Modelling in Biology

Module aims

The course mainly introduces and covers Nonlinear Dynamics, Networks in Biology and the basics of stochastic processes in Biology and Medicine.

Learning outcomes

Understand and explain the role of dynamics and the importance of nonlinearity Understand the different behaviours that Ordinary Differential Equation (ODE) models of order 1, 2, 3, and higher can exhibit Know and apply various mathematical analysis techniques (phase plane analysis, bifurcation analysis, linearisation and linear stability analysis) to characterise the behaviour of ODE models of various orders Establish a link between observations, modelling, mathematical formulation and computational/numerical analysis Appreciate the role and importance of variability in steady-state biological systems Appreciate the role and consequences of noise in dynamical biological systems Understand how technological and biological processes can be mapped onto networks Understand that network structure has consequences for system behaviour Analyse dynamical systems using the methods introduced in the lectures such as phase plane, global and local stability analysis, bifurcation analysis Understand which type of behaviour can be exhibited by ODE models depending on their order Use and apply computational modelling with ordinary and stochastic differential equations Analyse systems characterised by distributions of random variables Formulate and analyse models for simple discrete and continuous stochastic processes Analyse and compare properties of networks Translate descriptions and observations into mathematical models and computational schemes Use of MATLAB for scientific computing and for numerical analysis of mathematical models Building Ordinary Differential Equations and stochastic models from scratch Scientific programming using MATLAB Modelling a system from first principles Research report writing and graphing

Module syllabus

Introduction and background: linear vs. nonlinear. Phase plane analysis. Stochastic and deterministic models One-dimensional systems: fixed point analysis; global and local stability; bifurcations Two-dimensional systems: oscillations and limit cycles. Hopf bifurcation and Poincare-Bendixson theorem Three-dimensional systems: chaos and quasi-periodicity (very briefly) Randomness in biological systems (analysing stationary states in and out of thermodynamic equilibrium in biology) Introduction to Markov processes and stochastic dynamics (master equation, Fokker-Planck equation, stochastic differential equations - meaning and simple approaches to solution and simulation) Stochastic dynamics in biological systems (applying the above techniques to biological examples) Introduction to networks (basic metrics, null models). Emergent properties, the influence of networks on dynamics and the identification of motifs Examples throughout the course drawn from: Ecology, Physiology, Neuroscience, Biochemistry & Genetics Problem sheets and MATLAB exercises are provided to illustrate and deepen the introduced concepts

Pre-requisites

BIOE40004 Mathematics I BE2-HMATH2 Mathematics II A clear understanding of the mathematical concepts in BE2-HSCL Signals, Systems and Control will also help in deeply understanding this course and making connections with the concepts introduced therein. Calculus: differentiation, ordinary differential equations, partial differential equations, complex numbers, Taylor series expansion Linear algebra: matrix multiplication, eigenvalues and eigenvectors Probability: basic rules of probability, mean, variance, correlation, normal distribution, binomial distribution, calculating expectations using a distribution.

Teaching methods

Lectures: 18 hours
Labs: 5 hours
Tutorials: 4 hours

Assessments

Examinations:
●  Written exam: Modelling in Biology; 80% weighting
    

Courseworks:
●  Written report: MiB: 20%
 

Feedback : Feedback in person during labs and class tutorials and from assessment of coursework submission. This should enable fast feedback cycles for the students during the duration of the course. Feedback on summative coursework given as annotated marked scripts.

Module leaders

Professor Guy-Bart Stan