Estimating chain length for time delays in dynamical systems using profile likelihood.

MOTIVATION: Apparent time delays in partly observed, biochemical reaction networks can be modelled by lumping a more complex reaction into a series of linear reactions often referred to as the linear chain trick. Since most delays in biochemical reactions are no true, hard delays but a consequence of complex unobserved processes, this approach often more closely represents the true system compared with delay differential equations. In this paper, we address the question of how to select the optimal number of additional equations, i.e. the chain length (CL). RESULTS: We derive a criterion based on parameter identifiability to infer CLs and compare this method to choosing the model with a CL that leads to the best fit in a maximum likelihood sense, which corresponds to optimizing the Bayesian information criterion. We evaluate performance with simulated data as well as with measured biological data for a model of JAK2/STAT5 signalling and access the influence of different model structures and data characteristics. Our analysis revealed that the proposed method features a superior performance when applied to biological models and data compared with choosing the model that maximizes the likelihood. AVAILABILITY AND IMPLEMENTATION: Models and data used for simulations are available at https://github.com/Data2Dynamics/d2d and http://jeti.uni-freiburg.de/PNAS_Swameye_Data. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Model Based Targeting of IL-6-Induced Inflammatory Responses in Cultured Primary Hepatocytes to Improve Application of the JAK Inhibitor Ruxolitinib.

IL-6 is a central mediator of the immediate induction of hepatic acute phase proteins (APP) in the liver during infection and after injury, but increased IL-6 activity has been associated with multiple pathological conditions. In hepatocytes, IL-6 activates JAK1-STAT3 signaling that induces the negative feedback regulator SOCS3 and expression of APPs. While different inhibitors of IL-6-induced JAK1-STAT3-signaling have been developed, understanding their precise impact on signaling dynamics requires a systems biology approach. Here we present a mathematical model of IL-6-induced JAK1-STAT3 signaling that quantitatively links physiological IL-6 concentrations to the dynamics of IL-6-induced signal transduction and expression of target genes in hepatocytes. The mathematical model consists of coupled ordinary differential equations (ODE) and the model parameters were estimated by a maximum likelihood approach, whereas identifiability of the dynamic model parameters was ensured by the Profile Likelihood. Using model simulations coupled with experimental validation we could optimize the long-term impact of the JAK-inhibitor Ruxolitinib, a therapeutic compound that is quickly metabolized. Model-predicted doses and timing of treatments helps to improve the reduction of inflammatory APP gene expression in primary mouse hepatocytes close to levels observed during regenerative conditions. The concept of improved efficacy of the inhibitor through multiple treatments at optimized time intervals was confirmed in primary human hepatocytes. Thus, combining quantitative data generation with mathematical modeling suggests that repetitive treatment with Ruxolitinib is required to effectively target excessive inflammatory responses without exceeding doses recommended by the clinical guidelines.

Driving the Model to Its Limit: Profile Likelihood Based Model Reduction.

In systems biology, one of the major tasks is to tailor model complexity to information content of the data. A useful model should describe the data and produce well-determined parameter estimates and predictions. Too small of a model will not be able to describe the data whereas a model which is too large tends to overfit measurement errors and does not provide precise predictions. Typically, the model is modified and tuned to fit the data, which often results in an oversized model. To restore the balance between model complexity and available measurements, either new data has to be gathered or the model has to be reduced. In this manuscript, we present a data-based method for reducing non-linear models. The profile likelihood is utilised to assess parameter identifiability and designate likely candidates for reduction. Parameter dependencies are analysed along profiles, providing context-dependent suggestions for the type of reduction. We discriminate four distinct scenarios, each associated with a specific model reduction strategy. Iterating the presented procedure eventually results in an identifiable model, which is capable of generating precise and testable predictions. Source code for all toy examples is provided within the freely available, open-source modelling environment Data2Dynamics based on MATLAB available at http://www.data2dynamics.org/, as well as the R packages dMod/cOde available at https://github.com/dkaschek/. Moreover, the concept is generally applicable and can readily be used with any software capable of calculating the profile likelihood.

Optimal experiment design for model selection in biochemical networks.

BACKGROUND: Mathematical modeling is often used to formalize hypotheses on how a biochemical network operates by discriminating between competing models. Bayesian model selection offers a way to determine the amount of evidence that data provides to support one model over the other while favoring simple models. In practice, the amount of experimental data is often insufficient to make a clear distinction between competing models. Often one would like to perform a new experiment which would discriminate between competing hypotheses. RESULTS: We developed a novel method to perform Optimal Experiment Design to predict which experiments would most effectively allow model selection. A Bayesian approach is applied to infer model parameter distributions. These distributions are sampled and used to simulate from multivariate predictive densities. The method is based on a k-Nearest Neighbor estimate of the Jensen Shannon divergence between the multivariate predictive densities of competing models. CONCLUSIONS: We show that the method successfully uses predictive differences to enable model selection by applying it to several test cases. Because the design criterion is based on predictive distributions, which can be computed for a wide range of model quantities, the approach is very flexible. The method reveals specific combinations of experiments which improve discriminability even in cases where data is scarce. The proposed approach can be used in conjunction with existing Bayesian methodologies where (approximate) posteriors have been determined, making use of relations that exist within the inferred posteriors.

Parameter trajectory analysis to identify treatment effects of pharmacological interventions.

The field of medical systems biology aims to advance understanding of molecular mechanisms that drive disease progression and to translate this knowledge into therapies to effectively treat diseases. A challenging task is the investigation of long-term effects of a (pharmacological) treatment, to establish its applicability and to identify potential side effects. We present a new modeling approach, called Analysis of Dynamic Adaptations in Parameter Trajectories (ADAPT), to analyze the long-term effects of a pharmacological intervention. A concept of time-dependent evolution of model parameters is introduced to study the dynamics of molecular adaptations. The progression of these adaptations is predicted by identifying necessary dynamic changes in the model parameters to describe the transition between experimental data obtained during different stages of the treatment. The trajectories provide insight in the affected underlying biological systems and identify the molecular events that should be studied in more detail to unravel the mechanistic basis of treatment outcome. Modulating effects caused by interactions with the proteome and transcriptome levels, which are often less well understood, can be captured by the time-dependent descriptions of the parameters. ADAPT was employed to identify metabolic adaptations induced upon pharmacological activation of the liver X receptor (LXR), a potential drug target to treat or prevent atherosclerosis. The trajectories were investigated to study the cascade of adaptations. This provided a counter-intuitive insight concerning the function of scavenger receptor class B1 (SR-B1), a receptor that facilitates the hepatic uptake of cholesterol. Although activation of LXR promotes cholesterol efflux and -excretion, our computational analysis showed that the hepatic capacity to clear cholesterol was reduced upon prolonged treatment. This prediction was confirmed experimentally by immunoblotting measurements of SR-B1 in hepatic membranes. Next to the identification of potential unwanted side effects, we demonstrate how ADAPT can be used to design new target interventions to prevent these.

Applications of analysis of dynamic adaptations in parameter trajectories.

Metabolic profiling in combination with pathway-based analyses and computational modelling are becoming increasingly important in clinical and preclinical research. Modelling multi-factorial, progressive diseases requires the integration of molecular data at the metabolome, proteome and transcriptome levels. Also the dynamic interaction of organs and tissues needs to be considered. The processes involved cover time scales that are several orders of magnitude different. We report applications of a computational approach to bridge the scales and different levels of biological detail. Analysis of dynamic adaptations in parameter trajectories (ADAPTs) aims to investigate phenotype transitions during disease development and after a therapeutic intervention. ADAPT is based on a time-dependent evolution of model parameters to describe the dynamics of metabolic adaptations. The progression of metabolic adaptations is predicted by identifying necessary dynamic changes in the model parameters to describe the transition between experimental data obtained during different stages. To get a better understanding of the concept, the ADAPT approach is illustrated in a theoretical study. Its application in research on progressive changes in lipoprotein metabolism is also discussed.

Parameter adaptations during phenotype transitions in progressive diseases.

BACKGROUND: The study of phenotype transitions is important to understand progressive diseases, e.g., diabetes mellitus, metabolic syndrome, and cardiovascular diseases. A challenge remains to explain phenotype transitions in terms of adaptations in molecular components and interactions in underlying biological systems. RESULTS: Here, mathematical modeling is used to describe the different phenotypes by integrating experimental data on metabolic pools and fluxes. Subsequently, trajectories of parameter adaptations are identified that are essential for the phenotypical changes. These changes in parameters reflect progressive adaptations at the transcriptome and proteome level, which occur at larger timescales. The approach was employed to study the metabolic processes underlying liver X receptor induced hepatic steatosis. Model analysis predicts which molecular processes adapt in time after pharmacological activation of the liver X receptor. Our results show that hepatic triglyceride fluxes are increased and triglycerides are especially stored in cytosolic fractions, rather than in endoplasmic reticulum fractions. Furthermore, the model reveals several possible scenarios for adaptations in cholesterol metabolism. According to the analysis, the additional quantification of one cholesterol flux is sufficient to exclude many of these hypotheses. CONCLUSIONS: We propose a generic computational approach to analyze biological systems evolving through various phenotypes and to predict which molecular processes are responsible for the transition. For the case of liver X receptor induced hepatic steatosis the novel approach yields information about the redistribution of fluxes and pools of triglycerides and cholesterols that was not directly apparent from the experimental data. Model analysis provides guidance which specific molecular processes to study in more detail to obtain further understanding of the underlying biological system.