Mathematical Biology

This is a master course for mathematics students about mathematical methods to gain insight in the mechanisms underlying biological phenomena.

Teachers

Time and location

Lecture notes

The lecture notes for this course are still a work in progress. New chapters or enhanced versions of older ones will appear here for download.

• Syllabus

Setup of the course

• lectures which explain and illustrate the methods while referring to other sources for detailed accounts of the underlying mathematical theory
• assignments which provide training in modelling and in the use of the methods
• around April 24 students will have to choose a mathematical biology paper, and have to give both a written and oral presentation about it.

During the third hour and at home students work by themselves or in couples on assignments, using mostly pen and paper.

In the course, a lot of attention is paid to "translation": how do we get from biological information to a mathematical formulation of questions ? And what do the mathematical results tell us about biological phenomena ? In addition, the course aims to introduce general physical ideas about time scales and spatial scales and how these can be used to great advantage when performing a mathematical analysis.

Prerequisites: basic knowledge about linear algebra, analysis, ODE, stochastic processes. (The key point, however, is the attitude: students should be willing to quickly fill in gaps in background knowledge.)

Presently four assignments are planned:
Hand in on February 28 : Chapter 3
Hand in on March 6 : Section 4.4
Hand in on March 20 : Exercises 6.3.3, 6.3.4, 6.3.5, 6.3.6 (Bonus 6.3.7)
Hand in on April 13 : Exercise 6.6, numbers 12, 13, 14 and 15

You are expected to hand in a detailed and readable elaboration. During the third hour of every session you can work on the assignments and ask the lecturers for advice. We recommend that you plan at home the questions you want to ask during the next session.

Concerning the final project, the aim is to write and lecture such that in a coherent (and for the students of the course understandable) way a specific topic is introduced and analysed, with attention for both the biological and the mathematical aspects. Whether or not the material comes from one or several papers is irrelevant.
Here is a list of papers from which you can choose.

The planning is as follows :
April 24 : decide about the topic by making a choice from among the various papers (note that you may suggest a paper yourself, but in this case you'll need our approval)
May 1 : no meeting (work at home on the topic)
May 8 : meeting for feedback, advice. Formulate a plan to us on what you will cover in your essay/presentation. It will help if you send such a plan one or two days before May 8 by e-mail
May 15 : no meeting. Send us a preliminary text for constructive critique. Make sure to do this early enough to allow us some time for reading and formulating feedback
May 22 : presentations and handing in of final project (depending on the number of participants, we may need to continue in the afternoon or to start earlier than usual

List of subjects

1. Exploiting time scale differences: the quasi-steady-state-approximation
   • Michaelis Menten enzyme kinetics
   • Holling's functional response
   • excitable media: Fitzhugh-Nagumo
2. Phase plane analysis
   Essentially an assignment: students work in couples through a series of exercises about prey-predator interaction. In a lecture we explain some key notions, such as linearized stability and Poincare-Bendixon.
3. Diffusion
   (mainly linear theory; partly in the form of assignments)
   • various derivations of the diffusion equation
   • the fundamental solution, superposition
   • transport by diffusion: what distance in how much time?
   • separation of variables, eigenfunctions/modes
   • the asymptotic speed of propagation
4. Reaction-Diffusion (nonlinearity)
   • travelling waves
   • scalar equations do NOT generate stable patterns (in convex domains)
   • Turing instability
   • bifurcation theory
   • transition layers (excitable systems) (rotating spirals?)
   • Ginzburg-Landau (only if time permits; most likely it will not)
5. Miscellaneous topics
   • Age/size structured populations (including cell cycle models ?)
   • Chemotaxis
   • Adaptive dynamics
   • Branching processes
   and maybe additional topics, like infectious disease epidemiology, if time permits.
6. Final project
   The last four weeks the students will work on a final project : writing an essay on a paper of their choice, and preparing a presentation about this paper. A list of suitable papers will be provided, but students may also come up with their own suggestions for a topic and/or a paper (note : these are subject to approval). The aim is to write and lecture such that in a coherent (and for the students of the course understandable) way a specific topic is introduced and analysed, with attention for both the biological and the mathematical aspects.

Learning goals and grading

1. read and understand the research literature about (deterministic) models of biological phenomena
2. participate actively in projects that aim to model biological phenomena
3. derive mathematical equations from bookkeeping considerations
4. interpret mathematical results in the biological context that motivated the analysis; more precisely the point is that mathematical statements are translated into a relation between phenomena and the underlying mechanisms
5. use formal arguments (based on differences in the time- or spatial scale of various mechanisms) to simplify equations in a meaningful way
6. apply various analytical techniques to study phase portraits of planar ODEs representing ecological systems
7. derive and analyse linear diffusion equations and their solutions
8. apply bifurcation theory to study systems of nonlinear reaction- diffusion equations

Here we present the ``toetsmatrijs'' for this course The numbers 1-8 in the first column refer to the learning goals listed above.
H1 = first home assignment
H2 = second home assignment, etc.
essay = written essay
pr = presentation
The question marks indicate that, depending on the subject of the final project, these may or may not contribute substantially.

	H1	H2	H3	H4	essay	pr
1	-	-	-	-	60	40
2	10	10	10	10	40	20
3	20	20	-	-	50	10
4	10	10	10	10	30	30
5	100	-	-	-	?	?
6	100	-	-	-	?	?
7	-	100	-	-	?	?
8	-	-	50	50	?	?