Wagner Trees: Phylogeny Reconstruction(from an exercise by Dr. John Lundberg)
readings - introduction - exercise instructions - homework - instructor hints
Wagner trees (and networks) are estimates of evolutionary trees. The construction of a Wagner Tree begins with a table that lists a series of characters for each species. (This is called a character by taxon matrix.) In the table the states of characters are coded numerically. The Wagner Tree is a evolutionary tree that t requires the minimum number of evolutionary steps to explain the observed pattern. It is therefore the "most parsimonious" tree.
The steps in tree construction and a hypothetical example.The steps in the construction of a Wagner Tree are illustrated here with a hypothetical data set of 4 taxa and 12 characters ( Link to Character by Species Data Table ). The characters are already coded and an ancestor for the four taxa has been estimated and added to the initial data matrix.
Step 1. Calculate the table showing the differences between the species according to the formula:The difference between any two spp. is the sum of the absolute difference between all their character states
In explicit notation :
Link to Difference Matrix
Step 2. Form the first interval of the tree by connecting the ancestor to the taxon to which it is most similar.
The first interval has a length of 4, i.e. 4 evolutionary steps separate C from the ancestor. Check the data matrix to see that these steps involve characters 3, 6, 10 and 11.
Step 3. Select the next taxon to be placed on the tree.The next taxon to be placed is that one which is most similar to any interval of the tree.
Calculate the difference between each unplaced taxon and every interval according to the formula. (**If in another example there is a tie - it is not unreasonable to toss a coin to decide placement).
d (L, INT(J,K)) = d (L, J) + d (L,K) - d (J,K)
where d (L, INT(J,K)) is the difference between an unplaced taxon L and an interval composed of the taxa J and K.
d (L, J) and d (L,K) and d (J,K) are the differences between these taxa taken directly from the difference matrix.
Thus, in the example, there are three remaining unplaced taxa, A, B, and D.
d (A, INT(ANC,C))
= d(A, ANC) + d (A,C) - d (ANC - C)
= 7 + 11 - 4
d (B, INT(ANC,C))
= d(B, ANC) + d (B,C) - d (ANC - C)
= 9 + 11 - 4
d (D, INT(ANC,C))
= d(D, ANC) + d (D,C) - d (ANC - C)
= 6 + 8 - 4
Since D is the closest to the interval it is placed next.
Step 4. Construct a hypothetical intermediate ancestor.One of the remarkable things about this method is that it is possible to reconstruct what the common ancestor of D and C should have looked like!
Obviously, we do not actually know that it looked like this, and therefore it is a hypothetical intermediate ancestor, that is sometimes called a "Hypothetical Taxonomic Unit" or HTU. This HTU will be placed between the the unplaced taxon selected in Step 3 and the two members of the interval to which that unplaced taxon is closest. It is possible (in most cases probable) that the unplaced taxon will diverge from the lineage leading from ANCESTOR to C at an intermediate level instead of directly from ANCESTOR or C.
Thus, from, where exactly can we hypothesize that D diverged ? This is where parsimony plays a major role.
The states of an HTU are always based on the states of three taxonomic units (real or hypothetical). HTU's are constructed in such a way as to reduce the number of evolutionary steps implied in the final tree. The parsimony rules for HTU construction are as follows:
(a) If all three taxa have the same state for a character, the HTU will take that state.
Example - Character 1
If D does connect to the HTU and if the HTU had any state except 1, we would be suggesting an evolutionary event for which there is no evidence.
(b) If two of the three taxa have the same state, the HTU will take that state.
Example, Character 4
If D does connect to the HTU and if the HTU had state 1, we would be suggesting that 0 -->1 and that 1 --> 0 in going from ANC to C. As coded, we have no a priori evidence of reversal.
(c) If all three taxa have different states, the hypothetical taxonomic unit will take the intermediate state.
If D does connect to the HTU, what would we imply if the HTU had state 0 or state 2?
The HTU for D, C and ANC, called HTU 1, is different from any other taxonomic unit so far in the study and it is added to both the data and difference matrices. If the HTU turns out to be identical to C or ANC we could ignore it.
Connect the unplaced taxon to the real or hypothetical taxon from which it differs least.
In our hypothetical case,
How long is each interval? Identify the characters that change state in each interval.
Step 6.If any real taxon remains unplaced, return to step 3. Otherwise, stop. (* Note that in our hypothetical case we have now 2 remaining taxa. In the next step 3, these will have to be compared to each of the three intervals that currently exist in the tree. Our old first interval, ANC - C, was destroyed with the intercalation of HTU 1.).
Complete the construction of the tree based on the data at hand. The answer is:
Plot the position of all character state changes. Which characters undergo homoplasious evolution? List the derived states that are shared by groups of taxa and the derived states that are unique to single taxa.
Wagner Networks are an extension of Wagner Trees. The difference is that no ancestral states are estimated a priori, and therefore, there is no hypothetical common ancestor to serve as a starting point for construction. The first interval is made by connecting (by convention) the two most different real taxa. After this one proceeds as in tree construction. Wagner networks yield hypotheses on cladistic pattern, but not sequence. But, sometimes the pattern may help in inferring sequence.
ASSIGNMENT: Construct a Wagner Tree using the imaginary animals of the genus Lundbergia (see below).
Creatures (Lundbergia sp.) for Wagner Trees (for Students)
If you would like to make this a more challenging experiment, you can add other Lundbergia species to the current three species.
Data Matrices for Lundbergia (for Instructors)
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