantity x(1 ? x) summed over the generations during which the gene frequency in the whole population assumes a specified value. The independence of geographical structure for the latter two quantities is not exact if there is selection, but is a close approximation.The model is a geographically structured version of Moran's haploid overlapping generation model. The population consists of colonies connected genetically by migration. Each individual has the same negative exponential lifetime distribution. When an individual dies, it is immediately replaced by an individual born in the same colony with a probability proportional to the frequency and fitness of the type giving birth. In a diploid population the quantity x(1 ? x) is proportional to the heterozygosity.     

Origination of Pareto distribution in complex dynamic systems

The Pareto distribution, whose probability density function can be approximated at sufficiently great chi as rho(chi) - chi(-alpha), where alpha > or = 2, is of crucial importance from both the theoretical and practical point of view. The main reason is its qualitative distinction from the normal (Gaussian) distribution. Namely, the probability of high deviations appears to be significantly higher. The conception of the universal applicability of the Gauss law remains to be widely distributed despite the lack of objective confirmation of this notion in a variety of application areas. The origin of the Pareto distribution in dynamic systems located in the gaussian noise field is considered. A simple one-dimensional model is discussed where the system response in a rather wide interval of the variable can be quite precisely approximated by this distribution.     

Global mutations and local mutations have very different effects on evolution, illustrated by mixed strategies of asymmetric binary games

We study the evolutionary effect of rare mutations causing global changes in traits. We consider asymmetric binary games between two players. The first player takes two alternative options with probability x and 1−x; and the second player takes options with probability y and 1−y. Due to natural selection and recurrent mutation, the population evolves to have broad distributions of x and y. We analyze three cases showing qualitatively different dynamics, exemplified by (1) vigilance-intrusion game, (2) asymmetric hawk-dove game and (3) cleaner-client game. We found that the evolutionary outcome is strongly dependent upon the distribution of mutants’ traits, more than the mutation rates. For example in the vigilance-intrusion game, the evolutionary dynamics show a perpetual stable oscillation if mutants are always close to the parent (local-mutation mode), whilst the population converges to a stable equilibrium distribution if mutants can be quite different from the parent (global-mutation mode), even for extremely low mutation rate. When common local mutations and rare global mutations occur simultaneously, the evolutionary outcome is controlled by the latter.     

Letter: Notes on "heat loss from a Newtonian animal"

Parker, Baker &; Smith (1972) have demonstrated mathematically that given the evolution of sexual reproduction, disruptive selection for the production of either many small gametes or a few large gametes may occur, resulting in a stable polymorphism of “sperm” and “egg” producers. Their model for the evolution of anisogamy requires only that zygote fitness (F) increase steeply with increases in zygote volume (V) (for Foc∝ V^x, x must be greater than 1·5) and that a sufficiently broad range of zygote productivity-size variants exist in the population (the higher the value of x, the broader the range needed). They suggest that anisogamy is almost universal in multicellular organisms but relatively rare in unicellular organisms because only for the former is an investment in extra gametic reserves at the expense of the number of gametes produced likely to be worthwhile in terms of increasing the survival probability of the zygote. In this note a graphical analysis and evidence from the anisogamous Protista will be presented concerning this hypothesis.     

Chromosome numbers,characterization of chromosomal pairing during meiosis,origin and natural propagation in polyploid cytotypes (4x, 5x and 6x) of Agrimonia eupatoria L. (Rosaceae) in northwest Himalayas (India)

Despite the presence of intraspecific polyploidy (2x, 4x, 5x and 6x) in Agrimonia eupatoria, origin of these cytotypes has never been addressed adequately. The aim of the present study was to record the original chromosome counts and characterize chromosomal pairing during meiosis and microsporogenesis in the 5x cytotype, and discussing the hypothesis regarding the possible origin of polyploid cytotypes (4x, 5x and 6x) in the species. The geographical distribution pattern of cytotypes in the Indian Himalayas and elsewhere has also been analyzed. The present meiotic analysis revealed three chromosomes counts, the tetraploid (2n?=?4x?=?56), the pentaploid (2n?=?5x?=?70) and the hexaploid (2n?=?6x?=?84) cytotypes based on x?=?14. Meiotic course was perfectly normal in the 4x and 6x cytotypes resulting into high pollen fertility (94–100 %). Meiotic course in the imbalanced 5x cytotype has been found to be irregular characterized by the presence of high frequency of univalents at diakinesis and metaphase-I. Abnormal meiotic course contributed towards high pollen sterility (74–88 %). Even the apparently fertile/stained pollen grains were of irregular shape and of heterogeneous sizes. Meiotic behaviour of the 5x cytotype is like typical of allopolyploid. Individuals of 5x cytotype did not produce seeds and propagate vegetatively (root suckers) while 4x and 6x cytotypes exploited sexual (seeds) as well as vegetative means for propagation. Chromosomal pairing in pentaploid cytotype is like typical of an allopolyploid and we assume that it might have originated owing to natural inter-cytotype hybridization between 4x and 6x cytotypes in a mixed population. Analysis of geographical distribution pattern of cytotypes shows that Indian Himalayas represent the most cytotype-diverse region for A. eupatoria with the existence of all the four cytotypes (2x, 4x, 5x, 6x). This shows the dynamic nature of the species at chromosomal level in this part of the world.     

The behaviour of coupled biochemical oscillators as a model of contact inhibition of cellular division

Intercellular communication of molecules between normal cells by tight junctions, and lack of this in some cancer cells (Loewenstein), can explain contact inhibition of cellular division in tissues. A general theory has been based on assuming the continual rise and fall (intrinsic oscillation) of a key substance x in each cell, with the period of the cell cycle. Periods are asynchronous in different cells, and x is exchanged between cells in contact by diffusion. A reduction in the resultant amplitude of fluctuation of x results, so that it does not reach the threshold x_t required for division to ensue; hence contact inhibition.The mathematical model is defined in its simplest form, and the sets of differential equations for arrays of cells are solved, from the isolated cell to the cell in an infinite sheet. The relative probability of division, P, is computed by numerical analysis from the area of resultant curves of x that lies above the threshold x_t. P depends on four dimensionless parameters, the order of coupling n (the number of cells directly communicating with a given cell), the total number of cells N in the aggregate, the communication constant K, and x_t, as a fraction of the amplitude of the intrinsic oscillation. The degree of synchrony, measured by the coefficient of variation σ of the periods, is important. If σ < ± 4%, contact inhibition is much reduced. The theory predicts that a paradoxical “contact-facilitation” is possible for very small aggregates of cells. For a cell in an infinite sheet, the amplitude of oscillation of x is reduced approximately by the factor $\text{1}\text{nK}$. For normal cells K is probably > 1, for cancer cells that lack communication, K is probably «< 1. However, two other basic causes for lack of regulation of tissue growth (cancer) could be excessive intrinsic oscillation of x, cf. x_t, and partial or complete synchronization of groups of cells by some unknown mechanism.     

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.     

Non-equilibrium thermodynamics of marginal- and conditional-open chemical systems

Every open chemical system treated in this paper is restricted to the case involving a sequence of monomolecular reactions. Various kinds of probability distribution governing it are introduced according to the situations in which it is placed. The chemical system subject to marginal distribution is given the term marginal-open system MO. The open chemical system ō discussed by Nicolis and Babloyantz can be regarded as the limiting system of MO. For an open chemical system, itself in contact with an external reservoir of finite volume, the probability distribution conditioned on the marginal distribution for the external reservoir in an arbitrarily fixed state is more appropriate. Such an open chemical system is called a conditional-open system CO. However, in the case of the external reservoir of infinite volume, although it is not certainly trivial, another conditional probability distribution has to be proposed; it is derived on the hypothesis that the probability distribution for an arbitrary total number of molecules in the open chemical system is known. The open chemical system so specified is called conditional-open system $\text{C}\text{O}\text{?}$. It is shown that for each system $\text{MO, CO}\text{and}\text{C}\text{O}\text{?}$ the change of entropy starting from the steady state provides a Liapunov function under some conditions and that the steady state is asymptotically stable. The relation of the entropy change to non-equilibrium fluctuations of chemical components in each system is discussed in comparison with that in the corresponding open chemical system ō, for which the steady state surely exists and is always stable. It is shown that the concept of $\text{C}\text{O}\text{?}$ is useful for investigating the phenomenon of steady-state coupling.     

Potential of genotyping-by-sequencing for genomic selection in livestock populations

Background

Next-generation sequencing techniques, such as genotyping-by-sequencing (GBS), provide alternatives to single nucleotide polymorphism (SNP) arrays. The aim of this work was to evaluate the potential of GBS compared to SNP array genotyping for genomic selection in livestock populations.

Methods

The value of GBS was quantified by simulation analyses in which three parameters were varied: (i) genome-wide sequence read depth (x) per individual from 0.01x to 20x or using SNP array genotyping; (ii) number of genotyped markers from 3000 to 300 000; and (iii) size of training and prediction sets from 500 to 50 000 individuals. The latter was achieved by distributing the total available x of 1000x, 5000x, or 10 000x per genotyped locus among the varying number of individuals. With SNP arrays, genotypes were called from sequence data directly. With GBS, genotypes were called from sequence reads that varied between loci and individuals according to a Poisson distribution with mean equal to x. Simulated data were analyzed with ridge regression and the accuracy and bias of genomic predictions and response to selection were quantified under the different scenarios.

Results

Accuracies of genomic predictions using GBS data or SNP array data were comparable when large numbers of markers were used and x per individual was ~1x or higher. The bias of genomic predictions was very high at a very low x. When the total available x was distributed among the training individuals, the accuracy of prediction was maximized when a large number of individuals was used that had GBS data with low x for a large number of markers. Similarly, response to selection was maximized under the same conditions due to increasing both accuracy and selection intensity.

Conclusions

GBS offers great potential for developing genomic selection in livestock populations because it makes it possible to cover large fractions of the genome and to vary the sequence read depth per individual. Thus, the accuracy of predictions is improved by increasing the size of training populations and the intensity of selection is increased by genotyping a larger number of selection candidates.

Electronic supplementary material

The online version of this article (doi:10.1186/s12711-015-0102-z) contains supplementary material, which is available to authorized users.     

An analytical model for a short-term advantage for sex

An analytical treatment is given for a model of Maynard Smith in which a short-term advantage for sex and recombination is provided by the mechanism of sib-competition. Suppose the next generation is formed by the winners of a large number of contests. Suppose a number of parents each contribute M offspring to a given contest, but the offspring of an asexual parent are identical whereas those of a sexual parent are distributed with some variance. If M is large there is a high probability that a sexual offspring will have a high enough fitness to win the contest. Calculations show that values of M around 3 and 4 are generally enough for sexual behaviour to overcome its two-fold disadvantage.     

Hydroperoxide and carboxyl groups preferential location in oxidized biomembranes experimentally determined by small angle X-ray scattering: Implications in membrane structure

We report small angle X-ray scattering (SAXS) data from large unilamellar vesicles as model membranes composed of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocoline (POPC) and two oxidized species, namely its hydroperoxidized form POPC-OOH and 1-palmitoyl-2-azelaoyl-sn-glycero-3-phosphocholine (PazePC) lipid that has a carboxyl group at the end of its truncated sn-2 chain. The replacement of POPC by either POPC-OOH (POPC-OOH_xPOPC_1−x) or PazePC (PazePC_xPOPC_1−x), with oxidized lipid molar ratio x varying from 0.00 up to 1.00, permits to experimentally inspect changes in the membrane structural properties due to oxidation. The volume fraction distribution of each lipid chemical group along the bilayer is determined. The results quantify that 95% of the hydroperoxide group lies in the membrane polar moiety, near the carbonyl and phosphate groups, whereas just 5% of OOH group experiences the polar/apolar interface, for all values of x studied. In the case of PazePC up to x = 0.33, a bimodal distribution of the carboxyl group in the interior and polar regions of the lipid membrane is obtained, probably due to a dynamic movement of the shortened alkyl chain towards the water interface. The mean molecular area A gradually increases from 65.4 ± 0.4 Ų for POPC bilayers to 78 ± 2 Ų for pure POPC-OOH bilayers, whereas POPC-OOH membrane thickness resulted to be 20% thinner than the non-oxidized POPC membrane. For PazePC up to x = 0.33, A increases to 67 ± 2 Ų with 10% of membrane thinning. The SAXS results thus demonstrate how the lipid oxidation progress affects the membrane structural features, thus paving the way to better understand membrane damage under oxidative stress.     

Coalescence Times and FST Under a Skewed Offspring Distribution Among Individuals in a Population

Estimates of gene flow between subpopulations based on F_ST (or N_ST) are shown to be confounded by the reproduction parameters of a model of skewed offspring distribution. Genetic evidence of population subdivision can be observed even when gene flow is very high, if the offspring distribution is skewed. A skewed offspring distribution arises when individuals can have very many offspring with some probability. This leads to high probability of identity by descent within subpopulations and results in genetic heterogeneity between subpopulations even when Nm is very large. Thus, we consider a limiting model in which the rates of coalescence and migration can be much higher than for a Wright–Fisher population. We derive the densities of pairwise coalescence times and expressions for F_ST and other statistics under both the finite island model and a many-demes limit model. The results can explain the observed genetic heterogeneity among subpopulations of certain marine organisms despite substantial gene flow.     

From retrospective to predictive structure activity correlations.

Semiempirical molecular orbital (CNDO/2) calculations have been performed on 22 halogen or ?CH₃ ?CF₃ and ?OCH₃ substituted benzylamines which show PNMT inhibitory activities. In terms of the single composite variable I = |0·24μ_x+0·52μ_y| a rather good linear correlation was obtained for all the compounds.Applying the “leave-L-out” technique with random sampling, this linear correlation and another quadratic one (for 22 artificially chosen points) was re-established with the aid of the remaining compounds. They have shown only small differences in the calculated pI₅₀ values as compared to those obtained with the aid of the original correlation equation, if about half of the points were left out. This means that to obtain a stable correlation with high enough predictive power one should keep the ratio $\text{N}\text{p}$ (N number of points, p number of adjustable parameters in the equation) as large as possible.Deriving an expression for the upper bound of the probability of a chance correlation in the case of a single variable correlation equation, it has been shown that, for N equal to 22 and V (the number of variables in the variable pool) equal to 11, this probability is less than 0·01%.     

Age distribution of trees in stationary forest system

A statistical theory for the age distribution of spatially dominant trees in a stationary forest system is developed. The result depends whether or not mortality is spatially correlated, as well as whether or not the stand boundaries are pre-determined. In the case of spatially non-correlated mortality, the tree age distribution is an exponential with survival rate as the base. In the case of spatially correlated mortality within a stand with pre-determined boundaries, the age distribution within the stand is an exponential with natural base. For a small stand, the median life span of the stand is inversely proportional to the number of trees (n); the median life span in relation to stand closure time is inversely proportional to n ln(n). For a large stand, the stand life does not extend to the closure time.The behaviour of a forest system without fixed stand boundaries depends on the dimensionality of the system. In the case of a one-dimensional system, the longevity distribution is exponential, most of the trees however having the same longevity. Consequently, the probability density of tree age is constant. However, the probability mass of size of catastrophe destroying a particular tree is evenly distributed. This is due to trees being rapidly born on empty areas in the beginning of the life cycle, and clusters rapidly growing into larger ones close to the end of tree life.     

Protein dynamics: Mössbauer spectroscopy on deoxymyoglobin crystals

Mössbauer absorption experiments on ⁵⁷Fe of deoxygenated myoglobin crystals and on K₄⁵⁷Fe(CN)₆ dissolved in the water of metmyoglobin crystals were performed over a large temperature range. At low temperatures the mean square displacements, 〈x²〉, of the iron indicate solid-like behaviour of the whole system, whereas at higher temperatures protein-specific modes of motion contribute to 〈x²>. The protein dynamics are correlated with the mobility of the water within the protein crystals. A Brownian oscillator is used to model the protein-specific modes of motion measured at the ⁵⁷Fe nucleus. Three modes are necessary for understanding the Mössbauer spectrum. Two of them correspond to an extremely overdamped Brownian oscillator. The third mode can be understood as quasi-free diffusion. Whereas the protein molecule is frozen in conformational substates in the low temperature regime, it reaches transition states with a finite probability in the high temperature regime. The surface water mediates a possible trigger mechanism that switches on protein dynamics within a narrow temperature interval. Results from Mössbauer spectroscopy and from X-ray structure analysis are compared.     

Systematic analysis of residual density suggests that a major limitation in well‐refined X‐ray structures of proteins is the omission of ordered solvent

In the residual electron density map of a fully refined X‐ray protein model, there should be no peaks arising from modeling errors or missing atoms. Any residual peaks that do occur should be contributed by random residual intensity differences between the model and the data. If the model is incomplete (i.e., some atoms are missing), there will be more positive peaks than negative ones. On the other hand, if the model includes inappropriately located atoms, there will be an excess of negative peaks. In this study, random residual peaks are quantified using the probability density function P(x), which is defined as the probability for a peak having peak height between x and x + dx. It is found that P(x) is single‐exponential and symmetric for both positive and negative peaks. Thus, P(x) can be used to discriminate residual peaks contributed by random noise in complete models from residual peaks being attributable to modeling errors in incomplete models. For a number of representative structures in the PDB it is found that P(x) has far more large (greater than 5 sigma) positive peaks than large negative peaks. This excess of large positive peaks suggests that the main defect in these refined structures is the omission of ordered water molecules.     

Survivor's dilemma: Defend the group or flee?

We consider a survival game of gregarious individuals, in which the aim of the players is survival to reproductive age under predator attacks. The survivor’s dilemma (shortly: SVD) game consists in the following: a group member either surely survives alone by fleeing, while its defensive mate may be killed; or tries to save its mate’s life, risking to get killed. The dilemma is that, in every single attack, fleeing ensures maximal survival probability, but if its mate survives by fighting both, and they remain together, its risk to be killed at the next attack will be lower. We show that, if defense is successful enough, then the one-attack game is a prisoner’s dilemma (PD), where fleeing is the strict ESS. We have additively decomposed the SVD game, according to the survival of the group mate of the focal prey, into two games: the aim of the “collective game” is survival of the group of prey. Counter-wise, the aim of the “hostile game” is survival alone (focal prey survives and its mate is killed by the predator). We obtain the following results: if the attack number is large enough, the multi-attack SVD game is dominated by the “collective game” in the sense that each individual can ensure its own maximal survival probability by maximizing the group survival probability in each attack. In the hostile game, the only strict ESS is the fleeing strategy. In the collective game there are two different cases: either defense is a unique strict ESS, or the collective game is bistable, i.e. fleeing and defense are local strict ESS’s. If defense is the only strict ESS in the collective game, and the attack number is large enough, defense replaces fleeing strategy in the multi-attack SVD game. However, in the bistable case, defense cannot invade into the fleeing population. It is shown that, if the interaction between relatives is frequent enough, than defense can replace fleeing strategy, in spite of the fact that in the well-mixed population the collective game is bistable.