Survival probability of an immobile target surrounded by mobile traps

We study analytically, in one dimension, the survival probability $P_{s}(t)$ up to time $t$ of an immobile target surrounded by mutually noninteracting traps each performing a continuous-time random walk (CTRW) in continuous space. We consider a general CTRW with symmetric and continuous (but otherwise arbitrary) jump length distribution $f(\eta)$ and arbitrary waiting time distribution $\psi(\tau)$. The traps are initially distributed uniformly in space with density $\rho$. We prove an exact relation, valid for all time $t$, between $P_s(t)$ and the expected maximum $E[M(t)]$ of the trap process up to time $t$, for rather general stochastic motion $x_{\rm trap}(t)$ of each trap. When $x_{\rm trap}(t)$ represents a general CTRW with arbitrary $f(\eta)$ and $\psi(\tau)$, we are able to compute exactly the first two leading terms in the asymptotic behavior of $E[M(t)]$ for large $t$. This allows us subsequently to compute the precise asymptotic behavior, $P_s(t)\sim a\, \exp[-b\, t^{\theta}]$, for large $t$, with exact expressions for the stretching exponent $\theta$ and the constants $a$ and $b$ for arbitrary CTRW. By choosing appropriate $f(\eta)$ and $\psi(\tau)$, we recover the previously known results for diffusive and subdiffusive traps. However, our result is more general and includes, in particular, the superdiffusive traps as well as totally anomalous traps

Statistical topography of fitness landscapes

Fitness landscapes are generalized energy landscapes that play an important conceptual role in evolutionary biology. These landscapes provide a relation between the genetic configuration of an organism and that organism’s adaptive properties. In this work, global topographical features of these fitness landscapes are investigated using theoretical models. The resulting predictions are compared to empirical landscapes. It is shown that these landscapes allow, at least with respect to the properties considered, for a rough classification into two types. In the biologically relevant limit, these two types of landscapes show very different behavior. While empirical landscapes cannot be classified on purely theoretical grounds, some of the known landscapes are classified here on a case-by-case basis. The study of theoretical models leads to consider sequences of record events of independent but non-identically distributed random variables and correlations between these record events. While these considerations appear at first to be outside the main direction of this thesis, they lead to a record-base tool of data analysis that is applied to one of the data sets considered here

Evolutionary accessibility of mutational pathways

Functional effects of different mutations are known to combine to the total effect in highly nontrivial ways. For the trait under evolutionary selection (`fitness'), measured values over all possible combinations of a set of mutations yield a fitness landscape that determines which mutational states can be reached from a given initial genotype. Understanding the accessibility properties of fitness landscapes is conceptually important in answering questions about the predictability and repeatability of evolutionary adaptation. Here we theoretically investigate accessibility of the globally optimal state on a wide variety of model landscapes, including landscapes with tunable ruggedness as well as neutral `holey' landscapes. We define a mutational pathway to be accessible if it contains the minimal number of mutations required to reach the target genotype, and if fitness increases in each mutational step. Under this definition accessibility is high, in the sense that at least one accessible pathwayexists with a substantial probability that approaches unity as the dimensionality of the fitness landscape (set by the number of mutational loci) becomes large. At the same time the number of alternative accessible pathways grows without bound. We test the model predictions against an empirical 8-locus fitness landscape obtained for the filamentous fungus \textit{Aspergillus niger}. By analyzing subgraphs of the full landscape containing different subsets of mutations, we are able to probe the mutational distance scale in the empirical data. The predicted effect of high accessibility is supported by the empirical data and very robust, which we argue to reflect the generic topology of sequence spaces.Comment: 16 pages, 4 figures; supplementary material available on reques

Correlations between record events in sequences of random variables with a linear trend

The statistics of records in sequences of independent, identically distributed random variables is a classic subject of study. One of the earliest results concerns the stochastic independence of record events. Recently, records statistics beyond the case of i.i.d. random variables have received much attention, but the question of independence of record events has not been addressed systematically. In this paper, we study this question in detail for the case of independent, non-identically distributed random variables, specifically, for random variables with a linearly moving mean. We find a rich pattern of positive and negative correlations, and show how their asymptotics is determined by the universality classes of extreme value statistics.Comment: 19 pages, 12 figures; some typos in Sections 3.1 and 3.3. correcte

Quantitative analyses of empirical fitness landscapes

The concept of a fitness landscape is a powerful metaphor that offers insight into various aspects of evolutionary processes and guidance for the study of evolution. Until recently, empirical evidence on the ruggedness of these landscapes was lacking, but since it became feasible to construct all possible genotypes containing combinations of a limited set of mutations, the number of studies has grown to a point where a classification of landscapes becomes possible. The aim of this review is to identify measures of epistasis that allow a meaningful comparison of fitness landscapes and then apply them to the empirical landscapes to discern factors that affect ruggedness. The various measures of epistasis that have been proposed in the literature appear to be equivalent. Our comparison shows that the ruggedness of the empirical landscape is affected by whether the included mutations are beneficial or deleterious and by whether intra- or intergenic epistasis is involved. Finally, the empirical landscapes are compared to landscapes generated with the Rough Mt.\ Fuji model. Despite the simplicity of this model, it captures the features of the experimental landscapes remarkably well.Comment: 24 pages, 5 figures; to appear in Journal of Statistical Mechanics: Theory and Experimen

Reconstructing Late Holocene North Atlantic atmospheric circulation changes using functional paleoclimate networks

Obtaining reliable reconstructions of long-term atmospheric circulation changes in the North Atlantic region presents a persistent challenge to contemporary paleoclimate research, which has been addressed by a multitude of recent studies. In order to contribute a novel methodological aspect to this active field, we apply here evolving functional network analysis, a recently developed tool for studying temporal changes of the spatial co-variability structure of the Earth's climate system, to a set of Late Holocene paleoclimate proxy records covering the last two millennia. The emerging patterns obtained by our analysis are related to long-term changes in the dominant mode of atmospheric circulation in the region, the North Atlantic Oscillation (NAO). By comparing the time-dependent inter-regional linkage structures of the obtained functional paleoclimate network representations to a recent multi-centennial NAO reconstruction, we identify co-variability between southern Greenland, Svalbard, and Fennoscandia as being indicative of a positive NAO phase, while connections from Greenland and Fennoscandia to central Europe are more pronounced during negative NAO phases. By drawing upon this correspondence, we use some key parameters of the evolving network structure to obtain a qualitative reconstruction of the NAO long-term variability over the entire Common Era (last 2000 years) using a linear regression model trained upon the existing shorter reconstruction

Records and sequences of records from random variables with a linear trend

We consider records and sequences of records drawn from discrete time series of the form $X_{n}=Y_{n}+cn$, where the $Y_{n}$ are independent and identically distributed random variables and $c$ is a constant drift. For very small and very large drift velocities, we investigate the asymptotic behavior of the probability $p_n(c)$ of a record occurring in the $n$th step and the probability $P_N(c)$ that all $N$ entries are records, i.e. that $X_1 < X_2 < ... < X_N$. Our work is motivated by the analysis of temperature time series in climatology, and by the study of mutational pathways in evolutionary biology.Comment: 21 pages, 7 figure