Universiteit Leiden

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Research project

Mathematical analysis of systems pharmacology models

Systems pharmacology aims to apply techniques from systems biology to pharmacological models.

Piet Hein van der Graaf

Such models are often nonlinear and can hide interesting behaviour. Analysis of these models using mathematical techniques such as steady state analysis, phase-plane and bifurcation analysis, is expected to provide novel insights into the model behaviour.

We are currently investigating a nonlinear model of the alternate pathway of complement activation. The model includes a positive feedback and results in a blow-up type behaviour. We will be using dynamical systems analysis to study model behaviour to understand whether the blow-up is parameter dependent and whether it can be delayed or prevented.

Previously, we have worked with another nonlinear positive feedback model – namely the prolactin response to antipsychotic medication. The model exhibited an approach to one steady state under certain drug doses and an approach to another steady state for other doses. Our analysis showed that the model in question had two steady states and their stability was dependent on a certain parameter combination. We further showed that the saddle nature of the unstable steady state, combined with the ‘if condition’, were responsible for the approach to two different steady states under different conditions. This work highlighted that even seemingly simple models can exhibit counterintuitive behaviour and mathematical analysis is essential to provide insight into their behaviour.

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