Permutable entire functions and multiply connected wandering domains

Let f and g be permutable transcendental entire functions. We use a recent analysis of the dynamical behaviour in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of f and g are equal; in particular, we show that J(f)=J(g) provided that neither f nor g has a simply connected wandering domain in the fast escaping set

The clockfront and wavefront model revisited

The currently accepted interpretation of the clock and wavefront model of somitogenesis is that a posteriorly moving molecular gradient sequentially slows the rate of clock oscillations, resulting in a spatial readout of temporal oscillations. However, while molecular components of the clocks and wavefronts have now been identified in the pre-somitic mesoderm (PSM), there is not yet conclusive evidence demonstrating that the observed molecular wavefronts act to slow clock oscillations. Here we present an alternative formulation of the clock and wavefront model in which oscillator coupling, already known to play a key role in oscillator synchronisation, plays a fundamentally important role in the slowing of oscillations along the anterior–posterior (AP) axis. Our model has three parameters which can be determined, in any given species, by the measurement of three quantities: the clock period in the posterior PSM, somite length and the length of the PSM. A travelling wavefront, which slows oscillations along the AP axis, is an emergent feature of the model. Using the model we predict: (a) the distance between moving stripes of gene expression; (b) the number of moving stripes of gene expression and (c) the oscillator period profile along the AP axis. Predictions regarding the stripe data are verified using existing zebrafish data. We simulate a range of experimental perturbations and demonstrate how the model can be used to unambiguously define a reference frame along the AP axis. Comparing data from zebrafish, chick, mouse and snake, we demonstrate that: (a) variation in patterning profiles is accounted for by a single nondimensional parameter; the ratio of coupling strengths; and (b) the period profile along the AP axis is conserved across species. Thus the model is consistent with the idea that, although the genes involved in pattern propagation in the PSM vary, there is a conserved patterning mechanism across species

A modified Oster-Murray-Harris mechanical model of morphogenesis

There are two main modeling paradigms for biological pattern formation in developmental biology: chemical prepattern models and cell aggregation models. This paper focuses on an example of a cell aggregation model, the mechanical model developed by Oster, Murray, and Harris [Development, 78 (1983), pp. 83--125]. We revisit the Oster--Murray--Harris model and find that, due to the infinitesimal displacement assumption made in the original version of this model, there is a restriction on the types of boundary conditions that can be prescribed. We derive a modified form of the model which relaxes the infinitesimal displacement assumption. We analyze the dynamics of this model using linear and multiscale nonlinear analysis and show that it has the same linear behavior as the original Oster--Murray--Harris model. Nonlinear analysis, however, predicts that the modified model will allow for a wider range of parameters where the solution evolves to a bounded steady state. The results from both analyses are verified through numerical simulations of the full nonlinear model in one and two dimensions. The increased range of boundary conditions that are well-posed, as well as a wider range of parameters that yield bounded steady states, renders the modified model more applicable to, and more robust for, comparisons with experiments

Ab Initio Modeling of the Herpesvirus VP26 Core Domain Assessed by CryoEM Density

Efforts in structural biology have targeted the systematic determination of all protein structures through experimental determination or modeling. In recent years, 3-D electron cryomicroscopy (cryoEM) has assumed an increasingly important role in determining the structures of these large macromolecular assemblies to intermediate resolutions (6–10 Å). While these structures provide a snapshot of the assembly and its components in well-defined functional states, the resolution limits the ability to build accurate structural models. In contrast, sequence-based modeling techniques are capable of producing relatively robust structural models for isolated proteins or domains. In this work, we developed and applied a hybrid modeling approach, utilizing cryoEM density and ab initio modeling to produce a structural model for the core domain of a herpesvirus structural protein, VP26. Specifically, this method, first tested on simulated data, utilizes the cryoEM density map as a geometrical constraint in identifying the most native-like models from a gallery of models generated by ab initio modeling. The resulting model for the core domain of VP26, based on the 8.5-Å resolution herpes simplex virus type 1 (HSV-1) capsid cryoEM structure and mutational data, exhibited a novel fold. Additionally, the core domain of VP26 appeared to have a complementary interface to the known upper-domain structure of VP5, its cognate binding partner. While this new model provides for a better understanding of the assembly and interactions of VP26 in HSV-1, the approach itself may have broader applications in modeling the components of large macromolecular assemblies

What makes a house a home? Nest box use by West European hedgehogs (Erinaceus europaeus) is influenced by nest box placement, resource provisioning and site-based factors

Artificial refuges provided by householders and/or conservation practitioners potentially represent one mechanism for mitigating declines in the availability of natural nest sites used for resting, breeding and hibernating in urban areas. The effectiveness of such refuges for different species is, however, not always known. In this study, we conducted a questionnaire survey of UK householders to identify factors associated with the use of ground-level nest boxes for West European hedgehogs (Erinaceus europaeus), a species of conservation concern. Overall, the percentage of boxes used at least once varied with season and type of use: summer day nesting (35.5–81.3%), breeding (7.2–28.2%), winter day nesting (20.1–66.5%) and hibernation (21.7–58.6%). The length of time the box had been deployed, the availability of artificial food and front garden to back garden access significantly increased the likelihood that a nest box had been used for all four nesting types, whereas other factors related to placement within the garden (e.g., in a sheltered location, on hardstanding such as paving, distance from the house) and resource provisioning (bedding) affected only some nesting behaviours. The factors most strongly associated with nest box use were the provisioning of food and bedding. These data suggest, therefore, that householders can adopt simple practices to increase the likelihood of their nest box being used. However, one significant limitation evident within these data is that, for welfare reasons, householders do not routinely monitor whether their box has been used. Consequently, future studies need to adopt strategies which enable householders to monitor their boxes continuously. Ultimately, such studies should compare the survival rates and reproductive success of hedgehogs within artificial refuges versus more natural nest sites, and whether these are affected by, for example, the impact of nest box design and placement on predation risk and internal microclimate

Comparing non-invasive surveying techniques for elusive, nocturnal mammals: a case study of the West European hedgehog (Erinaceus europaeus)

Monitoring changes in populations is fundamental for effective management. The West European hedgehog (Erinaceus europeaus) is of conservation concern in the UK because of recent substantial declines. Surveying hedgehogs is, however, problematic because of their nocturnal, cryptic behaviour. We compared the effectiveness of three methods (infra-red thermal camera, specialist search dog, spotlight) for detecting hedgehogs in three different habitats. Significantly more hedgehogs were detected, and at greater distance, using the camera and dog than the spotlight in amenity grassland and pasture; no hedgehogs were detected in woodland. Increasing ground cover reduced detection distances, with most detections (59.6%) associated with bare soil or mown grass; the dog was the only method that detected hedgehogs in vegetation taller than the target species’ height. The additional value of surveying with a detection dog is most likely to be realised in areas where badgers (Meles meles), an intra-guild predator, are and/or where sufficient ground cover is present; both would allow hedgehogs to forage further from refuge habitats such as hedgerows. Further consideration of the effectiveness of detection dogs for finding hedgehogs in nests, as well as developing techniques for monitoring this species in woodland, is warranted

Baker's conjecture for functions with real zeros

Baker's conjecture states that a transcendental entire functions of order less than 1/2 has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits consisting of unbounded wandering domains whenever the order is less than 1. This raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1. Key ingredients of our proofs are new results in classical complex analysis with wider applications. These new results concern: the winding properties of the images of certain curves proved using extremal length arguments, growth estimates for entire functions, and the distribution of the zeros of entire functions of order less than 1

Parameter identifiability and model selection for partial differential equation models of cell invasion

When employing a mechanistic model to study biological systems, practical parameter identifiability is important for making predictions in a wide range of scenarios, as well as for understanding the mechanisms driving the system behaviour. We argue that parameter identifiability should be considered alongside goodness-of-fit and model complexity as criteria for model selection. To demonstrate, we use a profile likelihood approach to investigate parameter identifiability for four extensions of the Fisher--KPP model, given experimental data from a cell invasion assay. We show that more complicated models tend to be less identifiable, with parameter estimates being more sensitive to subtle differences in experimental procedures, and require more data to be practically identifiable. The results from identifiability analysis can inform model selection, as well as data collection and experimental design.Comment: 23 pages in main text, 21 pages in supplementary material