Math 210G Mathematics AppreciationDr. Joe LakeyLecture 3: Mathematics and Biology website: http://www.math.nmsu.edu/~jlakey/home.html phone: 505-646-2417 office: Science Hall 230
Eukaryotic cells • Typical animal cell • 1. Nucleolus • 2. Nucleus • 3. Ribosome • 4. Vesicle • 5. Endoplasmic reticulum • 6. Golgi apparatus • 7. Cytoskeleton • 8. Endoplasmic reticulum • 9. Mitochondrion • 10. Vacuole • 11. Cytosol • 12. Lysosome • 13. Centriole
Creation of new cells • Mother cell divides into two daughter cells. • growth in multicellular organisms • reproduction in unicellular organisms. • Prokaryotes: binary fission. • Eukaryotes: mitosis/meiosis and cytokinesis. • DNA replication is required using specialized
Morphogenesis • Beginning of shape • physical and mathematical processes + constraints biological growth • D'Arcy Thompson and Alan Turing (20th cent) • Postulated chemical signals + physico-chemical processes: diffusion, activation, and deactivation in cell and organ growth. • The fuller understanding …required discovery of DNA + development of molecular biology and biochemistry.
Mechanisms for morphogenesis • Molecular basis: morphogens : diffuse, carry signals that control cell differentiation decisions (transcription factor proteins, stem cells, cell adhesion molecules) • Cellular basis: sorting out – enables structure formation. Cells sort into clusters that maximize contact between cells of the same type. (differential cell adhesion). • Adhesion. Cells that sharing adhesion molecules separate from cells having different adhesion molecules.
Reaction-diffusion • in 1952 Alan Turing wrote a paper] proposing a reaction-diffusion model as the basis of the development of patterns such as the spots and stripes seen in animal skin. • ∂C/∂t = F(C) D∇2C
Phyllotaxis • Arrangement of leaves in some plants • obeys a number of subtle mathematical relationships • Numbers of petals typically 8,13, 21, 34, 55, or 89 etc • Florets in sunflower: two oppositely directed spirals: 55 clockwise, 34 counterclockwise. • 5+8=13; 8+13=21; 13+21=34 ETC.
Leonardo of Pisa (c1170 – c1250) • Liber Abaci (1202) • introduces Arabic numerals • growth of a hypothetical population of rabbits based on idealized assumptions: • “Fibonacci numbers”
Generating Fibonacci Sequence • f(0)=0 • f(1)=1 • f(n)=f(n-1)+f(n-2) for n>1 • EXERCISE: compute f(9)
A. H. Church (1904). On the Relation of Phyllotaxis to Mechanical Laws. Williams and Norgate, London. Florets in sunflower: two oppositely directed spirals: 55 clockwise, 34 counterclockwise
Ratios of alternate Fibonacci numbers • converges to φ-2 • φ: golden ratio • Ratios measure fraction of turn between successive leaves on the stalk of a plant: • elm and linden: ½ • beech and hazel: ⅓ • oak and apple: ⅖ • poplar and rose: ⅜, • willow and almond 5/13 etc.
Mathematical morphogenesis • morphogenesis link with demos • Free boundary problems • Adhesion problems • Transport problems
A Fibonacci spiral uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34 • Ratio of sides: “golden mean”
Sir D'Arcy Wentworth Thompson (1860-1948) St. Andrews, Scotland • “On Growth and Form” : biologists overemphasized evolution as the fundamental determinant of the form and structure of living organisms; underemphasized roles of physical laws and mechanics • Thompson pointed out example after example of correlations between biological forms and mechanical phenomena.
“The Comparison of Related Forms,” Thompson suggested differences in the forms of related animals could be described by means of mathematical transformations
Thompson's illustration of the transformation of Argyropelecus olfersi into Sternoptyx diaphana by applying a 70° shear mapping
Quoting from “On Growth and Form” • We are accustomed to think of magnitude as a purely relative matter. We call a thing big or little with reference to what it is won’t to be, as when we speak of a small elephant or a large rat; … and that Lilliput and Brobdingnag are all alike, according as we look at them through one end of the glass or the other. Gulliver himself declared, in Brobdingnag, that 'undoubtedly philosophers are in the right when they tell us that nothing is great and little otherwise than by comparison': and Oliver Heaviside used to say, in like manner, that there is no absolute scale of size in the Universe, for it is boundless towards the great and also boundless towards the small. It is of the essence of the Newtonian philosophy that we should be able to extend our concepts and deductions from the one extreme of magnitude to the other…
…. Nevertheless, in physical science the scale of absolute magnitude becomes a very real and important thing; and a new and deeper interest arises out of the changing ratio of dimensions when we come to consider the inevitable changes of physical relations with which it is bound up. The effect of scale depends not on a thing in itself, but in relation to its whole environment or milieu; it is in conformity with the thing's 'place in Nature', its field of action and reaction in the Universe... Men and trees, birds and fishes, stars and star-systems, have their appropriate dimensions, ... The scale of human observation and experience lies within the narrow bounds of inches, feet or miles, all measured in terms drawn from our own selves or our own doings. Scales which include light-years, parsecs, Angström units, or atomic and sub-atomic magnitudes, belong to other orders of things and other principles of cognition.
A common effect of scale is due to the fact that, of the physical forces, some act either directly at the surface of a body, or otherwise in proportion to its surface or area; while others, and above all gravity, act on all particles, internal and external alike, and exert a force which is proportional to the mass, and so usually to the volume of the body. A simple case: two weights hung by two similar wires… forces are proportional to their masses/volumes…areas of cross-section of wires are as the squares … therefore the stresses in the wires per unit area are not identical, but increase in the ratio of the linear dimensions, and the larger the structure the more severe the strain becomes: Force/Area ≈ length3/length2 and the less the wires are capable of supporting it.
In short, it often happens that of the forces in action in a system some vary as one power and some as another, of the masses, distances or other magnitudes involved; the 'dimensions' remain the same in our equations of equilibrium, but the relative values alter with the scale. This is known as the 'Principle of Similitude', or of dynamical similarity, and it and its consequences are of great importance. In handful of matter cohesion, capillarity, chemical affinity, electric charge are all potent; across the solar system gravitation rules supreme; in the mysterious region of the nebulae, it may haply be that gravitation grows negligible again.
Does capacity to lift weight increase proportionally to athletes weight?
Can King Kong exist? • Strength of bones and muscles: proportional to cross sectional area • Weight: proportional to their volume. • As an object (like a leg) gets larger, its strength will increase proportional to the width squared, but the weight will increase proportional to the width cubed. • In order to survive, an animal must be able to support its own weight. If a gorilla can support 2x its weight, how large is the largest possible gorilla?
25 feet in height • 20 to 60 tons • 7,500 pounds of food per day
Overweight Gold Medal Olympians by BMI • Shawn Crawford (USA) Sprinting 177cm, 81kg, BMI=26 • Mark Lewis-Francis (GB) Sprinting 183cm, 89kg, BMI=26 • Matthew Pinsent (GB) Rowing 196cm, 108kg, BMI=28 • James Cracknell (GB) Rowing 192cm, 100kg, BMI=27 • Ed Coode (GB) Rowing 193cm, 96kg, athlete BMI=26 • Steve Williams (GB) Rowing 189cm, 96kg, BMI=27) • David Cal (Spain) Canoeing 183cm, 91kg, BMI=27) • Khadjimourat Gatsalov (Russia) Wrestling 180cm, 96kg, BMI=30 • Artur Taymazov (Uzebekistan) Wrestling 189cm, 112kg, BMI=31 • Roman Sebrle (Czechoslovakia) Decathlon186cm, 88kg, BMI=25 • Ryan Bayley (Austria) Cycling 181cm, 84kg, BMI=26 • Odlanier Solis Fonte (Cuba) Boxing 180cm, 91kg, BMI=28 • Alexander Povetkin (Russia) Boxing 188cm, 91kg, BMI=26 • Ihar Makarau (Belarus) Judo 180cm, 100kg, BMI=31 • Yuriy Bilonog (Ukraine) Shot put 200cm, 135kg, BMI=34
Baffling “universality” • Explanation proposed in 1883: • * Organism diameter L • Area A≈ L2, Volume V≈ L3. • * Fixed density M / VL ≈ (Mass)1/3, • * Heat dissipation ≈ surface area, • total metabolic rate: R≈ L2 ≈ (Mass)2/3, • not quite ¾ !