Allman E.S., Rhodes J.A. Mathematical Models in Biology: An Introduction

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206 Constructing Phylogenetic Trees

d. Does your tree agree topologically with the one produced by

UPGMA and/or Neighbor Joining using the Jukes-Cantor dis-

tance?

5.4.11. In this problem, you will attempt to use the maximum parsimony

method to construct an unrooted tree for the simulated sequences d1,

d2, d3, d4, d5, and d6 in the data ﬁle seqdata.mat. Begin by

ﬁnding the informative sites as in the last problem.

a. What percentage of the sites are informative?

b. Compute the number of unrooted trees that must be examined if

we really consider all possibilities.

c. Use Neighbor Joining, with the log-det distance computed from

the full sequences, to get a tree that is a good starting point for a

search for the most parsimonious. Compute its parsimony score

using only the ﬁrst 10 informative sites.

d. Again, using only the ﬁrst 10 informative sites, compute parsimony

scores of at least 4 other trees that are similar to the one in part (c).

Can you ﬁnd a more parsimonious one?

e. How conﬁdent are you that the most parsimonious tree you found

is actually the most parsimonious? What percentage of the possible

trees did you compute parsimony scores for? What percentage of

the informative sites did you use?

5.5. Other Methods

There are actually many other approaches to phylogenetic tree construction.

The list of proposed methods is quite long and grows longer each year, as

researchers continue to investigate the problem.

In addition to distance methods and parsimony, there is a third major class

of approaches called Maximum Likelihood methods. The basic approach of

maximum likelihood is to ﬁrst specify a particular model of molecular evo-

lution (such as Jukes-Cantor, Kimura 2- or 3-parameter, or a more elaborate

one). Then, we consider a speciﬁc tree that is a candidate for relating our

taxa. Assuming our model of evolution and speciﬁc tree are correct, we can

calculate the probability that the DNA sequence in our data could have been

produced. This is called the likelihood of the tree, given our data. We repeat

this process for all other trees, getting a likelihood value for each. Then, we

choose the tree with the greatest likelihood as the tree we feel best ﬁts our data.

To many researchers, maximum likelihood approaches, which follow a

long-established tradition in statistics, offer the best hope for good tree

construction. However, they face several problems. First, they depend on

5.5. Other Methods 207

choosing a speciﬁc model of evolution, and if that model does not describe the

real process well, one could question their validity. Second, as with parsimony,

they require considering all possible trees, and so they require heavy computa-

tional effort. For each tree topology considered, a time-consuming calculation

is needed to ﬁnd optimal model parameters consistent with the data. If the

number of taxa is large, it is impossible to search through all possible trees, op-

timizing model parameters for each, and so in practice heuristic shortcuts are

taken. Although these seem to perform well in practice, maximum likelihood

remains more computationally intensive than other approaches.

Another way of thinking of phylogenetic tree construction methods is to

split them into two classes: those that pick a tree based on some optimality cri-

terion, and those that are algorithms that produce a tree. Both Maximum Parsi-

mony and Maximum Likelihood are based on optimality criteria, whereas the

distance methods discussed here are algorithmic. Some researchers argue that

optimality criteria methods are inherently superior, because they at least make

explicit on what the tree choice is based. However, since searching through a

large number of trees for an optimal one can be computationally infeasible,

computer implementations of parsimony and likelihood methods sometimes

begin by considering trees produced by an algorithm (such as Neighbor Join-

ing), and variants of it obtained by moving a few branches around.

One of the difﬁculties of picking a method to use is that you can ﬁnd good

arguments for and against them all. Nonetheless, the need to construct trees to

investigate biological problems is simply too great not to make use of them.

The cautious approach is to always use a number of different methods on your

data. Rather than trusting a single method to give an accurate tree, look to

see if different methods give roughly the same results. They often do, and if

they do not, it is worth investigating why they don’t. It is simply not enough

to just run a computer program on your data and accept the tree produced.

Even when a tree has been chosen by one method or another, it would be

desirable to quantify how conﬁdent we should be of it. A partial answer to this

can be given by the statistical technique of bootstrapping. In the bootstrap

procedure, the true data sequences are used to create a set of new pseudo-

replicate sequences of the same length. The bases at a particular site in the new

sequences are chosen to be the bases appearing in a randomly chosen site in

the original sequences. A tree is constructed for the phylogeny of the pseudo-

replicates and recorded. This procedure is then repeated many times, giving a

large collection of bootstrap trees. If a high percentage of the bootstrap trees

are in agreement with the one produced using the original data, then we may

be more conﬁdent of it.

208 Constructing Phylogenetic Trees

An important caveat on using boostrapping, however, is that the technique

only helps assess the effects on tree construction of variability within the

sequences. Boostrapping says nothing about the fundamental soundness of

the method by which we choose a tree – it only indicates how variability in

the data affects the outcome of the method.

Computer software is essential for using any of the methods on more than

a few taxa. Two widely used packages that implement a variety of methods

are PAUP* (Swofford, 2002) and PHYLIP (Felsenstein, 1993). If you have

access to either, exploring their capabilities is worthwhile.

5.6. Applications and Further Reading

Let’s return to the question of hominoid phylogeny that introduced this chap-

ter. What tree can be inferred from the mitochondrial DNA data? Although

we could give you an answer, we would rather you found it for yourself.

In the exercises, you will have a chance to apply some of the methods of

this chapter to the data, starting either with the raw sequences or with some

distances already computed from the sequences.

The analysis of the data in (Hayasaka et al., 1988) rests primarily on use

of the Neighbor Joining algorithm, as will the analysis you can easily do with

MATLAB. If you have access to software designed for maximum parsimony,

maximum likelihood, or other methods, we urge you to see if those methods

give similar results.

Also, keep in mind the analysis you do is based on one particular stretch

of DNA. Studies based on other orthologous sequences might give different

results. Furthermore, there are many approaches to phylogenetic inference

that are not sequence-based. The evidence of all should be weighed before

making too strong a statement about the hominoid phylogeny.

As methods of phylogenetic tree construction from DNA sequence data

have developed, they have been used to study a number of interesting ques-

tions. Even a quick survey of a general research journal like Science turns up

a large number of papers in which genetic sequences are used to investigate

the evolution of various species from a common ancestor. Here are just a few

examples of some recent applications.

1. Investigating whether the evolution of several species parallel one an-

other: For instance, the evolution of hosts and parasites can be studied

by constructing separate phylogenetic trees for each. The similarity

of the tree topologies can indicate whether the parasites evolved with

5.6. Applications and Further Reading 209

the host, or if parasites “jumped” from one host species to another

(Hafner et al., 1994). Likewise, trees for two symbiotic species, such

as fungus-growing ants and the fungus they grow, help indicate how

far back in evolutionary history the symbiotic partnership stretches

(Chapela et al., 1994; Hinkle et al., 1994).

2. Determining likely infection sources of human immunodeﬁciency

virus (HIV) by constructing trees from HIV sequences from a number

of infected individuals: There have been several forensic applications

of this, to the Florida Dentist AIDS cases (Altman, 1994; Ou et al.,

1992) and to the case of a doctor accused of intentionally injecting HIV

into a former lover (Vogel, 1997, 1998).

3. Studying whether genes have entered the genome of a species through

lateral transfer (Andersson et al., 2001; Salzberg et al., 2001): When

a tree is constructed from DNA sequences for a gene, it is really a

“gene tree” showing gene relationships that may or may not be the

same as taxa relationships. Because some human genes are believed to

have been obtained by lateral transfer from bacteria that infected us, for

certain genes we may appear to be more closely related to some bacteria

than other mammals. If a gene is suspected to have arisen in a eukaryote

through lateral transfer from bacteria, then a tree can be constructed

using gene sequences from both eukaryotes and bacteria. The clustering

pattern should help indicate whether genes were transferred laterally

or not.

4. Monitoring restrictions on whale hunting: DNA samples from whale

meat sold as food and from whales in the wild were used to construct

a tree, indicating not only the species of whales being sold, but even

the ocean of origin (Baker and Palumbi, 1994).

5. Investigating the “Out of Africa” hypothesis of human origins: The

clustering pattern on a tree constructed from human DNA sequences

from ethnic groups around the world should help indicate how human

populations are related and hence how and from where they spread

(Cann et al., 1987; Gibbons, 1992).

Because sequences used in most published research are readily accessible

via the internet in databases such as GenBank, it is possible to investigate a

dataset from these or other studies on your own.

Sequence-based phylogenetic methods are still being actively researched,

by biologists, statisticians, computer scientists, and mathematicians. There

are many problems, approaches, and techniques that we have not touched

on here. How DNA sequences are identiﬁed as good data on which to base

210 Constructing Phylogenetic Trees

a phylogeny, how those sequences are aligned, and how we might measure

the conﬁdence we should have in a tree are only three of the topics we have

ignored. For more comprehensive overviews, good references are (Hillis et al.,

1996) and (Li, 1997).

Problems

Before attempting these problems, type primatedata in MATLAB to gain

access to the sequences and distance arrays mentioned, all of which come from

(Hayasaka et al., 1988). Type who to see the names of the variables this m-ﬁle

creates.

5.6.1. The distance array Dist

primates is a 12 × 12 matrix, with dis-

tances computed from a 6-parameter model of base substitution.

The names of the taxa in the order of the matrix entries are in

Names

primates. Perform the neighbor-joining algorithm on this

data with the command

nj(Dist

primates,Names primates{:}).

Draw the resulting metric tree.

5.6.2. Use biological knowledge and your answer to the last problem to

draw a rooted topological tree that might describe the evolutionary

history of the ﬁve hominoids mentioned in the introduction.

5.6.3. How many possible unrooted topological trees might describe the

evolution of the 12 primates? How many possible rooted topological

trees might describe the evolution of the ﬁve hominoids of the chapter

introduction?

5.6.4. The commands

Names

hominoids=Names primates(1:5),

Dist

hominoids=Dist primates(1:5,1:5)

will extract the names and distances between the ﬁrst ﬁve primates,

the hominoids of the introduction of this chapter. Use the program nj

on the distance data for these ﬁve only, drawing the resulting metric

tree. Does the topology agree with that produced in Problem 5.6.1?

Does the metric structure agree? Explain how any discrepancies you

notice might have been produced.

5.6.5. Use the command Seq

hominoids=Seq primates([1:5],:)

to extract the sequences for the hominoids. Some of the sequences

have gaps, indicated by the “–” character. Sites where any sequence

5.6. Applications and Further Reading 211

has a gap should be removed before computing distances. The com-

mands

gaps=(Seq

hominoids =='-')

gapsites=find(sum(gaps))

Seq

nogaps=Seq hominoids

Seq

nogaps(:,gapsites)=[ ]

will ﬁnd and delete those sites. Using the gapless sequences, compute

the Jukes-Cantor, Kimura 2-parameter, and log-det distances. Recall

that [DJC, DK2, DLD]=distances(Seq

nogaps)will do

this easily.

a. How similar are these distances to those in the array

Dist

primates?

b. Use each distance array you produce to construct a tree by Neighbor

Joining. Are they all the same topologically? Metrically?

5.6.6. Investigate how reasonable the Jukes-Cantor and Kimura models of

base substitution are for the descent of the hominoids from a com-

mon ancestor. Do this by considering two sequences at a time, us-

ing compseq to calculate a frequency array of bases in the two

sequences. Then, compute base distributions for each sequence and

Markov matrices that would describe the evolution of one into the

other. Are these close to those of a Jukes-Cantor or Kimura model?

Does the choice of a different model in (Hayasaka et al., 1988) seem

necessary? Explain.

5.6.7. Repeat Problem 5.6.5, but use all 12 primate sequences. Which of the

distances do you think is most valid to use? Explain.

5.6.8. From the sequences of the hominoids, isolate the ﬁrst 10 informative

sites. Use these to compute the parsimony score (by hand) of each

of the trees at the beginning of this chapter, as well as the one with

neighbor pairs (chimpanzee, gorilla) and (orangutan, gibbon). Which

of the three is most parsimonious?

5.6.9. Repeat the last problem, but using 10 informative sites chosen to be

equally spaced among the informative sites. Do you think this choice

of informative sites should be more or less sound than that of the last

problem? Explain. (Obviously using all informative sites would be

preferable, but that cannot be done easily by hand, because there are

90 of them for these 5 taxa.)

5.6.10. If you have access to software that will attempt to ﬁnd the most

parsimonious tree, use it on the full sequences for the ﬁve primates.

212 Constructing Phylogenetic Trees

(Note: these sequences are in a sample data ﬁle distributed with (Swof-

ford, 2002).)

5.6.11. The vectors codingsites and noncodingsites contain the

indices of the coding and noncoding sites in the primate se-

quences. The coding sites can be extracted with the command

Seq

coding=Seq primates(:,codingsites).

a. Compute frequency arrays of bases in the coding sequences for the

primates by comparing sequences two at a time. Does the Jukes-

Cantor model or a Kimura model seem reasonable, or do you think

a different model would be needed?

b. Repeat part (a) for the noncoding regions of the sequences. Do

you think the same model might apply to both the coding and

noncoding regions? Explain, referring to the data.

5.6.12. Because the coding and noncoding sites might evolve differently, they

might lead to inferring different trees.

a. Using only the coding sites, and the log-det distance, ﬁnd the

Neighbor Joining tree for the 12 primates. Does it agree topo-

logically with the tree made the same way using all sites?

b. Using only the noncoding sites, and the log-det distance, ﬁnd the

Neighbor Joining tree for the 12 primates. Does it agree topologi-

cally with the tree made the same way using all sites?

Projects

1. Dental transmission of HIV

In 1990, it was reported in the CDC’s Morbidity and Mortality Weekly

Report that a young woman in Florida had most likely been infected

with HIV by her dentist. This conclusion was based primarily on a lack

of alternative explanations for the infection. The dentist, who was HIV-

positive, then publicly requested that his patients be tested. Altogether,

seven patients were found to be HIV-positive.

Of course, a patient who was HIV-positive was not necessarily infected

by the dentist. A large dental practice might be expected to have some

infected patients whose infection had nothing to do with their dental care.

An epidemiological investigation tried to assess other risk factors for the

patients. For some, nondental infection scenarios seemed likely, while for

others, nondental infection seemed unlikely. Because of the difﬁculties of

getting accurate answers from patients on high-risk behaviors, however,

the results of such an investigation cannot be considered conclusive.

5.6. Applications and Further Reading 213

Because no other possible dental infection cases had ever been recorded,

some doubt remained concerning the Florida cases.

In 1992, the paper (Ou et al., 1992) appeared in Science. This work

took a completely different approach using DNA evidence to try to es-

tablish the likelihood of a dental infection route for the patients. Because

HIV mutates so quickly into quasi-species, one would expect people re-

cently infected from a common source to have more similar quasi-species

than those whose common source of infection was more removed. The

researchers therefore decided to sequence the highly variable envelope

gene of HIV from each patient, the dentist, and some other HIV-infected

people living nearby who were not expected to have had any close con-

tact with the cases being studied (i.e., local controls). They then used the

sequences to construct a phylogenetic tree, and by the clustering pattern,

identiﬁed which patients they believed had been infected by the dentist.

Some of the DNA sequences in the paper have been downloaded from

GenBank for you to use. In MATLAB, run the m-ﬁle flhiv to read the

sequences. This will create sequences with names:

dnt, lc1, lc5, ptb, ptc, ptd.

These refer to the dentist, local control 1, local control 5, patient b, patient

c, and patient d in the Science paper. Although these sequences are already

aligned, they are of differing lengths, so you will have to ﬁnd the shortest

and cut off the ends of the others to compare them.

Construct phylogenetic trees using these sequences and draw con-

clusions as to which patients were likely to have been infected by the

dentist.

Suggestions

It is best to try several different tree construction methods.

In deciding to use UPGMA or Neighbor Joining (or perhaps both),

consider the assumptions these methods make.

In selecting a distance formula to use, make sure you look at the data to

see which model seems most appropriate. If different distance formulas

give different trees, which one are you most conﬁdent of? Why?

If you use a method that produces an unrooted tree, where should you

place the root?

Before using the method of maximum parsimony, compute how many

different trees would need to be considered if all were to be examined.

Because parsimony is not really practical to do by hand for a large

number of trees, you should use as many informative sights as you ﬁnd

214 Constructing Phylogenetic Trees

bearable and compute the parsimony of a handful of different trees. One

of these trees should be the one produced by a distance method, and the

others should be trees that you think might also be good candidates.

How conﬁdent do you feel about the validity of your results and why? If

you dismiss your work constructing trees as not being rigorous enough,

do you feel more conﬁdent in just accepting the word of the patients

involved about their HIV risk factors? Give an honest appraisal of how

valuable you think phylogenetic methods are.

Genetics

We have all observed that offspring tend to have physical traits in common

with their parents. In humans, similarity in hair color, eye color, height, and

build often quite clearly run in families. That selective breeding might enhance

traits must have been noticed long ago in our history, as domesticated animals

and crops have strongly developed features that we ﬁnd useful.

On the other hand, the traits of offspring are generally not completely

predictable from observing those of the parents. A child might have a trait,

such as hemophilia, that neither parent exhibits, though such a trait might

occur more commonly within one family than another. Thus, despite patterns

to inheritance, chance also appears to be involved. Creating a mathematical

model of heredity requires capturing both of these aspects.

The ﬁrst decisive step was taken by the Augustinian monk Gregor Mendel

in the latter half of the nineteenth century. Experimenting with some carefully

chosen traits in peas, he was led to propose what we now call a gene as the basic

unit of inheritance. Though it is perhaps surprising to the modern student, at

that time the gene was an entirely abstract concept, with no proposed physical

basis, such as the DNA sequences we now immediately imagine.

Recognizing the value of quantitative analysis, Mendel created a mathe-

matical model for the transmission of heritable traits, based on the concepts

of probability. His genius was in both identifying simple enough traits to be

able to formulate a good model and then modeling the inheritance of those

traits successfully. Though subsequent work has added many new features to

our models, and we now know much more about the chemical and biological

mechanisms behind genetics, Mendel’s simple model remains the basic core

of our understanding of how many organisms pass on traits to their offspring.

6.1. Mendelian Genetics

In 1865, Mendel presented his ﬁndings from breeding experiments with

garden peas (Mendel, 1866) to a small group of scientists in Br¨unn, in the

215