A Mathematical Model for Flight Guidance in Honeybee Swarms

When a colony of honeybees relocates to a new nest site, less than 5?% of the bees (the scout bees) know the location of the new nest. Nevertheless, the small minority of informed bees manages to provide guidance to the rest and the entire swarm is able to fly to the new nest intact. The streaker bee hypothesis, one of the several theories proposed to explain the guidance mechanism in bee swarms, seems to be supported by recent experimental observations. The theory suggests that the informed bees make high-speed flights through the swarm in the direction of the new nest, hence conspicuously pointing to the desired direction of travel. This work presents a mathematical model of flight guidance in bee swarms based on the streaker bee hypothesis. Numerical experiments, parameter studies, and comparison with experimental data are presented.     

A Nonlocal Continuum Model for Biological Aggregation

We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short-range dispersal. For the case of one spatial dimension, we study the steady states analytically and numerically. There exist strongly nonlinear states with compact support and steep edges that correspond to localized biological aggregations, or clumps. These steady-state clumps are reached through a dynamic coarsening process. In the limit of large population size, the clumps approach a constant density swarm with abrupt edges. We use energy arguments to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit. The energy result holds in higher dimensions as well, and is demonstrated via numerical simulations in two dimensions.     

A Multidimensional Multispecies Continuum Model for Heterogeneous Biofilm Development

We propose a multidimensional continuum model for heterogeneous growth of biofilm systems with multiple species and multiple substrates. The new model provides a deterministic framework for the study of the interactions between several spe1cies and their effects on biofilm heterogeneity. It consists of a system of partial differential equations derived on the basis of conservation laws and reaction kinetics. The derivation and key assumptions are presented. The assumptions used are a combination of those used in the established one dimensional model, due to Wanner and Gujer, and for the viscous fluid model, of Dockery and Klapper. The work of Wanner and Gujer in particular has been extensively used through the years, and thus this new model is an extension to several spatial dimensions of an already proven working model. The model equations are solved using numerical techniques, for purposes of simulation and verification. The new model is applied to two different biofilm systems in several spatial dimensions, one of which is equivalent to a system originally studied by Wanner and Gujer. Dimensionless formulations for these two systems are given, and numerical simulation results with varying initial conditions are presented. An erratum to this article can be found at     

A Continuum Neuronal Model for the Instigation and Propagation of Cortical Spreading Depression

Cortical spreading depression (CSD) waves can occur in the cortices of various brain structures and are associated with the spread of depression of the electroencephalogram signal. In this paper, we present a continuum neuronal model for the instigation and spreading of CSD. Our model assumes that the brain-cell microenvironment can be treated as a porous medium consisting of extra- and intracellular compartments. The main mechanisms in our model for the transport of ions into and out of neurons are cross-membrane ionic currents and (active) pumps, coupled with diffusion in the extracellular space. To demonstrate the applicability of our model, we have carried out extensive numerical simulations under different initial conditions and inclusion of various mechanisms. Our results show that CSD waves can be instigated by injecting cross-membrane ionic currents or by applying KCl in the extracellular space. Furthermore, the estimated speeds of CSD waves are within the experimentally observed range. Effects of specific ion channels, background ion concentrations, extracellular volume fractions, and cell swelling on the propagation speed of CSD are also investigated.     

A Continuum Mathematical Model of the Developing Murine Retinal Vasculature

Angiogenesis, the process of new vessel growth from pre-existing vasculature, is crucial in many biological situations such as wound healing and embryogenesis. Angiogenesis is also a key regulator of pathogenesis in many clinically important disease processes, for instance, solid tumour progression and ocular diseases. Over the past 10–20 years, tumour-induced angiogenesis has received a lot of attention in the mathematical modelling community and there have also been some attempts to model angiogenesis during wound healing. However, there has been little modelling work of vascular growth during normal development. In this paper, we describe an in silico representation of the developing retinal vasculature in the mouse, using continuum mathematical models consisting of systems of partial differential equations. The equations describe the migratory response of cells to growth factor gradients, the evolution of the capillary blood vessel density, and of the growth factor concentration. Our approach is closely coupled to an associated experimental programme to parameterise our model effectively and the simulations provide an excellent correlation with in vivo experimental data. Future work and development of this model will enable us to elucidate the impact of molecular cues upon vasculature development and the implications for eye diseases such as diabetic retinopathy and neonatal retinopathy of prematurity.     

The Laminar Cortex Model: A New Continuum Cortex Model Incorporating Laminar Architecture

Local field potentials (LFPs) are widely used to study the function of local networks in the brain. They are also closely correlated with the blood-oxygen-level-dependent signal, the predominant contrast mechanism in functional magnetic resonance imaging. We developed a new laminar cortex model (LCM) to simulate the amplitude and frequency of LFPs. Our model combines the laminar architecture of the cerebral cortex and multiple continuum models to simulate the collective activity of cortical neurons. The five cortical layers (layer I, II/III, IV, V, and VI) are simulated as separate continuum models between which there are synaptic connections. The LCM was used to simulate the dynamics of the visual cortex under different conditions of visual stimulation. LFPs are reported for two kinds of visual stimulation: general visual stimulation and intermittent light stimulation. The power spectra of LFPs were calculated and compared with existing empirical data. The LCM was able to produce spontaneous LFPs exhibiting frequency-inverse (1/ƒ) power spectrum behaviour. Laminar profiles of current source density showed similarities to experimental data. General stimulation enhanced the oscillation of LFPs corresponding to gamma frequencies. During simulated intermittent light stimulation, the LCM captured the fundamental as well as high order harmonics as previously reported. The power spectrum expected with a reduction in layer IV neurons, often observed with focal cortical dysplasias associated with epilepsy was also simulated.     

A Continuum Mechanics Model of Enzyme-Based Tissue Degradation in Cancer Therapies

We propose a mathematical model to describe enzyme-based tissue degradation in cancer therapies. The proposed model combines the poroelastic theory of mixtures with the transport of enzymes or drugs in the extracellular space. The effect of the matrix-degrading enzymes on the tissue composition and its mechanical response are accounted for. Numerical simulations in 1D, 2D and axisymmetric (3D) configurations show how an injection of matrix-degrading enzymes alters the porosity of a biological tissue. We eventually exhibit numerically the main consequences of a matrix-degrading enzyme pretreatment in the framework of chemotherapy: the removal of the diffusive hindrance to the penetration of therapeutic molecules in tumors and the reduction of interstitial fluid pressure which improves transcapillary transport. Both effects are consistent with previous biological observations.     

Hierarchy of Attractants for Honey Bee Swarms

Chemical signals influence the selection of potential nest cavities by honey bee reproductive swarms. Attractants for swarms include the odors of old dark honey bee brood combs, odors from noncomb hive materials and propolis, and Nasonov pheromone, the odor released from the Nasonov glands of worker bees. Based on crossover and choice test experiments, swarms were shown to prefer, among otherwise identical cavities, those cavities containing Nasonov pheromone over cavities with only comb or other hive odors, cavities containing old comb over those with only noncomb odors or propolis, and cavities containing noncomb odors or propolis over those without bee or hive odor. Synergy between odors was not observed; that is, comb and/or noncomb hive odors did not enhance the attractiveness of Nasonov pheromone. The data support a model based on a hierarchy of olfactory attractants used by honey bee swarms, in order of highest to lowest: Nasonov pheromone, comb odor, noncomb and propolis odors, and, finally, absence of bee- or hive-produced odor.     

A Preferred Curvature-Based Continuum Mechanics Framework for Modeling Embryogenesis

Mechanics plays a key role in the development of higher organisms. However, understanding this relationship is complicated by the difficulty of modeling the link between local forces generated at the subcellular level and deformations observed at the tissue and whole-embryo levels. Here we propose an approach first developed for lipid bilayers and cell membranes, in which force-generation by cytoskeletal elements enters a continuum mechanics formulation for the full system in the form of local changes in preferred curvature. This allows us to express and solve the system using only tissue strains. Locations of preferred curvature are simply related to products of gene expression. A solution, in that context, means relaxing the system’s mechanical energy to yield global morphogenetic predictions that accommodate a tendency toward the local preferred curvature, without a need to explicitly model force-generation mechanisms at the molecular level. Our computational framework, which we call SPHARM-MECH, extends a 3D spherical harmonics parameterization known as SPHARM to combine this level of abstraction with a sparse shape representation. The integration of these two principles allows computer simulations to be performed in three dimensions on highly complex shapes, gene expression patterns, and mechanical constraints. We demonstrate our approach by modeling mesoderm invagination in the fruit-fly embryo, where local forces generated by the acto-myosin meshwork in the region of the future mesoderm lead to formation of a ventral tissue fold. The process is accompanied by substantial changes in cell shape and long-range cell movements. Applying SPHARM-MECH to whole-embryo live imaging data acquired with light-sheet microscopy reveals significant correlation between calculated and observed tissue movements. Our analysis predicts the observed cell shape anisotropy on the ventral side of the embryo and suggests an active mechanical role of mesoderm invagination in supporting the onset of germ-band extension.     

Capturing the Dynamics of a Hybrid Multiscale Cancer Model with a Continuum Model

Cancer is a complex disease involving processes at spatial scales from subcellular, like cell signalling, to tissue scale, such as vascular network formation. A number of multiscale models have been developed to study the dynamics that emerge from the coupling between the intracellular, cellular and tissue scales. Here, we develop a continuum partial differential equation model to capture the dynamics of a particular multiscale model (a hybrid cellular automaton with discrete cells, diffusible factors and an explicit vascular network). The purpose is to test under which circumstances such a continuum model gives equivalent predictions to the original multiscale model, in the knowledge that the system details are known, and differences in model results can be explained in terms of model features (rather than unknown experimental confounding factors). The continuum model qualitatively replicates the dynamics from the multiscale model, with certain discrepancies observed owing to the differences in the modelling of certain processes. The continuum model admits travelling wave solutions for normal tissue growth and tumour invasion, with similar behaviour observed in the multiscale model. However, the continuum model enables us to analyse the spatially homogeneous steady states of the system, and hence to analyse these waves in more detail. We show that the tumour microenvironmental effects from the multiscale model mean that tumour invasion exhibits a so-called pushed wave when the carrying capacity for tumour cell proliferation is less than the total cell density at the tumour wave front. These pushed waves of tumour invasion propagate by triggering apoptosis of normal cells at the wave front. Otherwise, numerical evidence suggests that the wave speed can be predicted from linear analysis about the normal tissue steady state.     

Flash Expansion Threshold in Whirligig Swarms

In the selfish herd hypothesis, prey animals move toward each other to avoid the likelihood of being selected by a predator. However, many grouped animals move away from each other the moment before a predator attacks. Very little is known about this phenomenon, called flash expansion, such as whether it is triggered by one individual or a threshold and how information is transferred between group members. We performed a controlled experiment with whirligig beetles in which the ratio of sighted to unsighted individuals was systematically varied and emergent flash expansion was measured. Specifically, we examined: the percentage of individuals in a group that startled, the resulting group area, and the longevity of the flash expansion. We found that one or two sighted beetles in a group of 24 was not enough to cause a flash expansion after a predator stimulus, but four sighted beetles usually initiated a flash expansion. Also, the more beetles that were sighted the larger the resulting group area and the longer duration of the flash expansion. We conclude that flash expansion is best described as a threshold event whose adaptive value is to prevent energetically costly false alarms while quickly mobilizing an emergent predator avoidance response. This is one of the first controlled experiments of flash expansion, an important emergent property that has applications to understanding collective motion in swarms, schools, flocks, and human crowds. Also, our study is a convincing demonstration of social contagion, how the actions of one individual can pass through a group.     

A Novel Method for Tracking Individuals of Fruit Fly Swarms Flying in a Laboratory Flight Arena

The growing interest in studying social behaviours of swarming fruit flies, Drosophila melanogaster, has heightened the need for developing tools that provide quantitative motion data. To achieve such a goal, multi-camera three-dimensional tracking technology is the key experimental gateway. We have developed a novel tracking system for tracking hundreds of fruit flies flying in a confined cubic flight arena. In addition to the proposed tracking algorithm, this work offers additional contributions in three aspects: body detection, orientation estimation, and data validation. To demonstrate the opportunities that the proposed system offers for generating high-throughput quantitative motion data, we conducted experiments on five experimental configurations. We also performed quantitative analysis on the kinematics and the spatial structure and the motion patterns of fruit fly swarms. We found that there exists an asymptotic distance between fruit flies in swarms as the population density increases. Further, we discovered the evidence for repulsive response when the distance between fruit flies approached the asymptotic distance. Overall, the proposed tracking system presents a powerful method for studying flight behaviours of fruit flies in a three-dimensional environment.     

Interpreting Leaf Water Potential Measurements with a Model of the Soil-Plant-Atmosphere Continuum

A water flux model, which assumes that the dynamic functioning of the soil-plant-atmosphere continuum may be described by a series of steady states, was examined as a means for interpreting leaf water potential measurements in ‘Valencia’ orange trees (Citrus sinensis (L.) Osbeck). According to the model, leaf water potential should be related to transpirational flux, which in this experiment was estimated by the ratio of vapor pressure deficit of the atmosphere to leaf diffusion resistance (VPD/r_leaf). Leaf water potentials decreased in a specific relationship with increasing values of VPD/r_leaf provided that soil water was adequate and soil temperature was not too low, but regardless of season of the year or climatic or edaphic differences among 3 field locations. When soil water tensions exceeded 0.3 bar or when soil temperatures were lower than 15°C, deviations from the model occurred in the form of more negative leaf water potentials than predicted by VPD/r_leaf. The model predicts from simple measurements made on intact plants that these differences were due to the modification of flow resistances by cool temperatures and the modification of both resistances and the potential of water at the source in the case of soil water depletion. The model may be a useful tool for interpreting plant water potential data under contrasting environmental conditions.     

Continuum remodeling revisited

Recent research effort in bone remodeling has been directed toward describing interstitial fluid flow in the lacuno-canalicular system and its potential as a cellular stimulus. Regardless of the precise contents of the mechanotransduction “black box”, it seems clear that the fluid flow on which the remodeling is predicated cannot occur under static loading conditions. In an attempt to help continuum remodeling simulations catch up with cellular and subcellular research, this paper presents a simple, strain rate driven remodeling algorithm for density allocation and principal material direction rotations. An explicit finite element code was written and deployed on a supercomputer which discretizes the remodeling process and uses an objective hypoelastic constitutive law to simulate trabecular realignment. Results indicate that a target strain rate for this dynamic approach is |D _I ^*| = 1.7% per second which seems reasonable when compared to observed strain rates. Simulations indicate that a morpho-mechanically realistic three-dimensional bone can be synthesized by applying a few dynamic loads at the envelope of common daily physiological rates, even with no static loading component.     

A Continuum of Specialists and Generalists in Empirical Communities

Understanding the persistence of specialists and generalists within ecological communities is a topical research question, with far-reaching consequences for the maintenance of functional diversity. Although theoretical studies indicate that restricted conditions may be necessary to achieve co-occurrence of specialists and generalists, analyses of larger empirical (and species-rich) communities reveal the pervasiveness of coexistence. In this paper, we analyze 175 ecological bipartite networks of three interaction types (animal hosts–parasite, plant–herbivore and plant–pollinator), and measure the extent to which these communities are composed of species with different levels of specificity in their biotic interactions. We find a continuum from specialism to generalism. Furthermore, we demonstrate that diversity tends to be greatest in networks with intermediate connectance, and argue this is because of physical constraints in the filling of networks.     

Evolution of Self-Organized Task Specialization in Robot Swarms

Division of labor is ubiquitous in biological systems, as evidenced by various forms of complex task specialization observed in both animal societies and multicellular organisms. Although clearly adaptive, the way in which division of labor first evolved remains enigmatic, as it requires the simultaneous co-occurrence of several complex traits to achieve the required degree of coordination. Recently, evolutionary swarm robotics has emerged as an excellent test bed to study the evolution of coordinated group-level behavior. Here we use this framework for the first time to study the evolutionary origin of behavioral task specialization among groups of identical robots. The scenario we study involves an advanced form of division of labor, common in insect societies and known as “task partitioning”, whereby two sets of tasks have to be carried out in sequence by different individuals. Our results show that task partitioning is favored whenever the environment has features that, when exploited, reduce switching costs and increase the net efficiency of the group, and that an optimal mix of task specialists is achieved most readily when the behavioral repertoires aimed at carrying out the different subtasks are available as pre-adapted building blocks. Nevertheless, we also show for the first time that self-organized task specialization could be evolved entirely from scratch, starting only from basic, low-level behavioral primitives, using a nature-inspired evolutionary method known as Grammatical Evolution. Remarkably, division of labor was achieved merely by selecting on overall group performance, and without providing any prior information on how the global object retrieval task was best divided into smaller subtasks. We discuss the potential of our method for engineering adaptively behaving robot swarms and interpret our results in relation to the likely path that nature took to evolve complex sociality and task specialization.     

具垂直传播的林龄结构病虫害模型平衡态的稳定性

讨论连续使用杀虫剂和具有垂直传播的林龄结构病虫害模型,对模型的动力学性态进行了分析,得到模型的基本再生数,讨论了模型无病平衡态的稳定性和病虫害平衡态的不稳定性,分析了病虫害的机理参数对病害流行的影响.     

Interface of the Polarizable Continuum Model of Solvation with Semi-Empirical Methods in the GAMESS Program

An interface between semi-empirical methods and the polarized continuum model (PCM) of solvation successfully implemented into GAMESS following the approach by Chudinov et al (Chem. Phys. 1992, 160, 41). The interface includes energy gradients and is parallelized. For large molecules such as ubiquitin a reasonable speedup (up to a factor of six) is observed for up to 16 cores. The SCF convergence is greatly improved by PCM for proteins compared to the gas phase.