Mathematics in Biology

Mathematics in Biology
Introduction
Math in Bio?
Every attempt to employ mathematical methods in the study of biological questions must be considered profoundly irrational and contrary to the spirit of biology.
If mathematical analysis should ever hold a prominent place in biology -­ an aberration which is happily almost impossible -­ it would occasion a rapid and widespread degeneration of that science.
Auguste Comte, 1830
French philosoph, founder of s ociology
Source: http://www.cs.utah.edu/~crj/quotes.html
Math in Bio!
§ "mathematics ... was repugnant to me ... from my not being able to see any meaning in the early steps in algebra... This impatience was very foolish ... I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense".
(From Charles Darwin's Autobiography, 1876;; cited by R. May in Science 303:790-­793, 2004)
§ "The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them".
(By Sir William Henri Bragg, 1862-­1942;;
Physics Nobel Prize for X-­ray crystallography.)
Source: http://www.math.rutgers.edu/~sontag/336.html
A brief historical overview
Fibonacci
Leonardo of Pisa (best known as Fibonacci) (~1175 - ~1250)
• Italian mathematician, ("the most talented western mathematician of the Middle Ages"?)
• published "Liber Abaci " ("Book of Calculation") in 1202.
• Introduced arabic numbers in occident (more convenient than roman numbers!).
• Fibonacci numbers.
Liber Abaci (1202)
In his Liber Abaci, Fibonacci also posed, and solved, a problem
involving the growth of a population of rabbits based on
idealized assumptions. The solution, generation by generation,
was a sequence of numbers later known as Fibonacci numbers.
Source: wikipedia
Giovani Alfonso Borelli
Giovanni Alfonso Borelli (Italy, 1608-­1679) • Italian physiologist, physicist, astronom, philosoph, and mathematician.
• Father of biomechanics.
• Relates animals to machines and utilizes
mathematics to prove his theories.
• Published De Motu Animalium (1680).
Pope MH (2005) Giovanni Alfonso Borelli -­ the father of biomechanics. Spine (Phila Pa 1976). 30:2350-­5.
Thomas Robert Malthus
Thomas Robert Malthus (England, 1766-­1834)
• Reverend, political economist, interested in demography, developed (initiated) theories of population growth.
• Published "An Essay on the Principle of Population" (Six editions, from 1798 to 1826)
• "The power of population is indefinitely greater than the
power in the earth to produce subsistence for man".
• Proposed solution: celibacy (sexual abstinence) before marriage, postponement of marriage, especially in poor families (malthusianism -­ 1st step towards eugeny?)
Sources: wikipedia + Jacques van Helden (cours "Biologie & Société")
Pierre-­François Verhulst
Pierre-François Verhulst (Belgique, 1804-1849)
• Belgian mathematician, interested in number theory and in population dynamics
• Professor at Université Libre de Bruxelles from 1835 to 1840.
• Published: -­ "Notice sur la loi que la population poursuit dans son accroissement". Correspondance mathématique et physique 10:113-­121 (1838).
-­ "Recherches mathématiques sur la loi d'accroissement de la population". Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-­Lettres de Bruxelles 18:1-­42 (1845).
• Proposed the "logistic equation" (1838) as a more realistic alternative to the Malthus law.
• Using data on the belgian population in 1815, 1830 and 1845, he determined the 3 parameters of the logistic function and estimated to 6,6 millions the maximum (asymptotic) population in Belgium...
Source: wikipedia + wiki KULeuven
Pierre-­François Verhulst
Lotka & Volterra
Alfred James Lotka (US, 1880-­1949)
• mathematician, physical chemist, and statistician, famous for his work in population dynamics and energetics.
• published "Elements of Mathematical Biology" (1925) • known for his energetics perspectives of evolution. Lotka proposed that natural selection was, at its root, a struggle among organisms for available energy;; organisms that survive and prosper are those that capture and use energy at a rate and efficiency more effective than that of its competitors. Vito Volterra (Italy, 1860-­1940)
• Italian mathematician, known for his contributions to
mathematical biology and integral equations.
• Published "Leçons sur la théorie mathématique de la lutte pour la vie", Paris, 1931. Lotka and Volterra both (independently) proposed models
for competing populations and predator-prey systems.
Lotka & Volterra
Lotka & Volterra
Lotka (1925) Elements of Physical Biology, Lotka A.J. (1920)
PNAS 6: 410-­415
Lotka & Volterra
Evolution of predator and prey populations in time Schnell, Grima, Maini (2007), American Scientist 95, 134
Robert May
Robert May (Australian/UK, 1936-­present)
• interest in animal population dynamics and the relationship between complexity and stability in natural communities.
• development of theoretical ecology with application to the study of disease and biodiversity.
• selected publications:
May RM (1976) Simple mathematical models with very complicated dynamics, Nature 261: 459–467.
May RM (1988) How many species are there on Earth, Science 241:1441-­1449.
RM Anderson, RM May (1992) Infectious diseases of humans: dynamics and control, Oxford.
May RM (1994) Biological Diversity: Differences between Land and Sea, Phil Trans Royal Soc B: Biol Sciences 343:105-­111.
May RM (2001) Stability and Complexity in Model Ecosystems, Princeton University Press.
D'Arcy Thompson
D'Arcy Wentworth Thompson (Scotland, 1860-­1948)
W. Arthur (2006) Nature 7:401-­406
D'Arcy Thompson
W. Arthur (2006) Nature 7:401-­406
D'Arcy Thompson
D'Arcy Thompson philosophy
In his introduction he makes the following point: “…in dealing with the facts of embryology or the phenomena of inheritance, the common language of the books seems to deal too much with the material elements concerned.” And he goes on to explain that in his view biologists should place less emphasis on matter (such as a piece of embryonic tissue) and more on the forces that shape it.
W. Arthur (2006) Nature 7:401-­406
D'Arcy Thompson
Problems and limitations of the theory
Working as he was in the early twentieth century, D’Arcy Thompson was well within the era of evolutionary trees (a generalized one of which is, famously, the only picture in the whole of Darwin’s Origin of Species). But he was working well before the rigorous treatment of such trees that began with the advent of phylogenetic systematics in the mid-­twentieth century. This limitation shows in the fact that he was usually content to note that the morphologies of a group of related genera could be derived from each other by appropriate transformations, and he was not terribly concerned with mapping the genera to a phylogeny so that it became apparent which way round the transformations had taken place.
W. Arthur (2006) Nature 7:401-­406
D'Arcy Thompson
Problems and limitations of the theory
Direct versus indirect development. It is interesting that D’Arcy Thompson generally used, as examples, direct rather than indirect developers. He used many crustaceans as examples (FIG. 2), but no insects. Likewise, in the vertebrates, he used many fish (FIG. 3), but no amphibians. This is a limitation of the theory (so far anyway), but not a problem — indeed it was a wise strategy to limit his examples in this way. It is difficult enough to understand how a directly developing system evolves in quantitative terms, without adding the complexities of metamorphosis.
The main deficiency of the theory of transformations, from a genetic or developmental point of view, is that no causal mechanism was proposed for their occurrence. Of course, we cannot blame D’Arcy Thompson for a failure to incorporate ideas about transcription factors into his theory, as they were then unknown. Nor can we blame him, at least in the first edition of On Growth and Form, for omitting the embryological ideas of gradients, fields and morphogens, as these also awaited articulation. But there is one thing that we can and perhaps should blame him for — a neglect of juvenile morphologies. Notice that all the forms shown in here (and in the book of D'arcy Thompson) are those of adults. This concentration on adults is strange, given two facts. First, it must have been as obvious to D’Arcy Thompson as it is to biologists today that there is no way evolution can turn one sort of adult form into another except by modifying the course of development. Second, many of the chapters leading up to the one on transformations, such as the one on spirals mentioned above, did explicitly deal with both adult and juvenile morphology. It is as if the developmental and evolutionary parts of the book were disconnected from each other.
See also: "D'Arcy T hompson F antome de la Biologie -­ Des outils mathématiques et physiques pour expliquer les formes du vivant", by N. Witkowski, La Recherche (janvier 1998)
W. Arthur (2006) Nature 7:401-­406
Chemical kinetics
Chemical kinetics
Antoine-­Laurent de Lavoisier (1743-­1794): => quantitative measurements in chemistry + notion of stoichiometry
Claude Louis Berthollet (1748-­1822):
=> first qualitative form of the law of mass action
Maximilian Guldberg (1833-­1902) &Peter Waage (1839-­1900): => first quantitative expression of the law of mass action Svante August Arrhenius (1859-­1927) & Van’t Hoff (1852-­1911): => effect of temperature on reaction rate
Enzyme kinetics
Michaelis & Menten
Leonor Michaelis (Germany 1875-­ 1949) • Biochemist, physical chemist, and physician
• Professor at HU Berlin, Nagoya (Japon) and Rockefeller (USA)
•
NB: Besides his work on enzyme kinetics, he found that thioglycolic acid could dissolve keratin, a discovery that would come to have several implications in the cosmetic industry, including the hair permanent wave.
Maud Leonora Menten (Canada, 1879-­1960) • Physician who made significant contributions to enzyme kinetics and histochemistry. • Professor at Toronto, Chicago and Berlin
•
NB: She realized the first separation of proteins by electrophoresis in 1944.
Michaelis & Menten are known for the derivation of kinetic equations for enzyme-­catalyzed biochemical reactions
(they originally studied the invertase, which catalyzes the hydrolysis of sucrose into glucose and fructose).
Gene regulation
Jacob & Monod
Jacques Monod
(France, 1910-­1976)
François Jacob
(France, 1920-­2013)
§ Nobel prize in physiology / medicine 1965
§ Developement of a "model" describing the interactions between genes and proteins during transcription (gene regulation)
§ Monod also made important contributions to the field of enzymology with his theory of allostery in 1965 (with Jeffries Wyman and Jean-­Pierre Changeux)
Ronald Fisher
Ronald Aylmer Fisher (1890-­1962)
genetics, mathematics, statistical biology
Apart from his central role in the development of the field of statistics, Fisher was among the most prominent of twentieth century workers in the fields of genetics and statistical biology, and one of the primary founders of the field of population genetics.
His work included mathematical investigations that helped provide a statistical foundation for the emerging neo-­Darwinist synthesis: clarification of the notion of degrees of freedom, development of the maximum likelihood estimation concept and analysis of variance technique, and in general the working out of modern statistical principles of experimental design.
In genetics his 1930 book The Genetical Theory of Natural Selection, with its ground-­breaking treatment of the concepts of fitness and dominance, was a milestone work in that field.
Source: http://www.wku.edu/~smithch/chronob/FISH1890.htm
Hodgkin – Huxley Alan Lloyd Hodgkin
(1914-1998)
• Hodgkin and Huxley are both physiologists and biophysicists
• Nobel Prize in Physiology/Medicine (1963)
• Discovery of the basis propagation of nerve impulses (action potential) • Publications:
Hodgkin AL, Huxley AF (1939) Action potentials recorded from Inside a nerve fibre, Nature 144: 710-­711
Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 177:500-­544
Andrew Huxley
(1917-­2012) • Some years later, FitzHugh and Nagumo propose a simplified version of the Hodgkin-­Huxley model to describe the activation and deactivation dynamics of a spiking neuron, excitability and relaxation oscillations.
• Publications:
FitzHugh R (1955) Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophysics, 17:257-­-­278
FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophysical J. 1:445-­466
FitzHugh R. (1968) Motion picture of nerve impulse propagation using computer animation. Journal of Applied Physiology, 25:628-­630
Erwin Schrödinger
Erwin Schrödinger (Austrian, 1887-­1961)
• Physicist, contribution to quantum theory
• Nobel Prize (with Paul Dirac, 1933)
• Published "What is life?" (1946), where he introduced the concept of a complex molecule with the genetic code for living organisms. According to James D. Watson's memoire, "DNA, the Secret of Life", Schrödinger's book gave Watson the inspiration to search the gene, which led to the discovery of the DNA double helix structure in 1953. Max Delbrück
Max Delbrück (Germany/USA, 1906-­1981)
• Biophysicist
• Nobel Prize in physiology/medicine (1969) for "discoveries concerning the replication mechanism and the genetic structure of viruses".
• He stimulated physical scientists' interest into biology, especially as to basic research to physically explain genes.
• His inferences on genes'susceptibility to mutation was relied on by Erwin Schrödinger in his book "What Is Life?"
Nobel Lecture, Published in Science 1969
Source: wikipedia
Alan Turing
Alan Mathison Turing (UK, 1912-­1954)
• mathematician, logician, cryptanalyst, and computer scientist.
• contributed to the development of computer science, giving a formalisation of the concepts of "algorithm" and "computation" with the "Turing machine".
• father of computer science and artificial intelligence.
• interested in mathematical biology;; wrote a paper on the chemical basis of morphogenesis (cf. "Turing patterns"), and predicted oscillating chemical reactions such as the Belousov-­
Zhabotinsky reaction (discovered in the 1960s).
Alan Turing
Ilya Prigogine
Ilya Prigogine (Russian/Belgian, 1917-­2003)
• Physicist and chemist (ULB)
• Interested dissipative structures, thermodynamics of systems far from equilibrium complex systems, irreversibility, self-­organization.
• Nobel prize in Chemistry,1977 • Published (with G. Nicolis) "Self-­Organization in Non-­Equilibrium Systems", Wiley, 1977.
Arthur Winfree
Arthur Winfree (USA, 1942-­2002)
• Theoretical Biologist (developed mathematical models and theories for circadian clocks, cardiac rhythms, developmental processes, etc) • The Geometry of Biological Time (1980/2001)
Belousov-­Zhabotinsky reaction (in Petri dish)
Synthetic Biology
Synthetic Biology
Synthetic Biology
Michael Elowitz
Alexander van Oudenaarden
Jeff Hasty
Stan Leibler
James Collins
Ron Weiss