Team:Aberdeen Scotland/modeling/pde

Introduction

It was realised quite soon into the project that upon modelling the Inner Dynamics of the system, the individual modules could be collimated in to a single Partial Differential Equation – hereafter referred to as PDE – which would accurately describe the evolution of the system in space and time.

A PDE model was an excellent “next step” for our modelling effort – and although a physically meaningful model could not be generated; the following describes a concise modelling methodology which would merit implementation outside of iGEM – or indeed, in future iGEM competitions.

The Keller-Segel Chemotaxis system is a set of two coupled Nonlinear PDE’s; Parabolic except for a mixed Hyperbolic-Elliptic term in the cross-diffusive flux. The terms “Parabolic”, “Hyperbolic” and “Elliptic” refer to the number of boundary conditions the system must satisfy – the dependence on which defines (or “sets”) the behaviour of the system.

The Keller-Segel system was perfect for modelling our PicoPlumber modules, inasmuch as Quorum Sensing and Internal Dynamics could be effectively expressed in its formulation [1].

The following is an account of our researches into utilising these powerful Mathematical tools to describe our system.

One-Dimensional Diffusion Equation

As PDE’s are a challenging and actively-researched topic in Mathematics [4],[5]; our modelling effort began by solving a much simpler problem by way of introduction to the field.

The one-dimensional Heat Equation is the classic introduction to Partial Differential Equations. It was modelled by utilising an Implicit Finite Difference method of our own composition. This allowed us to attain an aptitude for the more difficult Keller-Segel system to come.

The Heat Equation and the Diffusion Equation of our chemo-attractant shared the same mathematical formulation – differing in fact, by the choice of the constant k – and so this introduction was still retained a Biological pertinence.

Keller-Segel System for Chemotaxis

As mentioned above, the Keller-Segel System for Chemotaxis is the set of coupled Nonlinear PDE's which describe chemotactic motion. The mixed Hyperbolic-Elliptic term endows the system with rich behaviour – but also renders it insoluble by elementary Numerical Methods[footnote1].

In searching for a more advanced numerical method, Discontinuous Galerkin Finite Elements were found in Systems Biology literature [3] to have successfully modelled the system and identified parametric instabilities.

In their award-winning paper [6], Hillen and Painter describe augmentations to the standard Keller-Segel System, hereafter referred to as the "Minimal Model". These augmentations account for effects such as density- and signal-dependent sensitivity, nonlinear diffusion and cell-growth & multiplication. The following documents the theoretical knowledge necessary to create such a numerical algorithm.

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