Adam Bobrowski

A semigroup representation and asymptotic behavior of certain statistics of the fisher-wright-moran coalescent

We derive new results giving mathematical properties of functions of allele frequencies under the time-continuous Fisher-Wright-Moran model with mutations of the general Markov-chain form. The matrix R(t) (possibly infinite) of the joint distributions of the types of a pair of alleles sampled from the population at time t , satisfies a matrix differential equation of the form d R(t) /d t = [ Q * R(t) + R(t)Q ] − [1 / (2 N ] R(t) + [1/(2 N )] Π(t) where Q is the intensity matrix of the Markov chain, II(t) is its diagonalized probability distribution, and N is the effective population size. This is the Lyapunov differential equation, known in control theory. Investigation of behavior of its solutions leads to consideration of tensor products of transition (Markov) semigroups. Semigroup theory methods allow proofs of asymptotic results for the model, also in the cases when the population size does not stay constant. If population is composed of a number of disjoint subpopulations, the asymptotics depend on the growth rate of the population. Special cases of the model include stepwise mutation models with and without allele size constraints, and with directional bias of mutations. Allele state changes caused by recombinatorial misalignment and more complex sequence conversion patterns also can be incorporated in this model. The methodology developed can also be applied to model coevolution of disease and marker loci, of further use for linkage disequilibrium mapping of disease genes.

ASYMPTOTIC BEHAVIOUR OF AN OPERATOR EXPONENTIAL RELATED TO BRANCHING RANDOM WALK MODELS OF DNA REPEATS

We study the asymptotic behaviour of an infinite system of differential equations describing the expectations in a branching random walk. The original stochastic formulation was employed to describe the process of evolution of reversible drug resistance in cancer cells. The problem is formulated as an operator exponential function in the space of absolutely summable sequences. Conditions are found for the asymptotic decay of the operator exponential function, using methods of the spectral theory of linear operators. A discussion is provided relating mathematical results to the behaviour of models of gene amplification and evolution of DNA repeats.

On moments-preserving cosine families and semigroups in C[0, 1]

We use the newly developed Lord Kelvin’s method of images (Bobrowski in J Evol Equ 10(3):663–675, 2010; Semigroup Forum 81(3):435–445, 2010) to show existence of a unique cosine family generated by a restriction of the Laplace operator in C[0, 1] that preserves the first two moments. We characterize the domain of its generator by specifying its boundary conditions. Also, we show that it enjoys inherent symmetry properties, and in particular that it leaves the subspaces of odd and even functions invariant. Furthermore, we provide information on long-time behavior of the related semigroup.

Families of Operators Describing Diffusion Through Permeable Membranes

We present generation and limit results for semigroups and cosine families for snapping out Brownian motion, a process modeling diffusion through permeable membranes.

Boundary Conditions in Evolutionary Equations in Biology

The talks are devoted to the most important examples of boundary conditions in evolutionary equations that model biological phenomena. The first notable one is the boundary condition in the so-called McKendrick equation, modeling births in an age-structured population. Currently, the McKendrick equation and its generalizations are often used as a building block of more complicated models, for example those involving quiescence or several linked populations. Analytically, the related boundary condition is still of importance, being at the same time of interesting form and having a clear biological meaning. Other boundary conditions of interest describe behavior of diffusion processes at the boundaries. As developed by W. Feller in the 1950, stochastic processes in population genetics, including the famous Wright’s diffusion being an approximation of the Wright–Fisher model of genetic drift, suggest boundary conditions that were not known before. The seminal works of W. Feller, A.D. Wentzell and P. Levy have led mathematicians and biologists to the general form of such boundary conditions, and to a thorough understanding of their probabilistic and analytical meaning.

An operator semigroup in mathematical genetics

1 Introduction.- 2 Genetic background.- 3 Motivating example.- 4 Mathematical tools.- 5 Master Equation.- 6 Epilogue.

On shape preserving semigroups

Motivated by positivity-, monotonicity-, and convexity preserving differential equations, we introduce a definition of shape preserving operator semigroups and analyze their fundamental properties. In particular, we prove that the class of shape preserving semigroups is preserved by perturbations and taking limits. These results are applied to partial delay differential equations.

Master Equation and Asymptotic Behavior of Its Solutions

We have found, under assumption of existence of the limit distribution of allelic states at pairs of chromosomes, explicit forms of the limit. Depending on the limit behavior of population size, we obtained different limit joint probability distributions of pairs. An interesting example of application of expression for the constant population size limit of the joint distribution is the model of microsatellite mutation with lower and upper bounds on the microsatellite size. From the viewpoint of microsatellite models, this is an unusual situation, since most of them, with a notable exception of Durretts’s model do not have a limit distribution of repeat count.

Motivating Example: Population Bottlenecks in the History of Modern Humans, Use of the Imbalance Index

A special case of our master equation, specialized for microsatellite mutations, was derived to study influence of bottlenecks in modern human history. Jorde and co-workers analyzed allele frequency distributions at 60 tetranucleotide loci in a worldwide survey of human populations. Kimmel and co-workers investigated whether there is imbalance between allele size variances and heterozygosity observed in these data, as analyzed by an imbalance index they introduced. Three major groups of population, Asians, Africans, and Europeans, were considered for this purpose. The analysis shows that the data are consistent with a population bottleneck in modern humanity history and gradual settling of Europe and Asia.

Time to the MRCA of a sample in a Wright–Fisher model with variable population size

Determining the expected distribution of the time to the most recent common ancestor of a sample of individuals may deliver important information about the genetic markers and evolution of the population. In this paper, we introduce a new recursive algorithm to calculate the distribution of the time to the most recent common ancestor of the sample from a population evolved by any conditional multinomial sampling model. The most important advantage of our method is that it can be applied to a sample of any size drawn from a population regardless of its size growth pattern. We also present a very efficient method to implement and store the genealogy tree of the population evolved by the Galton-Watson process. In the final section we present results applied to a simulated population with a single bottleneck event and to real populations of known size histories.