Clemens Kreutz

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.

Systems biology: experimental design

Experimental design has a long tradition in statistics, engineering and life sciences, dating back to the beginning of the last century when optimal designs for industrial and agricultural trials were considered. In cell biology, the use of mathematical modeling approaches raises new demands on experimental planning. A maximum informative investigation of the dynamic behavior of cellular systems is achieved by an optimal combination of stimulations and observations over time. In this minireview, the existing approaches concerning this optimization for parameter estimation and model discrimination are summarized. Furthermore, the relevant classical aspects of experimental design, such as randomization, replication and confounding, are reviewed.

Data-based identifiability analysis of non-linear dynamical models

MOTIVATION Mathematical modelling of biological systems is becoming a standard approach to investigate complex dynamic, non-linear interaction mechanisms in cellular processes. However, models may comprise non-identifiable parameters which cannot be unambiguously determined. Non-identifiability manifests itself in functionally related parameters, which are difficult to detect. RESULTS We present the method of mean optimal transformations, a non-parametric bootstrap-based algorithm for identifiability testing, capable of identifying linear and non-linear relations of arbitrarily many parameters, regardless of model size or complexity. This is performed with use of optimal transformations, estimated using the alternating conditional expectation algorithm (ACE). An initial guess or prior knowledge concerning the underlying relation of the parameters is not required. Independent, and hence identifiable parameters are determined as well. The quality of data at disposal is included in our approach, i.e. the non-linear model is fitted to data and estimated parameter values are investigated with respect to functional relations. We exemplify our approach on a realistic dynamical model and demonstrate that the variability of estimated parameter values decreases from 81 to 1% after detection and fixation of structural non-identifiabilities.

Gene expression profiling in polycythaemia vera: overexpression of transcription factor NF-E2

The molecular aetiology of polycythaemia vera (PV) remains unknown and the differential diagnosis between PV and secondary erythrocytosis (SE) can be challenging. Gene expression profiling can identify candidates involved in the pathophysiology of PV and generate a molecular signature to aid in diagnosis. We thus performed cDNA microarray analysis on 40 PV and 12 SE patients. Two independent data sets were obtained: using a two‐step training/validation design, a set of 64 genes (class predictors) was determined, which correctly discriminated PV from SE patients. Separately 253 genes were identified to be upregulated and 391 downregulated more than 1·5‐fold in PV compared with healthy controls (P < 0·01). Of the genes overexpressed in PV, 27 contained Sp1 sites: we therefore propose that altered activity of Sp1‐like transcription factors may contribute to the molecular aetiology of PV. One Sp1 target, the transcription factor NF‐E2 [nuclear factor (erythroid‐derived 2)], is overexpressed 2‐ to 40‐fold in PV patients. In PV bone marrow, NF‐E2 is overexpressed in megakaryocytes, erythroid and granulocytic precursors. It has been shown that overexpression of NF‐E2 leads to the development of erythropoietin‐independent erythroid colonies and that ectopic NF‐E2 expression can reprogram monocytic cells towards erythroid and megakaryocytic differentiation. Transcription factor concentration may thus control lineage commitment. We therefore propose that elevated concentrations of NF‐E2 in PV patients lead to an overproduction of erythroid and, in some patients, megakaryocytic cells/platelets. In this model, the level of NF‐E2 overexpression determines both the severity of erythrocytosis and the concurrent presence or absence of thrombocytosis.

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.

Addressing parameter identifiability by model-based experimentation.

Mathematical description of biological processes such as gene regulatory networks or signalling pathways by dynamic models utilising ordinary differential equations faces challenges if the model parameters like rate constants are estimated from incomplete and noisy experimental data. Typically, biological networks are only partially observed. Only a fraction of the modelled molecular species is measurable directly. This can result in structurally non-identifiable model parameters. Furthermore, practical non-identifiability can arise from limited amount and quality of experimental data. In the challenge of growing model complexity on one side, and experimental limitations on the other side, both types of non-identifiability arise frequently in systems biological applications often prohibiting reliable prediction of system dynamics. On theoretical grounds this article summarises how and why both types of non-identifiability arise. It exemplifies pitfalls where models do not yield reliable predictions of system dynamics because of non-identifiabilities. Subsequently, several approaches for identifiability analysis proposed in the literature are discussed. The aim is to provide an overview of applicable methods for detecting parameter identifiability issues. Once non-identifiability is detected, it can be resolved either by experimental design, measuring additional data under suitable conditions; or by model reduction, tailoring the size of the model to the information content provided by the experimental data. Both strategies enhance model predictability and will be elucidated by an example application. [Includes supplementary material].

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.

An error model for protein quantification

MOTIVATION Quantitative experimental data is the critical bottleneck in the modeling of dynamic cellular processes in systems biology. Here, we present statistical approaches improving reproducibility of protein quantification by immunoprecipitation and immunoblotting. RESULTS Based on a large data set with more than 3600 data points, we unravel that the main sources of biological variability and experimental noise are multiplicative and log-normally distributed. Therefore, we suggest a log-transformation of the data to obtain additive normally distributed noise. After this transformation, common statistical procedures can be applied to analyze the data. An error model is introduced to account for technical as well as biological variability. Elimination of these systematic errors decrease variability of measurements and allow for a more precise estimation of underlying dynamics of protein concentrations in cellular signaling. The proposed error model is relevant for simulation studies, parameter estimation and model selection, basic tools of systems biology. AVAILABILITY Matlab and R code is available from the authors on request. The data can be downloaded from our website www.fdm.uni-freiburg.de/~ckreutz/data.

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.