Towards the systematic simplification of mechanistic models

Mechanistic models used for prediction should be parsimonious, as models which are over-parameterised may have poor predictive performance. Determining whether a model is parsimonious requires comparisons with alternative model formulations with differing levels of complexity. However, creating alternative formulations for large mechanistic models is often problematic, and usually time-consuming. Consequently, few are ever investigated. In this paper, we present an approach which rapidly generates reduced model formulations by replacing a model’s variables with constants. These reduced alternatives can be compared to the original model, using data based model selection criteria, to assist in the identification of potentially unnecessary model complexity, and thereby inform reformulation of the model. To illustrate the approach, we present its application to a published radiocaesium plant-uptake model, which predicts uptake on the basis of soil characteristics (e.g. pH, organic matter content, clay content). A total of 1024 reduced model formulations were generated, and ranked according to five model selection criteria: Residual Sum of Squares (RSS), AICc, BIC, MDL and ICOMP. The lowest scores for RSS and AICc occurred for the same reduced model in which pH dependent model components were replaced. The lowest scores for BIC, MDL and ICOMP occurred for a further reduced model in which model components related to the distinction between adsorption on clay and organic surfaces were replaced. Both these reduced models had a lower RSS for the parameterisation dataset than the original model. As a test of their predictive performance, the original model and the two reduced models outlined above were used to predict an independent dataset. The reduced models have lower prediction sums of squares than the original model, suggesting that the latter may be overfitted. The approach presented has the potential to inform model development by rapidly creating a class of alternative model formulations, which can be compared.

Calcium oscillations

Changes in cellular calcium concentration control a wide range of physiological processes, from the subsecond release of synaptic neurotransmitters, to the regulation of gene expression over months or years. Calcium can also trigger cell death through both apoptosis and necrosis, and so the regulation of cellular calcium concentration must be tightly controlled through the concerted action of pumps, channels and buffers that transport calcium into and out of the cell cytoplasm. A hallmark of cellular calcium signalling is its spatiotemporal complexity: stimulation of cells by a hormone or neurotransmitter leads to oscillations in cytoplasmic calcium concentration that can vary markedly in time course, amplitude, frequency, and spatial range. In this chapter we review some of the biological roles of calcium, the experimental characterisation of complex dynamic changes in calcium concentration, and attempts to explain this complexity using computational models. We consider the "toolkit" of cellular proteins which influence calcium concentration, describe mechanistic models of key elements of the toolkit, and fit these into the framework of whole cell models of calcium oscillations and waves. Finally, we will touch on recent efforts to use stochastic modelling to elucidate elementary calcium signal events, and how these may evolve into global signals.