Thursday, December 4, 2014

Why do the Patriots host the Broncos 3 years in a row? [math]

Why do the Patriots host the Broncos 3 years in a row?

NFL Scheduling

Every once in a while, somebody will notice that a given NFL team will host another non-divisional team from the same conference three years in a row.  Today the Patriots are hosting the Broncos for the third year in a row.  That's so unfair!  Or so it would seem.  But it turns out that there are sound mathematical reasons that it works out this way. 

To explain the reasons, let's consider how the NFL schedule is generated.  The NFL has 32 teams, organized in two conferences, each of which has 4 divisions with 4 teams each.  And the season lasts 16 games long for each team.  This coincidence of numbers makes for a very elegant scheduling algorithm, which will disappear as soon as the NFL expands again.

Each team plays a mix of divisional games, intraconference games, and interconference games.  For starters, each team plays a home game and an away game against each of its three division rivals. 

The next consideration is the desire to have regular games against all the other teams in the league.  That's neatly arranged by having all of the teams in each division play all the other teams in each other division on a rotating basis.  For example, this season all of the teams in the AFC East play all of the teams in the NFC  North.  That rotation is on a four-year schedule, since there are four divisions in each conference.

In contrast, there is a three year rotation for interconference matchups.  This year, the AFC East plays the AFC West, while the AFC North plays the AFC South. Last year the AFC East played the AFC North.  Next year, they'll play the AFC South.  

So each team plays 6 games in the division, 4 games against all the teams of one other division in the same conference, and 4 games against all the teams in one division intra-conference.  The home-and-away matchups for division vs. division play alternates so no city will go too, too long without being able to see every other team.  That means every eighth season for interconference, every sixth season for intraconference.  These account for 14 of the 16 games played every season.

That leaves two slots to fill up the season.  The NFL fills the last two slots by having every team play one team from each of the other two divisions in the same conference, namely, the two teams that finish in the same respective place in the standings.  In recent years, the Patriots have finished in first place, and thus had to play the first place team from the other two divisions.

For many years, this led to the Patriots playing the Colts every season.  One season, when the Patriots finished in second, the Colts did the same thing, so they kept playing each other.  Now that Peyton Manning has moved to Denver, there is focus on that matchup.  And two years ago, back when Tim Tebow was their QB, the Broncos visited Foxboro to play the Pats.  Last year, they did the same thing.  

And a few weeks ago they hosted the Broncos again.

Why is that?  Well, let's look at how this can go.  The three other first place teams last season were the Broncos, the Bengals, and the Colts.  As I mentioned earlier, the Pats play all the teams of the AFC West this season.  That they host the Broncos is part of the divisional schedule.  Three years ago they went to Denver, and they'll go to Denver again three years from now.  And if they'd gone to Denver this year, they would have hosted Denver three years ago, so they still would have hosted Denver three years in a row.

Why does it happen?  It happens because the home-away switch has to interleave with the rotating division vs. division matchup.  Recall the list of the first-place teams: Patriots, Colts, Bengals, and Broncos.  Since the Patriots hosted the Bengals and visit Indy this season, the Broncos have to do the opposite, just to ensure all four teams have exactly 8 home games and exactly 8 road games.  This makes for a "cycle": Pats host Bengals, who host the Broncos, who host the Colts, who host the Pats.  A more compact notation  here would be (Pats, Bengals,  Broncos, Colts), where each team hosts the team after it.  Or, to use the geographic names, (East, North, West, South).  Last year was (East, West, North, South).   And the year before that was (East, West, South, North).  
If I keep East in the first slot, there are six ways to fill the other three slots.  And it turns out the NFL is on a 6-year rotating schedule in this respect.    Now if I keep East in the first slot, the third slot is in a regular rotation:

(East, *, South, *)
(East, *, North, *)
(East, *, West, *)
(East, *, South, *)
(East, *, North, *)
(East, *, West, *)

So, let's start with (East, West, South, North) and see what we're forced to do.  The next year we have (East, *, North, *).  If we want to avoid the (East, West, *, *) option, we have to use (East, South, North, West).  That gives us

(East, West, South, North)
(East, South, North, West)
and, if we want to again avoid a repeat
(East, North, West, South)
So far, so good, yes?  Well, at least as far as the East goes.  But notice that the South hosts the North two years in a row, the West hosts the South both times it is possible, and the North hosts the West two years in a row.

So, it's possible to schedule things so the East team doesn't have any repeat hosting, but doing so would force every other divisional matchup to be constant in terms of who's on the road and who's hosting.  You just cannot alternate a two-year rotation and a three-year rotation for everybody.  So what the league does is use all six possibilities.

(East, West, South, North)
(East, West, North, South)
(East, North, West, South)
and those three reversed
(East, North, South, West)
(East, South, North, West)
(East, South, West, North)

In other words, East@West, East@West, division vs. division, West@East, West@East, division vs. division.  And since the division vs. division matchups themselves alternate, either the third or sixth year is also going to be East@West, leaving us with three years in a row of East@West, followed by three years in a row of West@East.

Thursday, February 7, 2013

the driving forces behind science denial

OK, this is just going to be an opinion post, as opposed to a rigorously detailed exposition of any field of science.

The major fields for science denial these days are evolution and climate science.  Both fields have seen the development of think tanks with "scholars" generating science-like papers to generate ideas to be funneled into the public discourse.  It would be interesting to track the path of memes in this way, but I'll put that off for another day.

The motivating forces behind these two types of anti-science are quite different.  The anti-evolutionists have traditionally been religiously motivated.  While it is true that many people in the field of evolution still hold religious beliefs of one sort or another, I think it's fair to say that the main issue here is that the theory of evolution (and its cousin the theory of common descent) run into this resistance because they fly directly in the face of the creation stories of Christianity.  (Am I tipping my hand by using the word 'story'?  I think it's slightly more neutral than 'myth'.)

So, when it comes to evolution, religion is the 900 pound gorilla in the room.

When it comes to dealing with religion, there appear to be two main approaches.  There is the Richard Dawkins/PZ Myers approach, which simply goes after religion from the front, i.e. simultaneously fighting the culture war as well as trying to gain acceptance for the science.  Then there is the less confrontational approach, represented by people like Chris Mooney and Francis Collins, and by Stephen J. Gould in the past, who say that there is no reason to make a decision between religion and science, and that it's counter-productive to predicate acceptance of evolution on a victory in the culture war.

My personal opinion on this matter lies somewhere between these two simplified accounts of these strategies.  On the one hand, I don't feel a desire to fight an all-out war on religion simply to get science more widely accepted.  My perspective is that I come from a religious family (even though I am not personally religious any longer) and know from personal experience that there is no need to target religious beliefs if one's immediate goal is to get science accepted.  I said I come from a religious family, but I also come from a very scientifically literate family.  And aside from myself, there are several individuals in my family who are at least as well-versed in science as I am, and yet have no problem whatsoever reconciling science and their religious beliefs.

On the other hand, I don't think arguments along the line of Gould's "non-overlapping magesteria" are terribly useful. For one thing, I've never ever heard the word "magesteria" used except for in the context of Gould's argument.  (Aside: if I google "magesteria" I get 131k hits, while if I google "magesteria -gould" I only get 82.7k hits, which implies that a full third of the time this word is used, its in the context of Gould's argument.  I'm not sure "magesteria" is a more legitimate word than "lorax" or "sneetch."  But I digress.)

But aside from the awkward usage of an obscure word, I worry that trotting out Collins as an example of a "scientist that can be trusted" is not necessarily helpful towards the greater goal of getting people to understand empiricism in general.  Also, I tend to resent the implication that his opinion should be cared about more because he's religious.  At some point we need to recognize that the large majority of Ph.D. scientists are not, in fact, religious, and to the extent they are, they are religious in fairly minimal ways.  (I.e. Christian scientists are far less dogmatic than, say, Christian Scientists.  Hmm.  Mary Baker Eddy really didn't make it easy for us to discuss this topic, did she?)  So I would really like people to become a bit more evidence-based in their thinking.  The reason evolution holds up is because of the evidence, not because of the authority of Francis Collins.

Having said that, there is some reason to believe that the presence of Francis Collins in this debate is certainly a positive for people who fight for science.

On to climate change denial.  This is a field driven not by religious forces so much as it is driven by corporatists, esp. those in the energy industry (read: oil industry).  These people, like their predecessors in the tobacco industry, who spent decades resisting any and all publications pointing out the link between cigarette smoking and cancer, are intentionally trying to obfuscate scientific products solely for the purpose of their personal financial gain.  These people are, unfortunately, far more dangerous than evolution deniers and possibly more dangerous than tobacco science deniers.  All of the available research indicates not only that global warming is happening right now, but also that it's been accelerating in recent years, and that its effects are being profoundly underestimated.

The fact that climate change isn't so directly in opposition to traditionally held religious beliefs gives one hope that there will be more tractability on this issue.  After all, there really is no shortage of scientific evidence in the form of simple measurements of temperature to support the global warming theory.  To deny global warming, it's become increasingly necessary to allege that data is being forged as part of some world-wide conspiracy (presumably to plunder that fountain of wealth known as Grant Money).

So, while the threat posed by anti-science is much greater vis-a-vis global warming than it is in evolution, the presence of so much fairly obvious data (January days in the high 60s Farenheit) gives me hope that public opinion will be malleable here.

The problem is that there's a huge lag between turning public opinion and getting the government on track to address a problem in any meaningful way.  The American government has no policy in place for dealing with carbon emissions other than to whistle past the graveyard while blaming the Indians and Chinese.

Thursday, January 17, 2013

Posting more often

Starting to do a better job getting into the science policy community of DC.  And one piece of advice I've gotten is that I should blog more - just to get acclimated to the process and to produce a body of work that people can see.  So I think I'll keep going with this blog and try to dramatically increase the posting pace from the historical average of once per six months.  :)

But as always, we'll have to keep an eye on this Blogger interface, which isn't as WYSIWYG as it used to be.

Some topics I hope to write about soon

  • Global warming - what gets me about this topic is how so much of the criticism simply ignores the theoretical basis for the theories regarding global warming.  That global warming is happening is an inevitable consequence of the increase in CO2 content in the atmosphere.  To deny global warming is to deny everything we know about heat reflection.
  • Vaccines - actually I don't have much to say about the science here.  
  • Fisheries - I think this is a neglected topic.  
  • Charismatic megafauna - why I love pandas and polar bears and don't care so much if some beetle species go extinct.  
Beyond that there are many issues in energy, medicine, computer privacy, and whatnot that will come to mind.

Sunday, September 2, 2012

Phylogeny history and methodology

Phylogeny in Brief

 The term "phylogeny" refers to the study of the tree of life and the theory of common descent.  When considering a group of species living today, the term also refers to the tree structure defined by this group combined in a graphical relationship with all of their common ancestors.  In this post, I compare the basic classification methods of early biologists with the methods developed over recent decades inspired by the dramatic increase in availability of genetic sequence data.
A phylogeny is a tree structure that defines a history for the species of the past that are ancestral to a group of species living today.  For example, consider the great apes, consisting of humans (Homo), chimpanzees (Pan), gorillas (Gorilla), and orangutans (Pongo):

Ape Phylogeny

(Graphic borrowed from

We see that the common ancestor of humans and chimpanzees is placed in a position more recent than that of humans, chimpanzees, and gorillas.  In this phylogeny, the orangutan is the outgroup, the species whose nearest ancestor to the rest of the species predates the rest of the common ancestors, i.e., the species first to diverge from the rest of the group.
Before the study of phylogeny (and more generally, the study of evolution), it was widely believed that the species we see today always existed in their current form.  Fish were always fish, trees were always trees, bears were always bears, and humans were always humans.  Indeed, this interpretation seems simple and intuitive.  But Darwin's Theory of Evolution suggested that species could evolve  with new traits arising in populations as they increased the fitness of living beings. Also, new species could arise when a living species separated into two or more branches that started to evolve independently. One conclusion drawn from the study of phylogeny is that humans and apes have a common ancestor, a belief that becomes easily caricatured as “People came from monkeys!”
Strictly speaking, humans did not descend from “monkeys.”  Rather, the theory of evolution states that humans, apes, and other primates all had a common ancestral species several million years ago.  In the current work, I try to briefly explain how this counter-intuitive theory has become dominant in the field of evolutionary biology.   


Theory of Common Descent

With the publication of The Origin of Species, Charles Darwin presented a radically new idea describing the history of all life on this planet.  According to this theory, species would evolve: small changes in a species that were favored by the environment would come to dominate the species.  For example, a species of finch that lived on an island on the ground might find it more useful to have a short, fat beak.  With a shorter beak, the finch would find it easier to feed on seeds it could collect, and its survival rate would improve. A species of finch on a different island might find a habitat with more trees.  In such a habitat, a long, narrow beak might be better at reaching insects in hard-to-reach crevices.  On a third island, a third type of beak might be favored. 
It is important to note that the species doesn’t choose what type of beak it has, nor does it intentionally pass on genes containing the adaptation.  Rather, the entire process is passive.  The dominance of a new characteristic is implied by the survival rate alone.  Or to be more precise, the dominance of a gene in the population grows as a function of the gene’s propagation rate.  Once a gene establishes a higher survival rate in the population, its ascension to a position of dominance is a mathematical inevitability.
Genetic mutability is the observable fact that the genetic composition of a species can change over time.  (This fact is most readily observable with quickly evolving species such as bacteria and viruses.)  But mutability is only part of the theory of evolution.  Genetic mutability does not, by itself, imply common descent.  Consider, as an example, the appearance of uniforms in the sport of professional baseball.  Each franchise tends to make small variations in the appearance from year to year.  And one could trace these changes if one had access to older uniforms or photos or drawings thereof.  But the fact that each franchise’s uniforms can change does not, by itself, imply that there was a single franchise at the beginning of the baseball league.
This brings us to the concept of common descent.  When Darwin famously toured the Galapagos Islands on the HMS Beagle, he observed similar species of finches on various islands.  Struck by the similarities of many of their features, Darwin postulated that they had all descended from a common ancestral finch. But when the ancestral species settled on the several islands, the descendants diverged in the sense that different characteristics became dominant in different locations.
As species diverge, interbreeding between the various populations can become more and more difficult, for a variety of biological and geographical reasons.  After enough time passes, the original populations are no longer biologically compatible in a mating sense.  This process of divergence is called speciation, and the diverse descendant populations are called species.

The Tree of Life

The concept of speciation led to a revolution in how scientists viewed our natural history.  No longer did scientists view species as eternally fixed in form and function, but rather, variability was allowed into the theory.  Furthermore, the concept of species extinction was first raised by the systematic investigation of fossils, the discovery of which was proceeding apace in the 19th Century.
The diversity of life had been recognized since antiquity and, indeed, the possibility of meaningful classification was always apparent.  Humans have long recognized the differences between birds, mammals, and fish, for example, and have understood that certain groups of animals are more similar to each other than any is to species outside the group. Thus we have carnivores and ungulates, insects and crabs, lizards and snakes, etc.  It became apparent that a natural taxonomy could be constructed on all living species, where dividing points in the taxonomy represented major differences between groups of species.
With the introduction of the theory of common descent, such a classification drew a natural meaning in terms of the shared evolutionary history.  Let us consider all the descendants of some species that existed hundreds of millions of years ago. If we traced the species through time, we could model any speciation event as a node, with a path leading away from that event representing a new species. 

As the descendant species themselves radiate geographically and undergo their own speciation events, the new paths will lead to new nodes and new divergences.

If we trace this process throughout the millennia, the tree structure can become quite elaborate.  In 1866, Ernst Haeckel suggested the tree structure below, dividing living species into three kingdoms: Plants, Protists, and Animals.

Haeckel’s Tree of Life

Reconstruction Methods

So far, we’ve considered how evolution may have proceeded.  But we have no witnesses to the millennia of evolutionary history.  Well, sure we have some fossil records that may represent ancestral species to what we see today, but at best the fossil record is incomplete.  Today’s biologist will need to rely on the powers of inference to determine what the most likely explanation would be.
Note that I am using the language of likelihood here.  Science does not reveal absolute truths, but provides theories, along with a measure of confidence in the theory.  Any phylogeny is a theory of what kind evolutionary history is shared by a set of species that live today.  So if we are going to tackle the task of building a tree like the one above, we need to have a well-defined protocol for doing so, with a sound basis in scientific logic.
I will briefly present two general methods for building phylogenies.  First, we’ll consider the character-based method, which was used by Darwin and Haeckel and early taxonomists.  In this method, characteristics of types of living species (plant vs. animal, cold blooded vs. warm blooded, exoskeleton vs vertebrate, etc.) are used to divide the species.  Taxa are grouped in the tree based on the theory that major changes in structure happen rarely.  Thus the tree structure should reflect this principle of parsimony.
In the second method, molecular sequence data is used to generate a tree.  This approach is more mathematical than the character method.  At its core is a mathematical model of how DNA and protein sequences evolve through time, accumulating tiny changes in the code that do not affect protein function.  We then apply statistical theory and use the power of modern computing to find a tree structure that best fits the data we are considering.

Tree Topologies

To select a phylogeny, we need to select a) a tree topology, and b) branch lengths for the tree.  Essentially, the topology of the tree is its shape.  For our purposes, each branch will also have a length associated with it, representing the time between speciation events.  Generally speaking, the process of fitting a phylogeny to sequence data (or, indeed, to any kind of data), has two parts: 1) selecting a toplogy, and 2) selecting branch lengths.  For a given tree topology and a model of evolution, finding the optimal branch lengths is a straightforward (albeit computationally expensive) optimization problem. (Felsenstein.)
The computational issues in phylogeny fitting arise from the number of tree topologies, which increases exponentially with the number of taxa (taxa are the species living today forming the base of the tree).  Let us consider why this happens:
Let’s let n be the number of taxa under consideration. Then a phylogeny with n taxa will have (n-2) internal nodes and thus (2n-3) branches.  If we add another taxon to the tree, there are (2n-3) branches that could be split by the new branch leading to the new leaf, and each of these insertion points would lead to a different topology.  If we let T(n) be the number of topologies, we see that  be the number of topologies, we see that T(n+1) = (2n-3) T(n).  Since T(3) = 1, the formula for T(n) is T(n)=(2n-5)!! (Here the double factorial notation x!! is defined for any odd value of x to mean x!! = (x*(x-2)*(x-4)*…*3*1).  In other words, the number of possible topologies increases very quickly as the number of taxa increases. 
The obvious approach to finding an optimal phylogeny would be to find the optimal branch lengths for each topology.  But it quickly becomes computationally infeasible given the rapid growth of T(n).  An exhaustive search over all topologies is quickly infeasible as n increases.  For example, 10!! = 3840, which is a manageable number of trees for a program to evaluate, but 20!! = 3.72 * 109, and 30!! = 4.28 * 1016.  Thus, exhaustive searches are impracticable for all but the smallest phylogenies. 

Character-based methods

The tree below (graphic from shows a mammalian phylogeny based on a traditional, character-based method.  In such an approach, high-level phenotypes are presumed to be preserved.  Thus all species sharing a trait are believed to make a clade, i.e. a subtree with a common ancestor that excludes species that don’t have said trait.  For example, placental mammals are generally believed to comprise a clade in the phylogeny of mammals.  In the tree above, the egg-laying mammals (monotremes) and the pouch-using marsupials are outgroups to the rest of the tree, all of which are placental mammals.  Also highlighted in this tree are the ungulates: a collection of orders included all mammals with hooves, and some orders (such as cetaceans) believed to be derived from ungulates.

Unfortunately for taxonomists, most useful characters are not so easily defined in a way that makes them cladistics.  As with the cetaceans above, a sub-clade within a clade may lose the evidence of the ancestral character.  Also, there is the problem of parallel evolution.  Useful biological features may have arisen along separate lines.  (A simple example of this would be the trait of winged flight, which exists within birds, mammals, and insects.)

Genetic revolution

The discovery of the structure of DNA in the mid-20th century led to a new way to compare species.  By comparing the sequences of homologous proteins from different species, one could see that each species had witnessed functionally neutral mutations through history.  A typical gene has thousands of base pairs in its coding region, so the mutation of a small number of base pairs in the gene can often be neutral with regard to the gene’s expression. 
There are a number of different types of DNA mutations.  A single base pair may vary.  Also, an entire stretch of the genome may be duplicated and repeated, or it may drop from the genome altogether.  But, presuming not too many mutations have happened, it is possible to align sequences from regions that have been relatively conserved over time.

Sample Multiple Sequence Alignment

(Grahpic borrowed from In this graphic, the four letters A, C, G, and T represent the four DNA bases adenine, cytosine, guanine, and thymine.  The ‘-‘ characters or ‘gaps’ are inserted into the sequences in the alignment to indicate points where a sequence has lost that particular base pair from its sequence.  (Alternatively, in a column where most of the species have a gap, the interpretation is that some of the sequences have had an extra base pair inserted.)
If we focus on columns that have no gaps, we can model the changes in the sequence according to a fairly simple random model.

Random Walks

Let us suppose we have a tourist walking through a museum with exhibits in several different rooms.  We could track the progress of the tourist and make a note every fifteen minutes of his location in the museum.  This kind of process is known as a random walk, a special kind of random process.  If the museum had a proscribed order for viewing exhibits, this pattern might be predictable.  But let us suppose that the museum consists of unrelated exhibits, and that our visitor is choosing his next exhibit at whim.  Let’s also assume that the amount of time he might spend in a given exhibit is itself random.  Our list of rooms, observed over time, represents the output of a random process. This kind of process, where our observations are of chosen from a finite number of states, but the odds of change vary as time varies, is called a stochastic process.
The example of the museum visitor has a feature which we may not want for modeling evolution.  Namely, the museum visitor is likely to spend a predictable amount of time at each exhibit.  In other words, the longer a visitor has already spent at a certain exhibit, the more likely he is to move on.  DNA mutation is, however, memoryless in the sense that the probability of a mutation at a given time is independent of how long it has been since the previous mutation.  An example of a memoryless random system in real life is a roulette wheel. Whether one has seen the number ‘25’ show up five times in a row or whether the wheel has been spun 2,000 times without seeing a ‘25’, the probability of a ‘25’ on the next spin will be the same (1/38 on a well-balanced American wheel).
A stochastic process that is memoryless can be called (if a few other conditions also hold) a Markov process. With a Markov process, the probability of any state change can be expressed as a function of the amount of time elapsed between observations.  To relate to our museum visitor, if i  and j represent two rooms of the museum, and if t is an amount of time, then we can define pij(t) to be the probability that if the visitor starts in room i, the probability that he is in room j after t units of time.
Using the alignment as the next point of departure, it’s possible to develop a dissimilarity measure on the species based on which sites are witnessing mutations.  The general idea is that one can estimate how much evolutionary time has passed between a pair of species as a function of how similar (or rather, how dissimilar) their sequences are.  The more time that passes, the more divergence you’ll see between the pair.  When all of the sites are considered at once (i.e., when all of the columns of the alignment are considered), we obtain an estimate for pij(t).  As t approaches infinity, the values of pij(t) will converge to a limiting value, pj (which in fact does not depend on i.)  This fact allows us to make an estimate of the value of t that best fits the data.


Given the enormous number of tree topologies possible, there is a need for a way to select one of the topologies without testing all of them. Many computer algorithms have been designed for this task.  The most popular methods for doing this with very large data sets are distance methods.  They work by using dissimilarity information on the set of taxa to generate a metric (distance function) on the taxa.  Then, a tree is constructed such that the evolutionary distances implied by the tree serve as a good approximation to the observed distances.  Computer algorithms have been designed that can do this in computational time polynomial in n. 
The most popular distance method is an algorithm known as Neighbor-Joining.  It works recursively:

  1. By scanning the input matrix M, find a pair of taxa, x, y that are to be neighbors in the tree.
  2. Reduce the set of taxa by defining a new taxon x-y, to replace x and y.
  3. Calculate a new reduced matrix M’on the reduced set of taxa.
  4. Recursively solve the problem for M’ to build a tree T’.
  5. Build the tree T by replacing attatching the node x-y in T’ to the nodes x and y.

This entire process can be done in computational time proportional to n3.  Once an initial tree topology has been selected, this can serve as starting point for whatever statistically-based search process that may be desired.

Neighbor-Joining Agglomeration

Tree Rearrangements

With an initial topology T in hand, the tree builder now needs methods for making slight tweaks to the topology, to generate other topologies “close” to T for evaluation.  There are several methods for doing this:

  • Nearest neighbor interchange: in a binary tree (i.e., a tree where each internal node is met by exactly three branches), each internal branch defines four subtrees, two on each side of the branch.  A nearest neighbor interchange swaps one of the subtrees on one side of the branch for one of the subtrees on the other side of the branch.

 NNI rearrangement

  • Sub-tree pruning and regrafting: similar to the reinsertion method above, except that instead of re-inserting just one taxon, an entire subtree is removed and re-inserted.  In the figure below, the subtree S is moved past the other subtrees T1T2, … Tk.
SPR rearrangement   

  • Tree bisection and resection: an internal branch is deleted from the tree, creating two smaller trees. Then the trees are re-connected by grafting part of a branch from one tree to part of a branch from the other tree. (All pairs of insertion points are tested.)
TBR rearrangement

It is common to pick one of the methods above to generate a new list of topologies, which are then each evaluated.  If a new topology is deemed superior, it is then selected as the initial point for a new search.  The process is repeated until an optimal point is reached.

Likelihood Methods

This raises the question of how we score topologies.  The most common method is to use a likelihood approach.  In this approach, we consider the likelihood score of each topology: given a model M, data D, tree T and probability function P, the likelihood score is


i.e., it’s the probability of seeing the given data, given the model of evolution used. For a given topology, finding the branch lengths which optimize this function is itself a problem that can require a good deal of computation.  But at long as the number of topologies is limited in size, these computations can be done in a reasonable amount of time.
So, to summarize, the molecular approach is to

1.      Given an alignment of homologous DNA or protein sequences, build a distance matrix D generated by the pairwise dissimilarities.
2.      From D, build a tree T0 such that the metric implied by T0 serves as a good approximation to D
3.      Use T0 as the initial point and use toplogical rearrangements (NNIs, SPRs, or TBRs) to search through candidate tree topologies.  For each tree topology, find branch lengths that optimize the likelihood function, and choose the topology whose likelihood is optimal among the candidate trees.  Repeat search until no improvement in likelihood function is found.

It is true that the tree selected by Step 3 will not necessarily be optimal over all possible trees.  But the advantage of using likelihood as the method here is that relative strengths of alternative trees is implicitly expressed by the likelihood scores.  The likelihood score not only points to a topology, but provides a way to measure just how much confidence we should have in that topology.

Impact of molecular information

With the general framework outlined above, statisticians and computer scientists have devised a number of methods to evaluate many possible phylogenies for species living today.  In many cases, the data generated by sequencing proteins and DNA from living species has forced scientists to rethink their earlier phylogenies.  All sorts of phylogenetic questions have been revisisted, such as
  • The early history of apes: which species diverged first from the other two, humans, chimpanzees, or gorillas?  
  • How many kingdoms are there in the tree of life?  That is to say, since the plant, animal, and fungal kingdoms are well-defined, how do the various protists and bacteria fit into the mix?
  • How do extinct classes of species (such as dinosaurs) relate to those living today?  In particular, are reptiles the nearest living relatives of dinosaurs, or do the birds serve as a more likely candidate?
With the advances in sequencing technology in the past twenty years, and the progress of projects to sequence the entire genomes of humans and several other species, we now have enough data to evaluate questions like these with far more sophisticated statistical methods than were available in the past.  Thus, the profusion of data has led to much stronger statistical support for various hypotheses than were possible in even the recent past.
As a final point, consider what taxonomists now think to be the shape of the mammalian tree, and how the ungulates have been affected by the accumulation of genetic data.  The molecular data simply do not support the theory that ungulates form a clade.  Rather, the similar hoof shape is something that has separately evolved several times in the long history of mammals.


A basic primer on phylogenies

The foundational work on likelihood by Joe Felsenstein

More about Neighbor-Joining

Tuesday, February 1, 2011

What are the topics?

Let's see, as I see it, these days there are three major political topics where science has been put on the defensive by non-scientific interests. The three types of enemies are primarily

1) Industries run by people who have a vested interest in seeing that their dangerous products being treated as the dangers they are.

2) Religious conservatives who feel that their worldviews are threatened by the epistemology of science.

3) Charlatans perpetrating out-and-out frauds on the public (see Wakefield).

oh, and

4) Jenny McCarthy

The primary fronts these days are

1) Evolution.

It's truly sad how evolution continues to battle against ignorance. I've had a student from a local high school tell me that he's not supposed to "believe in evolution". What do I do? The scientific debate on this topic closed over a century ago. It's not a topic where there's any more scientific doubt.

It's also a field where I have some professional expertise.

2) Global warming.

Here the opposition is largely driven by a small number of oil industry billionaires. Sadly, they believe their own propaganda about the problems that an excess of carbon dioxide can cause. The strange thing here is that the data is publicly available and incontrovertible. We have a wealth of data in terms of actual measurements of recorded temperatures, not to mention historical data from ice cores, tree rings, etc.

The strange thing here is how people seek out stories about why they should continue to adhere to their pre-conceptions rather than allow their beliefs to be changed by evidence. I heard a discussion, on Ira Glass's "This American Life" on NPR, between a leading climate scientist and a 15-year old girl who's a Glenn Beck fan.

After the scientist spent 15 minutes elaborating all the different kinds of evidence, the girl informed Ira that, no, she hadn't changed her opinion at all. She still had her intellectual blocks in place. After all, wasn't it colder during the Middle Ages? A complete non-responsive answer that ignored everything she had just been told but allowed her to make a smart-sounding argument, even though it was completely irrelevant.

Some people are just stupid. There's no way around it.

On to...

3) Vaccines (& autism)

Yeah, this one isn't being driven so much by industry or by organized religion, but rather by a small number of charlatans and frauds, supported by some wealthy idiots (like the aforementioned Jenny McCarthy). A number of science blogs, including the estimable Orac, regularly rip the anti-vaccine crowd to shreds.

I probably won't post much about this. See Landru's blog for more info.

Other science policy topics that need consideration include

4) the dangers of tobacco

an oldie but a goodie. But after a few decades of obstructionism the pro-science side of this debate clearly has an upper hand.

5) the dangers of genetically modified food

This is an issue where a lot of the opposition comes across as hysterical ('Frankenfood!') but the agribusiness concerns continue to show a blase attitude.

I don't know quite what to think here, but I have to wonder where the bees have gone.

and, of course

6) Energy concerns

Any kind of consumption of fossil fuels relates to global warming, outlined above. But there are other concerns, such as the pollution from oil spills. This is an issue where I wish I felt I could trust the nuclear power industry. But I don't.

OK, all for today.