## Transcribed Text

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.

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.