Introduction to
mathematical/theoretical
biology
Lutz Brusch
Andreas Deutsch
Anja Voss-Böhme
?
Overview
 Goals
 Definition: what is mathematical/theoretical
biology?
 Modeling
 History
 Applications
 Overview of lecture
Goals: learn how...
to read mathematical modeling papers
 to analyze mathematical models
 to critically judge the assumptions and the
contributions of mathematical models whenever
you encounter them in your research
 to develop a mathematical model, i.e. to choose
an appropriate mathematical structure

What is mathematical biology?
Mathematical biology/biomathematics/ theoretical
biology is an interdisciplinary field of academic study
which models natural, biological processes using
mathematical techniques. It has both practical and
theoretical applications in biological research.
 The strength of biomathematics lies in the
quantification of specific values but also in the
identification of common structures and
patterns at different levels of biological
organisation.

A first mathematical model:
rabbit population growth
The original problem that Fibonacci investigated (in the
year 1202) was about how fast rabbits could breed in
ideal circumstances.
 Suppose a newly-born pair of rabbits, one male, one
female, are put in a field. Rabbits are able to mate at
the age of one month so that at the end of its second
month a female can produce another pair of rabbits.
Suppose that our rabbits never die and that the female
always produces one new pair (one male,
one female) every month from the second
month on. The puzzle that Fibonacci posed was...
 How many pairs will there be in one year?

Solution
1. At the end of the first month, they mate,
but there is still one only 1 pair.
2. At the end of the second month the
female produces a new pair, so now there
are 2 pairs of rabbits in the field.
3. At the end of the third month, the original female produces
a second pair, making 3 pairs in all in the field.
4. At the end of the fourth month, the original female has produced
yet another new pair, the female born two months ago produces
her first pair also, making 5 pairs.
The number of pairs of rabbits in the field at the start of
each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
Fibonacci numbers: an+1=an + an-1, with a1=a2 =1
Why is this interesting?
Fibonacci numbers appear e.g. in phyllotactic patterns
Here is a real sunflower with a9  55 and a10  89 spirals moving
to the right and to the left, respectively.
Problem:
hypothesis
Experiment:
measurement
Data:
* empirical
* simulated
Statistics
Data evaluation and comparison
Mathematical model:
Theory
Results:
* empirical
* simulated
* theoretical
Mathematical analysis
a proof is a demonstration that, given certain axioms, some statement of
interest is necessarily true. Proofs employ logic but usually include
some amount of natural language. Some common proof techniques are:
 Direct proof: where the conclusion is established by logically
combining the axioms, definitions and earlier theorems
 Proof by induction: where a base case is proved, and an induction rule
used to prove an (often infinite) series of other cases
 Proof by contradiction (also known as reductio ad absurdum): where it
is shown that if some property were true, a logical contradiction
occurs, hence the property must be false.
 Proof by construction: constructing a concrete example with a property
to show that something having that property exists.
 Proof by exhaustion: where the conclusion is established by dividing it
into a finite number of cases and proving each one separately

Example: Proof that sqrt(2) is irrational
Simulations: Game of life
The rules:
1. Survival, if living cell has 2 or 3 neighbours,
2. Death, if living cell has less than 2 or more than 3 neighbours ,
3. Birth, if dead cell has precisely 3 living neighbours.
Configuration
Simulation: Game of Life
I. What are mathematical models
good for?

Quantitative predictions
(based on functional
relationship):
N (t )  Lt N (0)
Stability analysis, asymptotic behavior,...
 Understanding of stochastic/deterministic effects

II. What are mathematical
models good for?
 Mathematical models can help to explain
cooperative behavior
?
The roots...1. Biology


Biology: term was introduced by Jean Baptiste de Lamarck
(1744-1825) and Gottfried Reinhold Treviranus (see e.g.
„Biology, or philosophy of vital nature“, G. R. Treviranus, 1802),
Cell: the word cell was introduced in the 17th century by the
English scientist Robert Hooke, it was not until 1839 that two
Germans, Matthias Schleiden and Theodor Schwann, proved
that the cell is the common structural unit of living things. The cell
concept provided impetus for progress in embryology, founded
by the Estonian scientist Karl Ernst von Baer
Status of biology end of 19th century
huge amounts of data (from expeditions into colonies
and new observations (due to new physical and
chemical techniques)
 disciplines widely separated (zoology, botany, ...).
Physiology (part of medical research) was trendy and
cell biology had emerged as a central discipline (Max
Verworn (1901); ...if physiology wants to explain the
elementary and general processes of life, it can do so
only as cellular physiology...)

Roots...2. Theoretical biology
 A plant biologist (Johannes Reinke) introduced
the concept/notion of theoretical biology:
A theoretical biology has so far merely not yet
been considered, at least not as a connected
discipline (Reinke, 1901)...
The task of a theoretical biology would be not
only to find out the origins of biological events,
but also to check the basic assumptions of our
biological thinking
Roots: 3. Further roots
Ludwig v. Bertalanffy: Introduction to theoretical
biology I and II, 1932, 1942
 Early environmentalist Jakob v. Uexküll (1864-1944):
„Theoretische Biologie“ (1920), Umwelt-InnenweltAußenwelt
 Physicist Nicolas Rashevsky: Bulletin of mathematical
biophysics (1934) (today: Bulletin of Mathematical
Biology, 1973)
 Scientific foundation in Leiden 1935: Acta
Biotheoretica

Journals
Biometry: Biometrika (1901), Biometrics Bulletin (1945),
Biometrical Journal (1959)
 Acta Biotheoretica (1935)
 Cybernetics: Cybernetica (1958),...
 Journal of Theoretical Biology (1961)
 Mathematical Biosciences (1967)
 Theoretical Population Biology (1970)
 BioSystems (1972)

Journals cont.
 Bulletin of Math. Biophys. (1939)
Bull. Math. Biol. (1973)
 Journal of Mathematical Biology (1974)
 Mathematical Medicine and Biology (1984)
 Comments on Theor. Biol. (1989)
 Journal of Biological Systems (1993)
 Theorie in den Biowissenschaften (1996)
Roots: ...4. Population genetics
The famous experiments of Mendel, and the fruitful
communication between experimental biologists and applied
mathematicians in the 1930s, marked the beginnings of
population genetics and were seminal for biomathematics. As
early as 1896, the British professor K. Person applied the now
standard statistical techniques of probability curves and
regression lines to genetic data. This was seemingly the first
proof of the existence of a mathematical law for biological events
(1900). See also model examples later this lecture „HARDYWEINBERG LAW“, „FUNDAMENTAL THEOREM ON NATURAL
SELECTION“
 William Bateson: introduced the notion „genetics“ for research
on Mendelian heredity of characters (Cambridge, 1905)
 William Johannsen: introduced the notion „gene“ as something in
the gametes, by which the properties of the developing organism
is or can be conditioned or co-determined (Copenhagen, 1909)

Roots: 5. What is life?
Oscar Hertwig (1900): Life is based on a peculiar
organisation of material with which are connected again
peculiar processes and functions, how they never can
be found in non-living nature,...,with each of the infinite
steps and forms of organisation there are produced new
kinds of effects („Wirkungsweisen“).
 Remark: early formulation of nowadays favored
definition of life as a complicated adaptive, regulatory,
dynamical system based on physico-chemical
mechanisms.
 E. Schrödinger: What is life? (Dublin 1944)

10
x
m
Earth
Microtubule
Radiolaria
Electron
-12
-9
-6
-3
0
3
Water
6
1014 eukaryotic cells
DNA
1015 prokaryotic cells
Myxobacterium
Human
New disciplines
Biology (ca. 1800)
 Theoretical biology (ca. 1900)
 Cybernetics (N. Wiener, 1948): relations between
machines and living nature
 Bioinformatics (ca. 1970): information-technical
techniques to store, analyze and display the information
contents of biological systems, ...
 System biology (H. Kitano, 2001): interdisciplinary
approach focusing on a wholistic understanding of
complex living systems based on an integration of
biological data

Mathematical problems in
biology: examples
Ecology/ethology: optimization of food search
 Evolution: evolutionary stable strategies,
reconstruction of phylogenetic trees
 Development: embryological pattern formation
 Epidemiology: spread of infectious diseases
 Molecular genetics: coding and sequence alignment
 Neurology: contrast enhancement in neural networks
 Physiology: regulation of glucose level in the blood
 Biotechnology: fermenter control
 .....

In this lecture focus on ...
 How is the variety of biological forms, shapes
and organisms created
(development/genetics/evolution)?
 How are organismic activities maintained
(physiology)?
Overview: lecture
 Introduction/history
 Model examples
 Difference and differential equations
 Partial differential equations
 Stochastic processes
 Cellular automata
References
 See website!
Model examples:
1. population growth
 Exp./logistic growth