Convergence to equilibrium in a genetic model with differential viability between the sexes

A single locus, diallelic selection model with female and male viability differences is studied. If the variables are ratios of allele frequencies in each sex, a 2-dimensional difference equation describes the model. Because of the strong monotonicity of the resulting map, every initial genotypic structure converges to an equilibrium structure assuming that no equilibrium has eigenvalues on the unit circle.Partially supported by funds provided by a Science and Education Grant to the USDA-Forest Service, Southeastern Forest Experiment Station, Population Genetics of Forest Trees Research Unit, Raleigh, USASupported by a grant from the Max Kade Foundation, New York, USA     

Dynamical analysis of density-dependent selection in a discrete one-island migration model

A system of non-linear difference equations is used to model the effects of density-dependent selection and migration in a population characterized by two alleles at a single gene locus. Results for the existence and stability of polymorphic equilibria are established. Properties for a genetically important class of equilibria associated with complete dominance in fitness are described. The birth of an unusual chaotic attractor is also illustrated. This attractor is produced when migration causes chaotic dynamics on a boundary of phase space to bifurcate into the interior of phase space, resulting in bistable genetic polymorphic behavior.     

Bifurcation analysis of a model for hormonal regulation of the menstrual cycle

A model for hormonal control of the menstrual cycle with 13 ordinary differential equations and 41 parameters is presented. Important changes in model behavior result from variations in two of the most sensitive parameters. One parameter represents the level of estradiol sufficient for significant synthesis of luteinizing hormone, which causes ovulation. By studying bifurcation diagrams in this parameter, an interval of parameter values is observed for which a unique stable periodic solution exists and it represent an ovulatory cycle. The other parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. Changes in this parameter affect the uniqueness interval defined with respect to the first parameter. Hopf, saddle-node and transcritical bifurcations are examined. To attain a normal ovulatory menstrual cycle in this model, a balance must be maintained between healthy development of the follicles and flexibility in estradiol levels needed to produce the surge in luteinizing hormone.     

Dynamics and bifurcation of a model for hormonal control of the menstrual cycle with inhibin delay

A system of 13 ordinary differential equations with 42 parameters is presented to model hormonal regulation of the menstrual cycle. For an excellent fit to clinical data, the model requires a 36 h time delay for the effect of inhibin on the synthesis of follicle stimulating hormone. Biological and mathematical reasons for this delay are discussed. Bifurcations with respect to changes in three important parameters are examined. One parameter represents the level of estradiol adequate for significant synthesis of luteinizing hormone. Bifurcation diagrams with respect to this parameter reveal an interval of parameter values for which a unique stable periodic solution exists and this solution represents a menstrual cycle during which ovulation occurs. The second parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. The third parameter is the time delay. Changes in the second parameter and the time delay affect the size of the uniqueness interval defined with respect to the first parameter. Saddle-node, transcritical and degenerate Hopf bifurcations are studied.     

Humoral antibody response to individual viral proteins after murine cytomegalovirus infection

The purpose of this study was to identify viral proteins that played an important role in the humoral immune response to murine cytomegalovirus (MCMV). Viral proteins were separated from a purified virus preparation on polyacrylamide gels, were blotted onto nitrocellulose strips, and were reacted with antisera collected from mice on various days post infection. No antibody response was detected in serum obtained 5 days post infection, but by 10 days there was a faint response to five different proteins. Thereafter, the number of proteins eliciting an antibody response, as well as the intensity of the response, increased with time so that by 42 days post infection a response to 13 major antigens was detected. This method provides a means of separating out important immunogens from the more than 30 different MCMV proteins originally identified by polyacrylamide gel electrophoresis. Such information may improve our understanding of the pathogenesis of MCMV infection as well as host immune responses to the virus.     

Genome mosaicism is conserved but not unique in Pseudomonas aeruginosa isolates from the airways of young children with cystic fibrosis

Pseudomonas aeruginosa strains from the chronic lung infections of cystic fibrosis (CF) patients are phenotypically and genotypically diverse. Using strain PAO1 whole genome DNA microarrays, we assessed the genomic variation in P. aeruginosa strains isolated from young children with CF (6 months to 8 years of age) as well as from the environment. Eighty-nine to 97% of the PAO1 open reading frames were detected in 20 strains by microarray analysis, while subsets of 38 gene islands were absent or divergent. No specific pattern of genome mosaicism defined strains associated with CF. Many mosaic regions were distinguished by their low G + C content; their inclusion of phage related or pyocin genes; or by their linkage to a vgr gene or a tRNA gene. Microarray and phenotypic analysis of sequential isolates from individual patients revealed two deletions of greater than 100 kbp formed during evolution in the lung. The gene loss in these sequential isolates raises the possibility that acquisition of pyomelanin production and loss of pyoverdin uptake each may be of adaptive significance. Further characterization of P. aeruginosa diversity within the airways of individual CF patients may reveal common adaptations, perhaps mediated by gene loss, that suggest new opportunities for therapy.     

Density-dependent selection migration model with non-monotone fitness functions

This paper studies the classical single locus, diallelic selection model with diffusion for a continuously reproducing population. The phase variables are population density and allele frequency (or allele density). The genotype fitness depend only on population density but include one-hump functions of the density variable. With mild assumptions on genotype fitnesses, we study the geometry of the nullclines and the asymptotic behavior of solutions of the selection model without diffusion. For the diffusion model with zero Neumann boundary conditions, we use this geometric information to show that if the initial data satisfy certain conditions then the corresponding solution to the reaction-diffusion equation converges to the spatially constant stable equilibrium which is closest to the initial data.Research partially supported by NSF grant DMS-8920597Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina