A two-step patterning process increases the robustness of periodic patterning in the fly eye

Complex periodic patterns can self-organize through dynamic interactions between diffusible activators and inhibitors. In the biological context, self-organized patterning is challenged by spatial heterogeneities (‘noise’) inherent to biological systems. How spatial variability impacts the periodic patterning mechanism and how it can be buffered to ensure precise patterning is not well understood. We examine the effect of spatial heterogeneity on the periodic patterning of the fruit fly eye, an organ composed of ～800 miniature eye units (ommatidia) whose periodic arrangement along a hexagonal lattice self-organizes during early stages of fly development. The patterning follows a two-step process, with an initial formation of evenly spaced clusters of ～10 cells followed by a subsequent refinement of each cluster into a single selected cell. Using a probabilistic approach, we calculate the rate of patterning errors resulting from spatial heterogeneities in cell size, position and biosynthetic capacity. Notably, error rates were largely independent of the desired cluster size but followed the distributions of signaling speeds. Pre-formation of large clusters therefore greatly increases the reproducibility of the overall periodic arrangement, suggesting that the two-stage patterning process functions to guard the pattern against errors caused by spatial heterogeneities. Our results emphasize the constraints imposed on self-organized patterning mechanisms by the need to buffer stochastic effects. Author summary Complex periodic patterns are common in nature and are observed in physical, chemical and biological systems. Understanding how these patterns are generated in a precise manner is a key challenge. Biological patterns are especially intriguing, as they are generated in a noisy environment; cell position and cell size, for example, are subject to stochastic variations, as are the strengths of the chemical signals mediating cell-to-cell communication. The need to generate a precise and robust pattern in this ‘noisy’ environment restricts the space of patterning mechanisms that can function in the biological setting. Mathematical modeling is useful in comparing the sensitivity of different mechanisms to such variations, thereby highlighting key aspects of their design.We use mathematical modeling to study the periodic patterning of the fruit fly eye. In this system, a highly ordered lattice of differentiated cells is generated in a two-dimensional cell epithelium. The pattern is first observed by the appearance of evenly spaced clusters of ～10 cells that express specific genes. Each cluster is subsequently refined into a single cell, which initiates the formation and differentiation of a miniature eye unit, the ommatidium. We formulate a mathematical model based on the known molecular properties of the patterning mechanism, and use a probabilistic approach to calculate the errors in cluster formation and refinement resulting from stochastic cell-to-cell variations (‘noise’) in different quantitative parameters. This enables us to define the parameters most influencing noise sensitivity. Notably, we find that this error is roughly independent of the desired cluster size, suggesting that large clusters are beneficial for ensuring the overall reproducibility of the periodic cluster arrangement. For the stage of cluster refinement, we find that rapid communication between cells is critical for reducing error. Our work provides new insights into the constraints imposed on mechanisms generating periodic patterning in a realistic, noisy environment, and in particular, discusses the different considerations in achieving optimal design of the patterning network.     

Event Ordering in Live-Cell Imaging Determined from Temporal Cross-Correlation Asymmetry

We use the temporal asymmetry of the cross-correlation function to determine the temporal ordering of spatially localized cellular events in live-cell multichannel fluorescence imaging. The analysis is well suited to noisy, stochastic systems where the temporal order may not be apparent in the raw data. The approach is applicable to any biochemical reaction not in chemical equilibrium, including protein complex assembly, sequential enzymatic processes, gene regulation, and other cellular signaling events. As an automated quantitative measure, this approach allows the data to be readily interpreted statistically with minimal subjective biases. We first test the technique using simulations of simple biophysical models with a definite temporal ordering. We then demonstrate the approach by extracting the temporal ordering of three proteins—actin, sorting nexin 9, and clathrin—in the endocytic pathway.     

Random sampling, abundance–extinction dynamics and niche-filtering immigration constraints explain the generation of species richness gradients

Aim  The paradigm that species' patterns of distribution, abundance and coexistence are the result of adaptations of the species to their niches has recently been challenged by evidence that similar patterns may be generated by simple random processes. We argue here that a better understanding of macroecological patterns requires an integration of both ecological and neutral stochastic approaches. We demonstrate the utility of such an integrative approach by testing the sampling hypothesis in a species–energy relationship of forest bird species.
Location  A Mediterranean biome in Catalonia, Spain.
Methods  To test the sampling hypothesis we designed a metacommunity model that reproduces the stochastic sampling from a regional pool to predict local species richness variation. Four conceptually different sampling procedures were evaluated.
Results  We showed that stochastic sampling processes predicted a substantial part (over 40%) of the observed variation in species richness, but left considerable variation unexplained. This remaining variation in species richness may be better understood as the result of alternative ecological processes. First, the sampling model explained more variation in species richness when the probability that a species colonises a new locality was assumed to increase with its niche width, suggesting that ecological differences between species matter when it comes to explaining macroecological patterns. Second, extinction risk was significantly lower for species inhabiting high-energy regions, suggesting that abundance–extinction processes play a significant role in shaping species richness patterns.
Main conclusions  We conclude that species–energy relationships may not simply be understood as a result of either ecological or random sampling processes, but more likely as a combination of both.     

上海市景观格局的人工神经网络(ANN)模型

定量分析城市景观的空间格局,深入研究景观格局的形成机制,将有助于理解城市景观的格局与过程,分析城市化的社会、经济和生态学后果以及制定更有效的景观管理策略。研究以城市景观生态学途径,应用基于GIS的景观格局分析与人工神经网络(ANN)相结合的方法定量分析上海市城市景观格局(1994年)及其变化规律,建立了能够较好地模拟上海市景观格局对居住区用地、道路密度、人口密度、城市发展历史与黄浦江等自然、社会、经济因素响应的人工神经网络。结果表明,人工神经网络方法适于研究城市化驱动因素与城市景观格局的非线性对应关系,为景观格局形成机制和景观空间结构与生态学过程相互关系的深入研究提供了一条有效、实用的研究途径。     

Evolution of dispersal in explicitly spatial metacommunities

We apply an evolutionary game theoretic approach to the evolution of dispersal in explicitly spatial metacommunities, using a flexible parametric class of dispersal kernels, namely 2Dt kernels, and study the resulting evolutionary dynamics and outcomes. We observe strong selective pressure on mean dispersal distance (i.e., the first moment), and weaker, but significant, one on the shape of dispersal kernel (i.e., higher moments). We investigate the effects of landscape topology and spatial heterogeneity on the resulting ‘optimal’ dispersal kernels. The shape—importantly the tail structure—and stability of evolutionarily optimal dispersal strategies are strongly affected by landscape topology or connectivity. Specifically, the results suggest that the optimal dispersal kernels in the river network topology have heavier tails and are stable, while those in the direct topology, where organisms are allowed to travel directly from one location to another, have relatively thin tails and may be unstable. We also find that habitat spatial heterogeneity enables coexistence and controls spatial distribution of distinct groups of dispersal strategies and that alteration in topology alone may not be sufficient to change such coexistence. This work provides a tool to translate environmental changes such as global climate change and human intervention into changes in dispersal behavior, which in turn may lead to important alterations of biodiversity and biological invasion patterns.     

Patterns of patchy spread in multi-species reaction–diffusion models

Spread of populations in space often takes place via formation, interaction and propagation of separated patches of high species density, without formation of continuous fronts. This type of spread is called a ‘patchy spread’. In earlier models, this phenomenon was considered to be a result of a pronounced environmental or/and demographic stochasticity. Recently, it was found that a patchy spread can arise in a fully deterministic predator–prey system and in models of infectious diseases; in each case the process takes place in a homogeneous environment. It is well recognized that the observed patterns of patchy spread in nature are a result of interplay between stochastic and deterministic factors. However, the models considering deterministic mechanism of patchy spread are developed and studied much less compared to those based on stochastic mechanisms. A further progress in the understanding of the role of deterministic factors in the patchy spread would be extremely helpful. Here we apply multi-species reaction–diffusion models of two spatial dimensions in a homogeneous environment. We demonstrate that patterns of patchy spread are rather common for the considered approach, in particular, they arise both in mutualism and competition models influenced by predation. We show that this phenomenon can occur in a system without a strong Allee effect, contrary to what was assumed to be crucial in earlier models. We show, as well, a pattern of patchy spread having significantly different speeds in different spatial directions. We analyze basic features of spatiotemporal dynamics of patchy spread common for the reaction–diffusion approach. We discuss in which ecosystems we would observe patterns of deterministic patchy spread due to the considered mechanism.     

The coevolution of two phytoplankton species on a single resource: Allelopathy as a pseudo-mixotrophy

Without the top-down effects and the external/physical forcing, a stable coexistence of two phytoplankton species under a single resource is impossible — a result well known from the principle of competitive exclusion. Here I demonstrate by analysis of a mathematical model that such a stable coexistence in a homogeneous media without any external factor would be possible, at least theoretically, provided (i) one of the two species is toxin producing thereby has an allelopathic effect on the other, and (ii) the allelopathic effect exceeds a critical level. The threshold level of allelopathy required for the coexistence has been derived analytically in terms of the parameters associated with the resource competition and the nutrient recycling. That the extra mortality of a competitor driven by allelopathy of a toxic species gives a positive feed back to the algal growth process through the recycling is explained. And that this positive feed back plays a pivotal role in reducing competition pressures and helping species succession in the two-species model is demonstrated. Based on these specific coexistence results, I introduce and explain theoretically the allelopathic effect of a toxic species as a ‘pseudo-mixotrophy’—a mechanism of ‘if you cannot beat them or eat them, just kill them by chemical weapons’. The impact of this mechanism of species succession by pseudo-mixotrophy in the form of alleopathy is discussed in the context of current understanding on straight mixotrophy and resource-species relationship among phytoplankton species.     

Using Moment Equations to Understand Stochastically Driven Spatial Pattern Formation in Ecological Systems

Spatial patterns in biological populations and the effect of spatial patterns on ecological interactions are central topics in mathematical ecology. Various approaches to modeling have been developed to enable us to understand spatial patterns ranging from plant distributions to plankton aggregation. We present a new approach to modeling spatial interactions by deriving approximations for the time evolution of the moments (mean and spatial covariance) of ensembles of distributions of organisms; the analysis is made possible by “moment closure,” neglecting higher-order spatial structure in the population. We use the growth and competition of plants in an explicitly spatial environment as a starting point for exploring the properties of second-order moment equations and comparing them to realizations of spatial stochastic models. We find that for a wide range of effective neighborhood sizes (each plant interacting with several to dozens of neighbors), the mean-covariance model provides a useful and analytically tractable approximation to the stochastic spatial model, and combines useful features of stochastic models and traditional reaction-diffusion-like models.     

Measuring complexity to infer changes in the dynamics of ecological systems under stress

Despite advances in our mechanistic understanding of ecological processes, the inherent complexity of real-world ecosystems still limits our ability in predicting ecological dynamics especially in the face of on-going environmental stress. Developing a model is frequently challenged by structure uncertainty, unknown parameters, and limited data for exploring out-of-sample predictions. One way to address this challenge is to look for patterns in the data themselves in order to infer the underlying processes of an ecological system rather than to build system-specific models. For example, it has been recently suggested that statistical changes in ecological dynamics can be used to infer changes in the stability of ecosystems as they approach tipping points. For computer scientists such inference is similar to the notion of a Turing machine: a computational device that could execute a program (the process) to produce the observed data (the pattern). Here, we make use of such basic computational ideas introduced by Alan Turing to recognize changing patterns in ecological dynamics in ecosystems under stress. To do this, we use the concept of Kolmogorov algorithmic complexity that is a measure of randomness. In particular, we estimate an approximation to Kolmogorov complexity based on the Block Decomposition Method (BDM). We apply BDM to identify changes in complexity in simulated time-series and spatial datasets from ecosystems that experience different types of ecological transitions. We find that in all cases, K_BDM complexity decreased before all ecological transitions both in time-series and spatial datasets. These trends indicate that loss of stability in the ecological models we explored is characterized by loss of complexity and the emergence of a regular and computable underlying structure. Our results suggest that Kolmogorov complexity may serve as tool for revealing changes in the dynamics of ecosystems close to ecological transitions.     

Analytic approximation of spatial epidemic models of foot and mouth disease

The effect of spatial heterogeneity in epidemic models has improved with computational advances, yet far less progress has been made in developing analytical tools for understanding such systems. Here, we develop two classes of second-order moment closure methods for approximating the dynamics of a stochastic spatial model of the spread of foot and mouth disease. We consider the performance of such ‘pseudo-spatial’ models as a function of R₀, the locality in disease transmission, farm distribution and geographically-targeted control when an arbitrary number of spatial kernels are incorporated. One advantage of mapping complex spatial models onto simpler deterministic approximations lies in the ability to potentially obtain a better analytical understanding of disease dynamics and the effects of control. We exploit this tractability by deriving analytical results in the invasion stages of an FMD outbreak, highlighting key principles underlying epidemic spread on contact networks and the effect of spatial correlations.     

A tale of two cycles – distinguishing quasi-cycles and limit cycles in finite predator–prey populations

Periodic predator – prey dynamics in constant environments are usually taken as indicative of deterministic limit cycles. It is known, however, that demographic stochasticity in finite populations can also give rise to regular population cycles, even when the corresponding deterministic models predict a stable equilibrium. Specifically, such quasi-cycles are expected in stochastic versions of deterministic models exhibiting equilibrium dynamics with weakly damped oscillations. The existence of quasi-cycles substantially expands the scope for natural patterns of periodic population oscillations caused by ecological interactions, thereby complicating the conclusive interpretation of such patterns. Here we show how to distinguish between quasi-cycles and noisy limit cycles based on observing changing population sizes in predator – prey populations. We start by confirming that both types of cycle can occur in the individual-based version of a widely used class of deterministic predator – prey model. We then show that it is feasible and straightforward to accurately distinguish between the two types of cycle through the combined analysis of autocorrelations and marginal distributions of population sizes. Finally, by confronting these results with real ecological time series, we demonstrate that by using our methods even short and imperfect time series allow quasi-cycles and limit cycles to be distinguished reliably.     

Pattern analysis in branching and axillary flowering sequences

In the architectural approach to the study of plants, a major issue is to analyse branching and axillary flowering patterns. Due to the structured expression of the branching process and the noisy character of the observed patterns, we propose an analysis framework which is both structural and probabilistic. Data take the form of sequences which naturally represent the underlying structural information of branching and axillary flowering patterns and allow the application of a large number of methods ranging from exploratory analysis to stochastic modeling. The primary aim of the proposed analysis methods is to reveal patterns not directly apparent in the data, and thus to deepen our biological understanding of the underlying mechanisms that control the branching and the axillary flowering of plants over time and space. The proposed approach is illustrated using a set of examples corresponding to different plant species and different biological or agronomic objectives.     

Deterministic limits to stochastic spatial models of natural enemies

Stochastic spatial models are becoming an increasingly popular tool for understanding ecological and epidemiological problems. However, due to the complexities inherent in such models, it has been difficult to obtain any analytical insights. Here, we consider individual-based, stochastic models of both the continuous-time Lotka-Volterra system and the discrete-time Nicholson-Bailey model. The stability of these two stochastic models of natural enemies is assessed by constructing moment equations. The inclusion of these moments, which mimic the effects of spatial aggregation, can produce either stabilizing or destabilizing influences on the population dynamics. Throughout, the theoretical results are compared to numerical models for the full distribution of populations, as well as stochastic simulations.     

An artificial neural network model of the landscape pattern in Shanghai metropolitan region,China

To characterize the urbanization pattern quantitatively,a study on the mechanisms of the landscape pattern formation could facilitate the understanding on urban landscape patterns and processes,the ecological and socioeconomic consequences of urbanization,as well as the establishment of more effective strategies for landscape management.In this study,we integrated a Geographic Information System (GIS)-based analysis on landscape pattern with an artificial neural network (ANN) to quantitatively characterize the urbanization pattern of the metropolitan area of Shanghai,China,and to establish an ANN model that could preferably simulate the responses of urban landscape pattern to the natural and socioeconomic factors such as residence area,road density,population density,urban development history and the Huangpu River as an element of economic change.Our results showed that the ANN model seems appropriate for studying the nonlinear relationship among the forcing factors of urbanization and the urban landscape patterns,which provided an effective and practical approach for further understanding the mechanisms of the landscape formation pattern and the reciprocal relationship between landscape spatial pattern and ecological process.     

Genetic Drift Suppresses Bacterial Conjugation in Spatially Structured Populations

Conjugation is the primary mechanism of horizontal gene transfer that spreads antibiotic resistance among bacteria. Although conjugation normally occurs in surface-associated growth (e.g., biofilms), it has been traditionally studied in well-mixed liquid cultures lacking spatial structure, which is known to affect many evolutionary and ecological processes. Here we visualize spatial patterns of gene transfer mediated by F plasmid conjugation in a colony of Escherichia coli growing on solid agar, and we develop a quantitative understanding by spatial extension of traditional mass-action models. We found that spatial structure suppresses conjugation in surface-associated growth because strong genetic drift leads to spatial isolation of donor and recipient cells, restricting conjugation to rare boundaries between donor and recipient strains. These results suggest that ecological strategies, such as enforcement of spatial structure and enhancement of genetic drift, could complement molecular strategies in slowing the spread of antibiotic resistance genes.     

Genetic Drift Suppresses Bacterial Conjugation in Spatially Structured Populations

Conjugation is the primary mechanism of horizontal gene transfer that spreads antibiotic resistance among bacteria. Although conjugation normally occurs in surface-associated growth (e.g., biofilms), it has been traditionally studied in well-mixed liquid cultures lacking spatial structure, which is known to affect many evolutionary and ecological processes. Here we visualize spatial patterns of gene transfer mediated by F plasmid conjugation in a colony of Escherichia coli growing on solid agar, and we develop a quantitative understanding by spatial extension of traditional mass-action models. We found that spatial structure suppresses conjugation in surface-associated growth because strong genetic drift leads to spatial isolation of donor and recipient cells, restricting conjugation to rare boundaries between donor and recipient strains. These results suggest that ecological strategies, such as enforcement of spatial structure and enhancement of genetic drift, could complement molecular strategies in slowing the spread of antibiotic resistance genes.     

Relative role of deterministic and stochastic determinants of soil animal community: a spatially explicit analysis of oribatid mites

1. Ecologists are debating the relative role of deterministic and stochastic determinants of community structure. Although the high diversity and strong spatial structure of soil animal assemblages could provide ecologists with an ideal ecological scenario, surprisingly little information is available on these assemblages. 2. We studied species-rich soil oribatid mite assemblages from a Mediterranean beech forest and a grassland. We applied multivariate regression approaches and analysed spatial autocorrelation at multiple spatial scales using Moran's eigenvectors. Results were used to partition community variance in terms of the amount of variation uniquely accounted for by environmental correlates (e.g. organic matter) and geographical position. Estimated neutral diversity and immigration parameters were also applied to a soil animal group for the first time to simulate patterns of community dissimilarity expected under neutrality, thereby testing neutral predictions. 3. After accounting for spatial autocorrelation, the correlation between community structure and key environmental parameters disappeared: about 40% of community variation consisted of spatial patterns independent of measured environmental variables such as organic matter. Environmentally independent spatial patterns encompassed the entire range of scales accounted for by the sampling design (from tens of cm to 100 m). This spatial variation could be due to either unmeasured but spatially structured variables or stochastic drift mediated by dispersal. Observed levels of community dissimilarity were significantly different from those predicted by neutral models. 4. Oribatid mite assemblages are dominated by processes involving both deterministic and stochastic components and operating at multiple scales. Spatial patterns independent of the measured environmental variables are a prominent feature of the targeted assemblages, but patterns of community dissimilarity do not match neutral predictions. This suggests that either niche-mediated competition or environmental filtering or both are contributing to the core structure of the community. This study indicates new lines of investigation for understanding the mechanisms that determine the signature of the deterministic component of animal community assembly.     

Coexistence and invasibility in a two-species competition model with habitat-preference

The outcome of competition among species is influenced by the spatial distribution of species and effects such as demographic stochasticity, immigration fluxes, and the existence of preferred habitats. We introduce an individual-based model describing the competition of two species and incorporating all the above ingredients. We find that the presence of habitat preference—generating spatial niches—strongly stabilizes the coexistence of the two species. Eliminating habitat preference—neutral dynamics—the model generates patterns, such as distribution of population sizes, practically identical to those obtained in the presence of habitat preference, provided an higher immigration rate is considered. Notwithstanding the similarity in the population distribution, we show that invasibility properties depend on habitat preference in a non-trivial way. In particular, the neutral model results more invasible or less invasible depending on whether the comparison is made at equal immigration rate or at equal distribution of population size, respectively. We discuss the relevance of these results for the interpretation of invasibility experiments and the species occupancy of preferred habitats.     

Biological invasions: deriving the regions at risk from partial measurements

We consider the problem of forecasting the regions at higher risk for newly introduced invasive species. Favourable and unfavourable regions may indeed not be known a priori, especially for exotic species whose hosts in native range and newly-colonised areas can be different. Assuming that the species is modelled by a logistic-like reaction-diffusion equation, we prove that the spatial arrangement of the favourable and unfavourable regions can theoretically be determined using only partial measurements of the population density: (1) a local ‘spatio-temporal’ measurement, during a short time period and, (2) a ‘spatial’ measurement in the whole region susceptible to colonisation. We then present a stochastic algorithm which is proved analytically, and then on several numerical examples, to be effective in deriving these regions.