Some Bounds for Almost Absorbing Birth and Death Processes with Catastrophes

2000 Mathematics Subject Classification: 60J27.

Generating Birth and Death Processes

Associated with an ordered sequence of an even number 2N of positive real numbers is a birth and death process (BDP) on {0, 1, 2,..., N} having these real numbers as its birth and death rates. We generate another birth and death process from this BDP on {0, 1, 2,..., 2N}. This can be further iterated. We illustrate with an example from tan(kz). In BDP, the decay parameter, viz., the largest non-zero eigenvalue is important in the study of convergence to stationarity. In this article, the smallest eigenvalue is found to be useful.

The pgip family in soybean and three other legume species: evidence for a birth-and-death model of evolution

Background Polygalacturonase-inhibiting proteins (PGIPs) are leucine-rich repeat (LRR) plant cell wall glycoproteins involved in plant immunity. They are typically encoded by gene families with a small number of gene copies whose evolutionary origin has been poorly investigated. Here we report the complete characterization of the full complement of the pgip family in soybean (Glycine max [L.] Merr.) and the characterization of the genomic region surrounding the pgip family in four legume species. Results BAC clone and genome sequence analyses showed that the soybean genome contains two pgip loci. Each locus is composed of three clustered genes that are induced following infection with the fungal pathogen Sclerotinia sclerotiorum (Lib.) de Bary, and remnant sequences of pgip genes. The analyzed homeologous soybean genomic regions (about 126 Kb) that include the pgip loci are strongly conserved and this conservation extends also to the genomes of the legume species Phaseolus vulgaris L., Medicago truncatula Gaertn. and Cicer arietinum L., each containing a single pgip locus. Maximum likelihood-based gene trees suggest that the genes within the pgip clusters have independently undergone tandem duplication in each species. Conclusions The paleopolyploid soybean genome contains two pgip loci comprised in large and highly conserved duplicated regions, which are also conserved in bean, M. truncatula and C. arietinum. The genomic features of these legume pgip families suggest that the forces driving the evolution of pgip genes follow the birth-and-death model, similar to that proposed for the evolution of resistance (R) genes of NBS-LRR-type.

The 5S rDNA family evolves through concerted and birth-and-death evolution in fish genomes: an example from freshwater stingrays

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Purifying selection and birth-and-death evolution in the ubiquitin gene family

Ubiquitin is a highly conserved protein that is encoded by a multigene family. It is generally believed that this gene family is subject to concerted evolution, which homogenizes the member genes of the family. However, protein homogeneity can be attained also by strong purifying selection. We therefore studied the proportion (pS) of synonymous nucleotide differences between members of the ubiquitin gene family from 28 species of fungi, plants, and animals. The results have shown that pS is generally very high and is often close to the saturation level, although the protein sequence is virtually identical for all ubiquitins from fungi, plants, and animals. A small proportion of species showed a low level of pS values, but these values appeared to be caused by recent gene duplication. It was also found that the number of repeat copies of the gene family varies considerably with species, and some species harbor pseudogenes. These observations suggest that the members of this gene family evolve almost independently by silent nucleotide substitution and are subjected to birth-and-death evolution at the DNA level.

Evolution by the birth-and-death process in multigene families of the vertebrate immune system

Concerted evolution is often invoked to explain the diversity and evolution of the multigene families of major histocompatibility complex (MHC) genes and immunoglobulin (Ig) genes. However, this hypothesis has been controversial because the member genes of these families from the same species are not necessarily more closely related to one another than to the genes from different species. To resolve this controversy, we conducted phylogenetic analyses of several multigene families of the MHC and Ig systems. The results show that the evolutionary pattern of these families is quite different from that of concerted evolution but is in agreement with the birth-and-death model of evolution in which new genes are created by repeated gene duplication and some duplicate genes are maintained in the genome for a long time but others are deleted or become nonfunctional by deleterious mutations. We found little evidence that interlocus gene conversion plays an important role in the evolution of MHC and Ig multigene families.

First things and last : the story of birth and death certificates.

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Weak Ergodicity of Mt /Mt /N /N + R Queue

2000 Mathematics Subject Classification: 60J27, 60K25.

Corrected mean-field models for spatially-dependent advection-diffusion-reaction phenomena

In the exclusion-process literature, mean-field models are often derived by assuming that the occupancy status of lattice sites is independent. Although this assumption is questionable, it is the foundation of many mean-field models. In this work we develop methods to relax the independence assumption for a range of discrete exclusion process-based mechanisms motivated by applications from cell biology. Previous investigations that focussed on relaxing the independence assumption have been limited to studying initially-uniform populations and ignored any spatial variations. By ignoring spatial variations these previous studies were greatly simplified due to translational invariance of the lattice. These previous corrected mean-field models could not be applied to many important problems in cell biology such as invasion waves of cells that are characterised by moving fronts. Here we propose generalised methods that relax the independence assumption for spatially inhomogeneous problems, leading to corrected mean-field descriptions of a range of exclusion process-based models that incorporate (i) unbiased motility, (ii) biased motility, and (iii) unbiased motility with agent birth and death processes. The corrected mean-field models derived here are applicable to spatially variable processes including invasion wave type problems. We show that there can be large deviations between simulation data and traditional mean-field models based on invoking the independence assumption. Furthermore, we show that the corrected mean-field models give an improved match to the simulation data in all cases considered.

Strong ergodicity of monotone transition functions

By revealing close links among strong ergodicity, monotone, and the Feller–Reuter–Riley (FRR) transition functions, we prove that a monotone ergodic transition function is strongly ergodic if and only if it is not FRR. An easy to check criterion for a Feller minimal monotone chain to be strongly ergodic is then obtained. We further prove that a non-minimal ergodic monotone chain is always strongly ergodic. The applications of our results are illustrated using birth-and-death processes and branching processes.

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Exercise sheet 1 pdf

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