What Can We Learn from Global Sensitivity Analysis of Biochemical Systems?

Most biological models of intermediate size, and probably all large models, need to cope with the fact that many of their parameter values are unknown. In addition, it may not be possible to identify these values unambiguously on the basis of experimental data. This poses the question how reliable predictions made using such models are. Sensitivity analysis is commonly used to measure the impact of each model parameter on its variables. However, the results of such analyses can be dependent on an exact set of parameter values due to nonlinearity. To mitigate this problem, global sensitivity analysis techniques are used to calculate parameter sensitivities in a wider parameter space. We applied global sensitivity analysis to a selection of five signalling and metabolic models, several of which incorporate experimentally well-determined parameters. Assuming these models represent physiological reality, we explored how the results could change under increasing amounts of parameter uncertainty. Our results show that parameter sensitivities calculated with the physiological parameter values are not necessarily the most frequently observed under random sampling, even in a small interval around the physiological values. Often multimodal distributions were observed. Unsurprisingly, the range of possible sensitivity coefficient values increased with the level of parameter uncertainty, though the amount of parameter uncertainty at which the pattern of control was able to change differed among the models analysed. We suggest that this level of uncertainty can be used as a global measure of model robustness. Finally a comparison of different global sensitivity analysis techniques shows that, if high-throughput computing resources are available, then random sampling may actually be the most suitable technique.     

The uncertainty in simulations by a Global Biome Model (BIOME3) to alternative parameter values

The sensitivity of a global biome model (BIOME3) to uncertainty in parameter values was investigated by testing the model's sensitivity to minimum and maximum parameter values obtained from an extensive literature search. Simulations were conducted replacing the default parameter value by each of the maximum and minimum values determined from the literature. In doing so, the aim was to identify those parameters where the use of an alternate (observed) value leads to a significant change in the simulation of plant functional types at a global scale, in order to identify those which are functionally important to the model. BIOME3 was found to be insensitive to changes in the majority of its parameters, providing a generally sound foundation for confidence in model simulations. However, there was considerable sensitivity shown to over a quarter of the parameters. Three main types of parameters led to a change in plant functional types distribution relative to the control simulation: (i) parameters affecting the photosynthesis parameterization; (ii) parameters affecting the evapotranspiration parameterization; and (iii) root distribution which affected both parts of the model. The main causes of sensitivity were changes in the photosynthesis parameters leading to differential changes in plant functional type's net primary productivity. This caused a shift in the competitive balance between specific plant functional types or between C3 and C4 plant types, and a consequent change in their global distribution. Changes to the evapotranspiration parameters and root distribution similarly affected net primary productivity and soil moisture, and often led to shifts in the competitive balance between grass and trees. Changes in the value for several poorly known parameters produced substantial changes in the distribution of plant functional types, and reduced the κ‐statistic to a large degree, indicating areas of potential uncertainty in the model. This suggests that great care must be taken in prescribing values to these parameters and provides guidance on which parameters need further attention in observational work.     

The effect of time-varying mortality and carbon assimilation on models of carbon allocation in annual plants

Summary Models of optimal carbon allocation schedules have influenced the way plant ecologists think about life history evolution, particularly for annual plants. The present study asks (1) how, within the framework of these models, are their predictions affected by within-season variation in mortality and carbon assimilation rates?; and (2) what are the consequences of these prediction changes for empirical tests of the models? A companion paper examines the basic assumptions of the models themselves. I conducted a series of numerical experiments with a simple carbon allocation model. Results suggest that both qualitative and quantitative predictions can sometimes be sensitive to parameter values for net assimilation rate and mortality: for some parameter values, both the time and size at onset of reproduction, as well as the number of reproductive intervals, vary considerably as a result of small variations in these parameters. For other parameter values, small variations in the parameters result in only small changes in predicted phenotype, but these have very large fitness consequences. Satisfactory empirical tests are thus likely to require much accuracy in parameter estimates. The effort required for parameter estimation imposes a practical constraint on empirical tests, making large multipopulation comparisons impractical. It may be most practical to compare the predicted and observed fitness consequences of variation in the timing of onset of reproduction.     

Bias and variance in model results associated with spatial scaling of measurements for parameterization in regional assessments

Models are central to global change analyses, but they are often parameterized using data that represent only a portion of heterogeneity in a region. This creates uncertainty in the results and constrains the reliability of model inferences. Our objective was to evaluate the uncertainty associated with differential scaling of parameterization data to model soil organic carbon stock changes as a function of US agricultural land use and management. Specifically, we compared analyses in which model parameters were derived from field experimental data that were scaled to the entire US vs. the same data scaled to climate regions within the country. We evaluated the effect of differential scaling on both bias and variance in model results. Model results had less variance by scaling data to the entire country because of a larger sample size for deriving individual parameter values, although there was a relatively large bias associated with this parameterization, estimated at 2.7 Tg C yr^?1. Even with the large bias, resulting confidence intervals from the two parameterizations had considerable overlap for the estimated national rate of SOC change (i.e. 77% overlap in those intervals). Consequently, the results were relatively similar when focusing on the uncertainty rather than solely on the mean estimate. In contrast, large biases created less overlap in confidence intervals for the change rates within individual climate regions, compared with the national estimates. For example, the overlap in resulting intervals from the two parameterizations was only 32% for the warm temperate moist region, with a corresponding bias of 3.1 Tg C yr^?1. These findings demonstrate that there is a greater risk of making erroneous inferences because of large biases if models are parameterized with broader scale information, such as an entire country, and then used to address impacts at a finer spatial scale, such as sub‐regions within a country. In addition, the study demonstrates a trade‐off between variance and bias in model results that depends on the scaling of data for model parameterization.     

Evaluating soil microbial carbon use efficiency explicitly as a function of cellular processes: implications for measurements and models

Carbon use efficiency (CUE), the proportion of carbon (C) consumed by microbes that is converted into biomass, is an important parameter for soil C models with explicit microbial controls. While often considered as a single parameter, CUE is an emergent property of multiple microbial processes, including assimilation efficiency, biomass-specific respiration, enzyme production, and respiratory costs of enzyme production. These processes occur over variable time scales and imply different fates for C, and the same emergent CUE value can result when C is allocated in fundamentally different ways (e.g. a high investment in enzyme production vs. a high assimilation cost). We developed a model that represents the individual processes underlying emergent CUE to test how shifts in microbial allocation alter equilibrium soil C pool sizes. We found that an increase in emergent CUE that results from a change in assimilation efficiency, biomass specific respiration, or respiration costs from enzyme production causes soil organic C (SOC) to decline, while the same change in emergent CUE resulting from a change in enzyme production causes SOC to increase. We also used the model to test the sensitivity of CUE from isotopic C tracer estimates to changes in microbial allocation processes. We found that these estimates do not account for the same microbial processes represented by emergent CUE in models. We propose that considering microbial processes explicitly rather than representing CUE as a single parameter can improve data-model integration. In addition, modeling microbial processes explicitly will account for a wider range of possible outcomes from shifts in microbial C allocation, such as when increased SOC results from increasing CUE.     

Properties of cell death models calibrated and compared using Bayesian approaches

Using models to simulate and analyze biological networks requires principled approaches to parameter estimation and model discrimination. We use Bayesian and Monte Carlo methods to recover the full probability distributions of free parameters (initial protein concentrations and rate constants) for mass‐action models of receptor‐mediated cell death. The width of the individual parameter distributions is largely determined by non‐identifiability but covariation among parameters, even those that are poorly determined, encodes essential information. Knowledge of joint parameter distributions makes it possible to compute the uncertainty of model‐based predictions whereas ignoring it (e.g., by treating parameters as a simple list of values and variances) yields nonsensical predictions. Computing the Bayes factor from joint distributions yields the odds ratio (～20‐fold) for competing ‘direct’ and ‘indirect’ apoptosis models having different numbers of parameters. Our results illustrate how Bayesian approaches to model calibration and discrimination combined with single‐cell data represent a generally useful and rigorous approach to discriminate between competing hypotheses in the face of parametric and topological uncertainty.     

水文过程及其尺度响应

水文循环机制涉及到各个尺度界面的水文过程关系.水文变量(降雨、径流等)随时间和空间变化很大,是与尺度响应的非线性过程.尺度转换是建立合适的参数去描述这些过程, 涉及到如何穿越不同尺度约束体系(水文过程)的限制.基于对水文响应及其尺度转换认识所建立起来的现行水文模型的主流是还原分析方法,采用小尺度演绎方法,过度参数化成为习以常规,导致无资料流域预测结果的很大不确定性.因此,尺度转换是水文过程研究所面临的主要难点和热点问题.水文模型只有建立在对尺度相关的水文过程深刻理解基础上,对于预测水文响应才是有效的.预测径流在时间和空间的结果是根据径流系统穿越尺度过程的动态分析得出的,把尺度分析转变于水文变量的谱分析,即通过水文变量的谱相分析,认识发生在不同尺度范围内潜在的秩序和规律,研究他们形成的机制,寻求决定水文过程规则的通用表达方式,开发尺度转换的方法.     

Global net biome CO2 exchange predicted comparably well using parameter–environment relationships and plant functional types

Accurate estimation and forecasts of net biome CO₂ exchange (NBE) are vital for understanding the role of terrestrial ecosystems in a changing climate. Prior efforts to improve NBE predictions have predominantly focused on increasing models' structural realism (and thus complexity), but parametric error and uncertainty are also key determinants of model skill. Here, we investigate how different parameterization assumptions propagate into NBE prediction errors across the globe, pitting the traditional plant functional type (PFT)-based approach against a novel top-down, machine learning-based “environmental filtering” (EF) approach. To do so, we simulate these contrasting methods for parameter assignment within a flexible model–data fusion framework of the terrestrial carbon cycle (CARDAMOM) at a global scale. In the PFT-based approach, model parameters from a small number of select locations are applied uniformly within regions sharing similar land cover characteristics. In the EF-based approach, a pixel's parameters are predicted based on underlying relationships with climate, soil, and canopy properties. To isolate the role of parametric from structural uncertainty in our analysis, we benchmark the resulting PFT-based and EF-based NBE predictions with estimates from CARDAMOM's Bayesian optimization approach (whereby “true” parameters consistent with a suite of data constraints are retrieved on a pixel-by-pixel basis). When considering the mean absolute error of NBE predictions across time, we find that the EF-based approach matches or outperforms the PFT-based approach at 55% of pixels—a narrow majority. However, NBE estimates from the EF-based approach are susceptible to compensation between errors in component flux predictions and predicted parameters can align poorly with the assumed “true” values. Overall, though, the EF-based approach is comparable to conventional approaches and merits further investigation to better understand and resolve these limitations. This work provides insight into the relationship between terrestrial biosphere model performance and parametric uncertainty, informing efforts to improve model parameterization via PFT-free and trait-based approaches.     

Deriving fine-scale models of human mobility from aggregated origin-destination flow data

The spatial dynamics of epidemics are fundamentally affected by patterns of human mobility. Mobile phone call detail records (CDRs) are a rich source of mobility data, and allow semi-mechanistic models of movement to be parameterised even for resource-poor settings. While the gravity model typically reproduces human movement reasonably well at the administrative level spatial scale, past studies suggest that parameter estimates vary with the level of spatial discretisation at which models are fitted. Given that privacy concerns usually preclude public release of very fine-scale movement data, such variation would be problematic for individual-based simulations of epidemic spread parametrised at a fine spatial scale. We therefore present new methods to fit fine-scale mathematical mobility models (here we implement variants of the gravity and radiation models) to spatially aggregated movement data and investigate how model parameter estimates vary with spatial resolution. We use gridded population data at 1km resolution to derive population counts at different spatial scales (down to ∼ 5km grids) and implement mobility models at each scale. Parameters are estimated from administrative-level flow data between overnight locations in Kenya and Namibia derived from CDRs: where the model spatial resolution exceeds that of the mobility data, we compare the flow data between a particular origin and destination with the sum of all model flows between cells that lie within those particular origin and destination administrative units. Clear evidence of over-dispersion supports the use of negative binomial instead of Poisson likelihood for count data with high values. Radiation models use fewer parameters than the gravity model and better predict trips between overnight locations for both considered countries. Results show that estimates for some parameters change between countries and with spatial resolution and highlight how imperfect flow data and spatial population distribution can influence model fit.     

The importance of assessing parameter sensitivity when using biophysical models: a case study using plethodontid salamanders

Landscapes are continually changing due to numerous assaults, including habitat alteration, anthropogenic disturbances, and climate change. Understanding how species will respond to these changes is of critical importance for conservation and management. Mechanistic models, such as biophysical models (BPMs), are an increasingly popular tool to predict how local population dynamics or species’ distributions may be altered in response to environmental and climate changes. By mechanistically modeling relationships between environmental conditions, physiology and behavior, it is possible to make accurate predictions about how species may respond. However, BPMs are often difficult to implement due to lack of appropriate, species-specific data that is biologically realistic or relevant. In this study, we present a BPM for the salamander Plethodon jordani and assess how adding more biological realism has potential to alter model predictions about annual energy budgets. Additionally, we conducted local and global sensitivity analyses to evaluate the importance of accurately specifying model parameter values and functional relationships. We found that the addition of biological realism resulted in greater model complexity as well as substantially different estimates of energy balance. Correct parameterization of biophysical models is also critical, as small changes in parameter values can result in disproportionately large changes in downstream model estimates. Our model highlights the overall importance of using ecologically relevant and specific data for input parameters, as well as careful assessment of parameter sensitivity. We encourage researchers to be aware of the data they are using to parameterize BPMs, and urge the collection of system-specific data that is relevant in spatial and temporal scale. We also recommend greater and more transparent use of sensitivity analyses to provide a better understanding of the model, as well as greater confidence in model predictions.     

基于模式识别的景观格局分析与尺度转换研究框架

景观格局分析和尺度转换是生态学和地理学研究的核心问题之一.尽管多年来投入了大量的精力,然而由于尺度效应的复杂性和尺度转换过程的不确定性,仍然没有找到合适的方法.目前关于尺度转换的研究存在两个误区,其一是过分重视空间尺度的转换,忽略了过程尺度的转换;其二是过分强调了对不同尺度间数量关系的外推与转换,忽略了不同尺度间生态规律的外推与转换.在系统分析了景观格局与尺度转换研究现状、存在问题和难点的基础上,提出了借用模式识别的原理和方法,针对特定的生态过程,开展景观格局分析与尺度转换的思路.认为尺度转换的关键在于通过识别不同尺度上影响生态过程的主导因子,找到各个尺度上“格局（环境因子空间组合）-过程”、尺度间“格局-格局”之间的对应关系,通过建立“环境-格局-过程”模式识别数据库,就能够建立不同尺度之间基于模式识别的尺度转换方法.     

Sensitivity Analysis of the APEX Model for Assessing Sustainability of Switchgrass Grown for Biofuel Production in Central Texas

Crop simulation models are increasingly being used to understand the feasibility of large-scale cellulosic biofuel production along with the multi-dimensional impacts on environmental sustainability. However, how the uncertainty in model parameters impacts model performance for sustainability is unclear. In this case study, sensitivity analyses were conducted for three switchgrass sustainability metrics: total biomass production, nitrogen loss, and soil carbon change using the APEX (Agricultural Policy/Environmental eXtender) model. Fifteen out of the 45 parameters (25 crop growth (CROP) parameters and 20 additional model parameters (PARM)) were identified as influential for the three sustainability metrics for three lowland genotypes (WBC, AP13, and KAN) across two locations (Temple, TX, and Austin, TX). Our sensitivity results showed that parameter importance was not dependent on the genotypes but depended on the variables of interest, and differed only slightly between locations. Influential belowground-related CROP and PARM parameters were identified for each sustainability metric, indicating that belowground-related parameters are just as important as commonly measured aboveground CROP parameters. Further investigation of the linear or non-linear relationships and the two-way interactions between each of the individual influential parameters with the three sustainability metrics reflected the functions and characteristics within the APEX model and the interrelations among different processes. Strong interactions between the most influential parameters for total biomass, nitrogen loss, and soil carbon change also highlighted the importance of accurately setting these parameters. Identification of influential model parameters for switchgrass sustainability may help guide field measurements and provide further understanding of the interrelated processes in the APEX model. Furthermore, future field experiments can be designed to measure these influential parameters and understand the non-linear relationships identified between influential parameters and response variables. More accurate model parameterization will help improve APEX model performance and our understanding of the possible underlying physiological mechanisms.     

Conditional inversion to estimate parameters from eddy-flux observations

Aims Data assimilation is a useful tool to extract information from large datasets of the net ecosystem exchange (NEE) of CO₂ obtained by eddy-flux measurements. However, the number of parameters in ecosystem models that can be constrained by eddy-flux data is limited by conventional inverse analysis that estimates parameter values based on one-time inversion. This study aimed to improve data assimilation to increase the number of constrained parameters.Methods In this study, we developed conditional Bayesian inversion to maximize the number of parameters to be constrained by NEE data in several steps. In each step, we conducted a Bayesian inversion to constrain parameters. The maximum likelihood estimates of the constrained parameters were then used as prior to fix parameter values in the next step of inversion. The conditional inversion was repeated until there were no more parameters that could be further constrained. We applied the conditional inversion to hourly NEE data from Harvard Forest with a physiologically based ecosystem model.Important findings Results showed that the conventional inversion method constrained 6 of 16 parameters in the model while the conditional inversion method constrained 13 parameters after six steps. The cost function that indicates mismatch between the modeled and observed data decreased with each step of conditional Bayesian inversion. The Bayesian information criterion also decreased, suggesting reduced information loss with each step of conditional Bayesian inversion. A wavelet analysis reflected that model performance under conditional Bayesian inversion was better than that under conventional inversion at multiple time scales, except for seasonal and half-yearly scales. In addition, our analysis also demonstrated that parameter convergence in a subsequent step of the conditional inversion depended on correlations with the parameters constrained in a previous step. Overall, the conditional Bayesian inversion substantially increased the number of parameters to be constrained by NEE data and can be a powerful tool to be used in data assimilation in ecology.     

Predicting invasions in Australia by a Neotropical shrub under climate change: the challenge of novel climates and parameter estimation

Aim  To test how well species distributions and abundance can be predicted following invasion and climate change when using only species distribution and abundance data to estimate parameters.
Location  Models were developed for the species' native range in the Americas and applied to Australia.
Methods  We developed a predictive model for an invasive neotropical shrub ( Parkinsonia aculeata) using a popular ecophysiological bioclimatic modelling technique (CLIMEX) fitted against distribution and abundance data in the Americas. The effect of uncertainty in model parameter estimates on predictions in Australia was tested. Alternative data sources were used when model predictions were sensitive to uncertainty in parameter estimates. The resulting best-fit model was run under two climate change scenarios.
Results  Of the 19 parameters used, 9 could not be fitted using data from the native range. However, only parameters that lowered temperature or increased moisture requirements for growth noticeably altered the model prediction in Australia. Differences in predictions were dramatic, and reflect climates in Australia that were not represented in the Americas (novel climates). However, these poorly fitted parameters could be fitted post hoc using alternative data sources prior to predicting responses to climate change.
Conclusions  Novel climates prevented the development of a predictive model which relied only on native-range distribution and abundance data because certain parameters could not be fitted. In fact, predictions were more sensitive to parameter uncertainty than to climate change scenarios. Where uncertainty in parameter estimates affected predictions, it could be addressed through the inclusion of alternative data sources. However, this may not always be possible, for example in the absence of post-invasion data.     

Systemic properties of ensembles of metabolic networks: application of graphical and statistical methods to simple unbranched pathways

MOTIVATION: Mathematical models are the only realistic method for representing the integrated dynamic behavior of complex biochemical networks. However, it is difficult to obtain a consistent set of values for the parameters that characterize such a model. Even when a set of parameter values exists, the accuracy of the individual values is questionable. Therefore, we were motivated to explore statistical techniques for analyzing the properties of a given model when knowledge of the actual parameter values is lacking. RESULTS: The graphical and statistical methods presented in the previous paper are applied here to simple unbranched biosynthetic pathways subject to control by feedback inhibition. We represent these pathways within a canonical nonlinear formalism that provides a regular structure that is convenient for randomly sampling the parameter space. After constructing a large ensemble of randomly generated sets of parameter values, the structural and behavioral properties of the model with these parameter sets are examined statistically and classified. The results of our analysis demonstrate that certain properties of these systems are strongly correlated, thereby revealing aspects of organization that are highly probable independent of selection. Finally, we show how specification of a given behavior affects the distribution of acceptable parameter values.     

A framework for representing microbial decomposition in coupled climate models

Accurate prediction of future atmospheric CO₂ concentrations is essential for evaluating climate change impacts on ecosystems and human societies. One major source of uncertainty in model predictions is the extent to which global warming will increase atmospheric CO₂ concentrations through enhanced microbial decomposition of soil organic carbon. Recent advances in microbial ecology could help reduce this uncertainty, but current global models do not represent direct microbial control over decomposition. Instead, all of the coupled climate models reviewed in the most recent Intergovernmental Panel on Climate Change (IPCC) report assume that decomposition is a first-order decay process, proportional to the size of the soil carbon pool. Here we argue for the development of a new generation of models that link decomposition directly to the size and activity of microbial communities in coupled global models. This process begins with the formulation and validation of fine-scale models that capture fundamental microbial mechanisms without excessive mathematical complexity. These mechanistic models must then be scaled up through an aggregation process and validated at ecosystem to global scales prior to incorporation into global climate models (GCMs). Parameterizing microbial models at the global scale is challenging because some microbial properties such as in situ extracellular enzyme activities are very difficult to measure directly. New parameter fitting procedures may therefore be needed to infer the values of important microbial variables. Validating decomposition models at the global scale is also a challenge, and has not yet been accomplished with the land carbon models embedded in current GCMs. Fortunately new global datasets on soil carbon stocks and fluxes offer promising opportunities to validate both existing land carbon models and new microbial models. If challenges in scaling, parameterization, and validation can be overcome, a new generation of microbially-based decomposition models could substantially improve predictions of carbon–climate feedbacks in the Earth system. Development of new models structures would also reduce any bias due to the assumption of first-order decomposition across all of the models currently referenced in reports of the IPCC.     

Separating foliar physiology from morphology reveals the relative roles of vertically structured transpiration factors within red maple crowns and limitations of larger scale models

A spatially explicit mechanistic model, MAESTRA, was used to separate key parameters affecting transpiration to provide insights into the most influential parameters for accurate predictions of within-crown and within-canopy transpiration. Once validated among Acer rubrum L. genotypes, model responses to different parameterization scenarios were scaled up to stand transpiration (expressed per unit leaf area) to assess how transpiration might be affected by the spatial distribution of foliage properties. For example, when physiological differences were accounted for, differences in leaf width among A. rubrum L. genotypes resulted in a 25% difference in transpiration. An in silico within-canopy sensitivity analysis was conducted over the range of genotype parameter variation observed and under different climate forcing conditions. The analysis revealed that seven of 16 leaf traits had a ≥5% impact on transpiration predictions. Under sparse foliage conditions, comparisons of the present findings with previous studies were in agreement that parameters such as the maximum Rubisco-limited rate of photosynthesis can explain ～20% of the variability in predicted transpiration. However, the spatial analysis shows how such parameters can decrease or change in importance below the uppermost canopy layer. Alternatively, model sensitivity to leaf width and minimum stomatal conductance was continuous along a vertical canopy depth profile. Foremost, transpiration sensitivity to an observed range of morphological and physiological parameters is examined and the spatial sensitivity of transpiration model predictions to vertical variations in microclimate and foliage density is identified to reduce the uncertainty of current transpiration predictions.     

The promise and pitfalls of β‐diversity in ecology and conservation

A key challenge in ecology and conservation is to determine how processes at different scales create variation in community composition (β‐diversity). In this issue, Oldén & Halme show that grazers increase β‐diversity through multiple processes at different scales. We discuss how β‐diversity can elucidate fundamental processes of community assembly, challenges in linking processes to patterns, and unresolved questions across scales.