Model reduction and a priori kinetic parameter identifiability analysis using metabolome time series for metabolic reaction networks with linlog kinetics

In this work, we present a time-scale analysis based model reduction and parameter identifiability analysis method for metabolic reaction networks. The method uses the information obtained from short term chemostat perturbation experiments. We approximate the time constant of each metabolite pool by their turn-over time and classify the pools accordingly into two groups: fast and slow pools. We performed a priori model reduction, neglecting the dynamic term of the fast pools. By making use of the linlog approximative kinetics, we obtained a general explicit solution for the fast pools in terms of the slow pools by elaborating the degenerate algebraic system resulting from model reduction. The obtained relations yielded also analytical relations between a subset of kinetic parameters. These relations also allow to realize an analytical model reduction using lumped reaction kinetics. After solving these theoretical identifiability problems and performing model reduction, we carried out a Monte Carlo approach to study the practical identifiability problems. We illustrated the methodology on model reduction and theoretical/practical identifiability analysis on an example system representing the glycolysis in Saccharomyces cerevisiae cells.     

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.

     

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.     

Construction of a large scale integrated map of macrophage pathogen recognition and effector systems

Background

Mathematical models provide abstract representations of the information gained from experimental observations on the structure and function of a particular biological system. Conferring a predictive character on a given mathematical formulation often relies on determining a number of non-measurable parameters that largely condition the model's response. These parameters can be identified by fitting the model to experimental data. However, this fit can only be accomplished when identifiability can be guaranteed.

Results

We propose a novel iterative identification procedure for detecting and dealing with the lack of identifiability. The procedure involves the following steps: 1) performing a structural identifiability analysis to detect identifiable parameters; 2) globally ranking the parameters to assist in the selection of the most relevant parameters; 3) calibrating the model using global optimization methods; 4) conducting a practical identifiability analysis consisting of two (a priori and a posteriori) phases aimed at evaluating the quality of given experimental designs and of the parameter estimates, respectively and 5) optimal experimental design so as to compute the scheme of experiments that maximizes the quality and quantity of information for fitting the model.

Conclusions

The presented procedure was used to iteratively identify a mathematical model that describes the NF-κB regulatory module involving several unknown parameters. We demonstrated the lack of identifiability of the model under typical experimental conditions and computed optimal dynamic experiments that largely improved identifiability properties.     

Multi-Target Analysis and Design of Mitochondrial Metabolism

Analyzing and optimizing biological models is often identified as a research priority in biomedical engineering. An important feature of a model should be the ability to find the best condition in which an organism has to be grown in order to reach specific optimal output values chosen by the researcher. In this work, we take into account a mitochondrial model analyzed with flux-balance analysis. The optimal design and assessment of these models is achieved through single- and/or multi-objective optimization techniques driven by epsilon-dominance and identifiability analysis. Our optimization algorithm searches for the values of the flux rates that optimize multiple cellular functions simultaneously. The optimization of the fluxes of the metabolic network includes not only input fluxes, but also internal fluxes. A faster convergence process with robust candidate solutions is permitted by a relaxed Pareto dominance, regulating the granularity of the approximation of the desired Pareto front. We find that the maximum ATP production is linked to a total consumption of NADH, and reaching the maximum amount of NADH leads to an increasing request of NADH from the external environment. Furthermore, the identifiability analysis characterizes the type and the stage of three monogenic diseases. Finally, we propose a new methodology to extend any constraint-based model using protein abundances.     

Exploration of the intercellular heterogeneity of topotecan uptake into human breast cancer cells through compartmental modelling

A mathematical multi-cell model for the in vitro kinetics of the anti-cancer agent topotecan (TPT) following administration into a culture medium containing a population of human breast cancer cells (MCF-7 cell line) is described. This non-linear compartmental model is an extension of an earlier single-cell type model and has been validated using experimental data obtained using two-photon laser scanning microscopy (TPLSM). A structural identifiability analysis is performed prior to parameter estimation to test whether the unknown parameters within the model are uniquely determined by the model outputs. The full model has 43 compartments, with 107 unknown parameters, and it was found that the structural identifiability result could not be established even when using the latest version of the symbolic computation software Mathematica. However, by assuming that a priori knowledge is available for certain parameters, it was possible to reduce the number of parameters to 81, and it was found that this (Stage Two) model was globally (uniquely) structurally identifiable. The identifiability analysis demonstrated how valuable symbolic computation is in this context, as the analysis is far too lengthy and difficult to be performed by hand.     

Review: To be or not to be an identifiable model. Is this a relevant question in animal science modelling?

What is a good (useful) mathematical model in animal science? For models constructed for prediction purposes, the question of model adequacy (usefulness) has been traditionally tackled by statistical analysis applied to observed experimental data relative to model-predicted variables. However, little attention has been paid to analytic tools that exploit the mathematical properties of the model equations. For example, in the context of model calibration, before attempting a numerical estimation of the model parameters, we might want to know if we have any chance of success in estimating a unique best value of the model parameters from available measurements. This question of uniqueness is referred to as structural identifiability; a mathematical property that is defined on the sole basis of the model structure within a hypothetical ideal experiment determined by a setting of model inputs (stimuli) and observable variables (measurements). Structural identifiability analysis applied to dynamic models described by ordinary differential equations (ODEs) is a common practice in control engineering and system identification. This analysis demands mathematical technicalities that are beyond the academic background of animal science, which might explain the lack of pervasiveness of identifiability analysis in animal science modelling. To fill this gap, in this paper we address the analysis of structural identifiability from a practitioner perspective by capitalizing on the use of dedicated software tools. Our objectives are (i) to provide a comprehensive explanation of the structural identifiability notion for the community of animal science modelling, (ii) to assess the relevance of identifiability analysis in animal science modelling and (iii) to motivate the community to use identifiability analysis in the modelling practice (when the identifiability question is relevant). We focus our study on ODE models. By using illustrative examples that include published mathematical models describing lactation in cattle, we show how structural identifiability analysis can contribute to advancing mathematical modelling in animal science towards the production of useful models and, moreover, highly informative experiments via optimal experiment design. Rather than attempting to impose a systematic identifiability analysis to the modelling community during model developments, we wish to open a window towards the discovery of a powerful tool for model construction and experiment design.     

Analysis of unique structural identifiability via submodels

The paper presents a sufficient and necessary condition for unique structural identifiability of linear compartmental models. By virtue of this result unique identifiability can be tested via the analysis of some submodels of the original model. Thus, the identifiability problem is reduced step by step to simpler and, finally, to rather trivial problems. In addition to the knowledge of the symbolic expression of the transfer-function matrix, the proposed method of full-rank submodels requires only some numerical rank determinations, and hence allows for a quick and interactive test for unique structural identifiability. The procedure also gives a lower bound on the number of different solutions.     

The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows.

A structural identifiability analysis is performed on a mathematical model for the coupled transmission of two classes of pathogen. The pathogens, classified as major and minor, are aetiological agents of mastitis in dairy cows that interact directly and via the immunological reaction in their hosts. Parameter estimates are available from experimental data for all but four of the parameters in the model. Data from a longitudinal study of infection are used to estimate these unknown parameters. A novel approach and application of structural identifiability analysis is combined in this paper with the estimation of cross-protection parameters using epidemiological data.     

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.     

Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences

In this paper methods from differential algebra are used to study the structural identifiability of biological and pharmacokinetics models expressed in state-space form and with a structure given by rational functions. The focus is on the examples presented and on the application of efficient, automatic methods to test for structural identifiability for various input-output experiments. Differential algebra methods are coupled with Gr?bner bases, Lie derivatives and the Taylor series expansion in order to obtain efficient algorithms. In particular, an upper bound on the number of derivatives needed for the Taylor series approach for a structural identifiability analysis of rational function models is given.     

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.     

Analysis of endogenous process behavior in activated sludge

In this article, an autonomous four-compartment model that describes the endogenous respiration in an aerobic biodegradation process is proposed and analyzed theoretically. First, the multi-time scale of the system's behavior, to be taken into account in subsequent analyses, is emphasized. Then, an identifiability and observability study, given measurements of MLVSS (mixed liquor volatile suspended solids) and respiration rate, is performed for use under practical circumstances, such as in state and parameter estimation. It appears that the process is observable, but not fully identifiable. Hence, for the identification of some of the model parameters, additional measurements or experiments, also indicated here, have to be performed. Furthermore, it is shown that, under quasi-steady state conditions which, in general, appear shortly after initialization of an endogenous respiration experiment, the model can be reduced significantly. Finally, results of parameter estimation from available data are presented and discussed.     

Nonparametric methods for survival/sacrifice experiments

In many carcinogenicity studies, the time to disease occurrence is not clinically observable; a survival/sacrifice experiment is considered for nonparametric inference about the rate of disease occurrence. A multistate model for disease development and death is considered and an algorithm of the EM type for maximum likelihood estimation is obtained. Questions of identifiability and estimability are addressed. Under the model, interval hazards for disease occurrence are identifiable for intervals defined by the sacrifice times. A score test is developed appropriate for the comparison of two groups with respect to disease development without need of any assumption concerning lethality of the disease concerned.     

Development of a compartmental model of human zinc metabolism: identifiability and multiple studies analyses

A compartmental model of zinc metabolism has been developed from stable isotope tracer studies of five healthy adults. Multiple isotope tracers were administered orally and intravenously, and the resulting enrichment was measured in plasma, erythrocytes, urine, and feces for as long as 3 wk. Data from total zinc measurements and model-independent calculations of various steady-state parameters were also modeled with the kinetic data. A structure comprised of 14 compartments and as many as 25 unknown kinetic parameters was developed to adequately model the data from each of the individual studies. The structural identifiability of the model was established using the GLOBI2 identifiability analysis software. Numerical identifiability of parameter estimates was evaluated using statistical data provided by SAAM. A majority of the model parameters was estimated with sufficient statistical certainty to be considered well determined. After the fitting of the model and data from the individual studies using SAAM/CONSAM, results were submitted to SAAM extended multiple studies analysis for aggregation into a single set of population parameters and statistics. The model was judged to be valid based on criteria described elsewhere.     

Identification in an anaerobic batch system: global sensitivity analysis, multi-start strategy and optimization criterion selection

Several mathematical models have been developed in anaerobic digestion systems and a variety of methods have been used for parameter estimation and model validation. However, structural and parametric identifiability questions are relatively seldom addressed in the reported AD modeling studies. This paper presents a 3-step procedure for the reliable estimation of a set of kinetic and stoichiometric parameters in a simplified model of the anaerobic digestion process. This procedure includes the application of global sensitivity analysis, which allows to evaluate the interaction among the identified parameters, multi-start strategy that gives a picture of the possible local minima and the selection of optimization criteria or cost functions. This procedure is applied to the experimental data collected from a lab-scale sequencing batch reactor. Two kinetic parameters and two stoichiometric coefficients are estimated and their accuracy was also determined. The classical least-squares cost function appears to be the best choice in this case study.     

Decomposition-based qualitative experiment design algorithms for a class of compartmental models.

Qualitative experiment design, to determine experimental input/output configurations that provide identifiability for specific parameters of interest, can be extremely difficult if the number of unknown parameters and the number of compartments are relatively large. However, the problem can be considerably simplified if the parameters can be divided into several groups for separate identification and the model can be decomposed into smaller submodels for separate experiment design. Model decomposition-based experiment design algorithms are proposed for a practical class of large-scale compartmental models representative of biosystems characterized by multiple input sources and unidirectional interconnectivity among subsystems. The model parameters are divided into three types, each of which is identified consecutively, in three stages, using simpler submodel experiment designs. Several practical examples are presented. Necessary and sufficient conditions for identifiability using the algorithm are also discussed.     

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.