What if animals were fractals? University of Utah ACCESS 2009

universal laws in biology? In 1917, D’Arcy Thompson began his book On Growth and Form with the quote: “chemistry… was a science but not Science… for that true Science lay in its relation to mathematics.” He then goes on to say: math + chemsitry = Science biology + fluffy = science

universal laws in biology? Do biological phenomena obey underlying universal laws of life that can be mathematized so that biology can be formulated as a predictive, quantitative science?  “Newton’s laws of biology”

allometric scaling laws Allometry is the study of changes in characteristics of organisms as body sizes grow. Can we quantify how body mass/size affect other physiological aspects such as metabolic rate, life span, heart rate, or population density? A typical allometric scaling law is usually written in the form of Y = Y 0 M b where Y is the biological variable of interest, M is the mass. Both Y 0 and b are numbers to be determined from experimental data, and the scaling exponent is of particular interest as it characterizes how Y specifically changes as the mass is varied.

size matters Metabolic rate: rate of energy consumption if the animals are at rest in a neutrally temperate environment with digestive system inactive (Wikipedia definition)

some examples Allometric scaling exponents for various biological variables as a function of mass: Scaling Exponent Metabolic rate Heart beat rate Life Span Radius of aortas/ tree trunks Genome length for unicellular organism Brain mass ¾ -¼ ¼ 3/8 ¼ ¾

scaling of heart rate and life span animalheart ratelife span (wild)# of heart beats mouse elephant gorilla 500 beats/min 28 beats/min 70 beats/min 2 years 60 years 30 years Calculate the number of heart beats among the following animals…

metabolic rate scaling law How should metabolic rates depend on mass? It may be the case that… All animals are made up of cells, so mass  number of cells Each cell is consuming energy at a certain rate so metabolic rate  mass FACT: Aerobic metabolism is fueled by oxygen, whose concentration in hemoglobin is fixed. Here is a thought: maybe there is a relationship between surface area used to dissipate heat/waste and the metabolism of the animal… metabolic rate (R)  surface area (SA) mass (M)  volume (V)

metabolic rate scaling law Compare a spherical mouse of radius r with a spherical cat who is 3 times larger. = = r 3r

A 10 lb. goose needs 300 calories per day to survive. What about a 160 lb person? metabolic rate scaling law

derivation of the ¾ exponent West, Brown and Enquist proposed a derivation of the ¾ scaling exponent based on the idea of space filling fractals filling up the body (Nature 276(4),1997).

derivation of the ¾ exponent Suppose the body is supplied by a network of tree-like structures. Let L be the length scale of the network. The volume V served by the entire network is proportional to L 3. V =  L 3

derivation of the ¾ exponent Let’s fill a ball with a branch. Find . (Hint: V  r 3) The volume served by the entire network is the sum of volumes served by each of the branches…  L 3 =  l 3

Unlike real fractals, the tree-like structure of the network will end somewhere. For the circulatory system, it ends at the capillary levels and for trees, at the leaf structures.  terminal nodes L 3 = Nl 3 The metabolic rate R should be proportional to N. Why? R = wN where w is the energy consumption of cells supplied by a terminal node. Then derivation of the ¾ exponent

what were we doing again? Remember we are trying to find R = R 0 M b. The mass should be proportional to the volume of the network. In particular, thing about the fluid flowing within the structure. V  M blood  M The amount of fluid within the structure must be conserved Amount flowing in = amount flowing out v in A in = v out A out where v is the average speed and A is the cross sectional area. Assume that the flow is steady.* Then v in = v out. What does the mean in terms of the cross sectional area?

respiratory system circulatory system umm…

From the assumption that the cross sectional area is independent of any sectional cut, where A = cross sectional area of the network  density of fluid  proportion of blood/fluid to body Also assume A =  N, where  is the cross sectional area of the terminal node.

The Final Stretch Let’s put everything together now to get the ¾ scaling exponent…

Conclusions? We have found that b = ¾, which matches our data…