arXiv subjects

Peter Pfaffelhuber

Publications and source records attributed to Peter Pfaffelhuber.

44 records · Page 3Linked to original sources

arXiv2011

Marked metric measure spaces

A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r) is a complete and separable metric space and mu is a probability measure on XxI for some Polish space I of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed I. It arises as state space in the construction of Markov processes which take values in random graphs, e.g. tree-valued dynamics describing randomly evolving genealogical structures in population models. We derive here the topological properties of the space of mmm-spaces needed to study convergence in distribution of random mmm-spaces. Extending the notion of the Gromov-weak topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish space. We give a characterization of tightness for families of distributions of random mmm- spaces and identify a convergence determining algebra of functions, called polynomials.

math.PR↗

arXiv2010

Selective sweeps for recessive alleles and for other modes of dominance

A selective sweep describes the reduction of linked genetic variation due to strong positive selection. If s is the fitness advantage of a homozygote for the beneficial allele and h its dominance coefficient, it is usually assumed that h=1/2, i.e. the beneficial allele is co-dominant. We complement existing theory for selective sweeps by assuming that h is any value in [0,1]. We show that genetic diversity patters under selective sweeps with strength s and dominance 0<h<1 are similar to co-dominant sweeps with selection strength 2hs. Moreover, we focus on the case h=0 of a completely recessive beneficial allele. We find that the length of the sweep, i.e. the time from occurrence until fixation of the beneficial allele, is of the order of sqrt(N/s) generations, if N is the population size. Simulations as well as our results show that genetic diversity patterns in the recessive case h=0 greatly differ from all other cases.

q-bio.PE↗

arXiv2010

The tree length of an evolving coalescent

A well-established model for the genealogy of a large population in equilibrium is Kingman's coalescent. For the population together with its genealogy evolving in time, this gives rise to a time-stationary tree-valued process. We study the sum of the branch lengths, briefly denoted as tree length, and prove that the (suitably compensated) sequence of tree length processes converges, as the population size tends to infinity, to a limit process with cadlag paths, infinite infinitesimal variance, and a Gumbel distribution as its equilibrium.

math.PR↗

arXiv2010

The Aldous-Shields model revisited (with application to cellular ageing)

In Aldous and Shields (1988), a model for a rooted, growing random binary tree was presented. For some c>0, an external vertex splits at rate c^(-i) (and becomes internal) if its distance from the root (depth) is i. For c>1, we reanalyse the tree profile, i.e. the numbers of external vertices in depth i=1,2,.... Our main result are concrete formulas for the expectation and covariance-structure of the profile. In addition, we present the application of the model to cellular ageing. Here, we assume that nodes in depth h+1 are senescent, i.e. do not split. We obtain a limit result for the proportion of non-senescent vertices for large h.

math.PR↗

arXiv2009

Asymptotics of a Brownian ratchet for Protein Translocation

Protein translocation in cells has been modelled by \emph{Brownian ratchets}. In such models, the protein diffuses through a nanopore. On one side of the pore, ratcheting molecules bind to the protein and hinder it to diffuse out of the pore. We study a Brownian ratchet by means of a reflected Brownian motion $(X_t)_{t\geq 0}$ with a changing reflection point $(R_t)_{t\geq 0}$. The rate of change of $R_t$ is $γ(X_t-R_t)$ and the new reflection boundary is distributed uniformly between $R_{t-}$ and $X_t$. The asymptotic speed of the ratchet scales with $γ^{1/3}$ and the asymptotic variance is independent of $γ$.

math.PR↗

arXiv2008

The pattern of genetic hitchhiking under recurrent mutation

Genetic hitchhiking describes evolution at a neutral locus that is linked to a selected locus. If a beneficial allele rises to fixation at the selected locus, a characteristic polymorphism pattern (so-called selective sweep) emerges at the neutral locus. The classical model assumes that fixation of the beneficial allele occurs from a single copy of this allele that arises by mutation. However, recent theory (Pennings and Hermisson, 2006a; Pennings and Hermisson, 2006b) has shown that recurrent beneficial mutation at biologically realistic rates can lead to markedly different polymorphism patterns, so called soft selective sweeps. We extend an approach that has recently been developed for the classical hitchhiking model (Schweinsbergand Durrett, 2005; Etheridge, Pfaffelhuber, Wakolbinger, 2006) to study the recurrent mutation scenario. We show that the genealogy at the neutral locus can be approximated (to leading orders in the selection strength) by a marked Yule process with immigration. Using this formalism, we derive an improved analytical approximation for the expected heterozygosity at the neutral locus at the time of fixation of the beneficial allele.

math.PR↗

arXiv2008

Convergence in distribution of random metric measure spaces: (Lambda-coalescent measure trees)

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov's idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows - provided the sequence is tight - from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Lambda-coalescents. We show that the Lambda-coalescent defines an infinite (random) metric measure space if and only if the so-called "dust-free"-property holds.

math.PR↗

arXiv2006

An approximate sampling formula under genetic hitchhiking

For a genetic locus carrying a strongly beneficial allele which has just fixed in a large population, we study the ancestry at a linked neutral locus. During this ``selective sweep'' the linkage between the two loci is broken up by recombination and the ancestry at the neutral locus is modeled by a structured coalescent in a random background. For large selection coefficients $α$ and under an appropriate scaling of the recombination rate, we derive a sampling formula with an order of accuracy of $\mathcal{O}((\log α)^{-2})$ in probability. In particular we see that, with this order of accuracy, in a sample of fixed size there are at most two nonsingleton families of individuals which are identical by descent at the neutral locus from the beginning of the sweep. This refines a formula going back to the work of Maynard Smith and Haigh, and complements recent work of Schweinsberg and Durrett on selective sweeps in the Moran model.

math.PR↗