An approximation of populations on a habitat with large carrying capacity.

We consider stochastic dynamics of a population which starts from a small colony on a habitat with large but limited carrying capacity. A common heuristics suggests that such population grows initially as a Galton-Watson branching process and then its size follows an almost deterministic path until reaching its maximum, sustainable by the habitat. In this paper we put forward an alternative and, in fact, more accurate approximation which suggests that the population size behaves as a special nonlinear transformation of the Galton-Watson process from the very beginning.

On the establishment of a mutant.

How long does it take for an initially advantageous mutant to establish itself in a resident population, and what does the population composition look like then? We approach these questions in the framework of the so called Bare Bones evolution model (Klebaner et al. in J Biol Dyn 5(2):147-162, 2011. https://doi.

Instantaneous mutation rate in cancer initiation and progression.

Background: Cancer is one of the leading causes for the morbidity and mortality worldwide. Although substantial studies have been conducted theoretically and experimentally in recent years, it is still a challenge to explore the mechanisms of cancer initiation and progression. The investigation for these problems is very important for the diagnosis of cancer diseases and development of treatment schemes.

What can be observed in real time PCR and when does it show?

Real time, or quantitative, PCR typically starts from a very low concentration of initial DNA strands. During iterations the numbers increase, first essentially by doubling, later predominantly in a linear way. Observation of the number of DNA molecules in the experiment becomes possible only when it is substantially larger than initial numbers, and then possibly affected by the randomness in individual replication.

Mathematical modelling for variations of inbreeding populations fitness with single and polygenic traits.

Background: Inbreeding mating has been widely accepted as the key mechanism to enhance homozygosity which normally will decrease the fitness of the population. Although this result has been validated by a large amount of biological data from the natural populations, a mathematical proof of these experimental discoveries is still not complete. A related question is whether we can extend the well-established result regarding the mean fitness from a randomly mating population to inbreeding populations.

On the establishment, persistence, and inevitable extinction of populations.

Comprehensive models of stochastic, clonally reproducing populations are defined in terms of general branching processes, allowing birth during maternal life, as for higher organisms, or by splitting, as in cell division. The populations are assumed to start small, by mutation or immigration, reproduce supercritically while smaller than the habitat carrying capacity but subcritically above it. Such populations establish themselves with a probability wellknown from branching process theory.

A mathematical model for eph/ephrin-directed segregation of intermingled cells.

Eph receptors, the largest family of receptor tyrosine kinases, control cell-cell adhesion/de-adhesion, cell morphology and cell positioning through interaction with cell surface ephrin ligands. Bi-directional signalling from the Eph and ephrin complexes on interacting cells have a significant role in controlling normal tissue development and oncogenic tissue patterning. Eph-mediated tissue patterning is based on the fine-tuned balance of adhesion and de-adhesion reactions between distinct Eph- and ephrin-expressing cell populations, and adhesion within like populations (expressing either Eph or ephrin).

A new model of time scheme for progression of colorectal cancer.

Background: tumourigenesis can be regarded as an evolutionary process, in which the transformation of a normal cell into a tumour cell involves a number of limiting genetic and epigenetic events. To study the progression process, time schemes have been proposed for studying the process of colorectal cancer based on extensive clinical investigations. Moreover, a number of mathematical models have been designed to describe this evolutionary process.

Stochasticity in the adaptive dynamics of evolution: the bare bones.

First a population model with one single type of individuals is considered. Individuals reproduce asexually by splitting into two, with a population-size-dependent probability. Population extinction, growth and persistence are studied.

On the path to extinction.

Populations can die out in many ways. We investigate one basic form of extinction, stable or intrinsic extinction, caused by individuals on the average not being able to replace themselves through reproduction. The archetypical such population is a subcritical branching process, i.

Random variation and concentration effects in PCR.

Even though the efficiency of the polymerase chain reaction (PCR) reaction decreases, analyses are made in terms of Galton-Watson processes, or simple deterministic models with constant replication probability (efficiency). Recently, Schnell and Mendoza have suggested that the form of the efficiency, can be derived from enzyme kinetics. This results in the sequence of molecules numbers forming a stochastic process with the properties of a branching process with population size dependence, which is supercritical, but has a mean reproduction number that approaches one.