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

10. [R]=m, [t]=s,[E]=kg*m2/s2, =kg/m3 R=tpEq k
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
Multiple points for r and t, so can test this scaling and get good estimate for E.

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

12. 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

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

14. Cruise speeds at sea level10 1 2 3
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 -5 -3 -1 1 3 5 7 10 9
Mass (grams)
Cruise speeds at sea level

15. Cruise speeds at sea level10 1 2 3
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 -5 -3 -1 1 3 5 7 10 9
Mass (grams)
Cruise speeds at sea level

16. Consider a simple explanation
A=Area W

17. Fits pretty well! Cruising speed (m/s) Mass (grams) 10 10 1 2 31 2 3
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 -5 -3 -1 1 3 5 7 10 9
Mass (grams)
Fits pretty well!

18. What do variations from nominal imply?10 1 2 3
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 -5 -3 -1 1 3 5 7 10 9
What do variations from nominal imply?
Mass (grams)

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

20. 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 ~ d2 sY
This implies d ~ L3/2
Or d/L ~ L1/2 ~ M1/6

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

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

23. A bit more quantitative…

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

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

26. 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?

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

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

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

30. Consider the Koch curve

31. Or the Sierpinski carpet

32. How long is the coast of Britain?

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

34. 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 ) : P 1 d æ du ö = h Ñ 2 u = h ç z ÷ x z dz è dz ø ) )( 4 1 ( 2 z r dx dP u - = h P L r Q D - = ) 8 ( 4 h p

35. ) ( L r K p Q Cost + = The power needed to create a flow in a tube2 L r K p Q
Cost + =
At optimal flow, costs are minimal C - 32 h L æ h ö 1 / 6 16 L o = Q 2 + 2 K = p rL r = ç ÷ Q 1 / 3 r p r 5 è p 2 K ø
Thus for an optimal system: 2 3
cost
Min.
KLr p Û

36. 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:
S r3 = konst
Cecil Murray,PNAS 12, 207 (1926).

37. This fits the experimental observation (here from a dog)
Science, (1990)

38. Murray’s Law in an artery

39. 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 r dp 4 h K h 1 / 2 æ ö t = - = - Q = - è w ø 2 dx p r 3 L
This is true over the whole length of the system.
Science, (1990)

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

41. But also via gene expression and protein synthesis
Nature, (2009)

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

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

44. Elastic moduli change with age

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

46. Boundary conditions change for flow in an elastic tube

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

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

49. Look at the heart rate of different animals

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

51. Metabolic rate is conveniently measured by oxygen consumption.

52. Plot the metabolic rate for many different animals

53. Works over many decades...

54. This has important medical ramifications in dosage of drugs:
Linearly extrapolating from the dosage for a cat (in mg/kg) to that of an Elephant
Science 138,1100 (1962)

55. A simple scaling argument:
Metabolic rate is mostly there to keep body temperature and thus heat
Heat loss is determined by the body’s surface
Thus metabolic rate
B ~ Q ~ A ~ L2 ~ V2/3 ~ M2/3
i.e. a power-law with an exponent of 2/3

56. For smaller animals this is pretty good
Mammals
Birds
J. theor. Biol, (2001)

57. To get to a regulatory mechanism, we need to know that capillaries are always the same size:
In fact that size is given by the comparison of diffusive and advective transport
rkap = 3 mm

58. Since capillaries have a certain size, the nutrient transport is determined only by the flow.
If there are too little (or too many) capillaries the flow needs to change and we get shearing forces creating new capillaries.
Thus, the metabolic rate of a cell determines the number of capillaries (and thus the overall metabolic rate).

59. One slightly couterintuitive conclusion: Each capillary feeds more cells in larger organisms
tissue
capillary

60. But in fact this is the case, cells in vivo have less consumption the bigger the animal (but constant in vitro…)

61. Visually: Cell structure depends on the size of the animal
Einzeller
Mensch

62. Resistance to flow is given by the number of capillariesZ
TOT 1 N
kap B -
Thus the blood pressure needed to pump through the network is
Dp = Q0 ZTOT ~ B 1 B -1 ~ M 0
Which does not depend on the animals’ mass! How does a giraffe get blood into the brain?

63. They actually do have higher blood pressure – the anatomy of the arteries is strikingly different...

64. Trees are different still...
three basic assumptions
Branching, hierarchical network that is space filling to feed all cells
Capillaries are invariant of plant size
Minimization of energy to send vital resources to the terminal units

65. Modelling the network of tubes

66. a = Vk+1/Vk = r2k+1lk+1/r2klk = b2g
Lets model the network of tubes solely via their splitting ratios (and all equal in one level of the network)
und
lk is the length of the k-th tube
rk is the radius of the k-th tube
The volume-ratio of tubes is therefore given by
a = Vk+1/Vk = r2k+1lk+1/r2klk = b2g

67. The network needs to be space filling to get nutrients to the whole body. So the volume services by the tubes in all levels is the same:
n is the splitting ratio

68. Xylem tansport in plants is made from individual tubes
Xylem tansport in plants is made from individual tubes. Thus the splittings are area preserving.
Thus we have for the volumes of the tubes:
a = b2g = n-4/3

69. S B ~ M 3/4 VB = nkVk = (na)k ~ V0 = Vkap a -N
Total volume is given by
VB = nkVk = (na)k ~ V0 = Vkap a -N S
given a = n-4/3 and invariance of capillaries, we get VB ~ Nkap4/3
Since Ncap ~ metabolic rate, B, and
VB ~ mass, M, we get
B ~ M 3/4

70. r0 = b -N rkap = Nkap1/2 rkap ~ (M3/4)1/2
Stemsize dependence
r0 = b -N rkap = Nkap1/2 rkap ~ (M3/4)1/2

71. Allows us to relate one level to the next.
All vessels of the same level can be considered identical. Define scale-free ratios
and
rk is radius of vessel at kth level
lk is length of vessel at kth level
Allows us to relate one level to the next.
The network is space-filling to reach the whole body
n is branching ratio

72. Can this help in understanding how an organism grows?
Energy input
Resting
metabolism
growth

73. Kleiber's rule tells us:
With the stationary
Solution:

74. Solving this gives a sigmiod curve:

75. Compare to experimental data for M and t

76. Scaling collapse:

77. The model also implies how much energy is used by an individual (tree)

78. This implies that population densities can be predicted via the usage of land for each tree as a function of tree size

79. This works pretty well

80. Recap Sec 3.3
Dimensional analysis and scaling arguments are very powerful tools in physics (that can be applied to biology)
Size changes have great influence on form and function.
Self regulatory mechanism govern the observed scaling relations.
Bone growth (and vessel growth) is greatly influenced (regulated) by mechanical forces
Cells of large animals have smaller metabolic rates – mostly due to heat conservation