A Brownian ratchet for protein translocation including dissociation of ratcheting sites - Depperschmidt, Andrej; Ketterer, N.; Pfaffelhuber, P. (Journal of Mathematical Biology, 2013)
We study a model for the translocation of proteins across membranes through a nanopore using a ratcheting mechanism. When the protein enters the nanopore it diffuses in and out of the pore according to a Brownian motion. Moreover, it is bound by ratcheting molecules which hinder the diffusion of the protein out of the nanopore, i. e. the Brownian motion is reflected such that no ratcheting molecule exits the pore. New ratcheting molecules bind at rate γ. Extending our previous approach (Depperschmidt and Pfaffelhuber in Stoch Processes Appl 120:901-925, 2010) we allow the ratcheting molecules to dissociate (at rate δ) from the protein (Model I). We also provide an approximate model (Model II) which assumes a Poisson equilibrium of ratcheting molecules on one side of the current reflection boundary. Using analytical methods and simulations we show that the speeds of both models are approximately the same. Our analytical results on Model II give the speed of translocation by means of a solution of an ordinary differential equation. This speed gives an approximation for the time it takes to translocate a protein of given length. © 2012 Springer-Verlag.
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Keywords: Cumulative process; Dissociation; Ratcheting mechanism; Reflected Brownian motion

A mixing tree-valued process arising under neutral evolution with recombination - Depperschmidt, Andrej; Pardoux, Etienne; Pfaffelhuber, Peter (Electronic Journal of Probability, 2015)
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Keywords: Ancestral recombination graph; Gromov-Hausdorff metric; Kingman coalescent; Tree-valued process

Asymptotic properties of certain diffusion ratchets with locally negative drift - Depperschmidt, Andrej; Sophia, Götz (ALEA : Latin American Journal of Probability and Mathematical Statistics, 2013)

Asymptotics of a Brownian ratchet for protein translocation - Depperschmidt, Andrej; Pfaffelhuber, Peter (Stochastic Processes and their Applications, 2010)
Protein translocation in cells has been modelled by 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 (Xt)t ≥ 0 with a changing reflection point (Rt)t ≥ 0. The rate of change of Rt is γ (Xt - Rt) and the new reflection boundary is distributed uniformly between Rt - and Xt. The asymptotic speed of the ratchet scales with γ1 / 3 and the asymptotic variance is independent of γ. © 2010 Elsevier B.V. All rights reserved.
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Keywords: Brownian motion; Cumulative process; Protein translocation; Ratchet mechanism; Reflected Brownian motion

Directed random walk on the backbone of an oriented percolation cluster - Birkner, Matthias; Cerny, Jiri; Depperschmidt, Andrej; Gantert, Nina (Electronic Journal of Probability, 2013)
We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the "ancestral lineage" of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) using a regeneration approach. Furthermore, we obtain a quenched central limit theorem (i.e. for almost any realisation of the cluster) via an analysis of joint renewals of two independent walks on the same cluster.
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Keywords: Central limit theorem in random environment; Dynamical random environment; Oriented percolation; Random walk; Supercritical cluster

Marked metric measure spaces - Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter (Electronic Communications in Probability, 2011)
A marked metric measure space (mmm-space) is a triple (X, r,μ), where (X, r) is a complete and separable metric space and μ is a probability measure on X × I 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 a 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 polynomial
UnivIS-Import:2015-03-09:Pub.2011.nat.dma.lma5.marked
Keywords: Gromov metric triples; Gromov-weak topology; Metric measure space; Population model; Prohorov metric

Path-properties of the tree-valued Fleming-Viot process - Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter (Electronic Journal of Probability, 2013)
We consider the tree-valued Fleming-Viot process, (X), with mutation and selection as studied in Depperschmidt, Greven and Pfaffelhuber (2012). This process models the stochastic evolution of the genealogies and (allelic) types under resampling, mutation and selection in the population currently alive in the limit of infinitely large populations. Genealogies and types are described by (isometry classes of) marked metric measure spaces. The long-time limit of the neutral tree-valued Fleming-Viot dynamics is an equilibrium given via the marked metric measure space associated with the Kingman coalescent. In the present paper we pursue two closely linked goals. First, we show that two well-known properties of the neutral Fleming-Viot genealogies at fixed time t arising from the properties of the dual, namely the Kingman coalescent, hold for the whole path. These properties are related to the geometry of the family tree close to its leaves. In particular we consider the number and the size of subfamilies whose individuals are not further than apart in the limit ∈ → 0. Second, we answer two open questions about the sample paths of the tree-valued Fleming-Viot process. We show that for all t > 0 almost surely the marked metric measure space Xt has no atoms and admits a mark function. The latter property means that all individuals in the tree-valued Fleming-Viot process can uniquely be assigned a type. All main results are proven for the neutral case and then carried over to selective cases via Girsanov's formula giving absolute continuit
Keywords: Kingman coalescent; Mutation; Path properties; Selection; Tree-valued fleming-viot process

Random walks in dynamic random environments and ancestry under local population regulation - Birkner, Matthias; Cerny, Jiri; Depperschmidt, Andrej (Electronic Journal of Probability, 2016)
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Keywords: Central limit theorem in random environment; Dynamical random environment; Logistic branching random walk; Oriented percolation; Random walk; Supercritical cluster

Some large deviations in Kingman's coalescent - Depperschmidt, Andrej; Pfaffelhuber, Peter; Scheuringer, Annika (Electronic Communications in Probability, 2015)
Kingman's coalescent is a random tree that arises from classical population genetic models such as the Moran model. Concerning the structure of the tree-top there are two well-known laws of large numbers: (i) The (shortest) distance, denoted by Tn, from the tree-top to the level when there are n lines in the tree satisfies almost surely; (ii) At time Tn, the population is naturally partitioned in exactly n families where individuals belong to the same family if they have a common ancestor at time Tn in the past. If denotes the relative size of the ith family, then almost surely. For both laws of large numbers we prove corresponding large deviations results. For (i), the rate of the large deviations is n and we can give the rate function explicitly. For (ii), the rate is n for downwards deviations and √n for upwards deviations. In both cases we give exact rate functions.
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Keywords: Kingman’s coalescent; Large deviations; Uniform spacings

Stochastic Evolution of genealogies of spatial populations: state description, characterization of dynamics and properties - Depperschmidt, Andrej; Greven, Andreas (arXiv, 2018)
We survey results on the description of stochastically evolving genealogies of populations and marked genealogies of multitype populations or spatial populations via tree-valued Markov processes on (marked) ultrametric measure spaces. In particular we explain the choice of state spaces and their topologies, describe the dynamics of genealogical Fleming-Viot and branching models by well-posed martingale problems, and formulate the typical results on the longtime behaviour. Furthermore we discuss the basic techniques of proofs and sketch as two key tools of analysis the different forms of duality and the Girsanov transformation.<br
Keywords: Stochastic evolution of genealogies of spatial populations: state description; characterization of dynamics and properties

Survival and complete convergence for a spatial branching system with local regulation - Birkner, Matthias; Depperschmidt, Andrej (Annals of Applied Probability, 2007)
We study a discrete time spatial branching system on ℤd with logistic-type local regulation at each deme depending on a weighted average of the population in neighboring demes. We show that the system survives for all time with positive probability if the competition term is small enough. For a restricted set of parameter values, we also obtain uniqueness of the nontrivial equilibrium and complete convergence, as well as long-term coexistence in a related two-type model. Along the way we classify the equilibria and their domain of attraction for the corresponding deterministic coupled map lattice on ℤd. © Institute of Mathematical Statistics, 2007.
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Keywords: Coexistence; Complete convergence; Regulated population; Survival

Tree-valued Fleming-Viot dynamics with mutation and selection - Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter (Annals of Applied Probability, 2012)
The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this treevalued enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions. The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanovtype theorem on marked metric measure spaces. To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming-Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming-Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case. © Institute of Mathematical Statistics, 201
Keywords: Ancestral selection graph; Coalescent; Duality; Fleming-Viot process; Genealogical tree; Girsanov theorem; Measurevalued diffusion; Metric measure space; Resampling; Tree-valued Fleming-Viot dynamics