University of Montpellier
Tropical geometry of biological systems
Mathematical modelling of biological systems is a daunting challenge. In order to cope realistically with the biochemistry of the cell, systems biology models include hundreds of variables. Ecological modelling also involves many variables representing different interacting organisms, undergoing migration and being spatially distributed. Modelling of infectious diseases and cancer gathers all these intricacies: it has to cope with population dynamics, complex single cell dynamics and spatial heterogeneity.
Denis Noble, a pioneer of multi-cellular modelling of human physiology, advocated the use of middle-out approaches in biological modelling. Middle-out is an alternative to bottom-up that tries to explain everything from first principles and to top-down that uses strongly simplified representations of reality. A middle-out model uses just enough details to render the essence of the overall organisation. Although this is potentially a very powerful principle, the general mathematical methods to put it into practice are still awaited.
Recently, we have used tropical geometry to extract the essence of biological systems and to simplify complex biological models. Tropical methods were applied to systems of differential equations with polynomial or rational right hand side and can be potentially applied also to partial differential equations. Tropical geometry methods exploit a property of biological systems called multiscaleness, summarized by two properties: i) the orders of magnitude of variables and timescales are widely distributed and ii) at a given time, only a small number of variables or components play a driving role, whereas large parts of the system have passive roles.
Tropical geometry is the natural framework for scaling and order of magnitude theory. We are using tropical geometry methods to identify sub-systems that are dominant in certain regions of the phase- and/or parameter-space of dynamical systems. These dominant sub-systems are rather simple, even for large systems. Tropical methods also provide the scaling needed to identify slow/fast decompositions and perform model order reduction in the framework of geometric singular perturbations theory. Interestingly, the dominance relations unravelled by tropical geometry can highlight approximate symmetries that are exploited for the simplification of biological systems.
Several tools for tropical simplification of biological systems are currently being developed and will be made available to the computer algebra and computational biology communities.