Introduction to mathematicaltheoretical biology

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Introduction to mathematical/theoretical biology

• Lutz Brusch
• Andreas Deutsch
• Anja Voss-Böhme

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Overview

• Goals
• Definition what is mathematical/theoretical
  biology?
• Modeling
• History
• Applications
• Overview of lecture

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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

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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.

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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?

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Solution

• At the end of the first month, they mate, but
  there is still one only 1 pair.
• At the end of the second month the female
  produces a new pair, so now there are 2 pairs of
  rabbits in the field.
• At the end of the third month, the original
  female produces a second pair, making 3 pairs in
  all in the field.
• 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, ...
an1an an-1, with a1a2 1
Fibonacci numbers
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Why is this interesting?
Fibonacci numbers appear e.g. in phyllotactic
patterns
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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

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Simulations Game of life
The rules1. 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

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I. What are mathematical models good for?

• Quantitative predictions(based on functional
  relationship)
• Stability analysis, asymptotic behavior,...
• Understanding of stochastic/deterministic effects

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II. What are mathematical models good for?

• Mathematical models can help to explain
  cooperative behavior

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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

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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...)

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Roots...2. Theoretical biology

• A plant biologist (Johannes Reinke) introduced
  the concept/notion of theoretical biologyA
  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

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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-Innenwelt-Außenwelt
• Physicist Nicolas Rashevsky Bulletin of
  mathematical biophysics (1934) (today Bulletin
  of Mathematical Biology, 1973)
• Scientific foundation in Leiden 1935 Acta
  Biotheoretica

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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)

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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)

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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 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)

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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)

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Earth
Electron
-12
-6
-9
-3
0
3
6

Human
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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

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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
• .....

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In this lecture focus on ...

• How is the variety of biological forms, shapes
  and organisms created (development/genetics/evol
  ution)?
• How are organismic activities maintained
  (physiology)?

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Overview lecture

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

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References

• See website!

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Model examples1. population growth

• Exp./logistic growth