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3.2. Allometric scaling laws

Dive into the world of allometry, investigating how biological structures change with size variations across populations and individual organisms. Discover the beauty of coordinate transformations to capture form changes and the simplicity of double logarithmic scales. From energy dependence in nuclear explosions to flight speeds and mass ratios in nature, embark on a journey through dimensional analysis, scaling arguments, and self-regulatory mechanisms like Wolff's Law. Unveil the connections between shape, size, and functionality in diverse biological systems, from leg bones to tree stems. Delve into the intricacies of fractal structures and their application in metabolic networks. Learn how Murray's Law governs optimal flow divisions in vessels and understand the balance between shearing forces and growth induction mechanisms. Explore the fascinating world of biomechanics and physiology through the lens of allometric scaling laws.

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3.2. Allometric scaling laws

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  1. 3.2. Allometric scaling laws

  2. Allometry (greek: allos = diferent; metros = measure): How does a part change when the total size is varied?

  3. Coordinatentransformations can capture changes in form

  4. This can also be looked at in the same individual

  5. But also for populations of different people to basically determine the ideal weight in terms of size...

  6. Plot this on a double logarithmic scale and it becomes simpler – and you can see where the BMI comes from…

  7. Independent dimensions: SI units Any quantity can be written as a power-law monomial in the independent units

  8. A (in)famous example: The energy of a nuclear explosion US government wanted to keep energy yield of nuclear blasts a secret. Pictures of nuclear blast were released in Life magazine Using Dimensional Analysis, G.I. Taylor determined energy of blast and government was upset because they thought there had been a leak of information

  9. Radius, R, of blast depends on time since explosion, t, energy of explosion, E, and density of medium, , that explosion expands into • [R]=m, [t]=s,[E]=kg*m2/s2, =kg/m3 • R=tpEq k q=1/5, k=-1/5, p=2/5

  10. Rowing speed for different numbers of Oarsmen Fdrag = r v2 l2 f(Re) from DA

  11. Power = r v3 l2 f(Re) ~ N N ~ Volume ~ l3 => N ~ v3 N2/3 => v ~ N1/9 Can be tested from results of olympic games in different rowing categories

  12. An example of a scaling argument – Flight speeds vs.mass

  13. 3 10 2 10 1 10 0 10 -5 -3 -1 1 3 5 7 9 10 10 10 10 10 10 10 10 Boeing 747 F-16 Beech Baron Cruising speed (m/s) goose sailplane starling eagle hummingbird house wren bee crane fly fruit fly dragonfly damsel fly Mass (grams) Cruise speeds at sea level

  14. 3 10 2 10 1 10 0 10 -5 -3 -1 1 3 5 7 9 10 10 10 10 10 10 10 10 Boeing 747 F-16 Beech Baron Cruising speed (m/s) goose sailplane starling eagle hummingbird house wren bee crane fly fruit fly dragonfly damsel fly Mass (grams) Cruise speeds at sea level

  15. Consider a simple explanation L A=Area W

  16. 3 10 2 10 1 10 0 10 -5 -3 -1 1 3 5 7 9 10 10 10 10 10 10 10 10 Boeing 747 F-16 Beech Baron Cruising speed (m/s) goose sailplane starling eagle hummingbird house wren bee crane fly fruit fly dragonfly damsel fly Mass (grams) Fits pretty well!

  17. 3 10 2 10 1 10 0 10 -5 -3 -1 1 3 5 7 9 10 10 10 10 10 10 10 10 What do variations from nominalimply? Boeing 747 Short wings, maneuverable F-16 Beech Baron Cruising speed (m/s) goose sailplane starling eagle hummingbird house wren bee Long wings, soaring and gliding crane fly fruit fly dragonfly damsel fly Mass (grams)

  18. More biological: how are shape and size connected? Elephant (6000 kg) Fox (5 kg)

  19. Simple scaling argument (Gallilei) Load is proportional to weight Weight is proportional to Volume ~ L3 Load is limited by yield stress and leg area; I.e. L3 ~ d2sY This implies d ~ L3/2 Or d/L ~ L1/2 ~ M1/6

  20. Only true for leg bones and land animals... Vogel, Comparative Biomechanics (2003)

  21. Bone calcification is dependent on applied stresses – self regulatory mechanism Wolff’s Law

  22. A bit more quantitative…

  23. Can also be seen in the legs of football players Food & Nutrition Research, 52 (2008)

  24. Similar for the size of the stem in trees – the bigger the tree the bigger its stem

  25. Another example: divisions in “fractal” systems (blood vessels) Metabolism works by nutrients, which are transported through pipes in a network. This forms a fractal structure, so what are fractals?

  26. A fractal looks the same on different magnifications...

  27. This is not particularly special, so does a cube...

  28. What’s special about fractals is that the “dimension” is not necessarily a whole number

  29. Consider the Koch curve

  30. Or the Sierpinski carpet

  31. How long is the coast of Britain?

  32. So what limits flows and shapes in blood vessels?

  33. 1 dP = - - 2 2 u ( )( r z ) h 4 dx p 4 r = - D Q ( ) P h 8 L Most vessels are laminar, i.e. governed by Poiseuille Flow • Take the Navier Stokes equation without external force and uniform flow along the tube u= u(r ) : ¶ æ ö P 1 d du = h Ñ = h 2 ç ÷ u z ¶ x z dz dz è ø

  34. = + 2 Cost Q p K ( r L ) p p 3 Û 2 Min. cost KLr 2 The power needed to create a flow in a tube At optimal flow, costs are minimal ¶ - C 32 L h 1 / 6 h æ ö 16 L = + = 2 o Q 2 K rL 0 p = 1 / 3 ç ÷ r Q ¶ 5 p r r 2 p K è ø Thus for an optimal system:

  35. What does that imply for the divisions? continuity Optimal Flow Q ~ r3 So on every level, the cube of the vessel size needs to be constant: Sr3 = konst Cecil Murray,PNAS 12, 207 (1926).

  36. This fits the experimental observation (here from a dog) Science, 249 992 (1990)

  37. Murray’s Law in an artery

  38. Again there’s a self-regulatory mechanism behind this. The shearing force on the vessel is constant if the size is given by the flow1/3 h K h 1 / 2 r dp 4 ö æ t = - = - = - Q è ø w p 3 2 dx r L This is true over the whole length of the system. Science, 249 992 (1990)

  39. Thus deviations from Q ~ r3 give shearing forces, inducing growth via e.g. K+ channels Nature, 331 168 (1988)

  40. But also via gene expression and protein synthesis Nature, 459 1131 (2009)

  41. This regulates the growth and leads to Murray’s Law Shearing force at a division

  42. These things are age dependent in humans (wall thickness and radius)

  43. Elastic moduli change with age

  44. An aorta (of a rabbit) has a non-linear stress-strain curve

  45. Boundary conditions change for flow in an elastic tube

  46. Then the flow is given by the Laplace-pressure

  47. Profile of blood speed (dog’s aorta)

  48. Look at the heart rate of different animals

  49. ...or the lifespan as a function of weight i.e. There’s only a constant number of heart beats

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