The study of animal foraging behaviour is of practical ecological importance 1 , and exemplifies the wider scientific problem of optimizing search strategies 2 .Lévy flights are random walks whose step lengths come from probability distributions with heavy power-law tails 3, 4 , such that clusters of short steps are connected by rare long steps. flight durations (time intervals between landing on the ocean) were then calculated as consecutive hours for which a bird remained dry, to a resolution of 1 h. It was assumed that birds landed on the water solely to feed, and that flight durations were thus indicative of distances between prey.Time series for 19 separate foraging trips 7 were pooled to give a total of 363 3 flights. The resulting log-log histogram of flight durations gave a straight line with a slope of approximately 2, and is reproduced in Supplementary Fig. 1 from the original raw data. The crux of the conclusion that the albatrosses were performing Lévy flights was that the slope of 2 implied the probability density function (pdf) of flight durations t (in hours), was 7, 10for t ≥ 1 h (leaving out the normalization constant). This is consistent with the Lévy flight definition that the tail of the pdf is of the power-law form t −µ , where 1 < µ ≤ 3 (though technically this is a Lévy walk 4,7,22 We first analyze a newer, larger, and higher resolution data set of albatross flight durations to test for Lévy flights. In 2004, 20 wandering albatrosses on BirdIsland were each fitted with a salt-water logger and a GPS device. The GPS data were too infrequent (at most one location h −1 ) to give distances between landings, but were needed to estimate each bird's departure time from Bird Island, in order to calculate the duration of the initial flight before first landing on the water (we calculated return flights similarly). The resulting data set of flight records was 4 pooled, as in ref. 7, yielding a total of 1416 flights to a resolution of 10 s (Fig. 1).The flights ≥ 1 h are clearly inconsistent with coming from the power law t −2 ascertained 7 for the 1992 data. Furthermore, data from a power law of any exponent (not just 2) would yield a straight line 23 , and this is clearly not the case.In fact, the flight durations t (in h) are consistent with coming from the shifted gamma distribution given by the pdfwhere y = t − 1/120 accounts for the assumed 30 s period before the bird searches for new food sources (see Methods), s = 0.31 is the shape parameter, r = 0.41 h −1 is the rate parameter, and Γ(·) is the gamma function. Equation (2) is valid for flights >30 s; for shorter flights we have f (t) = 0. The exponential term of (2) dominates for large t, implying Poisson behaviour, such that for long enough flights the birds essentially encounter prey randomly with a constant low probability.A Brownian random walker's displacement increases as t H where H = 1/2.If H > 1/2, we have "superdiffusion" as originally inferred in Fig. 2a The gamma distribution (2) has µ = 1 − s = 0.69. This is such a slow powerlaw ...

Colistin is an antibiotic of last resort, but has poor efficacy and resistance is a growing problem. Whilst it is well established that colistin disrupts the bacterial outer membrane by selectively targeting lipopolysaccharide (LPS), it was unclear how this led to bacterial killing. We discovered that MCR-1 mediated colistin resistance in Escherichia coli is due to modified LPS at the cytoplasmic rather than outer membrane. In doing so, we also demonstrated that colistin exerts bactericidal activity by targeting LPS in the cytoplasmic membrane. We then exploited this information to devise a new therapeutic approach. Using the LPS transport inhibitor murepavadin, we were able to cause LPS accumulation in the cytoplasmic membrane of Pseudomonas aeruginosa, which resulted in increased susceptibility to colistin in vitro and improved treatment efficacy in vivo. These findings reveal new insight into the mechanism by which colistin kills bacteria, providing the foundations for novel approaches to enhance therapeutic outcomes.

The aim of this study was to describe the anatomical locations of the femoral attachments of the anteromedial (AM) and posterolateral (PL) bundles of the anterior cruciate ligament (ACL). Twenty-two human cadaver knees with intact ACLs were used. The femoral attachments of the two bundles were identified, marked and photographed. They were measured and described in terms of the o'clock positions parallel to the femoral long axis and parallel to the roof of the intercondylar notch. The centres of the bundles were also measured in a high-low and a superficial-deep manner referencing from the centre of the posterior femoral condyle, and with respect to their positions within a measurement grid defined in this study. The bulk of the AM bundle was attached between the 9.30 and 11.30 o'clock positions and the PL bundle between the 8.30 and 10 o'clock positions. The AM and PL bundles were consistently found in specific zones of the measurement grid. Using the posterior condyle reference method, the centre of the AM bundle was at 68 ± 7% (range 57-78) in a shallow-deep direction and 55 ± 5% (44-62) in a high-low direction. The PL bundle was found at 56 ± 8% (40-73) in a shallow-deep direction, and 62 ± 7.0% (40-70) in a high-low direction. The attachment was oriented at 37° to the femoral long axis. The results from this study could be used to guide ACL reconstruction techniques.

This paper describes the anatomy of the posterior cruciate ligament (PCL) and the meniscofemoral ligaments (MFLs). The fibres of the PCL may be split into two functional bundles; the anterolateral bundle (ALB) and the posteromedial bundle (PMB), relating to their femoral attachments. The tibial attachment is relatively compact, with the ALB anterior to the PLB. These bundles are not isometric: the ALB is tightest in the mid-arc of knee flexion, the PMB is tight at both extension and deep flexion. At least one MFL is present in 93% of knees. On the femur, the anterior MFL attaches distal to the PCL, close to the articular cartilage; the posterior MFL attaches proximal to the PCL. They both attach distally to the posterior horn of the lateral meniscus. Their slanting orientation allows the MFLs to resist tibial posterior drawer.

Summary1. The size spectrum of an ecological community characterizes how a property, such as abundance or biomass, varies with body size. Size spectra are often used as ecosystem indicators of marine systems. They have been fitted to data from various sources, including groundfish trawl surveys, visual surveys of fish in kelp forests and coral reefs, sediment samples of benthic invertebrates and satellite remote sensing of chlorophyll. 2. Over the past decades, several methods have been used to fit size spectra to data. We document eight such methods, demonstrating their commonalities and differences. Seven methods use linear regression (of which six require binning of data), while the eighth uses maximum likelihood estimation. We test the accuracy of the methods on simulated data. 3. We demonstrate that estimated size-spectrum slopes are not always comparable between the seven regressionbased methods because such methods are not estimating the same parameter. We find that four of the eight tested methods can sometimes give reasonably accurate estimates of the exponent of the individual size distribution (which is related to the slope of the size spectrum). However, sensitivity analyses find that maximum likelihood estimation is the only method that is consistently accurate, and the only one that yields reliable confidence intervals for the exponent. 4. We therefore recommend the use of maximum likelihood estimation when fitting size spectra. To facilitate this, we provide documented R code for fitting and plotting results. This should provide consistency in future studies and improve the quality of any resulting advice to ecosystem managers. In particular, the calculation of reliable confidence intervals will allow proper consideration of uncertainty when making management decisions.

Summary 1.Ecologists are obtaining ever-increasing amounts of data concerning animal movement. A movement strategy that has been concluded for a broad variety of animals is that of Lévy flights, which are random walks whose step lengths come from probability distributions with heavy power-law tails. 2. The exponent that parameterizes the power-law tail, denoted μ , has repeatedly been found to be within the Lévy range of 1 < μ ≤ 3. Here, we use Monte Carlo simulations to show that the methods used to infer the value of μ are inaccurate. 3. The widely used method of simply logarithmically transforming a standard histogram of movement lengths has been shown elsewhere to be problematic. Here, we further demonstrate how poor it is, and show that it actually biases estimates of μ towards the Lévy range of 1 < μ ≤ 3, and can bias estimates towards the value of μ = 2. Thus, previous reports of animals undergoing Lévy flights, or of μ being close to the reported optimal value of μ = 2, may simply be a consequence of the bias generated by this method. 4.A technique that has been recently recommended is to logarithmically bin the data and then normalize the resulting histogram. We show that this technique also produces biased results, and suffers from similar problems as those just outlined, although to a lesser extent. 5. The proposed solution is to use likelihood. We find that calculating the maximum likelihood estimate of μ gives the most accurate results (having also tested the rank/frequency method). Likelihood has the further advantages of being the easiest method to implement, and of yielding accurate confidence intervals. Results are applicable to power-law distributions in general, and so are not restricted to inference of Lévy flights. 6. We also re-analyse a data set of grey seal movements that was originally reported to demonstrate Lévy flight behaviour. Using Akaike weights, we test four models, and find no evidence for Lévy flights. Overall, our results suggest that Lévy flights might not be as common as previously thought.

A surprisingly diverse variety of foragers have previously been concluded to exhibit movement patterns known as Lévy flights, a special type of random walk. These foragers range in size from microzooplankton in experiments to fishermen in the Pacific Ocean and the North Sea. The Lévy flight conclusion implies that all the foragers have similar scale-free movement patterns that can be described by a single dimensionless parameter, the exponent micro of a power-law (Pareto) distribution. However, the previous conclusions have been made using methods that have since been shown to be problematic: inaccurate techniques were used to estimate micro, and the power-law distribution was usually assumed to hold without testing any alternative hypotheses. Therefore, I address the open question of whether the previous data still support the Lévy flight hypothesis, and thus determine whether Lévy flights really are so ubiquitous in ecology. I present a comprehensive reanalysis of 17 data sets from seven previous studies for which Lévy flight behavior had been concluded, covering marine, terrestrial, and experimental systems from four continents. I use the modern likelihood and Akaike weights approach to test whether simple alternative models are more supported by the data than Lévy flights. The previously estimated values of the power-law exponent micro do not match those calculated here using the accurate likelihood approach, and almost all of them lie outside of the likelihood-based 95% confidence intervals. Furthermore, the original power-law Lévy flight model is overwhelmingly rejected for 16 out of the 17 data sets when tested against three other simple models. For one data set, the data are consistent with coming from a bounded power-law distribution (a truncated Lévy flight). For three other data sets, an exponential distribution corresponding to a simple Poisson process is suitable. Thus, Lévy flight movement patterns are not the common phenomena that was once thought, and are not suitable for use as ecosystem indicators for fisheries management, as has been proposed.

We investigate the dynamical behaviour of a simple plankton population model, which explicitly simulates the concentrations of nutrient, phytoplankton and zooplankton in the oceanic mixed layer. The model consists of three coupled ordinary differential equations. We use analytical and numerical techniques, focusing on the existence and nature of steady states and unforced oscillations (limit cycles) of the system. The oscillations arise from Hopf bifurcations, which are traced as each parameter in the model is varied across a realistic range. The resulting bifurcation diagrams are compared with those from our previous work, where zooplankton mortality was simulated by a quadratic function-here we use a linear function, to represent alternative ecological assumptions. Oscillations occur across broader ranges of parameters for the linear mortality function than for the quadratic one, although the two sets of bifurcation diagrams show similar qualitative characteristics. The choice of zooplankton mortality function, or closure term, is an area of current interest in the modelling community, and we relate our results to simulations of other models.