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.

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].

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.

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.

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.

Primary mouse hepatocytes for systems biology approaches: a standardized in vitro system for modelling of signal transduction pathways.

Complex cellular networks regulate regeneration, detoxification and differentiation of hepatocytes. By combining experimental data with mathematical modelling, systems biology holds great promises to elucidate the key regulatory mechanisms involved and predict targets for efficient intervention. For the generation of high-quality quantitative data suitable for mathematical modelling a standardised in vitro system is essential. Therefore the authors developed standard operating procedures for the preparation and cultivation of primary mouse hepatocytes. To reliably monitor the dynamic induction of signalling pathways, the authors established starvation conditions and evaluated the extent of starvation-associated stress by quantifying several metabolic functions of cultured primary hepatocytes, namely activities of glutathione-S-transferase, glutamine synthetase, CYP3A as well as secretion of lactate and urea into the culture medium. Establishment of constant metabolic activities after an initial decrease compared with freshly isolated hepatocytes showed that the cultured hepatocytes achieve a new equilibrium state that was not affected by our starving conditions. To verify the highly reproducible dynamic activation of signalling pathways in the in vitro system, the authors examined the JAK-STAT, SMAD, PI3 kinase, MAP kinase, NF-kappaB and Wnt/beta-catenin signalling pathways. For the induction of gp130, JAK1 and STAT3 phosphorylation IL6 was used, whereas TGFbeta was applied to activate the phosphorylation of SMAD1, SMAD2 and SMAD3. Both Akt/PKB and ERK1/2 phosphorylation were stimulated by the addition of hepatocyte growth factor. The time-dependent induction of a pool of signalling competent beta-catenin was monitored in response to the inhibition of GSK3beta. To analyse whether phosphorylation is actually leading to transcriptional responses, luciferase reporter gene constructs driven by multiple copies of TGFbeta-responsive motives were applied, demonstrating a dose-dependent increase in luciferase activity. Moreover, the induction of apoptosis by the TNF-like cytokine Fas ligand was studied in the in vitro system. Thus, the mouse hepatocyte in vitro system provides an important basis for the generation of high-quality quantitative data under standardised cell culture conditions that is essential to elucidate critical hepatocellular functions by the systems biology approach.

Quantitative data generation for systems biology: the impact of randomisation, calibrators and normalisers.

Systems biology is an approach to the analysis and prediction of the dynamic behaviour of biological networks through mathematical modelling based on experimental data. The current lack of reliable quantitative data, especially in the field of signal transduction, means that new methodologies in data acquisition and processing are needed. Here, we present methods to advance the established techniques of immunoprecipitation and immunoblotting to more accurate and quantitative procedures. We propose randomisation of sample loading to disrupt lane correlations and the use of normalisers and calibrators for data correction. To predict the impact of each method on improving the data quality we used simulations. These studies showed that randomisation reduces the standard deviation of a smoothed signal by 55% +/- 10%, independently from most experimental settings. Normalisation with appropriate endogenous or external proteins further reduces the deviation from the true values. As the improvement strongly depends on the quality of the normaliser measurement, a criteria-based normalisation procedure was developed. Our approach was experimentally verified by application of the proposed methods to time course data obtained by the immunoblotting technique. This analysis showed that the procedure is robust and can significantly improve the quality of experimental data.

Aperiodic flow-induced oscillations of collapsible tubes: a critical reappraisal.

The evidence for the aperiodic self-excited oscillations of flow-conveying collapsible tubes being mathematically chaotic is re-examined. Many cases which powerfully suggest nonlinear deterministic behaviour have not been recorded over time-spans which allow their exhaustive examination. The present investigation centred on a previously recorded robust and generic oscillation, but more recent and more discerning tests were applied. Despite hints that a low embedding dimension might suffice, the data appeared on most indices high-dimensional. A U-shaped return map was found and modelled using both radial basis functions and polynomials, but lack of detailed structure in the map prevented effective parameter estimation. On the basis of power-law rather than exponential divergence of nearby trajectories, and of inability to discriminate against behaviour which would also be manifested by a surrogate consisting of a noise-perturbed nonlinear periodic oscillator, it is concluded that the data do not support the idea that the aperiodicity in the particular oscillation examined is caused by deterministic chaos. There was evidence that the distributed nature of the physical system might underlie aspects of the high dimensionality. We advocate equally searching testing of any future candidate chaotic oscillations in the investigation of collapsed-tube flows.

How well can epileptic seizures be predicted? An evaluation of a nonlinear method.

The unpredictability of the occurrence of epileptic seizures contributes to the burden of the disease to a major degree. Thus, various methods have been proposed to predict the onset of seizures based on EEG recordings. A nonlinear feature motivated by the correlation dimension is a seemingly promising approach. In a previous study this method was reported to identify 'preictal dimension drops' up to 19 min before seizure onset, exceeding the variability of interictal data sets of 30-50 min duration. Here we have investigated the sensitivity and specificity of this method based on invasive long-term recordings from 21 patients with medically intractable partial epilepsies, who underwent invasive pre-surgical monitoring. The evaluation of interictal 24-h recordings comprising the sleep-wake cycle showed that only one out of 88 seizures was preceded by a significant preictal dimension drop. In a second analysis, the relation between dimension drops within time windows of up to 50 min before seizure onset and interictal periods was investigated. For false-prediction rates below 0.1/h, the sensitivity ranged from 8.3 to 38.3% depending on the prediction window length. Overall, the mean length and amplitude of dimension drops showed no significant differences between interictal and preictal data sets.

The seizure prediction characteristic: a general framework to assess and compare seizure prediction methods.

The unpredictability of seizures is a central problem for all patients suffering from uncontrolled epilepsy. Recently, numerous methods have been suggested that claim to predict from the EEG the onset of epileptic seizures. In parallel, new therapeutic devices are in development that could control upcoming seizures provided that their onset is known in advance. A reliable clinical application controlling seizures, consisting of a seizure prediction method and an intervention system, would improve patient quality of life. The question therefore arises as to whether the performance of the seizure prediction methods is already sufficient for clinical applications. The answer requires assessment criteria to judge and compare these methods, but recognized criteria still do not exist. Based on clinical, behavioral, and statistical considerations, we suggest the "seizure prediction characteristic" to evaluate seizure prediction methods. Results of this approach are exemplified by its application to the "dynamical similarity index" seizure prediction method using 582 hours of intracranial EEG data, including 88 seizures.