Mathematical modelling of carbohydrate degradation by human colonic microbiota

The human colon is an anaerobic ecosystem that remains largely unexplored as a result of its limited accessibility and its complexity. Mathematical models can play a central role for a better insight into its dynamics. In this context, this paper presents the development of a mathematical model of carbohydrate degradation. Our aim was to provide an in silico approach to contribute to a better understanding of the fermentation patterns in such an ecosystem. Our mathematical model is knowledge-based, derived by writing down mass-balance equations. It incorporates physiology of the intestine, metabolic reactions and transport phenomena. The model was used to study various nutritional scenarios and to assess the role of the mucus on the system behavior. Model simulations provided an adequate qualitative representation of the human colon. Our model is complementary to experimental studies on human colonic fermentation, which, of course, is not meant to replace. It may be helpful to gain insight on questions that are still difficult to elucidate by experimentation and suggest future experiments.     

Feasibility constraints on the elastic expansion of ecosystem models

Stable n-species ecosystem models may be expanded into larger (n + 1)-species stable models when an invading species is introduced. Such invasions are referred to as being successful due to the elasticity of the original community. Elasticity is dependent upon the interaction terms of both the original community members and the invading species. Feasibility constraints for elasticity and inelasticity are presented here for these terms in the context of a generalized ecosystem model where invasion causes only minor displacements in equilibrium population densities.     

Hybrid dynamic modeling of Escherichia coli central metabolic network combining Michaelis–Menten and approximate kinetic equations

The construction of dynamic metabolic models at reaction network level requires the use of mechanistic enzymatic rate equations that comprise a large number of parameters. The lack of knowledge on these equations and the difficulty in the experimental identification of their associated parameters, represent nowadays the limiting factor in the construction of such models. In this study, we compare four alternative modeling approaches based on Michaelis–Menten kinetics for the bi-molecular reactions and different types of simplified rate equations for the remaining reactions (generalized mass action, convenience kinetics, lin-log and power-law). Using the mechanistic model for Escherichia coli central carbon metabolism as a benchmark, we investigate the alternative modeling approaches through comparative simulations analyses. The good dynamic behavior and the powerful predictive capabilities obtained using the hybrid model composed of Michaelis–Menten and the approximate lin-log kinetics indicate that this is a possible suitable approach to model complex large-scale networks where the exact rate laws are unknown.     

A declarative constraint-based method for analyzing discrete genetic regulatory networks

Dynamical modeling has proven useful for understanding how complex biological processes emerge from the many components and interactions composing genetic regulatory networks (GRNs). However, the development of models is hampered by large uncertainties in both the network structure and parameter values. To remedy this problem, the models are usually developed through an iterative process based on numerous simulations, confronting model predictions with experimental data and refining the model structure and/or parameter values to repair the inconsistencies. In this paper, we propose an alternative to this generate-and-test approach. We present a four-step method for the systematic construction and analysis of discrete models of GRNs by means of a declarative approach. Instead of instantiating the models as in classical modeling approaches, the biological knowledge on the network structure and its dynamics is formulated in the form of constraints. The compatibility of the network structure with the constraints is queried and in case of inconsistencies, some constraints are relaxed. Common properties of the consistent models are then analyzed by means of dedicated languages. Two such languages are introduced in the paper. Removing questionable constraints or adding interesting ones allows to further analyze the models. This approach allows to identify the best experiments to be carried out, in order to discriminate sets of consistent models and refine our knowledge on the system functioning. We test the feasibility of our approach, by applying it to the re-examination of a model describing the nutritional stress response in the bacterium Escherichia coli.     

Incorrect Likelihood Methods Were Used to Infer Scaling Laws of Marine Predator Search Behaviour

Background

Ecologists are collecting extensive data concerning movements of animals in marine ecosystems. Such data need to be analysed with valid statistical methods to yield meaningful conclusions.

Principal Findings

We demonstrate methodological issues in two recent studies that reached similar conclusions concerning movements of marine animals (Nature 451∶1098; Science 332∶1551). The first study analysed vertical movement data to conclude that diverse marine predators (Atlantic cod, basking sharks, bigeye tuna, leatherback turtles and Magellanic penguins) exhibited “Lévy-walk-like behaviour”, close to a hypothesised optimal foraging strategy. By reproducing the original results for the bigeye tuna data, we show that the likelihood of tested models was calculated from residuals of regression fits (an incorrect method), rather than from the likelihood equations of the actual probability distributions being tested. This resulted in erroneous Akaike Information Criteria, and the testing of models that do not correspond to valid probability distributions. We demonstrate how this led to overwhelming support for a model that has no biological justification and that is statistically spurious because its probability density function goes negative. Re-analysis of the bigeye tuna data, using standard likelihood methods, overturns the original result and conclusion for that data set. The second study observed Lévy walk movement patterns by mussels. We demonstrate several issues concerning the likelihood calculations (including the aforementioned residuals issue). Re-analysis of the data rejects the original Lévy walk conclusion.

Conclusions

We consequently question the claimed existence of scaling laws of the search behaviour of marine predators and mussels, since such conclusions were reached using incorrect methods. We discourage the suggested potential use of “Lévy-like walks” when modelling consequences of fishing and climate change, and caution that any resulting advice to managers of marine ecosystems would be problematic. For reproducibility and future work we provide R source code for all calculations.     

Avian community assembly rules: The foliage-gleaning guild

Summary The theory of animal community organization has been dominated by general models based on the Lotka-Volterra equations. The predictions of these models are difficult to test in particular situations. Moreover, a great deal of ecological information is incommensurate with the data requirements of these models. A different approach to community organization addresses the ecosystem assembly problem. This problem is defined to be that of constructing an algorithm which assembles a subset of a species pool in a specified environment.A model of ecosystem assembly, based on generative grammars as used in theoretical linguistics, is described. It was constructed from and validated with data collected by D.H. Morse on a guild of foliage-gleaning birds inhabiting spruce forests on islands off the coast of Maine. The data were divided into two groups. One group, from the years 1967–1970, was used for model construction; the second group, from 1971–1975, was used to validate the model.The model has two major components. One component inserts species onto islands according to the microhabitat used by each species and the resources available on each island. A second component deletes those inserted species from islands on which they were not observed to occur during 1967–1970. This component is composed of deletion rules that remove species depending on (a) their sizes and resource requirements, (b) the sizes and resource requirements of other species present in the ecosystem, and (c) the structure of the vegetation on the islands. Model validation was performed by comparing the predicted distributions of species against observed distributions not used in model construction. Model accuracy for the later data (1971–1975) was slightly higher than for the earlier data (1967–1970), approximately 88% and 84%, respectively.The behavior of the model was investigated with several simulations. These included the effects of the removal of certain deletion rules and the effects of the application of the rules without regard to their order. Other simulations demonstrated the application of the model to the prediction of the effects of habitat manipulation and the removal of particular species from the species pool.     

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.     

Robust Forward Simulations of Recurrent Hitchhiking

Evolutionary forces shape patterns of genetic diversity within populations and contribute to phenotypic variation. In particular, recurrent positive selection has attracted significant interest in both theoretical and empirical studies. However, most existing theoretical models of recurrent positive selection cannot easily incorporate realistic confounding effects such as interference between selected sites, arbitrary selection schemes, and complicated demographic processes. It is possible to quantify the effects of arbitrarily complex evolutionary models by performing forward population genetic simulations, but forward simulations can be computationally prohibitive for large population sizes (>10⁵). A common approach for overcoming these computational limitations is rescaling of the most computationally expensive parameters, especially population size. Here, we show that ad hoc approaches to parameter rescaling under the recurrent hitchhiking model do not always provide sufficiently accurate dynamics, potentially skewing patterns of diversity in simulated DNA sequences. We derive an extension of the recurrent hitchhiking model that is appropriate for strong selection in small population sizes and use it to develop a method for parameter rescaling that provides the best possible computational performance for a given error tolerance. We perform a detailed theoretical analysis of the robustness of rescaling across the parameter space. Finally, we apply our rescaling algorithms to parameters that were previously inferred for Drosophila and discuss practical considerations such as interference between selected sites.     

Muscle Synergies Heavily Influence the Neural Control of Arm Endpoint Stiffness and Energy Consumption

Much debate has arisen from research on muscle synergies with respect to both limb impedance control and energy consumption. Studies of limb impedance control in the context of reaching movements and postural tasks have produced divergent findings, and this study explores whether the use of synergies by the central nervous system (CNS) can resolve these findings and also provide insights on mechanisms of energy consumption. In this study, we phrase these debates at the conceptual level of interactions between neural degrees of freedom and tasks constraints. This allows us to examine the ability of experimentally-observed synergies—correlated muscle activations—to control both energy consumption and the stiffness component of limb endpoint impedance. In our nominal 6-muscle planar arm model, muscle synergies and the desired size, shape, and orientation of endpoint stiffness ellipses, are expressed as linear constraints that define the set of feasible muscle activation patterns. Quadratic programming allows us to predict whether and how energy consumption can be minimized throughout the workspace of the limb given those linear constraints. We show that the presence of synergies drastically decreases the ability of the CNS to vary the properties of the endpoint stiffness and can even preclude the ability to minimize energy. Furthermore, the capacity to minimize energy consumption—when available—can be greatly affected by arm posture. Our computational approach helps reconcile divergent findings and conclusions about task-specific regulation of endpoint stiffness and energy consumption in the context of synergies. But more generally, these results provide further evidence that the benefits and disadvantages of muscle synergies go hand-in-hand with the structure of feasible muscle activation patterns afforded by the mechanics of the limb and task constraints. These insights will help design experiments to elucidate the interplay between synergies and the mechanisms of learning, plasticity, versatility and pathology in neuromuscular systems.     

Scalable Nanohelices for Predictive Studies and Enhanced 3D Visualization

Spring-like materials are ubiquitous in nature and of interest in nanotechnology for energy harvesting, hydrogen storage, and biological sensing applications.  For predictive simulations, it has become increasingly important to be able to model the structure of nanohelices accurately.  To study the effect of local structure on the properties of these complex geometries one must develop realistic models.  To date, software packages are rather limited in creating atomistic helical models.  This work focuses on producing atomistic models of silica glass (SiO₂) nanoribbons and nanosprings for molecular dynamics (MD) simulations. Using an MD model of “bulk” silica glass, two computational procedures to precisely create the shape of nanoribbons and nanosprings are presented.  The first method employs the AWK programming language and open-source software to effectively carve various shapes of silica nanoribbons from the initial bulk model, using desired dimensions and parametric equations to define a helix.  With this method, accurate atomistic silica nanoribbons can be generated for a range of pitch values and dimensions.  The second method involves a more robust code which allows flexibility in modeling nanohelical structures.  This approach utilizes a C++ code particularly written to implement pre-screening methods as well as the mathematical equations for a helix, resulting in greater precision and efficiency when creating nanospring models.  Using these codes, well-defined and scalable nanoribbons and nanosprings suited for atomistic simulations can be effectively created.  An added value in both open-source codes is that they can be adapted to reproduce different helical structures, independent of material.  In addition, a MATLAB graphical user interface (GUI) is used to enhance learning through visualization and interaction for a general user with the atomistic helical structures.  One application of these methods is the recent study of nanohelices via MD simulations for mechanical energy harvesting purposes.     

Representing Variable Habitat Quality in a Spatial Food Web Model

Why are marine species where they are? The scientific community is faced with an urgent need to understand aquatic ecosystem dynamics in the context of global change. This requires development of scientific tools with the capability to predict how biodiversity, natural resources, and ecosystem services will change in response to stressors such as climate change and further expansion of fishing. Species distribution models and ecosystem models are two methodologies that are being developed to further this understanding. To date, these methodologies offer limited capabilities to work jointly to produce integrated assessments that take both food web dynamics and spatial-temporal environmental variability into account. We here present a new habitat capacity model as an implementation of the spatial-temporal model Ecospace of the Ecopath with Ecosim approach. The new model offers the ability to drive foraging capacity of species from the cumulative impacts of multiple physical, oceanographic, and environmental factors such as depth, bottom type, temperature, salinity, oxygen concentrations, and so on. We use a simulation modeling procedure to evaluate sampling characteristics of the new habitat capacity model. This development bridges the gap between envelope environmental models and classic ecosystem food web models, progressing toward the ability to predict changes in marine ecosystems under scenarios of global change and explicitly taking food web direct and indirect interactions into account.     

Parameter optimization by using differential elimination: a general approach for introducing constraints into objective functions

Background

The investigation of network dynamics is a major issue in systems and synthetic biology. One of the essential steps in a dynamics investigation is the parameter estimation in the model that expresses biological phenomena. Indeed, various techniques for parameter optimization have been devised and implemented in both free and commercial software. While the computational time for parameter estimation has been greatly reduced, due to improvements in calculation algorithms and the advent of high performance computers, the accuracy of parameter estimation has not been addressed.

Results

We propose a new approach for parameter optimization by using differential elimination, to estimate kinetic parameter values with a high degree of accuracy. First, we utilize differential elimination, which is an algebraic approach for rewriting a system of differential equations into another equivalent system, to derive the constraints between kinetic parameters from differential equations. Second, we estimate the kinetic parameters introducing these constraints into an objective function, in addition to the error function of the square difference between the measured and estimated data, in the standard parameter optimization method. To evaluate the ability of our method, we performed a simulation study by using the objective function with and without the newly developed constraints: the parameters in two models of linear and non-linear equations, under the assumption that only one molecule in each model can be measured, were estimated by using a genetic algorithm (GA) and particle swarm optimization (PSO). As a result, the introduction of new constraints was dramatically effective: the GA and PSO with new constraints could successfully estimate the kinetic parameters in the simulated models, with a high degree of accuracy, while the conventional GA and PSO methods without them frequently failed.

Conclusions

The introduction of new constraints in an objective function by using differential elimination resulted in the drastic improvement of the estimation accuracy in parameter optimization methods. The performance of our approach was illustrated by simulations of the parameter optimization for two models of linear and non-linear equations, which included unmeasured molecules, by two types of optimization techniques. As a result, our method is a promising development in parameter optimization.

     

Evolutionary Sparse Learning for Phylogenomics

We introduce a supervised machine learning approach with sparsity constraints for phylogenomics, referred to as evolutionary sparse learning (ESL). ESL builds models with genomic loci—such as genes, proteins, genomic segments, and positions—as parameters. Using the Least Absolute Shrinkage and Selection Operator, ESL selects only the most important genomic loci to explain a given phylogenetic hypothesis or presence/absence of a trait. ESL models do not directly involve conventional parameters such as rates of substitutions between nucleotides, rate variation among positions, and phylogeny branch lengths. Instead, ESL directly employs the concordance of variation across sequences in an alignment with the evolutionary hypothesis of interest. ESL provides a natural way to combine different molecular and nonmolecular data types and incorporate biological and functional annotations of genomic loci in model building. We propose positional, gene, function, and hypothesis sparsity scores, illustrate their use through an example, and suggest several applications of ESL. The ESL framework has the potential to drive the development of a new class of computational methods that will complement traditional approaches in evolutionary genomics, particularly for identifying influential loci and sequences given a phylogeny and building models to test hypotheses. ESL’s fast computational times and small memory footprint will also help democratize big data analytics and improve scientific rigor in phylogenomics.     

Estimating the effects of non‐native species on nutrient recycling using species‐specific and general allometric models

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Emergent Global Patterns of Ecosystem Structure and Function from a Mechanistic General Ecosystem Model

Anthropogenic activities are causing widespread degradation of ecosystems worldwide, threatening the ecosystem services upon which all human life depends. Improved understanding of this degradation is urgently needed to improve avoidance and mitigation measures. One tool to assist these efforts is predictive models of ecosystem structure and function that are mechanistic: based on fundamental ecological principles. Here we present the first mechanistic General Ecosystem Model (GEM) of ecosystem structure and function that is both global and applies in all terrestrial and marine environments. Functional forms and parameter values were derived from the theoretical and empirical literature where possible. Simulations of the fate of all organisms with body masses between 10 µg and 150,000 kg (a range of 14 orders of magnitude) across the globe led to emergent properties at individual (e.g., growth rate), community (e.g., biomass turnover rates), ecosystem (e.g., trophic pyramids), and macroecological scales (e.g., global patterns of trophic structure) that are in general agreement with current data and theory. These properties emerged from our encoding of the biology of, and interactions among, individual organisms without any direct constraints on the properties themselves. Our results indicate that ecologists have gathered sufficient information to begin to build realistic, global, and mechanistic models of ecosystems, capable of predicting a diverse range of ecosystem properties and their response to human pressures.     

A method for lake ecosystem health assessment: an Ecological Modeling Method (EMM) and its application

Ecosystem health is a newly proposed concept that sets new goals for environmental management. Its definition, indexing and assessment methods are still being perfected. An Ecological Modeling Method (EMM) for lake ecosystem health assessment is proposed in this paper. The EMM's procedures are: (1) to analyze the ecosystem structure of a lake in order to determine the structure and complexity of the lake's ecological model; (2) to develop a model having ecological health indicators, by designing a conceptual diagram, establishing model equations, estimating model parameters and being integrated with ecological indicators; (3) to compare the simulated and observed values of important state variables and process rates (i.e. model calibration) in order to evaluate the applicability of the model to lake ecosystem health assessment; (4) to calculate ecosystem health indicators based on the developed model; and (5) to assess lake ecosystem health according to the values of the ecosystem health indicators. The EMM was applied, as a case study, to the ecosystem health assessment of a eutrophic Chinese lake (Lake Chao) between April 1987 and March 1988. A relative order of health states from poor to good was determined as follows: August–October 1987 > April–May 1987 > June–July 1987 > November–December 1987 > January–March 1988. These results compared quite favourably with the actual current conditions at Lake Chao. The EMM method, therefore, was suitable in assessing lake ecosystem health at Lake Chao.     

Estimating diurnal to annual ecosystem parameters by synthesis of a carbon flux model with eddy covariance net ecosystem exchange observations

We performed a synthetic analysis of Harvard Forest net ecosystem exchange of CO₂ (NEE) time series and a simple ecosystem carbon flux model, the simplified Photosynthesis and Evapo‐Transpiration model (SIPNET). SIPNET runs at a half‐daily time step, and has two vegetation carbon pools, a single aggregated soil carbon pool, and a simple soil moisture sub‐model. We used a stochastic Bayesian parameter estimation technique that provided posterior distributions of the model parameters, conditioned on the observed fluxes and the model equations. In this analysis, we estimated the values of all quantities that govern model behavior, including both rate constants and initial conditions for carbon pools. The purpose of this analysis was not to calibrate the model to make predictions about future fluxes but rather to understand how much information about process controls can be derived directly from the NEE observations. A wavelet decomposition enabled us to assess model performance at multiple time scales from diurnal to decadal. The model parameters are most highly constrained by eddy flux data at daily to seasonal time scales, suggesting that this approach is not useful for calculating annual integrals. However, the ability of the model to fit both the diurnal and seasonal variability patterns in the data simultaneously, using the same parameter set, indicates the effectiveness of this parameter estimation method. Our results quantify the extent to which the eddy covariance data contain information about the ecosystem process parameters represented in the model, and suggest several next steps in model development and observations for improved synthesis of models with flux observations.     

Modeling Imatinib-Treated Chronic Myelogenous Leukemia: Reducing the Complexity of Agent-Based Models

We develop a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia. Our model is based on replacing the recent agent-based model of Roeder et al. (Nat. Med. 12(10):1181–1184, 2006) by a system of deterministic difference equations. These difference equations describe the time-evolution of clusters of individual agents that are grouped by discretizing the state space. Hence, unlike standard agent-base models, the complexity of our model is independent of the number of agents, which allows to conduct simulation studies with a realistic number of cells. This approach also allows to directly evaluate the expected steady states of the system. The results of our numerical simulations show that our model replicates the averaged behavior of the original Roeder model with a significantly reduced computational cost. Our general approach can be used to simplify other similar agent-based models. In particular, due to the reduced computational complexity of our technique, one can use it to conduct sensitivity studies of the parameters in large agent-based systems.