Mathematical methods in biology

Contents:

This course gives a highly practical introduction to mathematical concepts and methods applied in the life sciences. We learn mathematics through solving problems of biological interest, with emphasis on applicable skills and hands-on experience.

The full course consists of four parts (each can be taken separately):

(1) Fundamentals (construction of simple models and basic calculus)
(2) Probability (handling stochastic phenomena, groundwork for statistics)
(3) Vectors and matrices (applied to population dynamics, quantitative genetics and statistics)
(4) Dynamic models (techniques to analyse models of population growth, reaction kinetics, etc.)

Parts 1 & 2 are given in the fall semester, parts 3 & 4 in spring. Each part takes one study period (seven weeks), 2 h interactive lectures and 2 h exercises per week.

The course is specifically tailored for biology students and assumes no background in mathematics. Both undergraduates and graduate students are welcome. Prior registration is not necessary.

Prerequisites:

You will need to use Excel or any comparable software capable of simple calculations and plotting.

Time and place (spring 2012):

Part 3: weeks 3 - 9 ( = first study period). First lecture: Friday 20 January 8.30
Part 4: weeks 11 - 18 ( = second study period, Easter break 5-11 April)

Lectures: Monday 12:15 - 13.45
Exercises: Friday 8:30 - 10.00

Viikki campus, Kehitystalo (Koetilantie 3, next to Gardenia; see map), room 117

Don't be misled by the unusual location: this course is intended for biology students. The best place would be Biokeskus 3, but it turned out that the nearest rooms in Viikki with real blackboards are in Kehitystalo!

Exam, course codes and credits:

Each part of the course gives separate credits (3 op per part).

Exam: problem-solving (in writing), the problems are similar to homework exercises. Everything may be used (books, notes, dictionary) but may not be shared during the exam. There is no need for laptops; nevertheless laptops can be used if so desired, but the internet connection must be switched off (download necessary files in advance). Exercise class activity decides marginal grades.

First exam of part 2: Friday 16 December 10.15-12.00 (replacing the last exercise class, in the lecture room)
Second exam of Part 2: 13 January 10.15 - 12.00, please come to my office A420 on the fourth floor of Exactum.

Course codes:

Part 1: 57381
Part 2: 57382
Part 3: 57383
Part 4: 57384

Course books:

There is no single textbook the course would follow closely. The books listed below are useful reference books also for later use, but you need not buy them to participate in this course successfully.

L. Edelstein-Keshet (1988) Mathematical models in biology. McGraw-Hill Education, ISBN 0075549506.
C. Neuhauser (2003) Calculus for biology and medicine. Prentice Hall, ISBN 0131234412.
S. Otto & T. Day (2007) A biologist's guide to mathematical modeling in ecology and evolution. Princeton University Press, ISBN 0691123446.
D. W. Jordan & P. Smith (2002) Mathematical techniques. Oxford University Press, ISBN 0199249725.
S. P. Ellner & J. Guckenheimer (2006) Dynamic models in biology. Princeton University Press, ISBN-10: 0691125899.

Lecturer:

Eva Kisdi (PhD in Biology). Office: Kumpula campus, EXACTUM, room A420

Further information:

eva.kisdi [at] helsinki.fi

Part 1

Exercises and projects

The solutions of projects will be read and commented by a fellow student in the exercise classes. Hence, please write down the logic, methods and results in such a way that others can understand it. For the exercises, it is enough if you can present the solution on the blackboard.

Exercises 1-4	solution	discussed on 16 Sept
Exercises 5-8	solution	discussed on 23 Sept
Exercises 9-13	solution	discussed on 30 Sept
Exercises 14-15	solution	discussed on 7 Oct
Exercises 16-17	solution	discussed on 14 Oct
Exercises 18-19	solution	discussed on 21 Oct

Project 1	solution	discussed on 7 Oct
Project 2	solution	discussed on 14 Oct
Project 3	solution	discussed on 21 Oct

Here are guidelines to evaluate each others' projects. Tell your partner what grade you would give to the project; these grades do not affect the course grade.

Some more exercises to practise differentiation

Lecture notes

Lecture notes can be downloaded as a single pdf file. This file is now complete and will not change in this year. If any correction is necessary, it will be listed here:

1. Missing word: p. 36, "In EACH next row of column c, put in the sum of the previous value of c and the change..."

Handouts and supporting files

Note #1:	Powers, exponents and logarithms
Note #2:	Radioactive decay and the number e
Note #3:	Rules of differentiation
Excel file	solving a system of differential equations numerically

Part 2

Exercises and projects

Exercises 1-7	solution	discussed on 11 Nov
Exercises 8-13	solution	discussed on 18 Nov
Exercises 14-18	solution	discussed on 25 Nov
Exercises 19-23	solution	discussed on 2 Dec (see also supporting file)
Exercise 24	solution	discussed on 12 Dec (!! on 9 December, lecture replaces exercise class, see above!!))
Exercises 25-26	solution	discussed on 12 Dec

Lecture notes

Unfortunately, full lecture notes are not yet available for this part. Only very brief notes will appear to outline the content of lectures: download them in one pdf file which will be updated after each lecture (consult the contents for new parts). If you had to miss a lecture, you will need to study the notes of someone present. As always, solving the homework exercises (and if necessary, studying the published solutions) is the main learning tool.

Handouts and supporting files

Numerical integration of the probability density function of the standard normal distribution

Part 3

Exercises and projects

Exercises 1-5	solution	discussed on 27 Jan
Exercises 6-10	solution	discussed on 3 Feb
Exercises 11-15	solution	discussed on 10 Feb

Lecture notes

Lecture notes can be downloaded as a single pdf file. This file will be continuously updated; check the table of contents for new parts.

Supporting files

Stable state distribution: Leslie matrix

Stable state distribution: Stochastic matrix