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1. Universal law of growth: In a growing organism, metabolism supplies energy to both maintain existing tissues and create new tissues by cell division. Let Y be the metabolic rate of an organism, Yc the metabolic rate of a single cell, Nc(t) the total number of cells at time t, mc the mass of a cell, and Ec the energy required to create a new cell. The cell properties, Ec, mc, and Yc are assumed to be constant and invariant with respect to the size of the organism. Thus: Y = YcNc + Ec d Nc dt . Let m be the total body mass of the organism at time t, and m = mcNc. (Note that Nc is the total number of cells in a body and is proportional to mass m, while the total number of capillaries NN is proportional to 3/4 power of m.) We have Y = Y0(m) 3/4 according to our lecture notes. a) Show that the above equation can be written as dm dt = am3/4 − bm with a = Y0mc/Ec and b = Yc/Ec. b) Let m = M be the mass of a matured organism, when it stops growing (i.e., dm/dt = 0). Find M, and show that the above equation can be rewritten as dm dt = am3/4  1 −  m M 1/4  c) Let r = (m/M) 1/4 , and R = 1 − r, Then the above equation becomes dR dt = −  a 4M1/4  R Solve this simple ordinary differential equation and show that a plot of ln(R(t)/R(0)) VS. the no-dimensional time at 4M1/4 should yield a straight line with a slope -1 for any organism regardless of its size. d) Based on this scaling for time t, argue that, for a mammal, the interval between heartbeats should scale with its size as M1/4 . 2. Read the attached paper by Etienne, Apol, and Olff. They gave a more general derivation of the 3/4 power law without the assumption of self-similarity. Rederive the 3/4 power scaling for the metabolic rate result under their more general condition.

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