Marcel Schilling

Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood

MOTIVATION Mathematical description of biological reaction networks by differential equations leads to large models whose parameters are calibrated in order to optimally explain experimental data. Often only parts of the model can be observed directly. Given a model that sufficiently describes the measured data, it is important to infer how well model parameters are determined by the amount and quality of experimental data. This knowledge is essential for further investigation of model predictions. For this reason a major topic in modeling is identifiability analysis. RESULTS We suggest an approach that exploits the profile likelihood. It enables to detect structural non-identifiabilities, which manifest in functionally related model parameters. Furthermore, practical non-identifiabilities are detected, that might arise due to limited amount and quality of experimental data. Last but not least confidence intervals can be derived. The results are easy to interpret and can be used for experimental planning and for model reduction. AVAILABILITY An implementation is freely available for MATLAB and the PottersWheel modeling toolbox at http://web.me.com/andreas.raue/profile/software.html. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

Covering a Broad Dynamic Range: Information Processing at the Erythropoietin Receptor

Seeing EPO The supply of red blood cells in mammals is controlled by the cytokine erythropoietin (EPO). In physiological situations, the concentration of EPO can change by 1000-fold. Becker et al. (p. 1404, published online 20 May) used a combination of mathematical modeling and experimental analysis to discern how cells can maintain a linear response to such a broad range of EPO concentrations. Critical features included internalization of EPO-bound receptors and subsequent degradation of the EPO ligand. Replenishment of receptors at the cell surface required a large supply of EPO receptors maintained in reserve inside the cell. These mechanisms allow cells to experience large increases in EPO concentration without becoming refractory to further stimulation. Modeling and experiments help to explain responsiveness of red blood cell precursors to very large changes in a proliferative signal. Cell surface receptors convert extracellular cues into receptor activation, thereby triggering intracellular signaling networks and controlling cellular decisions. A major unresolved issue is the identification of receptor properties that critically determine processing of ligand-encoded information. We show by mathematical modeling of quantitative data and experimental validation that rapid ligand depletion and replenishment of the cell surface receptor are characteristic features of the erythropoietin (Epo) receptor (EpoR). The amount of Epo-EpoR complexes and EpoR activation integrated over time corresponds linearly to ligand input; this process is carried out over a broad range of ligand concentrations. This relation depends solely on EpoR turnover independent of ligand binding, which suggests an essential role of large intracellular receptor pools. These receptor properties enable the system to cope with basal and acute demand in the hematopoietic system.

Lessons learned from quantitative dynamical modeling in systems biology.

Due to the high complexity of biological data it is difficult to disentangle cellular processes relying only on intuitive interpretation of measurements. A Systems Biology approach that combines quantitative experimental data with dynamic mathematical modeling promises to yield deeper insights into these processes. Nevertheless, with growing complexity and increasing amount of quantitative experimental data, building realistic and reliable mathematical models can become a challenging task: the quality of experimental data has to be assessed objectively, unknown model parameters need to be estimated from the experimental data, and numerical calculations need to be precise and efficient. Here, we discuss, compare and characterize the performance of computational methods throughout the process of quantitative dynamic modeling using two previously established examples, for which quantitative, dose- and time-resolved experimental data are available. In particular, we present an approach that allows to determine the quality of experimental data in an efficient, objective and automated manner. Using this approach data generated by different measurement techniques and even in single replicates can be reliably used for mathematical modeling. For the estimation of unknown model parameters, the performance of different optimization algorithms was compared systematically. Our results show that deterministic derivative-based optimization employing the sensitivity equations in combination with a multi-start strategy based on latin hypercube sampling outperforms the other methods by orders of magnitude in accuracy and speed. Finally, we investigated transformations that yield a more efficient parameterization of the model and therefore lead to a further enhancement in optimization performance. We provide a freely available open source software package that implements the algorithms and examples compared here.

Computational processing and error reduction strategies for standardized quantitative data in biological networks

High‐quality quantitative data generated under standardized conditions is critical for understanding dynamic cellular processes. We report strategies for error reduction, and algorithms for automated data processing and for establishing the widely used techniques of immunoprecipitation and immunoblotting as highly precise methods for the quantification of protein levels and modifications. To determine the stoichiometry of cellular components and to ensure comparability of experiments, relative signals are converted to absolute values. A major source for errors in blotting techniques are inhomogeneities of the gel and the transfer procedure leading to correlated errors. These correlations are prevented by randomized gel loading, which significantly reduces standard deviations. Further error reduction is achieved by using housekeeping proteins as normalizers or by adding purified proteins in immunoprecipitations as calibrators in combination with criteria‐based normalization. Additionally, we developed a computational tool for automated normalization, validation and integration of data derived from multiple immunoblots. In this way, large sets of quantitative data for dynamic pathway modeling can be generated, enabling the identification of systems properties and the prediction of targets for efficient intervention.

Theoretical and experimental analysis links isoform- specific ERK signalling to cell fate decisions

Cell fate decisions are regulated by the coordinated activation of signalling pathways such as the extracellular signal‐regulated kinase (ERK) cascade, but contributions of individual kinase isoforms are mostly unknown. By combining quantitative data from erythropoietin‐induced pathway activation in primary erythroid progenitor (colony‐forming unit erythroid stage, CFU‐E) cells with mathematical modelling, we predicted and experimentally confirmed a distributive ERK phosphorylation mechanism in CFU‐E cells. Model analysis showed bow‐tie‐shaped signal processing and inherently transient signalling for cytokine‐induced ERK signalling. Sensitivity analysis predicted that, through a feedback‐mediated process, increasing one ERK isoform reduces activation of the other isoform, which was verified by protein over‐expression. We calculated ERK activation for biochemically not addressable but physiologically relevant ligand concentrations showing that double‐phosphorylated ERK1 attenuates proliferation beyond a certain activation level, whereas activated ERK2 enhances proliferation with saturation kinetics. Thus, we provide a quantitative link between earlier unobservable signalling dynamics and cell fate decisions.

Division of labor by dual feedback regulators controls JAK2/STAT5 signaling over broad ligand range

Cellular signal transduction is governed by multiple feedback mechanisms to elicit robust cellular decisions. The specific contributions of individual feedback regulators, however, remain unclear. Based on extensive time‐resolved data sets in primary erythroid progenitor cells, we established a dynamic pathway model to dissect the roles of the two transcriptional negative feedback regulators of the suppressor of cytokine signaling (SOCS) family, CIS and SOCS3, in JAK2/STAT5 signaling. Facilitated by the model, we calculated the STAT5 response for experimentally unobservable Epo concentrations and provide a quantitative link between cell survival and the integrated response of STAT5 in the nucleus. Model predictions show that the two feedbacks CIS and SOCS3 are most effective at different ligand concentration ranges due to their distinct inhibitory mechanisms. This divided function of dual feedback regulation enables control of STAT5 responses for Epo concentrations that can vary 1000‐fold in vivo. Our modeling approach reveals dose‐dependent feedback control as key property to regulate STAT5‐mediated survival decisions over a broad range of ligand concentrations.

Data2Dynamics: a modeling environment tailored to parameter estimation in dynamical systems

UNLABELLED Modeling of dynamical systems using ordinary differential equations is a popular approach in the field of systems biology. Two of the most critical steps in this approach are to construct dynamical models of biochemical reaction networks for large datasets and complex experimental conditions and to perform efficient and reliable parameter estimation for model fitting. We present a modeling environment for MATLAB that pioneers these challenges. The numerically expensive parts of the calculations such as the solving of the differential equations and of the associated sensitivity system are parallelized and automatically compiled into efficient C code. A variety of parameter estimation algorithms as well as frequentist and Bayesian methods for uncertainty analysis have been implemented and used on a range of applications that lead to publications. AVAILABILITY AND IMPLEMENTATION The Data2Dynamics modeling environment is MATLAB based, open source and freely available at http://www.data2dynamics.org. CONTACT andreas.raue@fdm.uni-freiburg.de SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

Predictive mathematical models of cancer signalling pathways.

Abstract.  Bachmann J, Raue A, Schilling M, Becker V, Timmer J, Klingmüller U (German Cancer Research Center, Heidelberg; BIOSS Centre for Biological Signalling Studies, Freiburg; and University of Freiburg, Freiburg; Germany). Predictive mathematical models of cancer signalling pathways (Key Symposium). J Intern Med 2012; 271:155–165.

Dynamic Mathematical Modeling of IL13-Induced Signaling in Hodgkin and Primary Mediastinal B-Cell Lymphoma Allows Prediction of Therapeutic Targets

Primary mediastinal B-cell lymphoma (PMBL) and classical Hodgkin lymphoma (cHL) share a frequent constitutive activation of JAK (Janus kinase)/STAT signaling pathway. Because of complex, nonlinear relations within the pathway, key dynamic properties remained to be identified to predict possible strategies for intervention. We report the development of dynamic pathway models based on quantitative data collected on signaling components of JAK/STAT pathway in two lymphoma-derived cell lines, MedB-1 and L1236, representative of PMBL and cHL, respectively. We show that the amounts of STAT5 and STAT6 are higher whereas those of SHP1 are lower in the two lymphoma cell lines than in normal B cells. Distinctively, L1236 cells harbor more JAK2 and less SHP1 molecules per cell than MedB-1 or control cells. In both lymphoma cell lines, we observe interleukin-13 (IL13)-induced activation of IL4 receptor α, JAK2, and STAT5, but not of STAT6. Genome-wide, 11 early and 16 sustained genes are upregulated by IL13 in both lymphoma cell lines. Specifically, the known STAT-inducible negative regulators CISH and SOCS3 are upregulated within 2 hours in MedB-1 but not in L1236 cells. On the basis of this detailed quantitative information, we established two mathematical models, MedB-1 and L1236 model, able to describe the respective experimental data. Most of the model parameters are identifiable and therefore the models are predictive. Sensitivity analysis of the model identifies six possible therapeutic targets able to reduce gene expression levels in L1236 cells and three in MedB-1. We experimentally confirm reduction in target gene expression in response to inhibition of STAT5 phosphorylation, thereby validating one of the predicted targets.

Standardizing experimental protocols.

Systems biology aims at understanding the behavior of biological networks by mathematical modeling based on experimental data. However, frequently experimental data is derived from poorly defined cellular systems, the procedures of data generation are insufficiently documented and data processing is arbitrary. For the advancement of systems biology, standardization at multiple levels is essential. Several systems biology consortia have started by focusing on standardization of cellular systems and experimental procedures. Minimum information standards for the description of data sets and common languages for the description of biological pathways as well as for mathematical modeling are being developed. Standardization is required to facilitate data exchange between different research groups and finally the assembly of large integrated models providing novel biological insights.