Evolution, 4th Edition

412 CHAPTER 16

One widely used method for estimating phy-

logenies is based on likelihood. This is a gen-

eral statistical approach that is described in the

Appendix. Here we illustrate how likelihood is

used to estimate a phylogeny with a very simple

example. This is advanced material that can be

considered optional.

The data are from the example used earlier to

illustrate parsimony, with the DNA bases at three

sites in the genomes of three species (see Figure

16.12). The first step is to calculate the probability

that these data would be observed, given the

phylogenetic tree. The second step is to find

which of all possible phylogenies maximizes that

probability, which gives us the maximum likeli-

hood estimate for the phylogeny.

We begin by focusing on just the first of the three bases

(FIGURE 16.A1). To find the likelihood, we need to make

assumptions about how this base evolves. Here we make

the simple assumptions that the probability that a substitu-

tion occurs (that is, one base replaces another) is constant

in time and equal for all possible changes (for example,

from C to G, or from A to T). For the moment, we will also

assume that we know from using data from outgroups that

the base in the MRCA of these three species was an A.

The likelihood of the data depends on three things: the

topology (or branching order) of the tree, the lengths of

the branches (measured in millions of years), and the sub-

stitution rate (that is, the probability per million years that

one base will be replaced by another). The three possible

topologies are shown in Figure 16.A1. For each of them,

we find the lengths of the branches (t 1 and t 2 ) that make

the data most likely. We then choose the topology that has

the highest likelihood. For tree A, the likelihood turns out

to be given by this rather intimidating equation:

Here λ is the substitution rate. In this example, we’ll as-

sume that we know λ = 0.3, for example from a molecular

clock that has been calibrated for this gene in related spe-

cies. Other equations (which look quite similar) give the

likelihoods for trees B and C. We will not explain here how

this equation was derived, but the interested reader can

find a clear explanation on p. 194 of [15].

FIGURE 16.A2 shows a plot of the likelihood as the

branch lengths of tree A are varied. The maximum value of

the likelihood is reached when t 1 (the time from the root to

the speciation event between species 1 and species 2) is

2.7 million years, and when t 2 (the time from the speciation

event to the present) is 0. The estimate that t 2 = 0 makes

sense: it implies that the MRCA of species 1 and 2 also had

a T, and that there has been no time for either lineage to

have a substitution since then.

By evaluating Equation 16.A1 with t 1 = 2.7 and t 2 = 0, we

find that maximum value of the likelihood for tree A is 0.083.

Doing similar calculations for trees B and C show that their

maximum likelihoods are 0.016, which is about 5 times

smaller than the value for tree A. The data therefore suggest

that tree A is the actual phylogeny. We are not very confi-

dent in this conclusion, though. The difference in the two

likelihoods is not statistically significant, which is not surpris-

ing since the data come from just a single DNA base.

With more data, however, we become more certain

about the phylogeny. Data from two additional DNA

bases are shown in Figure 16.12. To make use of them, we

assume that the bases have evolved independently and

with the same substitution rate. In that case, we can simply

multiply the likelihoods for each base calculated from

Equation 16.A1 to find the overall likelihood of a given

phylogeny. The second base shows exactly the same evo-

L(t 1 ,t 2 ) =

x [3 + exp {

1

64

4

exp {– 3 (2t 1 ,3t 2 )λ}

4

3 (t 1 + t 2 )λ}]

x [3 + exp { 43 t 1 λ} – 2 exp{ 43 t 2 λ}+ exp{^43 (^ t 1 + 2t 2 )λ}– 2]^

BOX 16A

Estimating Trees with Likelihood

Futuyma Kirkpatrick Evolution, 4e

Sinauer Associates

Troutt Visual Services

Evolution4e_Box16.A1.ai Date 01-25-2017

Au: Are outgroups needed here?

t 2

t 1

A

T T A

Sp 1 Sp 2

Tree A

Sp 3

A

T A T

Sp 1 Sp 3 Sp 2

A

T T A

Sp 1 Sp 2

Tree B

Sp 3

Tree C

FIGURE 16.A1 The three possible topologies (shapes) for the phylog-

eny of three species and the base that they each have at a site in the

genome. We assume from using data from outgroup species we know

that the ancestor of the three species had an A, while species 1 and

species 2 have a T. The time from the tree’s root to the speciation event

is t 1 , and the time from that event to the present is t 2.

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