Finite-Time Lyapunov Exponents of Deep Neural Networks

We compute how small input perturbations affect the output of deep neural networks, exploring an analogy between deep feed-forward networks and dynamical systems, where the growth or decay of local perturbations is characterized by finite-time Lyapunov exponents. We show that the maximal exponent forms geometrical structures in input space, akin to coherent structures in dynamical systems. Ridges of large positive exponents divide input space into different regions that the network associates with different classes. These ridges visualize the geometry that deep networks construct in input space, shedding light on the fundamental mechanisms underlying their learning capabilities.

Inertia Induces Strong Orientation Fluctuations of Nonspherical Atmospheric Particles

The orientation of nonspherical particles in the atmosphere, such as volcanic ash and ice crystals, influences their residence times and the radiative properties of the atmosphere. Here, we demonstrate experimentally that the orientation of heavy submillimeter spheroids settling in still air exhibits decaying oscillations, whereas it relaxes monotonically in liquids. Theoretical analysis shows that these oscillations are due to particle inertia, caused by the large particle-fluid mass-density ratio. This effect must be accounted for to model solid particles in the atmosphere.

Lagrangian Supersaturation Fluctuations at the Cloud Edge

Evaporation of cloud droplets accelerates when turbulence mixes dry air into the cloud, affecting droplet-size distributions in atmospheric clouds, combustion sprays, and jets of exhaled droplets. The challenge is to model local correlations between droplet numbers, sizes, and supersaturation, which determine supersaturation fluctuations along droplet paths (Lagrangian fluctuations). We derived a statistical model that accounts for these correlations. Its predictions are in quantitative agreement with results of direct numerical simulations, and explain the key mechanisms at play.

Constraints on parameter choices for successful time-series prediction with echo-state networks

Echo-state networks are simple models of discrete dynamical systems driven by a time series. By selecting network parameters such that the dynamics of the network is contractive, characterized by a negative maximal Lyapunov exponent, the network may synchronize with the driving signal. Exploiting this synchronization, the echo-state network may be trained to autonomously reproduce the input dynamics, enabling time-series prediction. However, while synchronization is a necessary condition for prediction, it is not sufficient. Here, we study what other conditions are necessary for successful time-series prediction. We identify two key parameters for prediction performance, and conduct a parameter sweep to find regions where prediction is successful. These regions differ significantly depending on whether full or partial phase space information about the input is provided to the network during training. We explain how these regions emerge.

Heavy particles in a persistent random flow with traps

We study a one-dimensional model for heavy particles in a compressible fluid. The fluid-velocity field is modeled by a persistent Gaussian random function, and the particles are assumed to be weakly inertial. Since one-dimensional fluid-velocity fields are always compressible, the model exhibits spatial trapping regions where particles tend to accumulate. We determine the statistics of fluid-velocity gradients in the vicinity of these traps and show how this allows one to determine the spatial Lyapunov exponent and the rate of caustic formation. We compare our analytical results with numerical simulations of the model and explore the limits of validity of the theory. Finally, we discuss implications for higher-dimensional systems.

Alignment of Nonspherical Active Particles in Chaotic Flows

We study the orientation statistics of spheroidal, axisymmetric microswimmers, with shapes ranging from disks to rods, swimming in chaotic, moderately turbulent flows. Numerical simulations show that rodlike active particles preferentially align with the flow velocity. To explain the underlying mechanism, we solve a statistical model via the perturbation theory. We show that such an alignment is caused by correlations of fluid velocity and its gradients along particle paths combined with fore-aft symmetry breaking due to both swimming and particle nonsphericity. Remarkably, the discovered alignment is found to be a robust kinematical effect, independent of the underlying flow evolution. We discuss its possible relevance for aquatic ecology.

Stochastic unfolding of nanoconfined DNA: Experiments, model and Bayesian analysis

Nanochannels provide a means for detailed experiments on the effect of confinement on biomacromolecules, such as DNA. Here we introduce a model for the complete unfolding of DNA from the circular to linear configuration. Two main ingredients are the entropic unfolding force and the friction coefficient for the unfolding process, and we describe the associated dynamics by a non-linear Langevin equation. By analyzing experimental data where DNA molecules are photo-cut and unfolded inside a nanochannel, our model allows us to extract values for the unfolding force as well as the friction coefficient for the first time. In order to extract numerical values for these physical quantities, we employ a recently introduced Bayesian inference framework. We find that the determined unfolding force is in agreement with estimates from a simple Flory-type argument. The estimated friction coefficient is in agreement with theoretical estimates for motion of a cylinder in a channel. We further validate the estimated friction constant by extracting this parameter from DNA's center-of-mass motion before and after unfolding, yielding decent agreement. We provide publically available software for performing the required image and Bayesian analysis.

Clines on the seashore: The genomic architecture underlying rapid divergence in the face of gene flow

Adaptive divergence and speciation may happen despite opposition by gene flow. Identifying the genomic basis underlying divergence with gene flow is a major task in evolutionary genomics. Most approaches (e.g., outlier scans) focus on genomic regions of high differentiation. However, not all genomic architectures potentially underlying divergence are expected to show extreme differentiation. Here, we develop an approach that combines hybrid zone analysis (i.e., focuses on spatial patterns of allele frequency change) with system-specific simulations to identify loci inconsistent with neutral evolution. We apply this to a genome-wide SNP set from an ideally suited study organism, the intertidal snail Littorina saxatilis, which shows primary divergence between ecotypes associated with different shore habitats. We detect many SNPs with clinal patterns, most of which are consistent with neutrality. Among non-neutral SNPs, most are located within three large putative inversions differentiating ecotypes. Many non-neutral SNPs show relatively low levels of differentiation. We discuss potential reasons for this pattern, including loose linkage to selected variants, polygenic adaptation and a component of balancing selection within populations (which may be expected for inversions). Our work is in line with theory predicting a role for inversions in divergence, and emphasizes that genomic regions contributing to divergence may not always be accessible with methods purely based on allele frequency differences. These conclusions call for approaches that take spatial patterns of allele frequency change into account in other systems.

Distribution of label spacings for genome mapping in nanochannels

In genome mapping experiments, long DNA molecules are stretched by confining them to very narrow channels, so that the locations of sequence-specific fluorescent labels along the channel axis provide large-scale genomic information. It is difficult, however, to make the channels narrow enough so that the DNA molecule is fully stretched. In practice, its conformations may form hairpins that change the spacings between internal segments of the DNA molecule, and thus the label locations along the channel axis. Here, we describe a theory for the distribution of label spacings that explains the heavy tails observed in distributions of label spacings in genome mapping experiments.

Hairpins in the conformations of a confined polymer

If a semiflexible polymer confined to a narrow channel bends around by 180°, the polymer is said to exhibit a hairpin. The equilibrium extension statistics of the confined polymer are well understood when hairpins are vanishingly rare or when they are plentiful. Here, we analyze the extension statistics in the intermediate situation via experiments with DNA coated by the protein RecA, which enhances the stiffness of the DNA molecule by approximately one order of magnitude. We find that the extension distribution is highly non-Gaussian, in good agreement with Monte-Carlo simulations of confined discrete wormlike chains. We develop a simple model that qualitatively explains the form of the extension distribution. The model shows that the tail of the distribution at short extensions is determined by conformations with one hairpin.

One-Parameter Scaling Theory for DNA Extension in a Nanochannel

Experiments measuring DNA extension in nanochannels are at odds with even the most basic predictions of current scaling arguments for the conformations of confined semiflexible polymers such as DNA. We show that a theory based on a weakly self-avoiding, one-dimensional "telegraph" process collapses experimental data and simulation results onto a single master curve throughout the experimentally relevant region of parameter space and explains the mechanisms at play.

Statistical Model for the Orientation of Nonspherical Particles Settling in Turbulence

The orientation of small anisotropic particles settling in a turbulent fluid determines some essential properties of the suspension. We show that the orientation distribution of small heavy spheroids settling through turbulence can be accurately predicted by a simple Gaussian statistical model that takes into account particle inertia and provides a quantitative understanding of the orientation distribution on the problem parameters when fluid inertia is negligible. Our results open the way to a parametrization of the distribution of ice crystals in clouds, and potentially lead to an improved understanding of radiation reflection or particle aggregation through collisions in clouds.

Relative velocities in bidisperse turbulent suspensions

We investigate the distribution of relative velocities between small heavy particles of different sizes in turbulence by analyzing a statistical model for bidisperse turbulent suspensions, containing particles with two different Stokes numbers. This number, St, is a measure of particle inertia which in turn depends on particle size. When the Stokes numbers are similar, the distribution exhibits power-law tails, just as in the case of equal St. The power-law exponent is a nonanalytic function of the mean Stokes number St[over ¯], so that the exponent cannot be calculated in perturbation theory around the advective limit. When the Stokes-number difference is larger, the power law disappears, but the tails of the distribution still dominate the relative-velocity moments, if St[over ¯] is large enough.

Neutral processes forming large clones during colonization of new areas

In species reproducing both sexually and asexually clones are often more common in recently established populations. Earlier studies have suggested that this pattern arises due to natural selection favouring generally or locally successful genotypes in new environments. Alternatively, as we show here, this pattern may result from neutral processes during species' range expansions. We model a dioecious species expanding into a new area in which all individuals are capable of both sexual and asexual reproduction, and all individuals have equal survival rates and dispersal distances. Even under conditions that favour sexual recruitment in the long run, colonization starts with an asexual wave. After colonization is completed, a sexual wave erodes clonal dominance. If individuals reproduce more than one season, and with only local dispersal, a few large clones typically dominate for thousands of reproductive seasons. Adding occasional long-distance dispersal, more dominant clones emerge, but they persist for a shorter period of time. The general mechanism involved is simple: edge effects at the expansion front favour asexual (uniparental) recruitment where potential mates are rare. Specifically, our model shows that neutral processes (with respect to genotype fitness) during the population expansion, such as random dispersal and demographic stochasticity, produce genotype patterns that differ from the patterns arising in a selection model. The comparison with empirical data from a post-glacially established seaweed species (Fucus radicans) shows that in this case, a neutral mechanism is strongly supported.

Optical manipulation for studies of collisional dynamics of micron-sized droplets under gravity

A new experimental technique for creating and imaging collisions of micron-sized droplets settling under gravity is presented. A pair of glycerol droplets is suspended in air by means of two optical traps. The droplet relative velocities are determined by the droplet sizes. The impact parameter is precisely controlled by positioning the droplets using the two optical traps. The droplets are released by turning off the trapping light using electro-optical modulators. The motion of the sedimenting droplets is then captured by two synchronized high-speed cameras, at a frame rate of up to 63 kHz. The method allows the direct imaging of the collision of droplets without the influence of the optical confinement imposed by the trapping force. The method will facilitate efficient studies of the microphysics of neutral, as well as charged, liquid droplets and their interactions with light, electric field and thermodynamic environment, such as temperature or vapor concentration.

Angular Dynamics of a Small Particle in Turbulence

We compute the angular dynamics of a neutrally buoyant nearly spherical particle immersed in an unsteady fluid. We assume that the particle is small, that its translational slip velocity is negligible, and that unsteady and convective inertia are small perturbations. We derive an approximation for the torque on the particle that determines the first inertial corrections to Jeffery's equation. These corrections arise as a consequence of local vortex stretching and can be substantial in turbulence, where local vortex stretching is strong and closely linked to the irreversibility of turbulence.

Visualizing the Nonhomogeneous Structure of RAD51 Filaments Using Nanofluidic Channels

RAD51 is the key component of the homologous recombination pathway in eukaryotic cells and performs its task by forming filaments on DNA. In this study we investigate the physical properties of RAD51 filaments formed on DNA using nanofluidic channels and fluorescence microscopy. Contrary to the bacterial ortholog RecA, RAD51 forms inhomogeneous filaments on long DNA in vitro, consisting of several protein patches. We demonstrate that a permanent "kink" in the filament is formed where two patches meet if the stretch of naked DNA between the patches is short. The kinks are readily seen in the present microscopy approach but would be hard to identify using conventional single DNA molecule techniques where the DNA is more stretched. We also demonstrate that protein patches separated by longer stretches of bare DNA roll up on each other and this is visualized as transiently overlapping filaments. RAD51 filaments can be formed at several different conditions, varying the cation (Mg(2+) or Ca(2+)), the DNA substrate (single-stranded or double-stranded), and the RAD51 concentration during filament nucleation, and we compare the properties of the different filaments formed. The results provide important information regarding the physical properties of RAD51 filaments but also demonstrate that nanofluidic channels are perfectly suited to study protein-DNA complexes.

A universal mechanism generating clusters of differentiated loci during divergence-with-migration

Genome‐wide patterns of genetic divergence reveal mechanisms of adaptation under gene flow. Empirical data show that divergence is mostly concentrated in narrow genomic regions. This pattern may arise because differentiated loci protect nearby mutations from gene flow, but recent theory suggests this mechanism is insufficient to explain the emergence of concentrated differentiation during biologically realistic timescales. Critically, earlier theory neglects an inevitable consequence of genetic drift: stochastic loss of local genomic divergence. Here, we demonstrate that the rate of stochastic loss of weak local differentiation increases with recombination distance to a strongly diverged locus and, above a critical recombination distance, local loss is faster than local “gain” of new differentiation. Under high migration and weak selection, this critical recombination distance is much smaller than the total recombination distance of the genomic region under selection. Consequently, divergence between populations increases by net gain of new differentiation within the critical recombination distance, resulting in tightly linked clusters of divergence. The mechanism responsible is the balance between stochastic loss and gain of weak local differentiation, a mechanism acting universally throughout the genome. Our results will help to explain empirical observations and lead to novel predictions regarding changes in genomic architectures during adaptive divergence.

Statistical model for collisions and recollisions of inertial particles in mixing flows

Finding a quantitative description of the rate of collisions between small particles suspended in mixing flows is a long-standing problem. Here we investigate the validity of a parameterisation of the collision rate for identical particles subject to Stokes force, based on results for relative velocities of heavy particles that were recently obtained within a statistical model for the dynamics of turbulent aerosols. This model represents the turbulent velocity fluctuations by Gaussian random functions. We find that the parameterisation gives quantitatively good results in the limit where the "ghost-particle approximation" applies. The collision rate is a sum of two contributions due to "caustics" and to "clustering". Within the statistical model we compare the relative importance of these two collision mechanisms. The caustic formation rate is high when the particle inertia becomes large, and we find that caustics dominate the collision rate as soon as they form frequently. We compare the magnitude of the caustic contribution to the collision rate to the formation rate of caustics.

Preferential Sampling and Small-Scale Clustering of Gyrotactic Microswimmers in Turbulence

Recent studies show that spherical motile microorganisms in turbulence subject to gravitational torques gather in down-welling regions of the turbulent flow. By analyzing a statistical model we analytically compute how shape affects the dynamics, preferential sampling, and small-scale spatial clustering. We find that oblong organisms may spend more time in up-welling regions of the flow, and that all organisms are biased to regions of positive fluid-velocity gradients in the upward direction. We analyze small-scale spatial clustering and find that oblong particles may either cluster more or less than spherical ones, depending on the strength of the gravitational torques.

Finite-size corrections for confined polymers in the extended de Gennes regime

Theoretical results for the extension of a polymer confined to a channel are usually derived in the limit of infinite contour length. But experimental studies and simulations of DNA molecules confined to nanochannels are not necessarily in this asymptotic limit. We calculate the statistics of the span and the end-to-end distance of a semiflexible polymer of finite length in the extended de Gennes regime, exploiting the fact that the problem can be mapped to a one-dimensional weakly self-avoiding random walk. The results thus obtained compare favorably with pruned-enriched Rosenbluth method (PERM) simulations of a three-dimensional discrete wormlike chain model of DNA confined in a nanochannel. We discuss the implications for experimental studies of linear λ-DNA confined to nanochannels at the high ionic strengths used in many experiments.

Extension of nanoconfined DNA: Quantitative comparison between experiment and theory

The extension of DNA confined to nanochannels has been studied intensively and in detail. However, quantitative comparisons between experiments and model calculations are difficult because most theoretical predictions involve undetermined prefactors, and because the model parameters (contour length, Kuhn length, effective width) are difficult to compute reliably, leading to substantial uncertainties. Here we use a recent asymptotically exact theory for the DNA extension in the "extended de Gennes regime" that allows us to compare experimental results with theory. For this purpose, we performed experiments measuring the mean DNA extension and its standard deviation while varying the channel geometry, dye intercalation ratio, and ionic strength of the buffer. The experimental results agree very well with theory at high ionic strengths, indicating that the model parameters are reliable. At low ionic strengths, the agreement is less good. We discuss possible reasons. In principle, our approach allows us to measure the Kuhn length and the effective width of a single DNA molecule and more generally of semiflexible polymers in solution.

Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers

We numerically analyze the rotation of a neutrally buoyant spheroid in a shear flow at small shear Reynolds number. Using direct numerical stability analysis of the coupled nonlinear particle-flow problem, we compute the linear stability of the log-rolling orbit at small shear Reynolds number Re(a). As Re(a)→0 and as the box size of the system tends to infinity, we find good agreement between the numerical results and earlier analytical predictions valid to linear order in Re(a) for the case of an unbounded shear. The numerical stability analysis indicates that there are substantial finite-size corrections to the analytical results obtained for the unbounded system. We also compare the analytical results to results of lattice Boltzmann simulations to analyze the stability of the tumbling orbit at shear Reynolds numbers of order unity. Theory for an unbounded system at infinitesimal shear Reynolds number predicts a bifurcation of the tumbling orbit at aspect ratio λ(c)≈0.137 below which tumbling is stable (as well as log rolling). The simulation results show a bifurcation line in the λ-Re(a) plane that reaches λ≈0.1275 at the smallest shear Reynolds number (Re(a)=1) at which we could simulate with the lattice Boltzmann code, in qualitative agreement with the analytical results.

Erratum: Role of inertia for the rotation of a nearly spherical particle in a general linear flow [Phys. Rev. E 91, 053023 (2015)]

This corrects the article DOI: 10.1103/PhysRevE.91.053023.

Measurement of particle motion in optical tweezers embedded in a Sagnac interferometer

We have constructed a counterpropagating optical tweezers setup embedded in a Sagnac interferometer in order to increase the sensitivity of position tracking for particles in the geometrical optics regime. Enhanced position determination using a Sagnac interferometer has previously been described theoretically by Taylor et al. [Journal of Optics 13, 044014 (2011)] for Rayleigh-regime particles trapped in an antinode of a standing wave. We have extended their theory to a case of arbitrarily-sized particles trapped with orthogonally-polarized counter-propagating beams. The working distance of the setup was sufficiently long to optically induce particle oscillations orthogonally to the axis of the tweezers with an auxiliary laser beam. Using these oscillations as a reference, we have experimentally shown that Sagnac-enhanced back focal plane interferometry is capable of providing an improvement of more than 5 times in the signal-to-background ratio, corresponding to a more than 30-fold improvement of the signal-to-noise ratio. The experimental results obtained are consistent with our theoretical predictions. In the experimental setup, we used a method of optical levitator-assisted liquid droplet delivery in air based on commercial inkjet technology, with a novel method to precisely control the size of droplets.

Model for melting of confined DNA

When DNA molecules are heated they denature. This occurs locally so that loops of molten single DNA strands form, connected by intact double-stranded DNA pieces. The properties of this "melting" transition have been intensively investigated. Recently there has been a surge of interest in this question, in part caused by experiments determining the properties of partially bound DNA confined to nanochannels. But how does such confinement affect the melting transition? To answer this question we introduce and solve a model predicting how confinement affects the melting transition for a simple model system by first disregarding the effect of self-avoidance. We find that the transition is smoother for narrower channels. By means of Monte Carlo simulations we then show that a model incorporating self-avoidance shows qualitatively the same behavior and that the effect of confinement is stronger than in the ideal case.

Scaling regimes of a semiflexible polymer in a rectangular channel

We derive scaling relations for the extension statistics and the confinement free energy for a semiflexible polymer confined to a channel with a rectangular cross section. Our motivation is recent numerical results [Gupta et al., J. Chem. Phys. 140, 214901 (2014)] indicating that extensional fluctuations are quite different in rectangular channels compared to square channels. Our results are of direct relevance for interpreting current experiments on DNA molecules confined to nanochannels, as many experiments are performed for rectangular channels with large aspect ratios, while theoretical and simulation results are usually obtained for square channels.

Role of inertia for the rotation of a nearly spherical particle in a general linear flow

We analyze the angular dynamics of a neutrally buoyant, nearly spherical particle immersed in a steady general linear flow. The hydrodynamic torque acting on the particle is obtained by means of a reciprocal theorem, a regular perturbation theory exploiting the small eccentricity of the nearly spherical particle, and by assuming that inertial effects are small but finite.

Effect of weak fluid inertia upon Jeffery orbits

We consider the rotation of small neutrally buoyant axisymmetric particles in a viscous steady shear flow. When inertial effects are negligible the problem exhibits infinitely many periodic solutions, the "Jeffery orbits." We compute how inertial effects lift their degeneracy by perturbatively solving the coupled particle-flow equations. We obtain an equation of motion valid at small shear Reynolds numbers, for spheroidal particles with arbitrary aspect ratios. We analyze how the linear stability of the "log-rolling" orbit depends on particle shape and find it to be unstable for prolate spheroids. This resolves a puzzle in the interpretation of direct numerical simulations of the problem. In general, both unsteady and nonlinear terms in the Navier-Stokes equations are important.

Entrainment of heterogeneous glycolytic oscillations in single cells

Cell signaling, gene expression, and metabolism are affected by cell-cell heterogeneity and random changes in the environment. The effects of such fluctuations on cell signaling and gene expression have recently been studied intensively using single-cell experiments. In metabolism heterogeneity may be particularly important because it may affect synchronisation of metabolic oscillations, an important example of cell-cell communication. This synchronisation is notoriously difficult to describe theoretically as the example of glycolytic oscillations shows: neither is the mechanism of glycolytic synchronisation understood nor the role of cell-cell heterogeneity. To pin down the mechanism and to assess its robustness and universality we have experimentally investigated the entrainment of glycolytic oscillations in individual yeast cells by periodic external perturbations. We find that oscillatory cells synchronise through phase shifts and that the mechanism is insensitive to cell heterogeneity (robustness) and similar for different types of external perturbations (universality).

Confined polymers in the extended de Gennes regime

We show that the problem of describing the conformations of a semiflexible polymer confined to a channel can be mapped onto an exactly solvable model in the so-called extended de Gennes regime. This regime (where the polymer is neither weakly nor strongly confined) has recently been intensively studied experimentally and by means of computer simulations. The exact solution predicts precisely how the conformational fluctuations depend upon the channel width and upon the microscopic parameters characterizing the physical properties of the polymer.

Demography-adjusted tests of neutrality based on genome-wide SNP data

Tests of the neutral evolution hypothesis are usually built on the standard null model which assumes that mutations are neutral and the population size remains constant over time. However, it is unclear how such tests are affected if the last assumption is dropped. Here, we extend the unifying framework for tests based on the site frequency spectrum, introduced by Achaz and Ferretti, to populations of varying size. Key ingredients are the first two moments of the site frequency spectrum. We show how these moments can be computed analytically if a population has experienced two instantaneous size changes in the past. We apply our method to data from ten human populations gathered in the 1000 genomes project, estimate their demographies and define demography-adjusted versions of Tajima's D, Fay & Wu's H, and Zeng's E. Our results show that demography-adjusted test statistics facilitate the direct comparison between populations and that most of the differences among populations seen in the original unadjusted tests can be explained by their underlying demographies. Upon carrying out whole-genome screens for deviations from neutrality, we identify candidate regions of recent positive selection. We provide track files with values of the adjusted and unadjusted tests for upload to the UCSC genome browser.

The emergence of the rescue effect from explicit within- and between-patch dynamics in a metapopulation

Immigration can rescue local populations from extinction, helping to stabilize a metapopulation. Local population dynamics is important for determining the strength of this rescue effect, but the mechanistic link between local demographic parameters and the rescue effect at the metapopulation level has received very little attention by modellers. We develop an analytical framework that allows us to describe the emergence of the rescue effect from interacting local stochastic dynamics. We show this framework to be applicable to a wide range of spatial scales, providing a powerful and convenient alternative to individual-based models for making predictions concerning the fate of metapopulations. We show that the rescue effect plays an important role in minimizing the increase in local extinction probability associated with high demographic stochasticity, but its role is more limited in the case of high local environmental stochasticity of recruitment or survival. While most models postulate the rescue effect, our framework provides an explicit mechanistic link between local dynamics and the emergence of the rescue effect, and more generally the stability of the whole metapopulation.

Tumbling of small axisymmetric particles in random and turbulent flows

We analyze the tumbling of small nonspherical, axisymmetric particles in random and turbulent flows. We compute the orientational dynamics in terms of a perturbation expansion in the Kubo number, and obtain the tumbling rate in terms of Lagrangian correlation functions. These capture preferential sampling of the fluid gradients, which in turn can give rise to differences in the tumbling rates of disks and rods. We show that this is a weak effect in Gaussian random flows. But in turbulent flows persistent regions of high vorticity cause disks to tumble much faster than rods, as observed in direct numerical simulations [S. Parsa, E. Calzavarini, F. Toschi, and G. A. Voth, Phys. Rev. Lett. 109, 134501 (2012)]. For larger particles (at finite Stokes numbers), rotational and translational inertia affects the tumbling rate and the angle at which particles collide, due to the formation of rotational caustics.

Distribution of velocity gradients and rate of caustic formation in turbulent aerosols at finite Kubo numbers

In a one-dimensional model for a turbulent aerosol (inertial particles suspended in a random flow) we compute the distributions of particle-velocity gradients and the rate of caustic formation at finite but small Kubo numbers, Ku, for arbitrary Stokes numbers, St. Our results are consistent with those obtained earlier in the limit Ku→0 and St→∞ such that Ku(2)St remains constant. We show how finite-time correlations and nonergodic effects influence the inertial-particle dynamics at finite but small Kubo numbers.

Orientational correlations in confined DNA

We study how the orientational correlations of DNA confined to nanochannels depend on the channel diameter D by means of Monte Carlo simulations and a mean-field theory. This theory describes DNA conformations in the experimentally relevant regime where the Flory-de Gennes theory does not apply. We show how local correlations determine the dependence of the end-to-end distance of the DNA molecule upon D. Tapered nanochannels provide the necessary resolution in D to study experimentally how the extension of confined DNA molecules depends upon D. Our experimental and theoretical results are in qualitative agreement.

Evolutionary branching in a stochastic population model with discrete mutational steps

Evolutionary branching is analysed in a stochastic, individual-based population model under mutation and selection. In such models, the common assumption is that individual reproduction and life career are characterised by values of a trait, and also by population sizes, and that mutations lead to small changes ϵ in trait value. Then, traditionally, the evolutionary dynamics is studied in the limit ϵ→0. In the present approach, small but non-negligible mutational steps are considered. By means of theoretical analysis in the limit of infinitely large populations, as well as computer simulations, we demonstrate how discrete mutational steps affect the patterns of evolutionary branching. We also argue that the average time to the first branching depends in a sensitive way on both mutational step size and population size.

Metapopulation dynamics on the brink of extinction

We analyse metapopulation dynamics in terms of an individual-based, stochastic model of a finite metapopulation. We suggest a new approach, using the number of patches in the population as a large parameter. This approach does not require that the number of individuals per patch is large, neither is it necessary to assume a time-scale separation between local population dynamics and migration. Our approach makes it possible to accurately describe the dynamics of metapopulations consisting of many small patches. We focus on metapopulations on the brink of extinction. We estimate the time to extinction and describe the most likely path to extinction. We find that the logarithm of the time to extinction is proportional to the product of two vectors, a vector characterising the distribution of patch population sizes in the quasi-steady state, and a vector-related to Fisher's reproduction vector-that quantifies the sensitivity of the quasi-steady state distribution to demographic fluctuations. We compare our analytical results to stochastic simulations of the model, and discuss the range of validity of the analytical expressions. By identifying fast and slow degrees of freedom in the metapopulation dynamics, we show that the dynamics of large metapopulations close to extinction is approximately described by a deterministic equation originally proposed by Levins (1969). We were able to compute the rates in Levins' equation in terms of the parameters of our stochastic, individual-based model. It turns out, however, that the interpretation of the dynamical variable depends strongly on the intrinsic growth rate and carrying capacity of the patches. Only when the local growth rate and the carrying capacity are large does the slow variable correspond to the number of patches, as envisaged by Levins. Last but not least, we discuss how our findings relate to other, widely used metapopulation models.

Universal anomalous diffusion of weakly damped particles

We show that anomalous diffusion arises in two different models for the motion of randomly forced and weakly damped particles: one is a generalization of the Ornstein-Uhlenbeck process with a random force, which depends on position as well as time, the other is a generalization of the Chandrasekhar-Rosenbluth model of stellar dynamics, encompassing non-Coulombic potentials. We show that both models exhibit anomalous diffusion of position x and momentum p with the same exponents: (x{2})∼C{x}t{2} and (p{2})∼C{p}t{2/5}. We are able to determine the prefactors C{x}, C{p} analytically.

Distribution of relative velocities in turbulent aerosols

We compute the distribution of relative velocities for a one-dimensional model of heavy particles suspended in a turbulent flow, quantifying the caustic contribution to the moments of relative velocities. The same principles determine the corresponding caustic contribution in d spatial dimensions. The distribution of relative velocities Δv at small separations R acquires the universal form ρ(Δv,R)∼R(d-1)|Δv|(D(2)-2d) for large (but not too large) values of |Δv|. Here D(2) is the phase-space correlation dimension. Our conclusions are in excellent agreement with numerical simulations of particles suspended in a randomly mixing flow in two dimensions, and in quantitative agreement with published data on direct numerical simulations of particles in turbulent flows.

Climate change tendencies observable in the rainfall measurements since 1950 in the federal land of North Rhine-Westphalia and their consequences for urban hydrology

Climate change is present in climatological models - but did we already observe changes in the past measurement data? For the state of North Rhine Westphalia, the rainfall measurements since 1950 have been systematically analysed in order to find out whether there have already been trends and whether the behaviour of rainfall has changed in time. More than 600 station series have been screened for use in the project and quality controlled. Implausible data were discarded. For the analysis, standard values such as yearly sums, half-yearly sums, monthly sums, number of dry days, number of days with precipitation above a threshold, partial time series and extreme values statistics have been calculated and evaluated. Results show that also in the past 50 years, changes in precipitation regime could be observed. These changes have been regionally different. Consequences for urban hydrology include a development of more flexible design approaches.

Multiple regimes of diffusion

We consider the diffusion of independent particles experiencing random accelerations by a space- and time-dependent force as well as viscous damping. This model can exhibit several asymptotic behaviors, depending upon the limiting cases which are considered, some of which have been discussed in earlier work. Here, we explore the full space of dimensionless parameters and present an "asymptotic phase diagram" which delineates the limiting regimes.

Sequential Markov coalescent algorithms for population models with demographic structure

We analyse sequential Markov coalescent algorithms for populations with demographic structure: for a bottleneck model, a population-divergence model, and for a two-island model with migration. The sequential Markov coalescent method is an approximation to the coalescent suggested by McVean and Cardin, and by Marjoram and Wall. Within this algorithm we compute, for two individuals randomly sampled from the population, the correlation between times to the most recent common ancestor and the linkage probability corresponding to two different loci with recombination rate R between them. These quantities characterise the linkage between the two loci in question. We find that the sequential Markov coalescent method approximates the coalescent well in general in models with demographic structure. An exception is the case where individuals are sampled from populations separated by reduced gene flow. In this situation, the correlations may be significantly underestimated. We explain why this is the case.