Evolution__3rd_Edition

..
Horse teeth are an example of how
to measure evolutionary rates ...
. . . in “darwins”
CHAPTER 21 / Rates of Evolution 591
21.1 Rates of evolution can be expressed in “darwins,” as
illustrated by a study of horse evolution
The six to eight modern species of the horse family (Equidae) are the modern descendants
of a well known evolutionary lineage in the fossil record. The record extends back
through forms such as Merychippus and Mesohippus to Hyracotherium, which lived
55 million years ago and was once called Eohippus. Horses have characteristic teeth,
adapted to grind up plant material, and fossilized teeth provide the main evidence
that has been used to trace the history of horses. Early members of the lineage were
smaller on average than later forms, as Figure 21.1a illustrates. The Eocene ancestors
of modern horses were about the size of a dog, and the smallest was the size of a
cat. Their teeth were smaller too and had different shapes from modern horses. The
ancestor–descendant relations of the equid species are known reasonably well. The rate
of evolutionary change in the teeth can therefore be estimated by direct measurement,
in fossils from successive times within a lineage.
Horse teeth are classic subject matter in the study of evolutionary rates, and the most
comprehensive modern work on them is by MacFadden (1992). He measured four
properties of 408 tooth specimens, from 26 inferred ancestor–descendant pairs of
species (Figure 21.1b–d). The measure of rate used by MacFadden, and many other
paleontologists, was first suggested by Haldane (1949b). Suppose that a character has
been measured at two times, t 1 and t 2 ; t 1 and t 2 are expressed as times before the present,
in millions of years. t 1 might be 15.2 million years ago and t 2 14.2 million years ago (t 2 is
the more recent sample and has a shorter time to the present). The time interval
between the two samples can be written as ∆t = t 1 − t 2 , which is 1 million years if
t 1 = 15.2 and t 2 = 14.2. The average value of the character is defined as x 1 in the earlier
sample and x 2 in the later sample; we then take natural logarithms of x 1 and x 2 (the
natural logarithm is the log to base e where e ≈ 2.718, and it is symbolized by log or ln).
The evolutionary rate (r) then is:
ln x − ln x
r =
∆t
2 1
The rate is positive if the character is evolutionarily increasing and negative if it
is decreasing, but for many purposes the absolute rate of change, independent of the
sign, is what matters. Haldane defined a “darwin” as a unit to measure evolutionary
rates; 1 darwin is a change in the character by a factor of e (e ≈ 2.718) in 1 million
years. The formula above for r gives the rate in darwins provided that the time interval
is in millions of years. If, for example, x 1 = 1, x 2 = 2.718, and ∆t = 10 million years then
r = 0.1 darwins.
The reason for transforming the measurements logarithmically is to remove
spurious scaling effects. If logarithms were not taken, the rate of evolution of a character
would appear to speed up when it became larger even if its proportional rate
of change remained constant. With logarithmically transformed measurements, rates
of change can be compared between species of very different size, such as mice and
elephants.