Steady state analysis of influx and transbilayer distribution of ergosterol in the yeast plasma membrane

Mathematical modeling is now frequently used in outbreak investigations to understand underlying mechanisms of infectious disease dynamics, assess patterns in epidemiological data, and forecast the trajectory of epidemics. However, the successful application of mathematical models to guide public health interventions lies in the ability to reliably estimate model parameters and their corresponding uncertainty. Here, we present and illustrate a simple computational method for assessing parameter identifiability in compartmental epidemic models. We describe a parametric bootstrap approach to generate simulated data from dynamical systems to quantify parameter uncertainty and identifiability. We calculate confidence intervals and mean squared error of estimated parameter distributions to assess parameter identifiability. To demonstrate this approach, we begin with a low-complexity SEIR model and work through examples of increasingly more complex compartmental models that correspond with applications to pandemic influenza, Ebola, and Zika. Overall, parameter identifiability issues are more likely to arise with more complex models (based on number of equations/states and parameters). As the number of parameters being jointly estimated increases, the uncertainty surrounding estimated parameters tends to increase, on average, as well. We found that, in most cases, R0 is often robust to parameter identifiability issues affecting individual parameters in the model. Despite large confidence intervals and higher mean squared error of other individual model parameters, R0 can still be estimated with precision and accuracy. Because public health policies can be influenced by results of mathematical modeling studies, it is important to conduct parameter identifiability analyses prior to fitting the models to available data and to report parameter estimates with quantified uncertainty. The method described is helpful in these regards and enhances the essential toolkit for conducting model-based inferences using compartmental dynamic models.     

On the identifiability of metabolic network models

A major problem for the identification of metabolic network models is parameter identifiability, that is, the possibility to unambiguously infer the parameter values from the data. Identifiability problems may be due to the structure of the model, in particular implicit dependencies between the parameters, or to limitations in the quantity and quality of the available data. We address the detection and resolution of identifiability problems for a class of pseudo-linear models of metabolism, so-called linlog models. Linlog models have the advantage that parameter estimation reduces to linear or orthogonal regression, which facilitates the analysis of identifiability. We develop precise definitions of structural and practical identifiability, and clarify the fundamental relations between these concepts. In addition, we use singular value decomposition to detect identifiability problems and reduce the model to an identifiable approximation by a principal component analysis approach. The criterion is adapted to real data, which are frequently scarce, incomplete, and noisy. The test of the criterion on a model with simulated data shows that it is capable of correctly identifying the principal components of the data vector. The application to a state-of-the-art dataset on central carbon metabolism in Escherichia coli yields the surprising result that only $4$ out of $31$ reactions, and $37$ out of $100$ parameters, are identifiable. This underlines the practical importance of identifiability analysis and model reduction in the modeling of large-scale metabolic networks. Although our approach has been developed in the context of linlog models, it carries over to other pseudo-linear models, such as generalized mass-action (power-law) models. Moreover, it provides useful hints for the identifiability analysis of more general classes of nonlinear models of metabolism.     

Practical identifiability of biokinetic parameters of a model describing two-step nitrification in biofilms

Parameter estimation and model calibration are key problems in the application of biofilm models in engineering practice, where a large number of model parameters need to be determined usually based on experimental data with only limited information content. In this article, identifiability of biokinetic parameters of a biofilm model describing two-step nitrification was evaluated based solely on bulk phase measurements of ammonium, nitrite, and nitrate. In addition to evaluating the impact of experimental conditions and available measurements, the influence of mass transport limitation within the biofilm and the initial parameter values on identifiability of biokinetic parameters was evaluated. Selection of parameters for identifiability analysis was based on global mean sensitivities while parameter identifiability was analyzed using local sensitivity functions. At most, four of the six most sensitive biokinetic parameters were identifiable from results of batch experiments at bulk phase dissolved oxygen concentrations of 0.8 or 5 mg O(2)/L. High linear dependences between the parameters of the subsets (KO2,AOB,muAOB) and (KO2,NOB,muNOB) resulted in reduced identifiability. Mass transport limitation within the biofilm did not influence the number of identifiable parameters but, in fact, decreased collinearity between parameters, especially for parameters that are otherwise correlated (e.g., muAOB) and KO2,AOB, or muNOB and KO2,NOB). The choice of the initial parameter values had a significant impact on the identifiability of two parameter subsets, both including the parameters muAOB and KO2,AOB. Parameter subsets that did not include the subsets muAOB and KO2,AOB or muNOB and KO2,NOB were clearly identifiable independently of the choice of the initial parameter values.     

A PDMP model of the epithelial cell turn-over in the intestinal crypt including microbiota-derived regulations

Mechanistic models are a powerful tool to gain insights into biological processes. The parameters of such models, e.g. kinetic rate constants, usually cannot be measured directly but need to be inferred from experimental data. In this article, we study dynamical models of the translation kinetics after mRNA transfection and analyze their parameter identifiability. That is, whether parameters can be uniquely determined from perfect or realistic data in theory and practice. Previous studies have considered ordinary differential equation (ODE) models of the process, and here we formulate a stochastic differential equation (SDE) model. For both model types, we consider structural identifiability based on the model equations and practical identifiability based on simulated as well as experimental data and find that the SDE model provides better parameter identifiability than the ODE model. Moreover, our analysis shows that even for those parameters of the ODE model that are considered to be identifiable, the obtained estimates are sometimes unreliable. Overall, our study clearly demonstrates the relevance of considering different modeling approaches and that stochastic models can provide more reliable and informative results.

     

Cellular signaling identifiability analysis: A case study

Two primary purposes for mathematical modeling in cell biology are (1) simulation for making predictions of experimental outcomes and (2) parameter estimation for drawing inferences from experimental data about unobserved aspects of biological systems. While the former purpose has become common in the biological sciences, the latter is less common, particularly when studying cellular and subcellular phenomena such as signaling—the focus of the current study. Data are difficult to obtain at this level. Therefore, even models of only modest complexity can contain parameters for which the available data are insufficient for estimation. In the present study, we use a set of published cellular signaling models to address issues related to global parameter identifiability. That is, we address the following question: assuming known time courses for some model variables, which parameters is it theoretically impossible to estimate, even with continuous, noise-free data? Following an introduction to this problem and its relevance, we perform a full identifiability analysis on a set of cellular signaling models using DAISY (Differential Algebra for the Identifiability of SYstems). We use our analysis to bring to light important issues related to parameter identifiability in ordinary differential equation (ODE) models. We contend that this is, as of yet, an under-appreciated issue in biological modeling and, more particularly, cell biology.     

Global identifiability of latent class models with applications to diagnostic test accuracy studies: A Gröbner basis approach

Identifiability of statistical models is a fundamental regularity condition that is required for valid statistical inference. Investigation of model identifiability is mathematically challenging for complex models such as latent class models. Jones et al. used Goodman's technique to investigate the identifiability of latent class models with applications to diagnostic tests in the absence of a gold standard test. The tool they used was based on examining the singularity of the Jacobian or the Fisher information matrix, in order to obtain insights into local identifiability (ie, there exists a neighborhood of a parameter such that no other parameter in the neighborhood leads to the same probability distribution as the parameter). In this paper, we investigate a stronger condition: global identifiability (ie, no two parameters in the parameter space give rise to the same probability distribution), by introducing a powerful mathematical tool from computational algebra: the Gröbner basis. With several existing well-known examples, we argue that the Gröbner basis method is easy to implement and powerful to study global identifiability of latent class models, and is an attractive alternative to the information matrix analysis by Rothenberg and the Jacobian analysis by Goodman and Jones et al.     

The identifiability of tree topology for phylogenetic models, including covarion and mixture models.

For a model of molecular evolution to be useful for phylogenetic inference, the topology of evolutionary trees must be identifiable. That is, from a joint distribution the model predicts, it must be possible to recover the tree parameter. We establish tree identifiability for a number of phylogenetic models, including a covarion model and a variety of mixture models with a limited number of classes. The proof is based on the introduction of a more general model, allowing more states at internal nodes of the tree than at leaves, and the study of the algebraic variety formed by the joint distributions to which it gives rise. Tree identifiability is first established for this general model through the use of certain phylogenetic invariants.     

Structural sensitivity and resilience in a predator–prey model with density-dependent mortality

Numerous formulations with the same mathematical properties can be relevant to model a biological process. Different formulations can predict different model dynamics like equilibrium vs. oscillations even if they are quantitatively close (structural sensitivity). The question we address in this paper is: does the choice of a formulation affect predictions on the number of stable states? We focus on a predator–prey model with predator competition that exhibits multiple stable states. A bifurcation analysis is realized with respect to prey carrying capacity and species body mass ratio within range of values found in food web models. Bifurcation diagrams built for two type-II functional responses are different in two ways. First, the kind of stable state (equilibrium vs. oscillations) is different for 26.0–49.4% of the parameter values, depending on the parameter space investigated. Using generalized modelling, we highlight the role of functional response slope in this difference. Secondly, the number of stable states is higher with Ivlev's functional response for 0.1–14.3% of the parameter values. These two changes interact to create different model predictions if a parameter value or a state variable is altered. In these two examples of disturbance, Holling's disc equation predicts a higher system resilience. Indeed, Ivlev's functional response predicts that disturbance may trap the system into an alternative stable state that can be escaped from only by a larger alteration (hysteresis phenomena). Two questions arise from this work: (i) how much complex ecological models can be affected by this sensitivity to model formulation? and (ii) how to deal with these uncertainties in model predictions?     

Finding identifiable parameter combinations in nonlinear ODE models and the rational reparameterization of their input-output equations

When examining the structural identifiability properties of dynamic system models, some parameters can take on an infinite number of values and yet yield identical input-output data. These parameters and the model are then said to be unidentifiable. Finding identifiable combinations of parameters with which to reparameterize the model provides a means for quantitatively analyzing the model and computing solutions in terms of the combinations. In this paper, we revisit and explore the properties of an algorithm for finding identifiable parameter combinations using Gröbner Bases and prove useful theoretical properties of these parameter combinations. We prove a set of M algebraically independent identifiable parameter combinations can be found using this algorithm and that there exists a unique rational reparameterization of the input-output equations over these parameter combinations. We also demonstrate application of the procedure to a nonlinear biomodel.     

An algorithm for finding globally identifiable parameter combinations of nonlinear ODE models using Gröbner Bases

The parameter identifiability problem for dynamic system ODE models has been extensively studied. Nevertheless, except for linear ODE models, the question of establishing identifiable combinations of parameters when the model is unidentifiable has not received as much attention and the problem is not fully resolved for nonlinear ODEs. Identifiable combinations are useful, for example, for the reparameterization of an unidentifiable ODE model into an identifiable one. We extend an existing algorithm for finding globally identifiable parameters of nonlinear ODE models to generate the ‘simplest’ globally identifiable parameter combinations using Gröbner Bases. We also provide sufficient conditions for the method to work, demonstrate our algorithm and find associated identifiable reparameterizations for several linear and nonlinear unidentifiable biomodels.     

Identifiability and retrievability of unique parameters describing intrinsic Andrews kinetics

A key factor contributing to the variability in the microbial kinetic parameters reported from batch assays is parameter identifiability, i.e., the ability of the mathematical routine used for parameter estimation to provide unique estimates of the individual parameter values. This work encompassed a three-part evaluation of the parameter identifiability of intrinsic kinetic parameters describing the Andrews growth model that are obtained from batch assays. First, a parameter identifiability analysis was conducted by visually inspecting the sensitivity equations for the Andrews growth model. Second, the practical retrievability of the parameters in the presence of experimental error was evaluated for the parameter estimation routine used. Third, the results of these analyses were tested using an example data set from the literature for a self-inhibitory substrate. The general trends from these analyses were consistent and indicated that it is very difficult, if not impossible, to simultaneously obtain a unique set of estimates of intrinsic kinetic parameters for the Andrews growth model using data from a single batch experiment.     

Efficient parametric analysis of the chemical master equation through model order reduction

ABSTRACT: BACKGROUND: Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation. RESULTS: In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations. CONCLUSIONS: The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.     

Conceptual, methodological and computational issues concerning the compartmental modeling of a complex biological system: Postprandial inter-organ metabolism of dietary nitrogen in humans

A multi-compartmental model has been developed to describe dietary nitrogen (N) postprandial distribution and metabolism in humans. This paper details the entire process of model development, including the successive steps of its construction, parameter estimation and validation. The model was built using experimental data on dietary N kinetics in certain accessible pools of the intestine, blood and urine in healthy adults fed a [15N]-labeled protein meal. A 13-compartment, 21-parameter model was selected from candidate models of increasing order as being the minimum structure able to properly fit experimental data for all sampled compartments. Problems of theoretical identifiability and numerical identification of the model both constituted mathematical challenges that were difficult to solve because of the large number of unknown parameters and the few experimental data available. For this reason, new robust and reliable methods were applied, which enabled (i) a check that all model parameters could theoretically uniquely be determined and (ii) an estimation of their numerical values with satisfactory precision from the experimental data. Finally, model validation was completed by first verifying its a posteriori identifiability and then carrying out external validation.     

The structural identifiability of susceptible–infective–recovered type epidemic models with incomplete immunity and birth targeted vaccination

This paper considers the implications of a structural identifiability analysis on a series of fundamental three-compartment epidemic model structures, derived around the general SIR (susceptible–infective–recovered) framework. The models represent various forms of incomplete immunity acquired through natural infection, or from administration of a birth targeted vaccination programme. It is shown that the addition of a vaccination campaign has a negative effect on the structural identifiability of all considered models. In particular, the actual proportion of vaccination coverage achieved, an essential parameter, cannot be uniquely estimated from even ideal prevalence data.     

Parameter Identifiability and Redundancy: Theoretical Considerations

Background

Models for complex biological systems may involve a large number of parameters. It may well be that some of these parameters cannot be derived from observed data via regression techniques. Such parameters are said to be unidentifiable, the remaining parameters being identifiable. Closely related to this idea is that of redundancy, that a set of parameters can be expressed in terms of some smaller set. Before data is analysed it is critical to determine which model parameters are identifiable or redundant to avoid ill-defined and poorly convergent regression.

Methodology/Principal Findings

In this paper we outline general considerations on parameter identifiability, and introduce the notion of weak local identifiability and gradient weak local identifiability. These are based on local properties of the likelihood, in particular the rank of the Hessian matrix. We relate these to the notions of parameter identifiability and redundancy previously introduced by Rothenberg (Econometrica 39 (1971) 577–591) and Catchpole and Morgan (Biometrika 84 (1997) 187–196). Within the widely used exponential family, parameter irredundancy, local identifiability, gradient weak local identifiability and weak local identifiability are shown to be largely equivalent. We consider applications to a recently developed class of cancer models of Little and Wright (Math Biosciences 183 (2003) 111–134) and Little et al. (J Theoret Biol 254 (2008) 229–238) that generalize a large number of other recently used quasi-biological cancer models.

Conclusions/Significance

We have shown that the previously developed concepts of parameter local identifiability and redundancy are closely related to the apparently weaker properties of weak local identifiability and gradient weak local identifiability—within the widely used exponential family these concepts largely coincide.     

Structural identifiability of systems biology models: a critical comparison of methods

Analysing the properties of a biological system through in silico experimentation requires a satisfactory mathematical representation of the system including accurate values of the model parameters. Fortunately, modern experimental techniques allow obtaining time-series data of appropriate quality which may then be used to estimate unknown parameters. However, in many cases, a subset of those parameters may not be uniquely estimated, independently of the experimental data available or the numerical techniques used for estimation. This lack of identifiability is related to the structure of the model, i.e. the system dynamics plus the observation function. Despite the interest in knowing a priori whether there is any chance of uniquely estimating all model unknown parameters, the structural identifiability analysis for general non-linear dynamic models is still an open question. There is no method amenable to every model, thus at some point we have to face the selection of one of the possibilities. This work presents a critical comparison of the currently available techniques. To this end, we perform the structural identifiability analysis of a collection of biological models. The results reveal that the generating series approach, in combination with identifiability tableaus, offers the most advantageous compromise among range of applicability, computational complexity and information provided.     

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.     

Identifiability from parameter bounds. Structural and numerical aspects

Some a priori and a posteriori aspects of the identifiability problem for unidentifiable models are discussed. It is argued that the nation of identifiability from parameter bounds has a minor a priori structural relevance. The parameter bounds rationale may prove a useful a posteriori numerical notion. However, its practical potentiality needs careful evaluation, as the use of point estimates automatically builds into the model some hidden structural constraints. Examples are given.     

An ensemble of models of the acute inflammatory response to bacterial lipopolysaccharide in rats: results from parameter space reduction

In previous work, we developed an 8-state nonlinear dynamic model of the acute inflammatory response, including activated phagocytic cells, pro- and anti-inflammatory cytokines, and tissue damage, and calibrated it to data on cytokines from endotoxemic rats. In the interest of parsimony, the present work employed parametric sensitivity and local identifiability analysis to establish a core set of parameters predominantly responsible for variability in model solutions. Parameter optimization, facilitated by varying only those parameters belonging to this core set, was used to identify an ensemble of parameter vectors, each representing an acceptable local optimum in terms of fit to experimental data. Individual models within this ensemble, characterized by their different parameter values, showed similar cytokine but diverse tissue damage behavior. A cluster analysis of the ensemble of models showed the existence of a continuum of acceptable models, characterized by compensatory mechanisms and parameter changes. We calculated the direct correlations between the core set of model parameters and identified three mechanisms responsible for the conversion of the diverse damage time courses to similar cytokine behavior in these models. Given that tissue damage level could be an indicator of the likelihood of mortality, our findings suggest that similar cytokine dynamics could be associated with very different mortality outcomes, depending on the balance of certain inflammatory elements.