BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Johannes Borgqvist (University of Oxford\, Chalmers and GU) DTSTART:20231122T121500Z DTEND:20231122T130000Z DTSTAMP:20260417T003246Z UID:cam/13 DESCRIPTION:Title: Lie symmetries for constructing\, selecting and analysing mechanistic models in mathematical biology\nby Johannes Borgqvist (University of Oxford\, Chalmers and GU) as part of CAM seminar\n\nLecture held in MV:L14.\n\nAbs tract\nGiven the abundance of experimental data\, two of the most fundamen tal questions in mechanistic modelling of biological data concern model co nstruction and model selection. A common type of data is time series data describing how some quantity\, e.g. population size or protein abundance\, changes over time. Given such a time series\, it is often possible to con struct numerous candidate mechanistic models consisting of ordinary differ ential equations based on physical principles encoding distinct biological hypotheses. Worse still\, numerous candidate models of the same time seri es often describe the same data equally well\, and thus they cannot be dis tinguished based on their fit to data. In these situations it is therefore difficult to select one candidate model and thereby infer a biological me chanism underlying the biological data. In this presentation\, we tackle t he two fundamental problems of model construction and model selection by m eans of Lie symmetries (or simply just symmetries) of ordinary differentia l equations. These are\, simply put\, (one parameter pointwise) transforma tions known as $\\mathcal{C}^{\\infty}$ diffeomorphisms which map a soluti on curve to another solution curve. Symmetries are commonly used in mathem atical physics and they are the basis for numerous Nobel prizes but they a re almost unheard of in mathematical biology.\n\nTo solve the classical mo del selection problem\, we have developed and implemented a methodology fo r model selection based on symmetries. We implement this framework on actu al experimental data describing the age-related increase in cancer risk. I mportantly\, we infer experimentally validated hypotheses underlying diffe rent cancer types using the symmetry based framework which the standard me thodology based on model fitting fails to do. \n\nThereafter\, we switch f ocus to model construction in the context of travelling wave models of col lective cell migration. These models consists of a single second order ODE describing how the population density $u(z)$ changes with respect to a tr avelling wave variable $z=x-ct$ where the constant $c$ is referred to as t he wave speed. Moreover\, certain such models of reaction diffusion type a s well as other models with density dependent diffusion are known to have specific analytical solutions of a simple form for certain wave speeds. Th ese analytical solutions have been obtained by means of ansätze based on series expansions\, and using these methods it is difficult to define the class of models which have simple analytical solutions. To tackle this pro blem\, we consider a set of symmetries referred to as a Lie Algebra consis ting of two symmetries that has been used to find analytical solutions of a second order ODE encapsulating numerous oscillatory models such as the v an der Pol oscillator. Based on differential invariants\, we derive the mo st general class of models for which this Lie Algebra is manifest. Thereaf ter\, we implement Lie's algorithm based on step-wise integration in order to demonstrate how first integrals and (if possible) analytical solutions of all ODEs in our class of models are obtained. Using this general class of models\, we construct a sub-class of models characterised by the previ ously mentioned simple analytical solution. Lastly\, we demonstrate how th is sub-class encapsulates the previously known models with analytical solu tions and we quantify the action of the symmetries in this Lie algebra on these analytical solutions. In total\, this work demonstrates how classes of mechanistic models can be constructed based on mathematical properties encoded by a Lie Algebra in contrast to the standard way of model construc tion based on physical assumptions that are hard to validate.\n LOCATION:https://researchseminars.org/talk/cam/13/ END:VEVENT END:VCALENDAR