Modelling in Biology

Module aims

In this course you will be introduced to the mathematical and computational modelling techniques that are used to study biological systems. The course mainly introduces and covers Nonlinear Dynamics, Networks in Biology and the basics of Stochastic Processes in Biology. Examples throughout the course are drawn from: Ecology, Physiology, Neuroscience, Biophysics, Biochemistry & Genetics. This module has a strong practical focus allowing you to test out your understanding of the theory of these topics by developing your own models using Matlab or an equivalent software.

Learning outcomes

Upon successful completion of this module you will be able to: 

1) Explain the role of dynamics and the importance of nonlinearity

2) Describe the different behaviours that Ordinary Differential Equation (ODE) models of order 1, 2, 3, and higher can exhibit

3) Describe the role of variability and noise in steady-state and dynamical biological systems respectively

4) Describe common concepts in the application of networks in biology including how technological and biological processes can be mapped onto networks and the consequences of network structure on system behaviour 

5) Apply computational modelling with ordinary and stochastic differential equations

6) Analyse systems characterised by distributions of random variables

7) Formulate models for simple discrete and continuous stochastic processes

8) Apply various mathematical analysis techniques introduced in lectures (phase plane analysis, bifurcation analysis, linearisation and linear stability analysis) to characterise the behaviour of ODE models of various orders and dynamic systems

 

Module syllabus

In this module you will cover the following topics: 

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. 

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

Students will be taught over two terms. There are no in-person lectures in a conventional sense - instead, material is pre-recorded for students to work through, in combination with extensive notes and exercises. In-person sessions with GTAs and lecturers are used to cover questions arising from this independent work, demonstrate the application of ideas with worked examples, and facilitate students' practical work analysing these problems with MATLAB.  This approach is designed to maximise active student engagement with both the material and the educators. 

Lectures: 14 hours

Labs: 5 hours

Tutorials: 6 hours

Assessments

Examinations:

●  Exam: 80% weighting

    No type of previous exam answers or solutions will be available

Courseworks:

●  Written report: MiB part 1: Deterministic Nonlinear Dynamics; 10% weighting; One paper assessed coursework involving use of Matlab to answer questions from the first half of the course on Deterministic Systems described with Nonlinear ODEs

●  Written report: MIB part 2 ; 10% weighting

Module leaders

Professor Guy-Bart Stan