Numerical and graphical description of the information matrix in calibration experiments for state-space models

Clustering reveals limits of parameter identifiability in multi-parameter models of biochemical dynamics

Background

Compared to engineering or physics problems, dynamical models in quantitative biology typically depend on a relatively large number of parameters. Progress in developing mathematics to manipulate such multi-parameter models and so enable their efficient interplay with experiments has been slow. Existing solutions are significantly limited by model size.

Results

In order to simplify analysis of multi-parameter models a method for clustering of model parameters is proposed. It is based on a derived statistically meaningful measure of similarity between groups of parameters. The measure quantifies to what extend changes in values of some parameters can be compensated by changes in values of other parameters. The proposed methodology provides a natural mathematical language to precisely communicate and visualise effects resulting from compensatory changes in values of parameters. As a results, a relevant insight into identifiability analysis and experimental planning can be obtained. Analysis of NF- κB and MAPK pathway models shows that highly compensative parameters constitute clusters consistent with the network topology. The method applied to examine an exceptionally rich set of published experiments on the NF- κB dynamics reveals that the experiments jointly ensure identifiability of only 60 % of model parameters. The method indicates which further experiments should be performed in order to increase the number of identifiable parameters.

Conclusions

We currently lack methods that simplify broadly understood analysis of multi-parameter models. The introduced tools depict mutually compensative effects between parameters to provide insight regarding role of individual parameters, identifiability and experimental design. The method can also find applications in related methodological areas of model simplification and parameters estimation.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0205-8) contains supplementary material, which is available to authorized users.

Application of the International Water Association activated sludge models to describe aerobic sludge digestion

Batch and semi-continuous flow aerobic digesters were used to stabilize thickened waste-activated sludge at different initial conditions and mean solids retention times. Under dynamic conditions, total suspended solids, volatile suspended solids (VSS) and total and particulate chemical oxygen demand (COD and PCOD) were monitored in the batch reactors and effluent from the semi-continuous flow reactors. Activated Sludge Model (ASM) no. 1 and ASM no. 3 were applied to measured data (calibration data set) to evaluate the consistency and performances of models at different flow regimes for digester COD and VSS modelling. The results indicated that both ASM1 and ASM3 predicted digester COD, VSS and PCOD concentrations well (R2, Ra2 > or = 0.93). Parameter estimation concluded that compared to ASM1, ASM3 parameters were more consistent across different batch and semi-continuous flow runs with different operating conditions. Model validation on a data set independent from the calibration data successfully predicted digester COD (R2 = 0.88) and VSS (R2 = 0.94) concentrations by ASM3, while ASM1 overestimated both reactor COD (R2 = 0.74) and VSS concentrations (R2 = 0.79) after 15 days of aerobic batch digestion.

Systems biology: parameter estimation for biochemical models

Mathematical models of biological processes have various applications: to assist in understanding the functioning of a system, to simulate experiments before actually performing them, to study situations that cannot be dealt with experimentally, etc. Some parameters in the model can be directly obtained from experiments or from the literature. Others have to be inferred by comparing model results to experiments. In this minireview, we discuss the identifiability of models, both intrinsic to the model and taking into account the available data. Furthermore, we give an overview of the most frequently used approaches to search the parameter space.

Aerobic phosphorus release linked to acetate uptake in bio-P sludge: Process modeling using oxygen uptake rate

The main processes involved in enhanced biological phosphorus removal (EBPR) under anaerobic and subsequently aerobic conditions are widely described in the literature. Polyphosphate accumulating organisms (PAO) are the organisms responsible for this process. However, the mechanisms of PAO are not fully established yet under conditions that differ from the classical anaerobic/aerobic conditions. In this work, we made a comparison between the behavior of PAO under classical EBPR conditions and its behavior when consuming substrate under only aerobic conditions. In addition, oxygen uptake rate (OUR) was measured in the set of experiments under aerobic conditions to improve the characterization of the process. A kinetic and stoichiometric model based on Activated Sludge Model No.2 (ASM2) and including glycogen economy (AnOx model), calibrated for classical anaerobic/aerobic conditions, was not able to describe the experimental data since it underestimated the acetate consumption, the PHB storage, and the OUR. Two different hypotheses for describing the experimental measurements were proposed and modeled. Both hypotheses considered that PAO, under aerobic conditions, uptake acetate coupled to PHB storage, glycogen degradation, and phosphorus release as in anaerobic conditions. Moreover, the first hypothesis (PAO-hypothesis) considered that PAO were able to store acetate as PHB linked to oxygen consumption and the second one (OHO hypothesis) considered that this storage was due to ordinary heterotrophic organisms (OHO). Both hypotheses were evaluated by simulation extending the AnOx model with additional equations. The main differences observed were the predictions for PHB degradation during the famine phase and the OUR profile during both feast and famine phases. The OHO hypothesis described the experimental profiles more accurately than the PAO hypothesis.

Forward and backward uncertainty propagation: an oxidation ditch modelling example

In the field of water technology, forward uncertainty propagation is frequently used, whereas backward uncertainty propagation is rarely used. In forward uncertainty analysis, one moves from a given (or assumed) parameter subspace towards the corresponding distribution of the output or objective function. However, in the backward uncertainty propagation, one moves in the reverse direction, from the distribution function towards the parameter subspace. Backward uncertainty propagation, which is a generalisation of parameter estimation error analysis, gives information essential for designing experimental or monitoring programmes, and for tighter bounding of parameter uncertainty intervals. The procedure of carrying out backward uncertainty propagation is illustrated in this technical note by working example for an oxidation ditch wastewater treatment plant. Results obtained have demonstrated that essential information can be achieved by carrying out backward uncertainty propagation analysis.