Introduction to mathematical/theoretical

biology

Introduction to mathematical/theoretical

biology

Lutz BruschAndreas DeutschAnja Voss-Böhme

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OverviewOverview

Goals Definition: what is mathematical/theoretical

biology? Modeling History Applications Overview of lecture

Goals: learn how...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?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

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?

SolutionSolution1. 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, ...

an+1=an + an-1, with a1=a2 =1 Fibonacci numbers:

Why is this interesting? Why is this interesting?

Here is a real sunflower with and spirals moving to the right and to the left, respectively.

559 a 8910 a

Fibonacci numbers appear e.g. in phyllotactic patterns

Data:* empirical * simulated

Results:* empirical* simulated* theoretical

Experiment:measurement

Mathematical model:Theory

Data

acquisition

Experimental design

Statistics

Data evaluation and comparison

Simulation

Modeling

Analysis

(proof)

Problem:hypothesis

Mathematical analysisMathematical 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 lifeSimulations: 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?I. What are mathematical models good for? Quantitative predictions

(based on functional relationship):

Stability analysis, asymptotic behavior,... Understanding of stochastic/deterministic effects

)0()( NLtN t

II. What are mathematical models good for?

II. What are mathematical models good for?

Mathematical models can help to explain cooperative behavior

?

The roots...1. BiologyThe 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 centuryStatus 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 biologyRoots...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 rootsRoots: 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-Innenwelt-Außenwelt

Physicist Nicolas Rashevsky: Bulletin of mathematical biophysics (1934) (today: Bulletin of Mathematical Biology, 1973)

Scientific foundation in Leiden 1935: Acta Biotheoretica

JournalsJournals 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.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 geneticsRoots: ...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 „HARDY-WEINBERG 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?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)

Human

MyxobacteriumDNA

Micro-tubule

Radiolaria

Water

Electron

Earth

1410 eukaryotic cells

1510 prokaryotic cells

-12 -6-9 -3 0 3 6

x10 m

New disciplinesNew 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: examplesMathematical 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 ...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: lectureOverview: lecture

Introduction/history Model examples Difference and differential equations Partial differential equations Stochastic processes Cellular automata

ReferencesReferences

See website!

Model examples:1. population growthModel examples:1. population growth

Exp./logistic growth