Abstract: For a continuous-time branching random walk on the lattice $\mathbf{Z}^d$ with random (spatially homogeneous) branching rates, the quenched population moments $m_n$ ($n=1,2,\dots$) satisfy a chain of evolution equations driven by the Anderson operator with random potential and random source (the latter is a polynomial of the lower-order moments). We study a general non-homogeneous Cauchy problem with random data and, in particular, derive the Feynman-Kac representation for the solution. We then analyze the long-time asymptotics of the annealed moments $\langle m_n^p\rangle$ ($p\ge1$) obtained by averaging over the medium realizations. Under the assumption that the random potential has a Weibull type upper tail, the corresponding Lyapunov exponents $\lambda_{n,p}$ are computed. Our results show that the moments of all orders grow in a non- regular, intermittent fashion. An important implication is the identity $\lambda_{n,p}=\lambda_{1,np}$ suggesting that intermittency of the moments $m_n$ is in a sense reduced to that of the first-order moment $m_1$.

Abstract: We discuss the principal ideas of existing models for the processive motion of the motor proteins kinesin and Ncd. We find that in order to explain the unidirectionality of these two-headed motors, but also the different directions of walking for kinesin and Ncd, we cannot use one-dimensional models or quasi one-dimensional models of the ratchet potential type used so far. Instead we have to describe the two heads as extended objects in at least two-dimensional space. Our model of directed binding easily explains the generally excepted and experimentally supported hand-over-hand movement. It also accounts for the motion of kinesin and Ncd in two opposite directions as a result of their different relaxed state orientations. We establish the Langevin and Fokker-Planck description and try to solve these equations numerically for sets of parameter values that are consistent with experimental data. We also try to provide some insight, how to see chemical reactions, such as protein-protein binding, mechanically. Finally,we briefly discuss an extension of our model to another motor protein, namely myosin, which walks on actin filaments with a much broader probability distribution of step sizes compared to the kinesin.

Summary: We consider an interacting particle Markov process for Darwinian evolution in a discrete asexual population with non-constant population size, involving a linear birth rate, a density-dependent logistic death rate and a probability of mutation at each birth event. We introduce a renormalization parameter $K$ scaling the size of the population, which leads, when $K\rightarrow\infty$, to a deterministic dynamics. By combining in a non-standard way the limits of large population and of rare mutations, we show that a time scale separation between birth and death events and mutation events occurs and that the interacting particle microscopic process converges to the biological model of evolution known as the "trait substitution sequence of adaptive dynamics", which describes Darwinian evolution as a Markov jump process in the trait space.

Abstract: Numerous authors (Fisher, Haldane, Medawar, Williams, Hamilton, Charlesworth,...) have proposed evolutionary explanations of senescence and mortality that involve large numbers of mildly deleterious mutations that: * meander towards extinction in the population, * are constantly reintroduced, * have age-specific effects. We propose a mathematical framework for such explanations in the haploid case and in the diploid case under assumptions similar to the quasi-linkage equilibrium assumption of Barton and Turelli.

Abstract: Recursions for sampling distributions of allele configurations are derived for the situation when the genealogy of the underlying population is modelled by a coalescent process with simultaneous multiple collisions of ancestral lineages. Solutions of the recursion are only known for the two boundary cases when the coalescent process is either the Kingman coalescent or the star-shaped coalescent. For the Kingman coalescent, the recursion reduces to that known for the classical Ewens sampling formula. For the star-shaped coalescent, the solution is related to the hook composition structure. We show that, in all other cases, the sampling formulas are non-regenerative, which illustrates their complexity. The asymptotic behavior of the number K(n) of alleles (types) for large sample size n is studied, in particular for the star-shaped coalescent and the Bolthausen-Sznitman coalescent.

Abstract: Models of simply generated random trees and labelled variants thereof are discussed. We focus on certain counting parameters for which moment recurrences of underlying limit distributions can be derived. We discuss possible applications to problems motivated from the biological and physical sciences.

Abstract: For a finite measure $\Lambda$ on $[0,1]$, the $\Lambda$-coalescent is a coalescent process such that, whenever there are $b$ clusters, each $k$-tuple of clusters merges into one at rate $\int_0^1 x^{k-2} (1-x)^{b-k} \Lambda(dx)$. It has recently been shown that if $1 < \alpha < 2$, the $\Lambda$-coalescent in which $\Lambda$ is the Beta$(2-\alpha, \alpha)$ distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an $\alpha$-stable branching mechanism. Here we will review this result and then show how to use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove a limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks of the coalescents. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents.

Abstract: Experiments on the dynamics of vibrational fluctuations in myoglobin revealed an interesting behavioral cross-over occurring in the range of 180-200 K. In this temperature range the mean square displacement of atomic positions versus temperature sharply increases its slope indicating the dissociation of CO from the haeme group. In this talk I will discuss a theoretical model that provides a framework for the quantitative description of this phenomenon. The basis of our calculations is an assumption of an effective potential with multiple local minima. In particular we consider a quartic potential in place of the simple quadratic. We then use non-Gaussian statistics to obtain a relationship between the mean square displacement and model parameters. We compare our model to published experimental data and show that it can describe the data set using physically meaningful parameters which are fitted to the experimental data. In the process we verify the Gaussian approximation's applicability only to the low-temperature regime. In the high-temperature limit, however, deviations from the Gaussian approximation are due to the double-well nature of our effective potential. We find that the published datasets showing the thermal transition display the qualitative trends predicted by appropriate algebraic approximations to our predicted myoglobin behaviors. In addition, I will discuss the role of thermal fluctuations in the behaviour of other key functional proteins, in particular kinesin and tubulin. Particular emphasis will be placed on the behaviour of the water of hydration.

Abstract: Classical branching processes are trapped in the Malthusian dichotomy of extinction versus growth beyond all bounds. An attempt to combat this is to consider spatially distributed populations whose reproduction mechanism is regulated by a competition of individuals. In Feller's diffusion limit, this leads to branching diffusions with logistic growth and migration between colonies. For a large class of initial distributions, the interplay of migration, reproduction and competition takes the population to an "upper invariant measure" (Hutzenthaler 2005). Directed percolation arguments (Etheridge (2004)) show that this equilibrium must be non-trivial for sufficiently large super-criticality in the reproduction. One might be tempted to believe that even for a small supercriticality, a sufficiently mobile migration can prevent the system from suffering local extinction. This is not the case, as we prove by comparison with a mean field model. This latter result extends also to more general local reproduction regulations. The talk is based on joint work with Martin Hutzenthaler.