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.

We show that parameter estimation for pattern formation models can be efficiently performed using an evolution strategy (ES). As a case study we use a quantitative spatio-temporal model of the regulatory network for early development in Drosophila melanogaster. In order to estimate the parameters, the simulated results are compared to a time series of gene products involved in the network obtained with immunohistochemistry. We demonstrate that a (mu,lambda)-ES can be used to find good quality solutions in the parameter estimation. We also show that an ES with multiple populations is 5-140 times as fast as parallel simulated annealing for this case study, and that combining ES with a local search results in an efficient parameter estimation method.

Background: Inverse modelling of gene regulatory networks (GRNs) capable of simulating continuous spatio-temporal biological processes requires accurate data and a good description of the system. If quantitative relations between genes cannot be extracted from direct measurements, an efficient method to estimate the unknown parameters is mandatory. A model that has been proposed to simulate spatio-temporal gene expression patterns is the connectionist model. This method describes the quantitative dynamics of a regulatory network in space. The model parameters are estimated by means of model-fitting algorithms. The gene interactions are identified without making any prior assumptions concerning the network connectivity. As a result, the inverse modelling might lead to multiple circuits showing the same quantitative behaviour and it is not possible to identify one optimal circuit. Consequently, it is important to address the quality of the circuits in terms of model robustness.

BackgroundInference of gene regulatory networks (GRNs) requires accurate data, a method to simulate the expression patterns and an efficient optimization algorithm to estimate the unknown parameters. Using this approach it is possible to obtain alternative circuits without making any a priori assumptions about the interactions, which all simulate the observed patterns. It is important to analyze the properties of the circuits.FindingsWe have analyzed the simulated gene expression patterns of previously obtained circuits that describe gap gene dynamics during early Drosophila melanogaster embryogenesis. Using hierarchical clustering we show that amplitude variation and defects observed in the simulated gene expression patterns are linked to similar circuits, which can be grouped. Furthermore, analysis of the long-term dynamics revealed four main dynamical attractors comprising stable patterns and oscillatory patterns. In addition, we also performed a correlation analysis on the parameters showing an intricate correlation pattern.ConclusionsThe analysis demonstrates that the obtained gap gene circuits are not unique showing variable long-term dynamics and highly correlating scattered parameters. Furthermore, although the model can simulate the pattern up to gastrulation and confirms several of the known regulatory interactions, it does not reproduce the transient expression of all gap genes as observed experimentally. We suggest that the shortcomings of the model may be caused by overfitting, incomplete model description and/or missing data.