Use of Stochastic Models To Assess the Effect of Environmental Factors on Microbial Growth

We present a novel application of a stochastic ecological model to the study and analysis of microbial growth dynamics as influenced by environmental conditions in an extensive experimental data set. The model proved to be useful in bridging the gap between theoretical ideas in ecology and an applied problem in microbiology. The data consisted of recorded growth curves of Escherichia coli grown in triplicate in a base medium with all 32 possible combinations of five supplements: glucose, NH₄Cl, HCl, EDTA, and NaCl. The potential complexity of 2⁵ experimental treatments and their effects was reduced to 2² as just the metal chelator EDTA, the presumed osmotic pressure imposed by NaCl, and the interaction between these two factors were enough to explain the variability seen in the data. The statistical analysis showed that the positive and negative effects of the five chemical supplements and their combinations were directly translated into an increase or decrease in time required to attain stationary phase and the population size at which the stationary phase started. The stochastic ecological model proved to be useful, as it effectively explained and summarized the uncertainty seen in the recorded growth curves. Our findings have broad implications for both basic and applied research and illustrate how stochastic mathematical modeling coupled with rigorous statistical methods can be of great assistance in understanding basic processes in microbial ecology.     

On the lag phase and initial decline of microbial growth curves

The lag phase is generally thought to be a period during which the cells adjust to a new environment before the onset of exponential growth. Characterizing the lag phase in microbial growth curves has importance in food sciences, environmental sciences, bioremediation and in understanding basic cellular processes. The goal of this work is to extend the analysis of cell growth curves and to better estimate the duration of the lag phase. A non-autonomous model is presented that includes actively duplicating cells and two subclasses of non-duplicating cells. The growth curves depend on the growth and death rate of these three subpopulations and on the initial proportion of each. A deterministic and a stochastic model are both developed and give the same results. A notable feature of the model is the decline of cells during the early stage of the growth curve, and the range of parameters when this decline occurs is identified. A limited growth model is also presented that accounts for the lag, exponential growth and stationary phase of microbial growth curves.     

Temporal scaling of bacterial taxa is influenced by both stochastic and deterministic ecological factors

Microorganisms operate at a range of spatial and temporal scales acting as key drivers of ecosystem properties. Therefore, many key questions in microbial ecology require the consideration of both spatial and temporal scales. Spatial scaling, in particular the species-area relationship (SAR), has a long history in ecology and has recently been addressed in microbial ecology. However, the temporal analogue of the SAR, the species-time relationship, has received far less attention even in the science of general ecology. Here we focus upon the role of temporal scaling in microbial ecological patterns by coupling molecular characterization of bacterial communities in discrete island (bioreactor) systems with a macroecological approach. Our findings showed that the temporal scaling exponent (slope), and therefore taxa turnover of the bacterial taxa-time relationship decreased as selective pressure (industrial wastewater concentration) increased. Also, as the concentration of industrial wastewater increased across the bioreactors, we observed a gradual switch from stochastic community assembly to more deterministic (niche)-based considerations. The identification of broad-scale statistical patterns is particularly relevant to microbial ecology, as it is frequently difficult to identify individual species or their functions. In this study, we identify wide-reaching statistical patterns of diversity and show that they are shaped by the prevalent underlying ecological factors.     

Toward a stochastic formulation of microbial growth in relation to bioreactor performances: case study of an E. coli fed-batch process

A stochastic microbial growth model has been elaborated in the case of the culture of E. coli in fed-batch and scale-down reactors. This model is based on the stochastic determination of the generation time of the microbial cells. The determination of generation time is determined by choosing the appropriate value on a log-normal distribution. The appropriateness of such distribution is discussed and growth curves are obtained that show good agreement compared with the experimental results. The mean and the standard deviation of the log-normal distribution can be considered to be constant during the batch phase of the culture, but they vary when the fed-batch mode is started. It has been shown that the parameters related to the log-normal distribution are submitted to an exponential evolution. The aim of this study is to explore the bioreactor hydrodynamic effect on microbial growth. Thus, in a second time, the stochastic growth model has been reinforced by data coming from a previous stochastic bioreactor mixing model (1). The connection of these hydrodynamic data with the actual stochastic growth model has allowed us to explain the scale-down effect associated with the glucose concentration fluctuations. It is important to point out that the scale-down effect is induced differently according to the feeding strategy involved in the fed-batch experiments.     

微生物生态学理论框架

微生物是生态系统的重要组成部分,直接或间接地参与所有的生态过程。微生物生态学是基于微生物群体的科学,利用微生物群体DNA/RNA等标志物,重点研究微生物群落构建、组成演变、多样性及其与环境的关系,在生态学理论的指导和反复模型拟合下由统计分析得出具有普遍意义的结论。其研究范围从基因尺度到全球尺度。分子生物学技术的发展,使人们可以直接从基因水平上考查其多样性,从而使得对微生物空间分布格局及其成因的深入研究成为可能。进而可以从方法学探讨微生物生物多样性、分布格局、影响机制及其对全球变化的响应等。在微生物生态学研究中,群落构建与演化、分布特征(含植物-微生物相互关系)、执行群体功能的机理(生物地球化学循环等)、对环境变化的响应与反馈机理是今后需要关注的重点领域。概述了微生物生态学的概念,并初步提出其理论框架,在对比宏观生态学基础理论和模型的基础上,分析微生物多样性的研究内容、研究方法和群落构建的理论机制,展望了今后研究的重点领域。     

Ecological Application of Antibiotics as Respiratory Inhibitors of Bacterial Populations

Two terregenous and four marine bacterial isolates were treated with six antibiotics and antibiotic combinations. Comparisons made between responses of cells in early and late logarithmic and stationary growth phases indicated variable sensitivity to the agents. Bacteria in stationary and late log-phase cultures exhibited the greatest resistance, whereas the early log-phase cells exhibited greatest antibiotic susceptibility. We conclude that the tested antibiotics cannot be used for ecological purposes to delineate bacterial respiration in mixed microbial communities.     

Predicting stability of mixed microbial cultures from single species experiments: 1. Phenomenological model

The growth of mixed microbial cultures on mixtures of substrates is a fundamental problem of both theoretical and practical interest. On the one hand, the literature is abundant with experimental studies of mixed-substrate phenomena [T. Egli, The ecological and physiological significance of the growth of heterotrophic microorganisms with mixtures of substrates, Adv. Microbiol. Ecol. 14 (1995) 305-386]. On the other hand, a number of mathematical models of mixed-substrate growth have been analyzed in the last three decades. These models typically assume specific kinetic expressions for substrate uptake and biomass growth rates and their predictions are formulated in terms of parameters of the model. In this work, we formulate and analyze a general mathematical model of mixed microbial growth on mixtures of substitutable substrates. Using this model, we study the effect of mutual inhibition of substrate uptake rates on the stability of the equilibria of the model. Specifically, we address the following question: How much of the dynamics exhibited by two competing species can be inferred from single species data? We provide geometric criteria for stability of various types of equilibria corresponding to non-competitive exclusion, competitive exclusion, and coexistence of two competing species in terms of growth isoclines and consumption curves. A growth isocline is a curve in the plane of substrate concentrations corresponding to the zero net growth of a given species. In [G.T. Reeves, A. Narang, S.S. Pilyugin, Growth of mixed cultures on mixtures of substitutable substrates: The operating diagram for a structured model, J. Theor. Biol. 226 (2004) 143-157], we introduced consumption curves as sets of all possible combinations of substrate concentrations corresponding to balanced growth of a single microbial species. Both types of curves can be obtained in single species experiments.     

Quantitative patterns of development of community of <Emphasis Type="Italic">Pseudomonas aeruginosa</Emphasis> dissociants

The stationary phase of batch culture of Pseudomonas aeruginosa dissociants has been described by a variational model of consumption and growth. The generalized entropy functional was used as the objective function. The model parameters include the requirements of the dissociants for the main nutrients: carbon, nitrogen, and phosphorus. The variational model was used to calculate the limiting regions and microbial community composition during stationary growth for different initial combinations of the resources as a function of the limiting resources. A correspondence between the experimental data and model calculations has been demonstrated. A possibility to control the community structure is discussed.     

微生物生态研究中BIOLOG方法数据分析及R语言实现

微生物生态研究中,对微生物群落结构、群落特征以及其与环境因素的关系的揭示,一直受到广泛关注;适当的数据分析方法有助于更清晰地刻画微生物群落结构特征,明确其与环境因素的关系。结合实例,对微生物生态研究中基于BIOLOG微平板技术的数据分析方法进行梳理,分别介绍数据读取整理、特征指数计算、非限制性排序、限制性排序、聚类分析、环境向量拟合、蒙特尔检验等常用数据操作及生态分析方法;针对不同方法结论,结合研究目标和生态理论给出具有统计学意义的解释,并评价不同方法特点及适用场景;分析过程以R语言实现,并提供全部代码。结果表明,BIOLOG方法产生数据能从多个角度表征微生物群落功能特征,并结合环境指标梯度进行分析;但BIOLOG数据可能不满足正态性分布,在基于正态分布的分析前应提前进行检验;排序分析时应慎用主成分分析,可优先采用其他基于距离矩阵的排序方法;R语言能够简化BIOLOG数据读取及操作,易于完成各类统计分析。本研究能够对微生物生态研究者科学选择应用统计分析方法、提高数据处理效率提供直接参考。     

Ecology theory disentangles microbial dichotomies

Microbes are often discussed in terms of dichotomies such as copiotrophic/oligotrophic and fast/slow-growing microbes, defined using the characterisation of microbial growth in isolated cultures. The dichotomies are usually qualitative and/or study-specific, sometimes precluding clear-cut results interpretation. We can unravel microbial dichotomies as life history strategies by combining ecology theory with Monod curves, a laboratory mathematical tool of bacterial physiology that relates the specific growth rate of a microbe with the concentration of a limiting nutrient. Fitting of Monod curves provides quantities that directly correspond to key parameters in ecological theories addressing species coexistence and diversity, such as r/K selection theory, resource competition and community structure theory and the CSR triangle of life strategies. The resulting model allows us to reconcile the copiotrophic/oligotrophic and fast/slow-growing dichotomies as different subsamples of a life history strategy triangle that also includes r/K strategists. We also used the number of known carbon sources together with community structure theory to partially explain the diversity of heterotrophic microbes observed in metagenomics experiments. In sum, we propose a theoretical framework for the study of natural microbial communities that unifies several existing proposals. Its application would require the integration of metagenomics, metametabolomics, Monod curves and carbon source data.     

Demonstrating the suitability of genetic algorithms for driving microbial ecosystems in desirable directions

The functioning of natural microbial ecosystems is determined by biotic interactions, which are in turn influenced by abiotic environmental conditions. Direct experimental manipulation of such conditions can be used to purposefully drive ecosystems toward exhibiting desirable functions. When a set of environmental conditions can be manipulated to be present at a discrete number of levels, finding the right combination of conditions to obtain the optimal desired effect becomes a typical combinatorial optimisation problem. Genetic algorithms are a class of robust and flexible search and optimisation techniques from the field of computer science that may be very suitable for such a task. To verify this idea, datasets containing growth levels of the total microbial community of four different natural microbial ecosystems in response to all possible combinations of a set of five chemical supplements were obtained. Subsequently, the ability of a genetic algorithm to search this parameter space for combinations of supplements driving the microbial communities to high levels of growth was compared to that of a random search, a local search, and a hill-climbing algorithm, three intuitive alternative optimisation approaches. The results indicate that a genetic algorithm is very suitable for driving microbial ecosystems in desirable directions, which opens opportunities for both fundamental ecological research and industrial applications.     

Understanding evolutionary and ecological dynamics using a continuum limit

Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual‐based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright–Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi‐stationary distribution in a finite population.     

Application of multivariate statistical techniques in microbial ecology

Recent advances in high‐throughput methods of molecular analyses have led to an explosion of studies generating large‐scale ecological data sets. In particular, noticeable effect has been attained in the field of microbial ecology, where new experimental approaches provided in‐depth assessments of the composition, functions and dynamic changes of complex microbial communities. Because even a single high‐throughput experiment produces large amount of data, powerful statistical techniques of multivariate analysis are well suited to analyse and interpret these data sets. Many different multivariate techniques are available, and often it is not clear which method should be applied to a particular data set. In this review, we describe and compare the most widely used multivariate statistical techniques including exploratory, interpretive and discriminatory procedures. We consider several important limitations and assumptions of these methods, and we present examples of how these approaches have been utilized in recent studies to provide insight into the ecology of the microbial world. Finally, we offer suggestions for the selection of appropriate methods based on the research question and data set structure.     

Novel model‐based clustering reveals ecologically differentiated bacterial genomes across a large climate gradient

A pervasive challenge in microbial ecology is understanding the genetic level where ecological units can be differentiated. Ecological differentiation often occurs at fine genomic levels, yet it is unclear how to utilise ecological information to define ecotypes given the breadth of environmental variation among microbial taxa. Here, we present an analytical framework that infers clusters along genome‐based microbial phylogenies according to shared environmental responses. The advantage of our approach is the ability to identify genomic clusters that best fit complex environmental information whilst characterising cluster niches through model predictions. We apply our method to determine climate‐associated ecotypes in populations of nitrogen‐fixing symbionts using whole genomes, explicitly sampled to detect climate differentiation across a heterogeneous landscape. Although soil and plant host characteristics strongly influence distribution patterns of inferred ecotypes, our flexible statistical method enabled us to identify climate‐associated genomic clusters using environmental data, providing solid support for ecological specialisation in soil symbionts.     

Scaling up: how do exogenous fluctuations in individual-based resource competition models re-emerge in aggregated stochastic population models?

In applied population dynamics the choice of stochastic per capita growth function has implications for population viability analyses, management recommendations, and pest control. This model choice is often based on statistical criteria, mathematical tractability or personal preferences, and general ecological guidelines are either too vague or entirely missing. To identify such guidelines, it is important to understand how exogenous and endogenous factors interact at the individual level and re-emerge at the aggregated population level. We therefore study different types of resource competition (contest vs. scramble competition) and different types of exogenous fluctuations (food and weather fluctuations) at the individual level in a simple individual-based simulation model. We statistically fit the resulting time series to find out (1) which functional form of the growth function (‘hyperbolic’ or ‘exponential’) better describes contest and scramble competition and (2) whether the pattern of population fluctuations resulting from the simulations can be assigned to vertical, lateral or nonlinear perturbations in the stochastic growth function (a classification scheme suggested by Royama 1992, Analytical Population Dynamics, Chapman and Hall, London). We found that the same type of competition can result in ‘hyperbolic’ or ‘exponential’ functional forms, depending on the type of exogenous fluctuations. So it is the interplay between exogenous variability and endogenous resource competition that affects model performance. In contrast to the widespread assumption of vertical (additive) perturbations, our findings highlight the importance of (non-additive) lateral and nonlinear perturbations and their combinations with vertical perturbations. The choice of the stochastic growth function should therefore consider not only statistical criteria but also ecological guidelines. We derive such ecological guidelines from our analysis.     

The use of automated tubidimetric data for the construction of kinetic models

Summary An automated tubidimetric instrument (Bioscreen) was used to observe the growth response ofListeria monocytogenes to combinations of temperature (15–30°C), hydrogen-ion (0.1–21.9 m) (equivalent pH 4.66–7.0) and NaCl concentration (0.5–9.5% w/v). Compared to traditional plate count techniques, the technique allowed many more data points to be captured and replicates to be used, with less expenditure of effort. Optical density curves were filtered (smoothed) to minimize the effect of signal noise and the mean signal from uninoculated wells was subtracted to minimize the effect of signal draft. A novel procedure for fitting growth curves to optical density data has been developed. The procedure involves the use of the logistic function and a calibration equation for fitting, in a single step, in the dimension of optical density. This approach allowed the four parameters of the logistic equation to be derived at each set of experimental conditions. A quadratic response surface was then fitted to the curve parameters using temperature, NaCl and hydrogen-ion concentration as three independent variables. Predicted time to 1000-fold increase in cell numbers compared well to predictions from predictive microbial growth equations generated in other laboratories using traditional plate counting. We propose that this technique should be further evaluated as a method for generating data for modeling the kinetics of microbial growth.Mention of brand or firm names does not constitute an endorsement by the US Department of Agriculture over others of a similar nature not mentioned.     

Web of ecological interactions in an experimental gut microbiota

The dynamics of all ecosystems are dictated by intrinsic, density‐dependent mechanisms and by density‐independent environmental forcing. In spite of the importance of the gastrointestinal microbiota in health and disease, the ecology of this system remains largely unknown. Here, we take an ecological approach to gut microbial community analysis, with statistical modelling of time series data from chemostats. This approach removes effects of host forcing, allowing us to describe a network of intrinsic interactions determining the dynamic structure of an experimental gut microbiota. Surprisingly, the main colonization pattern in this simplified model system resembled that of the human infant gut, suggesting a potentially important role of density‐dependent interactions in the early gut microbiota. Knowledge of ecological structures in microbial systems may provide us with a means of controlling such systems by modifying the strength and nature of interactions among microbes and between the microbes and their environment.     

The need for stochastic replication of ecological neural networks

Artificial neural networks are becoming increasingly popular as predictive statistical tools in ecosystem ecology and as models of signal processing in behavioural and evolutionary ecology. We demonstrate here that a commonly used network in ecology, the three-layer feed-forward network, trained with the backpropagation algorithm, can be extremely sensitive to the stochastic variation in training data that results from random sampling of the same underlying statistical distribution, with networks converging to several distinct predictive states. Using a random walk procedure to sample error-weight space, and Sammon dimensional reduction of weight arrays, we demonstrate that these different predictive states are not artefactual, due to local minima, but lie at the base of major error troughs in the error-weight surface. We further demonstrate that various gross weight compositions can produce the same predictive state, suggesting the analogy of weight space as a 'patchwork' of multiple predictive states. Our results argue for increased inclusion of stochastic training replication and analysis into ecological and behavioural applications of artificial neural networks.     

Combining mechanism and drift in community ecology: a novel statistical mechanics approach

A key challenge for models of community ecology is to combine deterministic mechanism and stochastic drift in a systematic, transparent and tractable manner. Another challenge is to explain and unify different ecological patterns, hitherto modelled in isolation, within a single modelling framework. Here, we show that statistical mechanics provides an effective way to meet both challenges. We apply the statistical principle of maximum entropy (MaxEnt) to a simple resource-based, non-neutral model of a plant community. In contrast to previous ecological applications of MaxEnt, our use of MaxEnt emphasises its theoretical basis in the combinatorics of sampling frequencies, an approach that clarifies its ecological interpretation. In this approach, mechanism and drift are identified, respectively, with ecological resource constraints and entropy maximization. We obtain realistic predictions for species abundance distributions as well as contrasting stability-diversity relationships at community and population levels. The model also predicts critical behaviour that may provide a basis for understanding desertification and other ecological tipping points. Our results complement and extend previous ecological applications of MaxEnt to new areas of community ecology, and further illustrate MaxEnt as a powerful yet simple modelling tool for combining mechanism and drift in a way that unifies disparate ecological patterns.