Reducing 'noise' from biological models
When non-stochastic or deterministic effects such as radiation dose are in play, severity varies with dose up to a threshold value. Classical deterministic mathematical modelling becomes invalid when concentration of a chemical species is low. This is because of 'noise' with stochastic effects, a scenario very common in biological systems. Stochastic effects occur by chance and are typical in models of cancer and genetic effects. The STOCHDETBIOMODEL (Stochastic and deterministic modelling of biological and biochemical phenomena with applications to circadian rhythms and pattern formation) project investigated theoretical problems connected with stochastic and deterministic modelling of biological systems. Stochastic modelling is an invaluable tool but problems arise when there are changes in model parameters that lead to a change in model behaviour, bifurcation. Theoretical problems took in model reduction and bifurcation analysis of stochastic differential equations, the high computational cost of stochastic models and interconnection of deterministic and stochastic approaches using tensor-structured parametric analysis (TPA). In particular, the researchers analysed the robustness of the model as applied to circadian rhythms even in instances where diffusion is added to the model. The analysis of TPA was based on recently proposed low-parametric tensor-structured representations of classical matrices and vectors. TPA has been implemented in Matlab and the codes are available. Furthermore, application to circadian rhythms used introduction of delays to quasi steady state assumptions. This approach yielded a simplified system that accurately agrees with the original system not only qualitatively but also quantitatively. The researchers tailored the correct size of delays for a particular model of circadian rhythms. The results of the research have been widely disseminated via eight conferences and publication of three peer-reviewed papers. Collaboration between universities from UK, USA and China was very fruitful for all concerned. As the knowledge base of biological systems widens, there is an increasing need to employ modelling techniques to represent their dynamics. Stochastic and deterministic modelling is applicable to a whole range of phenomena including collective behaviour or insects, movement of bacteria to a chemical stimulus and gene regulatory networks.
Keywords
Biological model, circadian, skin pattern, non-stochastic, stochastic, tensor-structured parametric analysis, collective behaviour
