Bisexual branching processes in a genetic context: the extinction problem for Y-linked genes

In this paper the inheritance of a Y-linked gene with alleles R and r in a population with both females and males is modelled using a two-type bisexual branching process. It is assumed that the reproductive distribution associated with the R allele can differ from the associated with the r allele and that females prefer to mate with a male having the R allele rather than with a male with the r allele. Under these assumptions, we provide some conditions for the extinction and/or survival of both alleles in the population. These conditions depend on the magnitudes of the average number of females and males per mating unit. Moreover, the almost sure extinction of the r allele is independent of the behaviour of the R allele. On the other hand, the survival of the R allele with positive probability may depend strongly on the reproductive behaviour of the other allele. Theoretical results are illustrated by means of simulated examples and some open problems are proposed to the reader as conjectures.     

Bisexual branching processes in a genetic context: rates of growth for Y-linked genes

A multitype bisexual branching process is considered to model the behaviour of a Y-linked gene with two genotypes in a two-sex population. It is assumed perfect fidelity mating with preference of females for the males carrying certain allele of the gene. Under these assumptions, we study the rate of growth of each genotype on the event of non-extinction. The rate of growth of a genotype may depend on whether the other survives or becomes extinct and, in general, both genotype frequencies grow at different rates. We also investigate conditions for the simultaneous explosion of both genotypes to have positive or zero probability.     

Limiting genotype frequencies of Y-linked genes through bisexual branching processes with blind choice

The limiting genotype growth rates and the limiting genotype frequencies of Y-linked genes are studied in a two-sex monogamous population. To this end, the evolution of the numbers of females, males, and mating units of each genotype is modelled by a multitype bisexual branching process in which it assumed that the gene has no influence on the mating process. It is deduced from this model that the average numbers of female and male descendants per mating unit of a genotype determine its growth rate, which does not depend on the behaviour of the other genotypes. Hence, the dominant genotype is found. Conditions for the simultaneous survival of genotypes to have positive probability are also investigated. Finally, the main results are illustrated by means of examples.     

A model describing the effect of sex-reversed YY fish in an established wild population: The use of a Trojan Y chromosome to cause extinction of an introduced exotic species

A novel means of inducing extinction of an exotic fish population is proposed using a genetic approach to shift the ratio of male to females within a population. In the proposed strategy, sex-reversed fish containing two Y chromosomes are introduced into a normal fish population. These YY fish result in the production of a disproportionate number of male fish in subsequent generations. Mathematical modeling of the system following introduction of YY fish at a constant rate reveals that female fish decline in numbers over time, leading to eventual extinction of the population.     

On the time to extinction for a two-type version of Bartlett's epidemic model

We are interested in how the addition of type heterogeneities affects the long time behaviour of models for endemic diseases. We do this by analysing a two-type version of a model introduced by Bartlett under the restriction of proportionate mixing. This model is used to describe diseases for which individuals switch states according to susceptible-->infectious-->recovered and immune, where the immunity is life-long. We describe an approximation of the distribution of the time to extinction given that the process is started in the quasi-stationary distribution, and we analyse how the variance and the coefficient of variation of the number of infectious individuals depends on the degree of heterogeneity between the two types of individuals. These are then used to derive an approximation of the time to extinction. From this approximation we conclude that if we increase the difference in infectivity between the two types the expected time to extinction decreases, and if we instead increase the difference in susceptibility the effect on the expected time to extinction depends on which part of the parameter space we are in, and we can also obtain non-monotonic behaviour. These results are supported by simulations.     

Accounting for roughness of circular processes: using Gaussian random processes to model the anisotropic spread of airborne plant disease

Variables with values in the circle or indexed by the circle have been studied in order to investigate questions in ecology, epidemiology, climatology and oceanography for example. To model circular variables with rough behaviors, the use of Gaussian random processes (GRPs) can be particularly convenient as will be seen in this paper. The roughness of a GRP being mainly determined by its correlation function, a circular correlation function convenient for rough processes is proposed. These mathematical tools are applied to describe the anisotropic spread of an airborne plant disease from a point source: a hierarchical model including two circular GRPs is built and used to analyze data coming from a field experiment. This random-effect model is fitted to data using a Monte-Carlo expectation-maximization (MCEM) algorithm.     

Gamma frailty model for linkage analysis with application to interval-censored migraine data

For many diseases, it seems that the age at onset is geneticallyinfluenced. Therefore, the age-at-onset data are often collectedin order to map the disease gene(s). The ages are often (right)censored or truncated, and therefore, many standard techniquesfor linkage analysis cannot be used. In this paper, we presenta correlated frailty model for censored survival data of siblings.The model is used for testing heritability for the age at onsetand linkage between the loci and the gene(s) that influence(s)the survival time. The model is applied to interval-censoredmigraine twin data. Heritability (obtained from the frailtiesrather than actual onset times) was estimated as 0.42; thisvalue was highly significant. The highest lod score, a scoreof 1.9, was found at the end of chromosome 19.     

Necessary conditions for a minimal model of receptor to show adaptive response over a wide range of levels of stimulus

Sensory systems respond to temporal changes in the stimulus and adapt to the new level when it persists, this pattern of response being maintained in a wide range of levels of stimulus. Here we use a simple model of adaptation developed by Segel et al. (J. Theor. Biol. 120 (1986) 151-179) and extended by Hauri and Ross (Biophys. J. 68 (1995) 708-722) to study the conditions in which it shows wide range of response. The model consists of a receptor that switches between a variable number of states, either by mass action law or by covalent modification. Using a global optimization procedure, we have optimized the adaptive response of the alternatives of the model with different number of states. We find that it is impossible to obtain a wide range of response if the receptor switches between states following mass-action laws, irrespective of the number of states. Instead, a wide range (of five orders of magnitude of ligand concentration) can be obtained if the receptor switches between several states by irreversible covalent modification, in agreement with previous models. Therefore, in this model, expenditure of energy to maintain a large number of covalent modification cycles operating outside equilibrium is necessary to achieve a wide range of response. The optimal values of the parameters present similar patterns to those reported for specific receptors, but there is no quantitative agreement. For instance, ligand affinity varies several orders of magnitude between the different states of the receptor, what is unlikely to be fulfilled by real systems. To see if the minimal model can show adaptive response and range with quantitatively plausible parameter values a sub-optimal receptor was studied, finding that adaptive response of high intensity can still be obtained in at least three orders of magnitude.