Introduction to mathematical biology : modelling and concepts Lutz Brusch Andreas Deutsch Anja Voss-Böhme ? Overview Definition: what is mathematical biology? Modelling History Applications Goals Overview of lecture 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. Striking mathematical models Malthus (1798, population growth) Fisher (1930, population genetics) Turing (1952, development) Hodgkin-Huxley (1952, neurophysiology) Segel (1971, development): Dictyostelium: Excitable dynamics cAMP ... Dictyostelium discoideum Signal: cAMP Chemotaxis: Mechanism: 1. cells secrete cAMP upon stimulation by (i) starvation or (ii) cAMP 2. cells react to cAMP by preferably moving towards large signal concentration Two time scales: fast signal diffusion, slow cell migration Dictyostelium: modelling/simulation (courtesy of S. Maree, Utrecht) 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. 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? Fibonacci numbers point to a general structure in biology, e.g. they 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 I. What are mathematical models good for? Quantitative predictions (based on functional relationship): N (t ) L N (0) t Stability analysis, asymptotic behavior,... Understanding of stochastic/deterministic effects II. What are mathematical models good for? Mathematical models can help to explain cooperative behavior, in particular spatio-temporal pattern formation ? cell 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 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 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: 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 Roots: ...4. Population genetics The experiments of Mendel, and the communication between experimental biologists and applied mathematicians in the 1930s, marked the beginnings of population genetics. In 1896, the British K. Person applied the now standard statistical techniques of probability curves and regression lines to genetic data. This was the first proof of the existence of a mathematical law for biological events (1900). 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) M. Eigen/ P. Schuster: hypercycles (1979) Roots: 6. Development Turing 1952 Wolpert 1969 Segel 1971 Meinhardt/Gierer 1972 ... 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) Conference ECMTB05: European Conference on Mathematical and Theoretical Biology Dresden, Germany, July 18-22, 2005 (www.ecmtb05.org) 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 Evolution: evolutionary stable strategies, reconstruction of phylogenetic trees Development: origin of multicellularity, logic of signaling networks, embryological pattern formation Ecology/ethology: maintenance/origin of sex, optimization of food search 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 ..... Goals: learn how... to read mathematical modelling 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 In this lecture focus on ... Development What are modelling problems? What are the underlying concepts? Overview: lecture 1. Introduction 2. Diffusion 3. Gradients 4. Turing mech./waves 5. Oscillations 6. Chaos 7. Fluctuations and noise 8. Self-organization 9. Networks 10. Scaling 11. Model validation/ data & model References See website! Model examples: 1. population growth Exp./logistic growth 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 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