Cellular metabolism continuously processes an enormous range of external compounds into endogenous metabolites and is as such a key element in human physiology. The multifaceted physiological role of the metabolic network fulfilling the catalytic conversions can only be fully understood from a whole-body perspective where the causal interplay of the metabolic states of individual cells, the surrounding tissue and the whole organism are simultaneously considered. We here present an approach relying on dynamic flux balance analysis that allows the integration of metabolic networks at the cellular scale into standardized physiologically-based pharmacokinetic models at the whole-body level. To evaluate our approach we integrated a genome-scale network reconstruction of a human hepatocyte into the liver tissue of a physiologically-based pharmacokinetic model of a human adult. The resulting multiscale model was used to investigate hyperuricemia therapy, ammonia detoxification and paracetamol-induced toxication at a systems level. The specific models simultaneously integrate multiple layers of biological organization and offer mechanistic insights into pathology and medication. The approach presented may in future support a mechanistic understanding in diagnostics and drug development.

BackgroundMathematical modeling and analysis have become, for the study of biological and cellular processes, an important complement to experimental research. However, the structural and quantitative knowledge available for such processes is frequently limited, and measurements are often subject to inherent and possibly large uncertainties. This results in competing model hypotheses, whose kinetic parameters may not be experimentally determinable. Discriminating among these alternatives and estimating their kinetic parameters is crucial to improve the understanding of the considered process, and to benefit from the analytical tools at hand.ResultsIn this work we present a set-based framework that allows to discriminate between competing model hypotheses and to provide guaranteed outer estimates on the model parameters that are consistent with the (possibly sparse and uncertain) experimental measurements. This is obtained by means of exact proofs of model invalidity that exploit the polynomial/rational structure of biochemical reaction networks, and by making use of an efficient strategy to balance solution accuracy and computational effort.ConclusionsThe practicability of our approach is illustrated with two case studies. The first study shows that our approach allows to conclusively rule out wrong model hypotheses. The second study focuses on parameter estimation, and shows that the proposed method allows to evaluate the global influence of measurement sparsity, uncertainty, and prior knowledge on the parameter estimates. This can help in designing further experiments leading to improved parameter estimates.

BackgroundApoptosis is a form of programmed cell death essential for the maintenance of homeostasis and the removal of potentially damaged cells in multicellular organisms. By binding its cognate membrane receptor, TNF receptor type 1 (TNF-R1), the proinflammatory cytokine Tumor Necrosis Factor (TNF) activates pro-apoptotic signaling via caspase activation, but at the same time also stimulates nuclear factor κB (NF-κB)-mediated survival pathways. Differential dose-response relationships of these two major TNF signaling pathways have been described experimentally and using mathematical modeling. However, the quantitative analysis of the complex interplay between pro- and anti-apoptotic signaling pathways is an open question as it is challenging for several reasons: the overall signaling network is complex, various time scales are present, and cells respond quantitatively and qualitatively in a heterogeneous manner.ResultsThis study analyzes the complex interplay of the crosstalk of TNF-R1 induced pro- and anti-apoptotic signaling pathways based on an experimentally validated mathematical model. The mathematical model describes the temporal responses on both the single cell level as well as the level of a heterogeneous cell population, as observed in the respective quantitative experiments using TNF-R1 stimuli of different strengths and durations. Global sensitivity of the heterogeneous population was quantified by measuring the average gradient of time of death versus each population parameter. This global sensitivity analysis uncovers the concentrations of Caspase-8 and Caspase-3, and their respective inhibitors BAR and XIAP, as key elements for deciding the cell's fate. A simulated knockout of the NF-κB-mediated anti-apoptotic signaling reveals the importance of this pathway for delaying the time of death, reducing the death rate in the case of pulse stimulation and significantly increasing cell-to-cell variability.ConclusionsCell ensemble modeling of a heterogeneous cell population including a global sensitivity analysis presented here allowed us to illuminate the role of the different elements and parameters on apoptotic signaling. The receptors serve to transmit the external stimulus; procaspases and their inhibitors control the switching from life to death, while NF-κB enhances the heterogeneity of the cell population. The global sensitivity analysis of the cell population model further revealed an unexpected impact of heterogeneity, i.e. the reduction of parametric sensitivity.

Summary: Often competing hypotheses for biochemical networks exist in the form of different mathematical models with unknown parameters. Considering available experimental data, it is then desired to reject model hypotheses that are inconsistent with the data, or to estimate the unknown parameters. However, these tasks are complicated because experimental data are typically sparse, uncertain, and are frequently only available in form of qualitative if–then observations. ADMIT (Analysis, Design and Model Invalidation Toolbox) is a MatLabTM-based tool for guaranteed model invalidation, state and parameter estimation. The toolbox allows the integration of quantitative measurement data, a priori knowledge of parameters and states, and qualitative information on the dynamic or steady-state behavior. A constraint satisfaction problem is automatically generated and algorithms are implemented for solving the desired estimation, invalidation or analysis tasks. The implemented methods built on convex relaxation and optimization and therefore provide guaranteed estimation results and certificates for invalidity.Availability: ADMIT, tutorials and illustrative examples are available free of charge for non-commercial use at http://ifatwww.et.uni-magdeburg.de/syst/ADMIT/Contact: stefan.streif@ovgu.de

This work introduces a unified framework for model invalidation and parameter estimation for nonlinear systems. We consider a model given by implicit nonlinear difference equations that are polynomial in the variables. Experimental data is assumed to be available as possibly sparse, uncertain, but (set-)bounded measurements. The derived approach is based on the reformulation of the invalidation and parameter/state estimation tasks into a set-based feasibility problem. Exploiting the polynomial structure of the considered model class, the resulting non-convex feasibility problem is relaxed into a convex semi-definite one, for which infeasibility can be efficiently checked. The parameter/state estimation task is then reformulated as an outer-bounding problem. In comparison to other methods, we check for feasibility of whole parameter/state regions. The practicability of the proposed approach is demonstrated with two simple biological example systems.

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Production of bio-pharmaceuticals in cell culture, such as mammalian cells, is challenging. Mathematical models can provide support to the analysis, optimization, and the operation of production processes. In particular, unstructured models are suited for these purposes, since they can be tailored to particular process conditions. To this end, growth phases and the most relevant factors influencing cell growth and product formation have to be identified. Due to noisy and erroneous experimental data, unknown kinetic parameters, and the large number of combinations of influencing factors, currently there are only limited structured approaches to tackle these issues. We outline a structured set-based approach to identify different growth phases and the factors influencing cell growth and metabolism. To this end, measurement uncertainties are taken explicitly into account to bound the time-dependent specific growth rate based on the observed increase of the cell concentration. Based on the bounds on the specific growth rate, we can identify qualitatively different growth phases and (in-)validate hypotheses on the factors influencing cell growth and metabolism. We apply the approach to a mammalian suspension cell line (AGE1.HN). We show that growth in batch culture can be divided into two main growth phases. The initial phase is characterized by exponential growth dynamics, which can be described consistently by a relatively simple unstructured and segregated model. The subsequent phase is characterized by a decrease in the specific growth rate, which, as shown, results from substrate limitation and the pH of the medium. An extended model is provided which describes the observed dynamics of cell growth and main metabolites, and the corresponding kinetic parameters as well as their confidence intervals are estimated. The study is complemented by an uncertainty and outlier analysis. Overall, we demonstrate utility of set-based methods for analyzing cell growth and metabolism under conditions of uncertainty.

Analysis and safety considerations of chemical and biological processes frequently require an outer approximation of the set of all feasible steady-states. Nonlinearities, uncertain parameters, and discrete variables complicate the calculation of guaranteed outer bounds. In this paper, the problem of outer-approximating the region of feasible steady-states, for processes described by uncertain nonlinear differential algebraic equations including discrete variables and discrete changes in the dynamics, is adressed. The calculation of the outer bounding sets is based on a relaxed version of the corresponding feasibility problem. It uses the Lagrange dual problem to obtain certificates for regions in state space not containing steady-states. These infeasibility certificates can be computed efficiently by solving a semidefinite program, rendering the calculation of the outer bounding set computationally feasible. The derived method guarantees globally valid outer bounds for the steady-states of nonlinear processes described by differential equations. It allows to consider discrete variables, as well as switching system dynamics. The method is exemplified by the analysis of a simple chemical reactor showing parametric uncertainties and large variability due to the appearance of bifurcations characterising the ignition and extinction of a reaction.