Regression Analysis for Constraining Free Parameters in Electrophysiological Models of Cardiac Cells

A major challenge in computational biology is constraining free parameters in mathematical models. Adjusting a parameter to make a given model output more realistic sometimes has unexpected and undesirable effects on other model behaviors. Here, we extend a regression-based method for parameter sensitivity analysis and show that a straightforward procedure can uniquely define most ionic conductances in a well-known model of the human ventricular myocyte. The model''s parameter sensitivity was analyzed by randomizing ionic conductances, running repeated simulations to measure physiological outputs, then collecting the randomized parameters and simulation results as “input” and “output” matrices, respectively. Multivariable regression derived a matrix whose elements indicate how changes in conductances influence model outputs. We show here that if the number of linearly-independent outputs equals the number of inputs, the regression matrix can be inverted. This is significant, because it implies that the inverted matrix can specify the ionic conductances that are required to generate a particular combination of model outputs. Applying this idea to the myocyte model tested, we found that most ionic conductances could be specified with precision (R² > 0.77 for 12 out of 16 parameters). We also applied this method to a test case of changes in electrophysiology caused by heart failure and found that changes in most parameters could be well predicted. We complemented our findings using a Bayesian approach to demonstrate that model parameters cannot be specified using limited outputs, but they can be successfully constrained if multiple outputs are considered. Our results place on a solid mathematical footing the intuition-based procedure simultaneously matching a model''s output to several data sets. More generally, this method shows promise as a tool to define model parameters, in electrophysiology and in other biological fields.     

Parameter Sensitivity Analysis of Stochastic Models Provides Insights into?Cardiac Calcium Sparks

We present a parameter sensitivity analysis method that is appropriate for stochastic models, and we demonstrate how this analysis generates experimentally testable predictions about the factors that influence local Ca²⁺ release in heart cells. The method involves randomly varying all parameters, running a single simulation with each set of parameters, running simulations with hundreds of model variants, then statistically relating the parameters to the simulation results using regression methods. We tested this method on a stochastic model, containing 18 parameters, of the cardiac Ca²⁺ spark. Results show that multivariable linear regression can successfully relate parameters to continuous model outputs such as Ca²⁺ spark amplitude and duration, and multivariable logistic regression can provide insight into how parameters affect Ca²⁺ spark triggering (a probabilistic process that is all-or-none in a single simulation). Benchmark studies demonstrate that this method is less computationally intensive than standard methods by a factor of 16. Importantly, predictions were tested experimentally by measuring Ca²⁺ sparks in mice with knockout of the sarcoplasmic reticulum protein triadin. These mice exhibit multiple changes in Ca²⁺ release unit structures, and the regression model both accurately predicts changes in Ca²⁺ spark amplitude (30% decrease in model, 29% decrease in experiments) and provides an intuitive and quantitative understanding of how much each alteration contributes to the result. This approach is therefore an effective, efficient, and predictive method for analyzing stochastic mathematical models to gain biological insight.     

Uninformative Parameters and Model Selection Using Akaike's Information Criterion

Abstract: As use of Akaike's Information Criterion (AIC) for model selection has become increasingly common, so has a mistake involving interpretation of models that are within 2 AIC units (ΔAIC ≤ 2) of the top-supported model. Such models are <2 ΔAIC units because the penalty for one additional parameter is +2 AIC units, but model deviance is not reduced by an amount sufficient to overcome the 2-unit penalty and, hence, the additional parameter provides no net reduction in AIC. Simply put, the uninformative parameter does not explain enough variation to justify its inclusion in the model and it should not be interpreted as having any ecological effect. Models with uninformative parameters are frequently presented as being competitive in the Journal of Wildlife Management, including 72% of all AIC-based papers in 2008, and authors and readers need to be more aware of this problem and take appropriate steps to eliminate misinterpretation. I reviewed 5 potential solutions to this problem: 1) report all models but ignore or dismiss those with uninformative parameters, 2) use model averaging to ameliorate the effect of uninformative parameters, 3) use 95% confidence intervals to identify uninformative parameters, 4) perform all-possible subsets regression and use weight-of-evidence approaches to discriminate useful from uninformative parameters, or 5) adopt a methodological approach that allows models containing uninformative parameters to be culled from reported model sets. The first approach is preferable for small sets of a priori models, whereas the last 2 approaches should be used for large model sets or exploratory modeling.     

Modeling and Analysis of the Molecular Basis of Pain in Sensory Neurons

Intracellular calcium dynamics are critical to cellular functions like pain transmission. Extracellular ATP plays an important role in modulating intracellular calcium levels by interacting with the P2 family of surface receptors. In this study, we developed a mechanistic mathematical model of ATP-induced P2 mediated calcium signaling in archetype sensory neurons. The model architecture, which described 90 species connected by 162 interactions, was formulated by aggregating disparate molecular modules from literature. Unlike previous models, only mass action kinetics were used to describe the rate of molecular interactions. Thus, the majority of the 252 unknown model parameters were either association, dissociation or catalytic rate constants. Model parameters were estimated from nine independent data sets taken from multiple laboratories. The training data consisted of both dynamic and steady-state measurements. However, because of the complexity of the calcium network, we were unable to estimate unique model parameters. Instead, we estimated a family or ensemble of probable parameter sets using a multi-objective thermal ensemble method. Each member of the ensemble met an error criterion and was located along or near the optimal trade-off surface between the individual training data sets. The model quantitatively reproduced experimental measurements from dorsal root ganglion neurons as a function of extracellular ATP forcing. Hypothesized architecture linking phosphoinositide regulation with P2X receptor activity explained the inhibition of P2X-mediated current flow by activated metabotropic P2Y receptors. Sensitivity analysis using individual and the whole system outputs suggested which molecular subsystems were most important following P2 activation. Taken together, modeling and analysis of ATP-induced P2 mediated calcium signaling generated qualitative insight into the critical interactions controlling ATP induced calcium dynamics. Understanding these critical interactions may prove useful for the design of the next generation of molecular pain management strategies.     

Differentially private distributed logistic regression using private and public data

Background

Privacy protecting is an important issue in medical informatics and differential privacy is a state-of-the-art framework for data privacy research. Differential privacy offers provable privacy against attackers who have auxiliary information, and can be applied to data mining models (for example, logistic regression). However, differentially private methods sometimes introduce too much noise and make outputs less useful. Given available public data in medical research (e.g. from patients who sign open-consent agreements), we can design algorithms that use both public and private data sets to decrease the amount of noise that is introduced.

Methodology

In this paper, we modify the update step in Newton-Raphson method to propose a differentially private distributed logistic regression model based on both public and private data.

Experiments and results

We try our algorithm on three different data sets, and show its advantage over: (1) a logistic regression model based solely on public data, and (2) a differentially private distributed logistic regression model based on private data under various scenarios.

Conclusion

Logistic regression models built with our new algorithm based on both private and public datasets demonstrate better utility than models that trained on private or public datasets alone without sacrificing the rigorous privacy guarantee.

     

Derivation and analysis of new stand-level mortality models based on existing growth equations

Considering that, the temporal trend in stocking, expressed as number of trees per unit area, is the opposite of that of growth, and that both trajectories are sigmoidal, we derived a temporal trajectory of density decrease by reversing the temporal trend of a generalized growth function. We derived and analysed twelve stand-level mortality models by using four data sets from monospecific even-aged stands. Stand dominant height rather than stand age was incorporated as an indicator of the growth stage and a careful examination of the models conformity with the essential logical properties of the stand-level survival models was conducted. We first tested the models adequacy and general predictive performance by fitting them to parameterization data sets and subsequently assessing them with validation data. The regression equations were re-fitted afterwards over the total data sets to make use of all available information in the final parameter estimates. Nine model formulations were successfully fitted and four of them were the most adequate in describing stand density decrease with dominant height growth. The site-specific effect was incorporated in the newly derived models through the predictor variable and the stand-specific starting density was accounted for through a specific model parameter. These new dominant height-dependent mortality equations can be considered for inclusion in the framework of stand-level growth models as transition functions.     

Refining predictions of climate change impacts on plant species distribution through the use of local statistics

Bioclimate envelope models are often used to predict changes in species distribution arising from changes in climate. These models are typically based on observed correlations between current species distribution and climate data. One limitation of this basic approach is that the relationship modelled is assumed to be constant in space; the analysis is global with the relationship assumed to be spatially stationary. Here, it is shown that by using a local regression analysis, which allows the relationship under study to vary in space, rather than conventional global regression analysis it is possible to increase the accuracy of bioclimate envelope modelling. This is demonstrated for the distribution of Spotted Meddick in Great Britain using data relating to three time periods, including predictions for the 2080s based on two climate change scenarios. Species distribution and climate data were available for two of the time periods studied and this allowed comparison of bioclimate envelope model outputs derived using the local and global regression analyses. For both time periods, the area under the receiver operating characteristics curve derived from the analysis based on local statistics was significantly higher than that from the conventional global analysis; the curve comparisons were also undertaken with an approach that recognised the dependent nature of the data sets compared. Marked differences in the future distribution of the species predicted from the local and global based analyses were evident and highlight a need for further consideration of local issues in modelling ecological variables.     

Fuzzy Stochastic Petri Nets for Modeling Biological Systems with Uncertain Kinetic Parameters

Stochastic Petri nets (SPNs) have been widely used to model randomness which is an inherent feature of biological systems. However, for many biological systems, some kinetic parameters may be uncertain due to incomplete, vague or missing kinetic data (often called fuzzy uncertainty), or naturally vary, e.g., between different individuals, experimental conditions, etc. (often called variability), which has prevented a wider application of SPNs that require accurate parameters. Considering the strength of fuzzy sets to deal with uncertain information, we apply a specific type of stochastic Petri nets, fuzzy stochastic Petri nets (FSPNs), to model and analyze biological systems with uncertain kinetic parameters. FSPNs combine SPNs and fuzzy sets, thereby taking into account both randomness and fuzziness of biological systems. For a biological system, SPNs model the randomness, while fuzzy sets model kinetic parameters with fuzzy uncertainty or variability by associating each parameter with a fuzzy number instead of a crisp real value. We introduce a simulation-based analysis method for FSPNs to explore the uncertainties of outputs resulting from the uncertainties associated with input parameters, which works equally well for bounded and unbounded models. We illustrate our approach using a yeast polarization model having an infinite state space, which shows the appropriateness of FSPNs in combination with simulation-based analysis for modeling and analyzing biological systems with uncertain information.     

Are Assumptions about the Model Type Necessary in Reaction-Diffusion Modeling? A FRAP Application

At present, fluorescence recovery after photobleaching (FRAP) data are interpreted using various types of reaction-diffusion (RD) models: the model type is usually fixed first, and corresponding model parameters are inferred subsequently. In this article, we describe what we believe to be a novel approach for RD modeling without using any assumptions of model type or parameters. To the best of our knowledge, this is the first attempt to address both model-type and parameter uncertainties in inverting FRAP data. We start from the most general RD model, which accounts for a flexible number of molecular fractions, all mobile, with different diffusion coefficients. The maximal number of possible binding partners is identified and optimal parameter sets for these models are determined in a global search of the parameter-space using the Simulated Annealing strategy. The numerical performance of the described techniques was assessed using artificial and experimental FRAP data. Our general RD model outperformed the standard RD models used previously in modeling FRAP measurements and showed that intracellular molecular mobility can only be described adequately by allowing for multiple RD processes. Therefore, it is important to search not only for the optimal parameter set but also for the optimal model type.     

Simulating Daily and Half-Hourly Fluxes of Forest Carbon Dioxide and Water Vapor Exchange with a Simple Model of Light and Water Use

For the large-scale application of simple, aggregated models, it is important to be able to link the values of model parameters to easily measurable ecosystem characteristics. However, the aggregation of model inputs and outputs over time and space can hamper this linkage. In this paper, two temporal versions of the same simple carbon dioxide (CO₂) and water exchange model, based on the concepts of water- and light-use efficiencies, were used to simulate the half-hourly and daily CO₂ and water exchange of a Douglas fir forest (Pseudotsuga menziesii (Mirb.) Franco) in the Netherlands for 2 years, before and after a thinning. We tested the performance of the models and the interpretability of changes in optimized parameter values, due to the thinning, in terms of ecosystem functioning. The performance of the half-hourly model was satisfactory, whereas the performance of the daily model was high for water exchange but clearly lower for CO₂ exchange. A comparison of the model parameters before and after the thinning showed that the coefficients of the half-hourly model could be separated into more physiologically determined and stand-determined characteristics, but this separation was not clear for the daily model. These results show that if the temporal resolution of the model is high enough, the effects of a major ecosystem manipulation, such as thinning, can be detected and interpreted using eddy flux data and a very simple biophysical model. The model parameters have an unambiguous interpretation and can be inferred from basic ecosystem observables, such as leaf area index (LAI) and aboveground biomass. A sensitivity analysis found strong correlations between parameter sets with similar model performance. For any comparison of the parameter values of different studies, ranges of parameter values and their correlations should be presented rather than one optimized value. Received 2 May 2001; accepted 15 February 2002.     

Evaluating the Use of Global Sensitivity Analysis in Dynamic MFA

Dynamic material flow analysis (MFA) provides information about material usage over time and consequent changes in material stocks and flows. In order to understand the effect of limited data quality and model assumptions on MFA results, the use of sensitivity analysis methods in dynamic MFA studies has been on the increase. So far, sensitivity analysis in dynamic MFA has been conducted by means of a one‐at‐a‐time method, which tests parameter perturbations individually and observes the outcomes on output. In contrast to that, variance‐based global sensitivity analysis decomposes the variance of the model output into fractions caused by the uncertainty or variability of input parameters. The present study investigates interaction and time‐delay effects of uncertain parameters on the output of an archetypal input‐driven dynamic material flow model using variance‐based global sensitivity analysis. The results show that determining the main (first‐order) effects of parameter variations is often sufficient in dynamic MFA because substantial effects attributed to the simultaneous variation of several parameters (higher‐order effects) do not appear for classical setups of dynamic material flow models. For models with time‐varying parameters, time‐delay effects of parameter variation on model outputs need to be considered, potentially boosting the computational cost of global sensitivity analysis. Finally, the implications of exploring the sensitivities of model outputs with respect to parameter variations in the archetypical model are used to derive model‐ and goal‐specific recommendations on choosing appropriate sensitivity analysis methods in dynamic MFA.     

Dimension reduction of multivariate outputs of socio-environmental agent-based models

Often, socio-environmental agent-based models (ABMs) are driven by a host of parameters, and their outputs are similarly multidimensional or vastly high-dimensional. While this complex data and its inter-relationships may be rendered tractable, the task is far from trivial. In this paper, we study the multidimensional outcome space of the socio-environmental, land-use RHEA ABM (Risks and Hedonics in an Empirical Agent-based Land Market), specifically the inter-distances among the outcome measures, and their reducibility using several well-known dimension reduction techniques, variants of multidimensional scalings. In testing the efficacy of several reduction algorithms, we temporally characterize the model’s reducibility while exposing changes in behavior across a wide parameter space that can signal sudden or gradual shifts and possible critical transitions. Our findings reveal that the ABM’s signature reducibility trends exhibit idiosyncrasies and unexpected non-linearity as well as discontinuities. These non-linearities and discontinuities are indicative of both gradual and sudden shifts, signaling potential propensity to internal perturbations induced by parameter settings. Additionally, we related the outcome space via their inter-distances to the multidimensional input parameter space, effectively assessing outcome “reducibility to model controls”. This analysis reveals that the model’s sensitivity to parameters is not only temporally dependent, but also can be partitioned by them, some of which suppress variability in this reduction to model controls, raising questions regarding the extent and structure of endogeneity that yields the distinct temporal trends in the relationship between inputs and outputs and their connections to outcome reducibility.     

Sensitivity analysis of respiratory parameter uncertainties: impact of criterion function form and constraints

A sensitivity analysis based on weighted least-squares regression is presented to evaluate alternative methods for fitting lumped-parameter models to respiratory impedance data. The goal is to maintain parameter accuracy simultaneously with practical experiment design. The analysis focuses on predicting parameter uncertainties using a linearized approximation for joint confidence regions. Applications are with four-element parallel and viscoelastic models for 0.125- to 4-Hz data and a six-element model with separate tissue and airway properties for input and transfer impedance data from 2-64 Hz. The criterion function form was evaluated by comparing parameter uncertainties when data are fit as magnitude and phase, dynamic resistance and compliance, or real and imaginary parts of input impedance. The proper choice of weighting can make all three criterion variables comparable. For the six-element model, parameter uncertainties were predicted when both input impedance and transfer impedance are acquired and fit simultaneously. A fit to both data sets from 4 to 64 Hz could reduce parameter estimate uncertainties considerably from those achievable by fitting either alone. For the four-element models, use of an independent, but noisy, measure of static compliance was assessed as a constraint on model parameters. This may allow acceptable parameter uncertainties for a minimum frequency of 0.275-0.375 Hz rather than 0.125 Hz. This reduces data acquisition requirements from a 16- to a 5.33- to 8-s breath holding period. These results are approximations, and the impact of using the linearized approximation for the confidence regions is discussed.     

The equilibrium assumption in estimating the parameters of metapopulation models

1.  The construction of a predictive metapopulation model includes three steps: the choice of factors affecting metapopulation dynamics, the choice of model structure, and finally parameter estimation and model testing.
2.  Unless the assumption is made that the metapopulation is at stochastic quasi-equilibrium and unless the method of parameter estimation of model parameters uses that assumption, estimates from a limited amount of data will usually predict a trend in metapopulation size.
3.  This implicit estimation of a trend occurs because extinction-colonization stochasticity, possibly amplified by regional stochasticity, leads to unequal numbers of observed extinction and colonization events during a short study period.
4.  Metapopulation models, such as those based on the logistic regression model, that rely on observed population turnover events in parameter estimation are sensitive to the implicit estimation of a trend.
5.  A new parameter estimation method, based on Monte Carlo inference for statistically implicit models, allows an explicit decision about whether metapopulation quasi-stability is assumed or not.
6. Our confidence in metapopulation model parameter estimates that have been produced from only a few years of data is decreased by the need to know before parameter estimation whether the metapopulation is in quasi-stable state or not.
7. The choice of whether metapopulation stability is assumed or not in parameter estimation should be done consciously. Typical data sets cover only a few years and rarely allow a statistical test of a possible trend. While making the decision about stability one should consider any information about the landscape history and species and metapopulation characteristics.     

Evaluation of habitat suitability index models by global sensitivity and uncertainty analyses: a case study for submerged aquatic vegetation

Habitat suitability index (HSI) models are commonly used to predict habitat quality and species distributions and are used to develop biological surveys, assess reserve and management priorities, and anticipate possible change under different management or climate change scenarios. Important management decisions may be based on model results, often without a clear understanding of the level of uncertainty associated with model outputs. We present an integrated methodology to assess the propagation of uncertainty from both inputs and structure of the HSI models on model outputs (uncertainty analysis: UA) and relative importance of uncertain model inputs and their interactions on the model output uncertainty (global sensitivity analysis: GSA). We illustrate the GSA/UA framework using simulated hydrology input data from a hydrodynamic model representing sea level changes and HSI models for two species of submerged aquatic vegetation (SAV) in southwest Everglades National Park: Vallisneria americana (tape grass) and Halodule wrightii (shoal grass). We found considerable spatial variation in uncertainty for both species, but distributions of HSI scores still allowed discrimination of sites with good versus poor conditions. Ranking of input parameter sensitivities also varied spatially for both species, with high habitat quality sites showing higher sensitivity to different parameters than low‐quality sites. HSI models may be especially useful when species distribution data are unavailable, providing means of exploiting widely available environmental datasets to model past, current, and future habitat conditions. The GSA/UA approach provides a general method for better understanding HSI model dynamics, the spatial and temporal variation in uncertainties, and the parameters that contribute most to model uncertainty. Including an uncertainty and sensitivity analysis in modeling efforts as part of the decision‐making framework will result in better‐informed, more robust decisions.     

基于过程模型CROBAS的全局灵敏度分析方法比较

过程机理模型在开发过程中常受限于生理学参数无法直接或准确测量.全局灵敏度分析可以评估模型预测结果对于生理学参数变化的响应,为模型结构改进、数据收集和参数校准提供参考.本研究基于过程模型CROBAS,以华山松为例,选取模型中描述树木结构关系的10个参数,以树高和各器官生物量的Nash-Sutcliffe效率(NSE)为目...     

parallelMCMCcombine: An R Package for Bayesian Methods for Big Data and Analytics

Recent advances in big data and analytics research have provided a wealth of large data sets that are too big to be analyzed in their entirety, due to restrictions on computer memory or storage size. New Bayesian methods have been developed for data sets that are large only due to large sample sizes. These methods partition big data sets into subsets and perform independent Bayesian Markov chain Monte Carlo analyses on the subsets. The methods then combine the independent subset posterior samples to estimate a posterior density given the full data set. These approaches were shown to be effective for Bayesian models including logistic regression models, Gaussian mixture models and hierarchical models. Here, we introduce the R package parallelMCMCcombine which carries out four of these techniques for combining independent subset posterior samples. We illustrate each of the methods using a Bayesian logistic regression model for simulation data and a Bayesian Gamma model for real data; we also demonstrate features and capabilities of the R package. The package assumes the user has carried out the Bayesian analysis and has produced the independent subposterior samples outside of the package. The methods are primarily suited to models with unknown parameters of fixed dimension that exist in continuous parameter spaces. We envision this tool will allow researchers to explore the various methods for their specific applications and will assist future progress in this rapidly developing field.     

A unified approach to the Richards-model family for use in growth analyses: Why we need only two model forms

This paper advances a unified approach to the modeling of sigmoid organismal growth. There are numerous studies on growth, and there have been several proposals and applications of candidate models. Still, a lack of interpretation of the parameter values persists and, consequently, differences in growth patterns have riddled this field. A candidate regression model as a tool should be able to assess and compare growth-curve shapes, systematically and precisely. The Richards models constitute a useful family of growth models that amongst a multitude of parameterizations, re-parameterizations and special cases, include familiar models such as the negative exponential, the logistic, the Bertalanffy and the Gompertz. We have reviewed and systemized this family of models. We demonstrate that two specific parameterizations (or re-parameterizations) of the Richards model are able to substitute, and thus to unify all other forms and models. This unified-Richards model (with its two forms) constitutes a powerful tool for an interpretation of important characteristics of observed growth patterns, namely, [I] maximum (relative) growth rate (i.e., slope at inflection), [II] age at maximum growth rate (i.e., time at inflection), [III] relative mass or length at maximum growth rate (i.e., relative value at an inflection), [IV] value at age zero (i.e., birth, hatching or germination), and [V] asymptotic value (i.e., adult weight or length). These five parameters can characterize uniquely any sigmoid-growth data. To date most studies only compare what is referred to as the “growth-rate constant” or simply “growth rate” (k). This parameter can be interpreted as neither relative nor actual growth rate, but only as a parameter that affects the slope at inflection. We fitted the unified-Richards and five other candidate models to six artificial data sets, generated from the same models, and made a comparison based on the corrected Akaike’s Information Criterion (AICc). The outcome may in part be the result of the random generation of data points. Still, in conclusion, the unified-Richards model performed consistently well for all data sets, despite the penalty imposed by the AICc.