Humans, it would seem, have a great love of categorizing, organizing, and pigeon-holing things. This love affair extends to life-forms, of course – we have been attempting to group and name plants, animals, and insects as far back as 1500 BC[footnoteRef:1]. By studying the relationships of things, we can better understand behaviors and characteristics important to agriculture, medicine, animal husbandry – and of course, evolution itself. [1: Manktelow, M. (2010) History of Taxonomy] From your basic biology classes, you should remember that the act of classifying organisms is called taxonomy. The science that studies how those organisms evolved – and are related to one another - is called phylogeny. In the early days of the scientific method, organisms were compared by their morphology – their physical structure and characteristics. While this works to a certain extent (and it was all we had to go on before we had DNA sequencing techniques), it caused some honestly hilarious pairings. For example, there's a ruminant primate (monkeys and cows are not in fact directly related) – and if you compare the morphology of an octopus' eye to that of humans, you can see that they must be closely related! With the advent of DNA sequencing, scientists were able to go directly "to the source" for information on evolutionary history (phylogeny). Thanks to molecules like the small ribosomoal subunit (16S in prokaryotes and 18S in eukaryotes), we have excellent unique identifiers for species. You'll learn more about the molecular biology of how this works in other courses; for purposes of this class we are more interested in how that sequence data is used to reconstruct the evolutionary history of species. The Data To reconstruct phylogeny and create a phylogenetic tree, we start with a Multiple Sequence Alignment (MSA). Illustrated below is a small section of an alignment of the 18S gene from several species: You can see substitutions as well as indels in this small sample. This information can then be used to both identify and group the species taxonomically in a variety of ways. Let's take a look at three of the most common methods of creating phylogenetic trees – Distance, Parsimony, and Bayesian. DISTANCE One of the simplest and oldest methods, the distance approach is still used today. It works by simply computing a distance matrix for each possible pairing of sequences. For example, given the following three sequences: S1 aactc S2 aagtc S3 tagtt We can count the substitutions between each pair and generate a matrix: S1 S2 S3 S1 - 1 3 S2 1 - 2 S3 3 2 - Notice that this forms two "triangles", where the upper triangle is the mirror of the lower (e.g, S1 vs S2 is shown in two places, and it's the same value). Also note that comparisons of the same sequences (S3 vs S3) are just a "dash". This is the simplest possible form of distance matrix calculation. From this, we can actually start drawing a phylogenetic tree – for exa ...

1. Humans, it would seem, have a great love of categorizing,
organizing, and pigeon-holing things. This love affair extends
to life-forms, of course – we have been attempting to group and
name plants, animals, and insects as far back as 1500
BC[footnoteRef:1]. By studying the relationships of things, we
can better understand behaviors and characteristics important to
agriculture, medicine, animal husbandry – and of course,
evolution itself. [1: Manktelow, M. (2010) History of
Taxonomy]
From your basic biology classes, you should remember that the
act of classifying organisms is called taxonomy. The science
that studies how those organisms evolved – and are related to
one another - is called phylogeny.
In the early days of the scientific method, organisms were
compared by their morphology – their physical structure and
characteristics. While this works to a certain extent (and it was
all we had to go on before we had DNA sequencing techniques),
it caused some honestly hilarious pairings. For example, there's
a ruminant primate (monkeys and cows are not in fact directly
related) – and if you compare the morphology of an octopus' eye
to that of humans, you can see that they must be closely related!
With the advent of DNA sequencing, scientists were able to go
directly "to the source" for information on evolutionary history
(phylogeny). Thanks to molecules like the small ribosomoal
subunit (16S in prokaryotes and 18S in eukaryotes), we have
excellent unique identifiers for species. You'll learn more about
the molecular biology of how this works in other courses; for
purposes of this class we are more interested in how that
sequence data is used to reconstruct the evolutionary history of

2. species.
The Data
To reconstruct phylogeny and create a phylogenetic tree, we
start with a Multiple Sequence Alignment (MSA). Illustrated
below is a small section of an alignment of the 18S gene from
several species:
You can see substitutions as well as indels in this small sample.
This information can then be used to both identify and group the
species taxonomically in a variety of ways. Let's take a look at
three of the most common methods of creating phylogenetic
trees – Distance, Parsimony, and Bayesian.
DISTANCE
One of the simplest and oldest methods, the distance approach
is still used today. It works by simply computing a distance
matrix for each possible pairing of sequences. For example,
given the following three sequences:
S1 aactc
S2 aagtc
S3 tagtt
We can count the substitutions between each pair and generate a
matrix:
S1
S2
S3

3. S1
-
1
3
S2
1
-
2
S3
3
2
-
Notice that this forms two "triangles", where the upper triangle
is the mirror of the lower (e.g, S1 vs S2 is shown in two places,
and it's the same value). Also note that comparisons of the
same sequences (S3 vs S3) are just a "dash".
This is the simplest possible form of distance matrix
calculation. From this, we can actually start drawing a
phylogenetic tree – for example, S1 and S2 are closer to each
other than they are to S3, but S3 is closer to S2 than it is to S1,
so we could come up with this tree topology:
This is a "rooted" tree drawn with proportional branch lengths –
meaning the distances correspond to the length of the lines. S3
is closer to S2 than S1, S2 is closer to S1 than S3!
As I mentioned above, this is a very old and simple approach.
It is, however, still used today, primarily because the
calculations are very easy and fast, which means that you can
easily use it to compute phylogenetic trees for large numbers of
species – something difficult to do with the other methods we'll
talk about.

4. The problem with the distance approach is that it is very
simplistic – it doesn't take into account any sort of evolutionary
model of change, and it assumes that all mutations are equal ly
likely. The first problem (the evolutionary model) cannot be
addressed by distance methods – but we can tweak the distance
method by applying a Mutation Model to provide information
with regards to mutation.
Mutational Models
There are several models of mutation that can be added to the
distance method. The simple method above, where all
mutations are assumed to be equally likely, is called the Jukes -
Cantor method. The most popular model is the Kimura 2-
parameter model, which assigns different values for transitions
() and transversions ():
This looks like a Markov model, doesn't it? That's because it is
– a simple, 2 parameter Markov model for evolution that is used
to weight the calculations when generating the distance matrix
from MSA.
It is important to note that substitutions are the only element in
the MSA that distance phylogeny takes into account – indels are
disregarded. Yet another reason why the distance method is
"simple" – and ultimately less accurate at recreating the actual
evolutionary paths. Let's move on to a method that does
attempt to recreate the actual evolutionary history of the species
(more commonly referred to as "taxa") in question.
MAXIMUM PARSIMONY

5. Parsimony is defined as "the scientific principle that things are
usually connected or behave in the simplest or most economical
way, especially with reference to alternative evolutionary
pathways." Maximum parsimony, then, means maximizing that
simplicity. What parsimony algorithms are designed to do is to
recreate the actual evolutionary history of the organisms being
analyzed with relation to each other in a fashion that minimizes
the number of steps required to traverse the entire tree –
meaning minimizing the number of evolutionary changes.
The information that parsimony algorithms use to infer the
evolutionary history are informative sites. These are columns
in the alignment that have more than one character (e.g., A as
well as C), each of which has to appear more than once. They
are called informative because by having that similarity to at
least one other sequence, they help inform the process of
inferring the ancestral states at the nodes of the tree. You
should recall that the tips of a phylogenetic tree are the
currently extant taxa; the root is the common ancestor, and the
middle nodes represent the species that existed at one time but
are now extinct. These ancestral node sequence states are
inferred using the informative sites.
We aren't going to spend too much time on maximum parsimony
here, since the statistics involved are not complex and involve
the same sort of substitution models that distance methods do; I
do want to point out that computationally, these methods have
to be heuristic rather than exhaustive – there are too many
possible trees once you have, say, 30 taxa, to look at all
possible tree configurations[footnoteRef:2], so these algorithms
take a variety of shortcuts to find a "best" tree – primarily
branch swapping to see if more parsimonious trees (with fewer
steps, or changes, required) can be found. [2: See
https://rdrr.io/cran/ape/man/howmanytrees.html for an example
including code you can use to calculate it!]

6. Let's move on to a more statistically-oriented method –
Maximum Likelihood.
MAXIMUM LIKELIHOOD
Maximum Likelihood was, for a long time, considered the
"third" method of building trees (after distance and parsimony).
As you may have guessed, it's based on the statistical concept of
maximum likelihood estimation, or MLE. MLE estimates the
parameters of a probability distribution by maximizing a
likelihood function such that the observed data is most probable
(or likely). A simpler way of saying this is that MLE evaluates
parameters (e.g., a phylogenetic tree structure) and determines
how likely it is that those parameters derive from the given data
(e.g., sequence data). This sounds backwards – you start with a
tree, then calculate the probability that the tree "fits" the data –
but it's actually very little different from the heuristic branch
swapping that happens in a parsimony analysis, where the tree
is modified to see if it fits better. We can define this as:
P(X|)
We can read this as "what is the probability of X given "; in this
case X always represents the observed data (the sequence
alignment) and represents the parameters of the model (the tree
topology as well as the evolutionary model selected by the
user). The goal of the algorithms that perform MLE
calculations is to find a value of that maximizes P (the
probability of X given ). As with parsimony analysis, the
number of possible trees is astronomically high once you exceed
a certain number of taxa, which makes these algorithms very
compute-time intensive. Similarly, there are heuristic
approaches that use a "starting" tree and simply optimize results

7. based on the evolutionary model chosen to find an optimal (but
probably not "best") tree. This is done by summing the
likelihood at each site in the alignment, with the assumption
that the sites evolve independently (a Markov chain-like
model). To derive the likelihood for any given site, the
algorithms calculate the probability of every possible
reconstruction of ancestral states given the chosen model of
substitution. Then, a branch-swapping step is performed
(similar to the parsimony approach above), but instead of
optimizing for the minimum number of changes overall, MLE
methods optimize the Likelihood calculations.
Evolution probably doesn't support the Markov chain model
fully, since a mutation at one site in a protein-coding gene may
cause missense or nonsense mutations – so there are
evolutionary constraints involved (individuals with nonsense or
missense mutations may be selected against, depending on how
detrimental the mutation is). Nonetheless, these methods work
sufficiently well.
Let's briefly look at one more method – Bayesian Inference of
Phylogeny.
BAYESIAN
As you may have guessed, the Bayesian method of phylogenetic
reconstruction is an inferential probabilistic method based on
Bayes' theorem. Similar to the MLE method, it attempts to
solve for the likelihood (posterior probability) that a given tree
matches the data (and evolutionary models) provided. It does
so, however, using the Bayes formula rather than a maximum
likelihood probability. Underlying this is a Markov Chain
Monte Carlo algorithm, where the probability distributions
describe the uncertainty of the unknowns (e.g., the tree
topology and the evolutionary model parameters). Bayes

8. theorem is used to calculate the posterior distribution of much
as MLE used the likelihood calculations:
The probability here, f(|D), is also called the likelihood, but
don't let that confuse you – it's the posterior probability based
on Bayesian inference.
One big (and positive) difference of Bayesian inference in this
case is that it makes definitive probabilistic statements about
the parameters – it gives us a value, the credibility interval, or
CI, that the parameter predicted is the true parameter,
something that is impossible with classical
statistics[footnoteRef:3]. [3: Classical statistics treats
parameters as unknown constants and cannot derive them de
novo]
FINAL THOUGHTS
The most common question any professor will ever hear about
this topic is "which method of phylogenetic reconstruction
should I use?" The answer (as you might have expected) is, "it
depends". Do you need to reconstruct the phylogeny for more
than 30 or so taxa? Then distance is the only approach that will
finish before the heat-death of the universe (at least until
quantum computing is a real thing[footnoteRef:4]). If you are
looking at fewer than 32 taxa? My advice has always been to
do as many methods as you can and compare the trees – identify
the common branches/nodes and draw what conclusions you
can. The software called Mr. Bayes (which is – you guessed it
– a Bayesian method) has become tremendously popular in the
past decade, but PAUP (a maximum parsimony method) and
PHYLIP (various approaches, but best at distance) are still very
heavily used. [4: And yes, I'm familiar with the D-Wave
adiabatic computer. It's not quite ready for prime time yet, at

9. least not for bioinformatics.]
That's it for this week – be sure to check in to the discussion
forums and post answers to the questions posed!
S2
S1S3
1. A company charting its profits notices that the relationship
between the number of
units sold, x, and the profit, P, is linear. If 190 units sold results
in $380 profit and
240 units sold results in $2980 profit, write the profit function
for this company.
P = _________
Find the marginal profit.
$____________
2. A company distributes college logo sweatshirts and sells
them for $45 each. The total
cost function is linear, and the total cost for 90 sweatshirts is
$3951, whereas the
total cost for 260 sweatshirts is $5991.
(a) Write the equation for the revenue function R(x).
R(x) = ___________

10. (b) Write the equation for the total cost function C(x).
C(x) = ____________
(c) Find the break-even quantity.
x = __________ sweatshirts
3. Suppose a certain home improvement outlet knows that the
monthly demand for
framing studs is 2,500 when the price is $4.25 each but that the
demand is 3,700
when the price is $3.89 each. Assuming that the demand
function is linear, write its
equation. Use p for price (in dollars) and q for quantity.
________________
4. It has been estimated that a certain stream can support 88,000
fish if it is
pollution-free. It has further been estimated that for each ton of
pollutants in the
stream, 1500 fewer fish can be supported. Assuming that the
relationship is linear,
write the equation that gives the population of fish p in terms of
the tons of
pollutants x.
________________
5. An electric utility company determines the monthly bill for a
residential customer by
adding an energy charge of 7.34 cents per kilowatt-hour to its
base charge of $18.39
per month. Write an equation for the monthly charge y in terms

11. of x, the number of
kilowatt-hours used. (Let y be measured in dollars.)
________________
6. Suppose that from 2020 to 2060, the number of females in the
U.S. under the age of
18, in millions, can be modeled by
N = 0.140x + 38.4
where x is the number of years after 2020.
(a) Viewing N as a function of x, what is the slope m of the
graph of this function?
m = _________
(b) What does the model predict the population of females under
18 (in millions)
will be in 2040?
__________million.
7.
8. A concert promoter needs to make $79,600 from the sale of
1740 tickets. The
promoter charges $40 for some tickets and $60 for the others.
Let x represent the
number of $40 tickets and y represent the number of $60
tickets.
(a) Write an equation that states that the sum of the tickets sold

12. is 1740.
______________
(b) Write an expression for how much money is received from
the sale of $40
tickets?
______________
(c) Write an expression for how much money is received from
the sale of $60
tickets?
_________________
(d) Write an equation that states that the total amount received
from the sale is
$79,600
___________________
(e) Solve the equations simultaneously to find how many tickets
of each type must
be sold to yield the $79,600.
x= ____________
y= ____________
9. If the demand for a pair of shoes is given by 2p + 5q = 200
and the supply function
for it is p − 2q = 10, compare the quantity demanded and the
quantity supplied

13. when the price is $90.
quantity demanded ___________ pairs of shoes
quantity supplied ____________ pairs of shoes
10. Find the market equilibrium point for the following demand
and supply functions.
Demand: p = −4q + 312
Supply: p= 6q + 1
(q, p) = ( __________ )
11. Find the equilibrium point for the following supply and
demand functions.
Demand: p = −4q + 220
Supply: p= 16q + 20
(q, p) = ( __________ )
12. Retailers will buy 45 Wi-Fi routers from a wholesaler if the
price is $10 each but only
20 if the price is $85. The wholesaler will supply 56 routers at
$46 each and 70 at
$50 each. Assuming that the supply and demand functions are
linear, find the market
equilibrium point.
(q, p) = ( __________ )
13.

14. (b) How long is it until the building is completely depreciated
(its value is zero)?
__________months
(c) The point (70, 225,000) lies on the graph. Explain what this
means.
The point (70, 225,000) means that after ________ months the
value of the
building will be $ _____________ .