Genetic code,hamming distance and stochastic matrices

In this paper we use the Gray code representation of the genetic code C = 00, U = 10, G = 11 and A = 01 (C pairs with G, A pairs with U) to generate a sequence of genetic code-based matrices. In connection with these code-based matrices, we use the Hamming distance to generate a sequence of numerical matrices. We then further investigate the properties of the numerical matrices and show that they are doubly stochastic and symmetric. We determine the frequency distributions of the Hamming distances, building blocks of the matrices, decomposition and iterations of matrices. We present an explicit decomposition formula for the genetic code-based matrix in terms of permutation matrices, which provides a hypercube representation of the genetic code. It is also observed that there is a Hamiltonian cycle in a genetic code-based hypercube.     

Marginal distributions of genetic coalgebras

We consider the backward evolution of a particular type of Mendelian genetic system whose transition probabilities give place to the so-called coalgebras with genetic realization and describe the equilibrium states of such mathematical objects and therefore those of the genetic system. We exploit the relationship between the genetic coalgebras modeling the transference of the genetic inheritance and cubic stochastic matrices of type (1, 2) studying first the ergodicity of these matrices in terms of the stationary probability distributions of the bivariate Markov chains defined by their accompanying matrices. Then we apply the obtained results to describe the equilibrium states of coalgebras with genetic realization.     

Mean square exponential and robust stability of stochastic discrete-time genetic regulatory networks with uncertainties

This paper aims to analyze global robust exponential stability in the mean square sense of stochastic discrete-time genetic regulatory networks with stochastic delays and parameter uncertainties. Comparing to the previous research works, time-varying delays are assumed to be stochastic whose variation ranges and probability distributions of the time-varying delays are explored. Based on the stochastic analysis approach and some analysis techniques, several sufficient criteria for the global robust exponential stability in the mean square sense of the networks are derived. Moreover, two numerical examples are presented to show the effectiveness of the obtained results.     

Population structure in the Connecticut Valley: II. A comparison of multidimensional scaling solutions of migration matrices and isonymy

In a previous paper (Swedlund et al., 1984) we have described the population structure of the historical Connecticut River Valley of Massachusetts in terms of matrimonial migration matrices. Using procedures described by Morton (1973), Harpending and Jenkins (1974), Jorde (1980), and others the exchanges between subdivisions which make up the matrices are made column stochastic and analyzed to predict genetic kinship. Subsequently the kinship estimates within and between subdivisions can be interpreted as genetic covariance and compared to the actual geographic distances between the respective subdivisions using a principal components analysis. In the present paper we extend these results by applying nonmetric multidimensional scaling to the migration matrices, and to isonymy matrices based on the same communities. We demonstrate that the multidimensional scaling configurations of marital migration represent the actual geographic relationships between the communities quite effectively for this particular case study from historical Massachusetts. Moreover, we argue that while these migration data may provide good estimates of social and genetic exchange between the subdivisions, surname analysis may also be informative of processes not revealed in the migration matrices alone.     

Mean square stability of uncertain stochastic BAM neural networks with interval time-varying delays

The robust asymptotic stability analysis for uncertain BAM neural networks with both interval time-varying delays and stochastic disturbances is considered. By using the stochastic analysis approach, employing some free-weighting matrices and introducing an appropriate type of Lyapunov functional which takes into account the ranges for delays, some new stability criteria are established to guarantee the delayed BAM neural networks to be robustly asymptotically stable in the mean square. Unlike the most existing mean square stability conditions for BAM neural networks, the supplementary requirements that the time derivatives of time-varying delays must be smaller than 1 are released and the lower bounds of time varying delays are not restricted to be 0. Furthermore, in the proposed scheme, the stability conditions are delay-range-dependent and rate-dependent/independent. As a result, the new criteria are applicable to both fast and slow time-varying delays. Three numerical examples are given to illustrate the effectiveness of the proposed criteria.     

A tale of two matrices: multivariate approaches in evolutionary biology

Two symmetric matrices underlie our understanding of microevolutionary change. The first is the matrix of nonlinear selection gradients (gamma) which describes the individual fitness surface. The second is the genetic variance-covariance matrix (G) that influences the multivariate response to selection. A common approach to the empirical analysis of these matrices is the element-by-element testing of significance, and subsequent biological interpretation of pattern based on these univariate and bivariate parameters. Here, I show why this approach is likely to misrepresent the genetic basis of quantitative traits, and the selection acting on them in many cases. Diagonalization of square matrices is a fundamental aspect of many of the multivariate statistical techniques used by biologists. Applying this, and other related approaches, to the analysis of the structure of gamma and G matrices, gives greater insight into the form and strength of nonlinear selection, and the availability of genetic variance for multiple traits.     

Genetic,phenotypic, and environmental correlations in black medic,Medicago lupulina L., grown in three different environments

We have investigated the relationship between phenotypic and genetic correlations among a large number of quantitative traits (36) in three different environments in order to determine their degree of disparity and whether phenotypic correlations could be substituted for their genetic counterparts whatever the environment. We also studied the influence of the environment on genetic and phenotypic correlations. Twenty accessions (full-sib families) ofMedicago luPulina were grown in three environments. In two of these two levels of environmental stress were generated by harvesting plants at flowering and by growing plants in competition with barley, respectively. A third environment, with no treatment, was used as a control with no stress. Average values of pod and shoot weight indicate that competition induces the highest level of stress. The genetic and phenotypic correlations among the 36 traits were compared. Significant phenotypic correlations were obtained easily, while there was no genetic variation for 1 or the 2 characters being correlated. The large positive correlation between the genetic and phenotypic correlation matrices indicated a good proportionality between genetic and phenotypic correlations matrices but not their similarity. In a given environment, when only those traits with a significant genetic variance were taken into account, there were still differences between genetic and phenotypic correlations, even when levels of significance for phenotypic correlations were lowered. Consequently, it is dangerous to substitute phenotypic correlations for genetic correlations. The number of traits that showed genetic variability increased with increasing environmental stress, consequently the number of significant genetic correlations also increased with increasing environmental stress. In contrast, the number of significant phenotypic correlations was not influnced by the environment. The structures of both phenotypic and genetic matrices, however, depended on the environment, and not in the same way for both matrices.     

Population divergence of genetic (co)variance matrices in a subdivided plant species,Brassica cretica

Abstract The present study of Brassica cretica had two objectives. First, we compared estimates of population structure (Q_st) for seven phenotypic characters with the corresponding measures for allozyme markers (F_st) to evaluate the supposition that genetic drift is a major determinant of the evolutionary history of this species. Secondly, we compared the genetic (co)variance ( G ) matrices of five populations to examine whether a long history of population isolation is associated with large, consistent differences in the genetic (co)variance structure. Differences between estimates of F_st and Q_st were too small to be declared significant, indicating that stochastic processes have played a major role in the structuring of quantitative variation in this species. Comparison of populations using the common principal component (CPC) method rejected the hypothesis that the G matrices differed by a simple constant of proportionality: most of the variation involved principal component structure rather than the eigenvalues. However, there was strong evidence for proportionality in comparisons using the method of percentage reduction in mean‐square error (MSE), at least when characters with unusually high (co)variance estimates were included in the analyses. Although the CPC and MSE methods provide different, but complementary, views of G matrix variation, we urge caution in the use of proportionality as an indicator of whether genetic drift is responsible for divergence in the G matrix.     

VARIATION IN GENETIC ARCHITECTURE OF CALLING SONG AMONG POPULATIONS OF ALLONEMOBIUS SOCIUS,A. FASCIATUS,AND A HYBRID POPULATION: DRIFT OR SELECTION?

Predictions using quantitative genetic models generally assume that the variance-covariance matrices remain constant over time. This assumption is based on the supposition that selection is generally weak and hence variation lost through selection can be replaced by new mutations. Whether this is generally true can only be ascertained from empirical studies. Ideally for such a study we should be able to make a prediction concerning the relative strength of selection versus genetic drift. If the latter force is prevalent then the variance-covariances matrices should be proportional to each other. Previous studies have indicated that females in the two sibling cricket species Allonemobius socius and A. fasciatus do not discriminate between males of the two species by their calling song. Therefore, differences between the calling song of the two males most likely result from drift rather than sexual selection. We test this hypothesis by comparing the genetic architecture of calling song of three populations of A. fasciatus with two populations of A. socius. We found no differences among populations within species, but significant differences in the G (genetic) and P (phenotypic) matrices between species, with the matrices being proportional as predicted under the hypothesis of genetic drift. Because of the proportional change in the (co)variances no differences between species are evident in the heritabilities or genetic correlations. Comparison of the two species with a hybrid population from a zone of overlap showed highly significant nonproportional variation in genetic architecture. This variation is consistent with a general mixture of two separate genomes or selection. Qualitative conclusions reached using the phenotypic matrices are the same as those reached using the genetic matrices supporting the hypothesis that the former may be used as surrogate measures of the latter.     

The neutral theory in an infinite population

Selective substitutions at one locus induce stochastic dynamics at a linked neutral locus that resemble genetic drift even when the population size is infinite. This new stochastic force, which is called genetic draft, causes genetic variation at the neutral locus to decrease with population size and the rate of deleterious substitution to increase with population size. The fact that heterozygosities in natural populations are only weakly dependent on population size suggests that genetic draft may be a much more important stochastic force than genetic drift in natural populations. Some of the mathematical properties of genetic draft are explored.     

An analysis of G matrix variation in two closely related cricket species,Gryllus firmus and G. pennsylvanicus

An important issue in evolutionary biology is understanding the pattern of G matrix variation in natural populations. We estimated four G matrices based on the morphological traits of two cricket species, Gryllus firmus and G. pennsylvanicus, each reared in two environments. We used three matrix comparison approaches, including the Flury hierarchy, to improve our ability to perceive all aspects of matrix variation. Our results demonstrate that different methods perceive different aspects of the matrices, which suggests that, until more is known about these methods, future studies should use several different statistical approaches. We also found that the differences in G matrices within a species can be larger than the differences between species. We conclude that the expression of the genetic architecture can vary with the environment and that future studies should compare G matrices across several environments. We also conclude that G matrices can be conserved at the level of closely related species.     

Evolution in silico and in vitro: the RNA model.

Theoretical concepts and experiments dealing with the evolution of molecules in vitro reached a state that allows for direct applications to the design of biomolecules with predefined properties. RNA evolution in vitro represents a basis for the development of a new and comprehensive model of evolution, focusing on the phenotype and its fitness relevant properties. Relations between genotypes and phenotypes are described by mappings from genotype space onto a space of phenotypes, which are many-to-one and thus give ample room for neutrality as expressed by the existence of extended neutral networks in genotype space. The RNA model reduces genotype-phenotype relations to mappings from sequences into secondary structures of minimal free energies and allows for derivation of otherwise inaccessible quantitative results. Continuity and discontinuity in evolution are defined through a new notion of accessibility in phenotype space that provides a basis for straight forward interpretation of computer simulations on RNA optimization; furthermore, it reveals the constructive role of random genomic drift in the search for phenotypes of higher fitness. The effects of population size on the course of evolutionary optimization can be predicted quantitatively by means of a simple stochastic model based on a birth-anddeath process with immigration.     

Robust filtering for stochastic genetic regulatory networks with time-varying delay

This paper addresses the robust filtering problem for a class of linear genetic regulatory networks (GRNs) with stochastic disturbances, parameter uncertainties and time delays. The parameter uncertainties are assumed to reside in a polytopic region, the stochastic disturbance is state-dependent described by a scalar Brownian motion, and the time-varying delays enter into both the translation process and the feedback regulation process. We aim to estimate the true concentrations of mRNA and protein by designing a linear filter such that, for all admissible time delays, stochastic disturbances as well as polytopic uncertainties, the augmented state estimation dynamics is exponentially mean square stable with an expected decay rate. A delay-dependent linear matrix inequality (LMI) approach is first developed to derive sufficient conditions that guarantee the exponential stability of the augmented dynamics, and then the filter gains are parameterized in terms of the solution to a set of LMIs. Note that LMIs can be easily solved by using standard software packages. A simulation example is exploited in order to illustrate the effectiveness of the proposed design procedures.     

Geographic variation in the G matrices of wild populations of the barn swallow

In this paper, we present an analysis of genetic variation in three wild populations of the barn swallow, Hirundo rustica. We estimated the P, E, and G matrices for six linear morphological measurements and tested for variation among populations using the Flury hierarchical method and the jackknife followed by MANOVA method. Because of nonpositive-definite matrices, we had to employ 'bending' to analyse the G and E matrices with the Flury method. Both statistical methods agree in finding that the P and G matrices are significantly different but comparison between the analysis of the P matrices and pairwise analyses of the P, E, and G matrices suggests caution in interpreting the Flury results concerning differences in matrix structure. The significant variation among the populations in the G matrices appears to be due in large measure to the most geographically distant population.     

Quantitative Genetics,Pleiotropy, and Morphological Integration in the Dentition of <Emphasis Type="Italic">Papio hamadryas</Emphasis>

Variation in the mammalian dentition is highly informative of adaptations and evolutionary relationships, and consequently has been the focus of considerable research. Much of the current research exploring the genetic underpinnings of dental variation can trace its roots to Olson and Miller’s 1958 book Morphological Integration. These authors explored patterns of correlation in the post-canine dentitions of the owl monkey and Hyopsodus, an extinct condylarth from the Eocene. Their results were difficult to interpret, as was even noted by the authors, due to a lack of genetic information through which to view the patterns of correlation. Following in the spirit of Olson and Miller’s research, we present a quantitative genetic analysis of dental variation in a pedigreed population of baboons. We identify patterns of genetic correlations that provide insight to the genetic architecture of the baboon dentition. This genetic architecture indicates the presence of at least three modules: an incisor module that is genetically independent of the post-canine dentition, and a premolar module that demonstrates incomplete pleiotropy with the molar module. We then compare this matrix of genetic correlations to matrices of phenotypic correlations between the same measurements made on museum specimens of another baboon subspecies and the Southeast Asian colobine Presbytis. We observe moderate significant correlations between the matrices from these three primate taxa. From these observations we infer similarity in modularity and hypothesize a common pattern of genetic integration across the dental arcade in the Cercopithecoidea.     

Ordered genotypes: an extended ITO method and a general formula for genetic covariance

Traditionally, the stochastic ITO transition matrices provide a simple general method for obtaining the joint genotype distribution and genotypic correlations between any specified pair of noninbred relatives. The ITO method has been widely used in modern genetic analysis; however, since it was originally derived for unordered genotypes, it is not very useful in some new applications -- for example, when one is modeling genomic imprinting and must keep track of the parental origin of alleles. To address these new, emerging problems, here we extend the ITO method to handle ordered genotypes. Our extended method is applied to calculate the covariance in unilineal and bilineal relatives under genomic imprinting, and some generalized linear functions of the transition matrices are given. Since the ITO method is limited to biallelic loci and to unilineal and bilineal relatives, we derive a general formula for calculating the genetic covariance using ordered genotypes for any type of relative pair.     

An overall description of retinotopic mapping in the cat's visual cortex areas 17, 18, and 19

Mathematical functions are derived which model the retinotopic mapping in the cat's visual cortical areas 17, 18, and 19. All three mappings are simple modifications of a complex power function with an exponent of 0.43. This function is decomposed so as to give an intermediate stage which is common to all three mappings and can be regarded as a model of the lateral geniculate nucleus mapping. The influence of retinotopic mapping on visual receptive fields was studied. The results show that a dependence of the receptive field properties on the position in the visual field is to be expected.     

Offspring risk and sibling risk for multilocus traits

The recurrence risk of a trait in a relative of type R is the probability that an individual who is in relationship of type R to an affected proband has the trait. It is intuitively clear that closer relationships lead to higher recurrence risks. However, no exact analysis of this phenomenon has been presented for multilocus traits. We prove a theorem that shows how recurrence risks are influenced by the degree of closeness of the relationship R. For example, our theorem implies that sibling risk is always higher than offspring risk. The loci influencing the trait are assumed to be autosomal and unlinked, but arbitrary epistasis between the loci is allowed. We give a detailed proof of the theorem by using stochastic matrices. A shorter proof based on the additive and dominance genetic variances is also sketched. Additionally, we also give some empirical results and discuss generalizations of the theorem.     

Effects of population size and isolation on the genetic structure of the East African mountain white‐eye Zosterops poliogaster (Aves)

Habitat size, quality and isolation determine the genetic structure and diversity of populations and may influence their evolutionary potential and vulnerability to stochastic events. Small and isolated populations are subject to strong genetic drift and can lose much of their genetic diversity due to stochastic fixation and loss of alleles. The mountain white‐eye Zosterops poliogaster, a cloud forest bird species, is exclusively found in the high mountains of East Africa. We analysed 13 polymorphic microsatellites for 213 individuals of this species that were sampled at different points in time in three mountain massifs differing in habitat size, isolation and habitat degradation. We analysed the genetic differentiation among mountain populations and estimated the effective population sizes. Our results indicate three mountain‐specific genetic clusters. Time cohorts did not show genetic divergences, suggesting that populations are large enough to prevent strong drift effects. Effective population sizes were higher in larger and geographically interconnected habitat patches. Our findings underline the relevance of ecological barriers even for mobile species and show the importance of investigating different estimators of population size, including both approaches based on single and multiple time‐points of sampling, for the inference of the demographic status of a population. © 2015 The Linnean Society of London, Biological Journal of the Linnean Society, 2015, 114 , 828–836.