# Mathematical methods in biology, fall 2007 to spring 2008

### Lecturer

### Scope

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

### Type

Advanced studies.

### Time and place (spring 2008):

Part 3: weeks 3 - 9 ( = first study period of the spring semester)

Part 4: weeks 11 - 18 ( = second study period of the spring semester, with Easter break)

Lectures:

Tuesday 12:15 - 13:45

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

Exercises:

Thursday 12:15 - 13:45

17 Jan, 24 Jan, 31 Jan, 7 Feb (first four weeks): Viikki D-rakennus LS (see map)

17 Apr: Viikki E-rakennus LS

All other times: Viikki Kehitystalo 117 (same as lectures)

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

### 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 theory (handling stochastic phenomena, groundwork for statistics)

(3) Vectors and matrices (applied to population dynamics, quantitative genetics and statistics)

(4) Dynamical systems (techniques to analyse models of population growth, reaction kinetics, etc.)

Parts 1 & 2 are given in the fall semester of 2007, parts 3 & 4 in spring 2008. 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.

### Exams

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.

### Bibliography

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 possess 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.

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.

### Feedback

You are very welcome to give feedback in person or via email. You can give feedback also anonymously via a departmental webform any time during the course.

At the end of the semester, PLEASE fill in the anonymous standard course questionnary. THANKS!

### 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.

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#### Handouts

Note #1: Powers, exponents and logarithms |

Note #2: Radioactive decay and the number e |

Note #3: Rules of differentiation |

Note #4: Bimolecular reactions |

#### Figures

Logistic growth with diffusion (Fisher's equation) |

### Part 2

#### 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.

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#### Handouts

Note #5: Geometric series: How to compute the sum of infinitely many terms

Note #6: The cumulative distribution function of the standard normal distribution

#### Figures

Membrane channels: dwell times

Binomial distribution: approaching normal

### Part 3

#### 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.

discussed on 24 Jan |
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#### Handouts

Note #7: Lagrange multiplier method

### Part 4

#### Exercises and projects

Remember that analytical solutions are not always available: Use numerical methods when necessary. Excel is sufficient.

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#### Handouts

Note #8: Linear stability analysis in two dimensions

Note #9: Divergence

Quadratic map

Plots of iterated maps: pdf

Cobweb diagrams: Java applet by Andy Burbanks, Loughborough University

Bifurcation diagram: Java applet by Richard Dallaway

Lyapunov exponents: pdf