1. CSE 5290: Algorithms for Bioinformatics Fall 2011
Suprakash Datta
Office: CSEB 3043
Phone: ext 77875
Course page:
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CSE 5290, Fall 2011

2. Next
Clustering
Some of the following slides are based on slides by the authors of our text.
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3. Clustering Arises in many different domains
No single “best algorithm” exists
Cluster shapes, sizes, heterogeneity depends on the data semantics
Complexities of algorithms vary
Dimensionality is a BIG problem
Distance metrics used are critical
Objective function? validation criteria?
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4. Clustering as learning
Machine learning
Supervised
Classifiers, function learning
Unsupervised
Clustering, some neural nets, model fitting
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5. Applications of Clustering
Viewing and analyzing vast amounts of biological data as a whole set can be perplexing
It is easier to interpret the data if they are partitioned into clusters combining similar data points.
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6. Inferring Gene Functionality
Researchers want to know the functions of newly sequenced genes
Simply comparing the new gene sequences to known DNA sequences often does not give away the function of gene
For 40% of sequenced genes, functionality cannot be ascertained by only comparing to sequences of other known genes
Microarrays allow biologists to infer gene function even when sequence similarity alone is insufficient to infer function.
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7. Microarrays and Expression Analysis
Microarrays measure the activity (expression level) of the genes under varying conditions/time points
Expression level is estimated by measuring the amount of mRNA for that particular gene
A gene is active if it is being transcribed
More mRNA usually indicates more gene activity
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8. Microarray Experiments
Produce cDNA from mRNA (more stable)
Attach phosphor to cDNA to see when a particular gene is expressed
Different color phosphors are available to compare many samples at once
Hybridize cDNA over the micro array
Scan the microarray with a phosphor-illuminating laser
Illumination reveals transcribed genes
Scan microarray multiple times for the different color phosphors
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9. Using Microarrays
Track the sample over a period of time to see gene expression over time
Track two different samples under the same conditions to see the difference in gene expressions
Each box represents one gene’s expression over time
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10. Using Microarrays (contd)
Green: expressed only from control
Red: expressed only from experimental cell
Yellow: equally expressed in both samples
Black: NOT expressed in either control or experimental cells
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11. Microarray Data
Microarray data are usually transformed into an intensity matrix (below)
The intensity matrix allows biologists to make correlations between diferent genes (even if they are dissimilar) and to understand how genes functions might be related
Time:
Time X
Time Y
Time Z
Gene 1 10 8
Gene 2 9
Gene 3 4
8.6 3
Gene 4 7
Gene 5 1 2
Intensity (expression level) of gene at measured time
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12. Clustering of Microarray Data
Plot each datum as a point in N-dimensional space
Make a distance matrix for the distance between every two gene points in the N-dimensional space
Genes with a small distance share the same expression characteristics and might be functionally related or similar.
Clustering reveal groups of functionally related genes
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13. Clustering of Microarray Data - contd
Clusters
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14. Homogeneity and Separation Principles
Homogeneity: Elements within a cluster are close to each other
Separation: Elements in different clusters are further apart from each other
…clustering is not an easy task!
Given these points a clustering algorithm might make two distinct clusters as follows
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15. Bad Clustering
This clustering violates both Homogeneity and Separation principles
Close distances from points in separate clusters
Far distances from points in the same cluster
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16. Good Clustering
This clustering satisfies both Homogeneity and Separation principles
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17. Clustering Techniques
Agglomerative: Start with every element in its own cluster, and iteratively join clusters together
Divisive: Start with one cluster and iteratively divide it into smaller clusters
Hierarchical: Organize elements into a tree, leaves represent genes and the length of the paths between leaves represents the distances between genes. Similar genes lie within the same subtrees
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18. Hierarchical Clustering
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19. Hierarchical Clustering: Example
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20. Hierarchical Clustering: Example
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21. Hierarchical Clustering: Example
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22. Hierarchical Clustering: Example
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23. Hierarchical Clustering: Example
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24. Hierarchical Clustering (cont’d)
Hierarchical Clustering is often used to reveal evolutionary history
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25. Hierarchical clustering
Can be top down or bottom up
Well-known algorithms (available in R):
Top-down: DIANA
Bottom-up: AGNES
Graph based algorithms
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26. Hierarchical Clustering Algorithm
Hierarchical Clustering (d , n)
Form n clusters each with one element
Construct a graph T by assigning one vertex to each cluster
while there is more than one cluster
Find the two closest clusters C1 and C2
Merge C1 and C2 into new cluster C with |C1| +|C2| elements
Compute distance from C to all other clusters
Add a new vertex C to T and connect to vertices C1 and C2
Remove rows and columns of d corresponding to C1 and C2
Add a row and column to d corresponding to the new cluster C
return T
The algorithm takes a nxn distance matrix d of pairwise distances between points as an input.
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27. Hierarchical Clustering Algorithm
Hierarchical Clustering (d , n)
Form n clusters each with one element
Construct a graph T by assigning one vertex to each cluster
while there is more than one cluster
Find the two closest clusters C1 and C2
Merge C1 and C2 into new cluster C with |C1| +|C2| elements
Compute distance from C to all other clusters
Add a new vertex C to T and connect to vertices C1 and C2
Remove rows and columns of d corresponding to C1 and C2
Add a row and column to d corrsponding to the new cluster C
return T
Different ways to define distances between clusters may lead to different clusterings
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28. Hierarchical Clustering: Recomputing Distances
dmin(C, C*) = min d(x,y)
for all elements x in C and y in C*
Distance between two clusters is the smallest distance between any pair of their elements
davg(C, C*) = (1 / |C*||C|) ∑ d(x,y)
Distance between two clusters is the average distance between all pairs of their elements
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29. Squared Error Distortion
Given a data point v and a set of points X,
define the distance from v to X
d(v, X)
as the (Euclidean) distance from v to the closest point from X.
Given a set of n data points V={v1…vn} and a set of k points X, define the
Squared Error Distortion
d(V,X) = ∑d(vi, X)2 / n < i < n
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30. K-Means Clustering Problem: Formulation
Input: A set, V, consisting of n points and a parameter k
Output: A set X consisting of k points (cluster centers) that minimizes the squared error distortion d(V,X) over all possible choices of X
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31. 1-Means Clustering Problem: an Easy Case
Input: A set, V, consisting of n points
Output: A single point x (cluster center) that minimizes the squared error distortion d(V,x) over all possible choices of x
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32. 1-Means Clustering Problem: an Easy Case
Input: A set, V, consisting of n points
Output: A single point x (cluster center) that minimizes the squared error distortion d(V,x) over all possible choices of x
1-Means Clustering problem is easy. However, it becomes very difficult (NP-complete) for more than one center.
An efficient heuristic method for K-Means clustering is the Lloyd algorithm
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33. K-Means Clustering: Lloyd Algorithm
Arbitrarily assign the k cluster centers
while the cluster centers keep changing
Assign each data point to the cluster Ci corresponding to the closest cluster representative (center) (1 ≤ i ≤ k)
After the assignment of all data points, compute new cluster representatives according to the center of gravity of each cluster, that is, the new cluster center is
∑v / |C | for all v in C for every cluster C
*This may lead to merely a locally optimal clustering.
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34. x1 x2 x3
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35. x1 x2 x3
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36. x1 x3 x2
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37. x1 x2 x3
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38. Conservative K-Means Algorithm
Lloyd algorithm is fast but in each iteration it moves many data points, not necessarily causing better convergence.
A more conservative method would be to move one point at a time only if it improves the overall clustering cost
The smaller the clustering cost of a partition of data points. the better that clustering
Different methods (e.g., the squared error distortion) can be used to measure this clustering cost
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39. K-Means “Greedy” Algorithm
ProgressiveGreedyK-Means(k)
Select an arbitrary partition P into k clusters
while forever
bestChange  0
for every cluster C
for every element i not in C
if moving i to cluster C reduces its clustering cost
if (cost(P) – cost(Pi  C) > bestChange
bestChange  cost(P) – cost(Pi  C)
i*  I
C*  C
if bestChange > 0
Change partition P by moving i* to C*
else
return P
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40. Clique Graphs
A clique is a graph with every vertex connected to every other vertex
A clique graph is a graph where each connected component is a clique
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41. Transforming an Arbitrary Graph into a Clique Graphs
A graph can be transformed into a
clique graph by adding or removing edges
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42. Corrupted Cliques Problem
Input: A graph G
Output: The smallest number of additions and removals of edges that will transform G into a clique graph
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43. Distance Graphs Turn the distance matrix into a distance graph
Genes are represented as vertices in the graph
Choose a distance threshold θ
If the distance between two vertices is below θ, draw an edge between them
The resulting graph may contain cliques
These cliques represent clusters of closely located data points!
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44. Transforming Distance Graph into Clique Graph
The distance graph (threshold θ=7) is transformed into a clique graph after removing the two highlighted edges
After transforming the distance graph into the clique graph, the dataset is partitioned into three clusters
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45. Heuristics for Corrupted Clique Problem
Corrupted Cliques problem is NP-Hard, some heuristics exist to approximately solve it:
CAST (Cluster Affinity Search Technique): a practical and fast algorithm:
CAST is based on the notion of genes close to cluster C or distant from cluster C
Distance between gene i and cluster C:
d(i,C) = average distance between gene i and all genes in C
Gene i is close to cluster C if d(i,C)< θ and distant otherwise
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46. Other strategies Parametric clustering Fuzzy clustering 11/24/2018
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47. CAST Algorithm CAST(S, G, θ) P  Ø while S ≠ Ø
v  vertex of maximal degree in the distance graph G
C  {v}
while a close gene i not in C or distant gene i in C exists
Find the nearest close gene i not in C and add it to C
Remove the farthest distant gene i in C
Add cluster C to partition P
S  S \ C
Remove vertices of cluster C from the distance graph G
return P
S – set of elements, G – distance graph, θ - distance threshold
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48. Parametric clustering
“Explain” the data using a mathematical model
Estimate the model parameters from the data
E.g. Use a mixture of Gaussians, estimate means, covariances and weights of Gaussian components
Many packages in R, e.g. mclust
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49. Soft vs hard clustering
Cluster memberships are probabilities rather than boolean variables
Cluster overlaps are allowed
Can get hard clusters easily
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50. Fuzzy clustering
Defines membership probabilities for non-parametric algorithms
Implicit parametrization?
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51. Next Phylogenetic trees
Some of the following slides are based on slides by the authors of our text.
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52. Evolution and DNA Analysis: the Giant Panda Riddle
For roughly 100 years scientists were unable to figure out which family the giant panda belongs to
Giant pandas look like bears but have features that are unusual for bears and typical for raccoons, e.g., they do not hibernate
In 1985, Steven O’Brien and colleagues solved the giant panda classification problem using DNA sequences and algorithms
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53. Evolutionary Tree of Bears and Raccoons
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54. Evolutionary Trees: DNA-based Approach
40 years ago: Emile Zuckerkandl and Linus Pauling brought reconstructing evolutionary relationships with DNA into the spotlight
In the first few years after Zuckerkandl and Pauling proposed using DNA for evolutionary studies, the possibility of reconstructing evolutionary trees by DNA analysis was hotly debated
Now it is a dominant approach to study evolution.
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55. Who are closer?
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56. Human-Chimpanzee Split?
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57. Chimpanzee-Gorilla Split?
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58. Three-way Split?
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59. Out of Africa Hypothesis
Around the time the giant panda riddle was solved, a DNA-based reconstruction of the human evolutionary tree led to the Out of Africa Hypothesis that claims our most ancient ancestor lived in Africa roughly 200,000 years ago
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60. Human Evolutionary Tree (cont’d)
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61. The Origin of Humans: ”Out of Africa” vs Multiregional Hypothesis
Humans evolved in the last two million years as a single species. Independent appearance of modern traits in different areas
Humans migrated out of Africa mixing with other humanoids on the way
There is a genetic continuity from Neanderthals to humans
Out of Africa:
Humans evolved in Africa ~150,000 years ago
Humans migrated out of Africa, replacing other shumanoids around the globe
There is no direct descendence from Neanderthals
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62. mtDNA analysis supports “Out of Africa” Hypothesis
African origin of humans inferred from:
African population was the most diverse
(sub-populations had more time to diverge)
The evolutionary tree separated one group of Africans from a group containing all five populations.
Tree was rooted on branch between groups of greatest difference.
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63. Evolutionary Trees How are these trees built from DNA sequences?
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64. Evolutionary Trees How are these trees built from DNA sequences?
leaves represent existing species
internal vertices represent ancestors
root represents the oldest evolutionary ancestor
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65. Rooted and Unrooted Trees
In the unrooted tree the position of the root (“oldest ancestor”) is unknown. Otherwise, they are like rooted trees
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66. Distances in Trees Edges may have weights reflecting:
Number of mutations on evolutionary path from one species to another
Time estimate for evolution of one species into another
In a tree T, we often compute
dij(T) - the length of a path between leaves i and j
dij(T) – tree distance between i and j
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67. Distance in Trees: examplej
d1,4 = = 68
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68. Distance Matrix
Given n species, we can compute the n x n distance matrix Dij
Dij may be defined as the edit distance between a gene in species i and species j, where the gene of interest is sequenced for all n species.
Dij – edit distance between i and j
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69. Edit Distance vs. Tree Distance
Given n species, we can compute the n x n distance matrix Dij
Dij may be defined as the edit distance between a gene in species i and species j, where the gene of interest is sequenced for all n species.
Dij – edit distance between i and j
Note the difference with
dij(T) – tree distance between i and j
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70. Fitting Distance Matrix
Given n species, we can compute the n x n distance matrix Dij
Evolution of these genes is described by a tree that we don’t know.
We need an algorithm to construct a tree that best fits the distance matrix Dij
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71. Fitting Distance Matrix
Fitting means Dij = dij(T)
Lengths of path in an (unknown) tree T
Edit distance between species (known)
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72. Reconstructing a 3 Leaved Tree
Tree reconstruction for any 3x3 matrix is straightforward
We have 3 leaves i, j, k and a center vertex c
Observe:
dic + djc = Dij
dic + dkc = Dik
djc + dkc = Djk
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73. Reconstructing a 3 Leaved Tree (cont’d)
dic + djc = Dij
+ dic + dkc = Dik
2dic + djc + dkc = Dij + Dik
2dic + Djk = Dij + Dik
dic = (Dij + Dik – Djk)/2
Similarly,
djc = (Dij + Djk – Dik)/2
dkc = (Dki + Dkj – Dij)/2
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74. Trees with > 3 Leaves An tree with n leaves has 2n-3 edges
This means fitting a given tree to a distance matrix D requires solving a system of “n choose 2” equations with 2n-3 variables
This is not always possible to solve for n > 3
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75. Additive Distance Matrices
Matrix D is ADDITIVE if there exists a tree T with dij(T) = Dij
NON-ADDITIVE otherwise
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76. Distance Based Phylogeny Problem
Goal: Reconstruct an evolutionary tree from a distance matrix
Input: n x n distance matrix Dij
Output: weighted tree T with n leaves fitting D
If D is additive, this problem has a solution and there is a simple algorithm to solve it
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77. Using Neighboring Leaves to Construct the Tree
Find neighboring leaves i and j with parent k
Remove the rows and columns of i and j
Add a new row and column corresponding to k, where the distance from k to any other leaf m can be computed as:
Dkm = (Dim + Djm – Dij)/2
Compress i and j into k, iterate algorithm for rest of tree
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78. Finding Neighboring Leaves
To find neighboring leaves we simply select a pair of closest leaves.
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79. Finding Neighboring Leaves
To find neighboring leaves we simply select a pair of closest leaves.
WRONG
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80. Finding Neighboring Leaves
Closest leaves aren’t necessarily neighbors
i and j are neighbors, but (dij = 13) > (djk = 12)
Finding a pair of neighboring leaves is
a nontrivial problem!
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81. Degenerate Triples
A degenerate triple is a set of three distinct elements 1≤i,j,k≤n where Dij + Djk = Dik
Element j in a degenerate triple i,j,k lies on the evolutionary path from i to k (or is attached to this path by an edge of length 0).
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82. Looking for Degenerate Triples
If distance matrix D has a degenerate triple i,j,k then j can be “removed” from D thus reducing the size of the problem.
If distance matrix D does not have a degenerate triple i,j,k, one can “create” a degenerative triple in D by shortening all hanging edges (in the tree).
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83. Shortening Hanging Edges to Produce Degenerate Triples
Shorten all “hanging” edges (edges that connect leaves) until a degenerate triple is found
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84. Finding Degenerate Triples
If there is no degenerate triple, all hanging edges are reduced by the same amount δ, so that all pair-wise distances in the matrix are reduced by 2δ.
Eventually this process collapses one of the leaves (when δ = length of shortest hanging edge), forming a degenerate triple i,j,k and reducing the size of the distance matrix D.
The attachment point for j can be recovered in the reverse transformations by saving Dij for each collapsed leaf.
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85. Reconstructing Trees for Additive Distance Matrices
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86. AdditivePhylogeny Algorithm
AdditivePhylogeny(D)
if D is a 2 x 2 matrix
T = tree of a single edge of length D1,2
return T
if D is non-degenerate
δ = trimming parameter of matrix D
for all 1 ≤ i ≠ j ≤ n
Dij = Dij - 2δ
else
δ = 0
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87. AdditivePhylogeny (cont’d)
Find a triple i, j, k in D such that Dij + Djk = Dik
x = Dij
Remove jth row and jth column from D
T = AdditivePhylogeny(D)
Add a new vertex v to T at distance x from i to k
Add j back to T by creating an edge (v,j) of length 0
for every leaf l in T
if distance from l to v in the tree ≠ Dl,j
output “matrix is not additive”
return
Extend all “hanging” edges by length δ
return T
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88. The Four Point Condition
AdditivePhylogeny provides a way to check if distance matrix D is additive
An even more efficient additivity check is the “four-point condition”
Let 1 ≤ i,j,k,l ≤ n be four distinct leaves in a tree
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89. The Four Point Condition (cont’d)
Compute: 1. Dij + Dkl, 2. Dik + Djl, 3. Dil + Djk 2 3 1
2 and 3 represent the same number: the length of all edges + the middle edge (it is counted twice)
1 represents a smaller number: the length of all edges – the middle edge
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90. The Four Point Condition: Theorem
The four point condition for the quartet i,j,k,l is satisfied if two of these sums are the same, with the third sum smaller than these first two
Theorem : An n x n matrix D is additive if and only if the four point condition holds for every quartet 1 ≤ i,j,k,l ≤ n
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91. Least Squares Distance Phylogeny Problem
If the distance matrix D is NOT additive, then we look for a tree T that approximates D the best:
Squared Error : ∑i,j (dij(T) – Dij)2
Squared Error is a measure of the quality of the fit between distance matrix and the tree: we want to minimize it.
Least Squares Distance Phylogeny Problem: finding the best approximation tree T for a non-additive matrix D (NP-hard).
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92. UPGMA: Unweighted Pair Group Method with Arithmetic Mean
UPGMA is a clustering algorithm that:
computes the distance between clusters using average pairwise distance
assigns a height to every vertex in the tree, effectively assuming the presence of a molecular clock and dating every vertex
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93. UPGMA’s Weakness
The algorithm produces an ultrametric tree : the distance from the root to any leaf is the same
UPGMA assumes a constant molecular clock: all species represented by the leaves in the tree are assumed to accumulate mutations (and thus evolve) at the same rate. This is a major pitfall of UPGMA.
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94. UPGMA’s Weakness: Example2 3 4 1
Correct tree
UPGMA
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95. Clustering in UPGMA Given two disjoint clusters Ci, Cj of sequences,1
dij = ––––––––– {p Ci, q Cj}dpq
|Ci|  |Cj|
Note that if Ck = Ci  Cj, then distance to another cluster Cl is:
dil |Ci| + djl |Cj|
dkl = ––––––––––––––
|Ci| + |Cj|
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96. UPGMA Algorithm Initialization: Assign each xi to its own cluster Ci
Define one leaf per sequence, each at height 0
Iteration:
Find two clusters Ci and Cj such that dij is min
Let Ck = Ci  Cj
Add a vertex connecting Ci, Cj and place it at height dij /2
Delete Ci and Cj
Termination:
When a single cluster remains
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97. UPGMA Algorithm (cont’d)1 4 3 2 5
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98. Alignment Matrix vs. Distance Matrix
Sequence a gene of length m nucleotides in n species to generate an…
n x m alignment matrix
CANNOT be transformed back into alignment matrix because information was lost on the forward transformation
Transform into…
n x n distance matrix
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99. Character-Based Tree Reconstruction
Better technique:
Character-based reconstruction algorithms use the n x m alignment matrix
(n = # species, m = #characters)
directly instead of using distance matrix.
GOAL: determine what character strings at internal nodes would best explain the character strings for the n observed species
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100. Character-Based Tree Reconstruction (cont’d)
Characters may be nucleotides, where A, G, C, T are states of this character. Other characters may be the # of eyes or legs or the shape of a beak or a fin.
By setting the length of an edge in the tree to the Hamming distance, we may define the parsimony score of the tree as the sum of the lengths (weights) of the edges
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101. Parsimony Approach to Evolutionary Tree Reconstruction
Applies Occam’s razor principle to identify the simplest explanation for the data
Assumes observed character differences resulted from the fewest possible mutations
Seeks the tree that yields lowest possible parsimony score - sum of cost of all mutations found in the tree
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102. Parsimony and Tree Reconstruction
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103. Small Parsimony Problem
Input: Tree T with each leaf labeled by an m-character string.
Output: Labeling of internal vertices of the tree T minimizing the parsimony score.
We can assume that every leaf is labeled by a single character, because the characters in the string are independent.
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104. Weighted Small Parsimony Problem
A more general version of Small Parsimony Problem
Input includes a k * k scoring matrix describing the cost of transformation of each of k states into another one
For Small Parsimony problem, the scoring matrix is based on Hamming distance
dH(v, w) = 0 if v=w
dH(v, w) = 1 otherwise
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105. Scoring Matrices A T G C 1 A T G C 3 4 9 2 Small Parsimony Problem
Weighted Parsimony Problem A T G C 1 A T G C 3 4 9 2
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106. Unweighted vs. Weighted
Small Parsimony Scoring Matrix: A T G C 1
Small Parsimony Score: 5
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107. Unweighted vs. Weighted
Weighted Parsimony Scoring Matrix: A T G C 3 4 9 2
Weighted Parsimony Score: 22
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108. Weighted Small Parsimony Problem: Formulation
Input: Tree T with each leaf labeled by elements of a k-letter alphabet and a k x k scoring matrix (ij)
Output: Labeling of internal vertices of the tree T minimizing the weighted parsimony score
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109. Sankoff’s Algorithm
Check children’s every vertex and determine the minimum between them
An example
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110. Sankoff Algorithm: Dynamic Programming
Calculate and keep track of a score for every possible label at each vertex
st(v) = minimum parsimony score of the subtree rooted at vertex v if v has character t
The score at each vertex is based on scores of its children:
st(parent) = mini {si( left child ) + i, t} +
minj {sj( right child ) + j, t}
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111. Sankoff Algorithm (cont.)
Begin at leaves:
If leaf has the character in question, score is 0
Else, score is 
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112. Sankoff Algorithm (cont.)
st(v) = mini {si(u) + i, t} + minj{sj(w) + j, t}
si(u)
i, A
sum A T  3 G 4 C 9
si(u)
i, A
sum A T G C
si(u)
i, A
sum A T  3 G 4 C 9
sA(v) = 0
sA(v) = mini{si(u) + i, A} + minj{sj(w) + j, A}
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113. Sankoff Algorithm (cont.)
st(v) = mini {si(u) + i, t} + minj{sj(w) + j, t}
sj(u)
j, A
sum A  T 3 G 4 C 9
sj(u)
j, A
sum A  T 3 G 4 C 9
sj(u)
j, A
sum A T G C
sA(v) = 0
sA(v) = mini{si(u) + i, A} + minj{sj(w) + j, A}
+ 9 = 9
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114. Sankoff Algorithm (cont.)
st(v) = mini {si(u) + i, t} + minj{sj(w) + j, t}
Repeat for T, G, and C
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115. Sankoff Algorithm (cont.)
Repeat for right subtree
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116. Sankoff Algorithm (cont.)
Repeat for root
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117. Sankoff Algorithm (cont.)
Smallest score at root is minimum weighted parsimony score
In this case, 9 – so label with T
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118. Sankoff Algorithm: Traveling down the Tree
The scores at the root vertex have been computed by going up the tree
After the scores at root vertex are computed the Sankoff algorithm moves down the tree and assign each vertex with optimal character.
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119. Sankoff Algorithm (cont.)
9 is derived from 7 + 2
So left child is T,
And right child is T
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120. Sankoff Algorithm (cont.)
And the tree is thus labeled…
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121. Fitch’s Algorithm Solves Small Parsimony problem
Dynamic programming in essence
Assigns a set of letter to every vertex in the tree.
If the two children’s sets of character overlap, it’s the common set of them
If not, it’s the combined set of them.
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122. Fitch’s Algorithm (cont’d)
An example: a c t a
{a,c}
{t,a} c t a a a a a a
{a,c}
{t,a} a a a c t a c t
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123. Fitch Algorithm
1) Assign a set of possible letters to every vertex, traversing the tree from leaves to root
Each node’s set is the combination of its children’s sets (leaves contain their label)
E.g. if the node we are looking at has a left child labeled {A, C} and a right child labeled {A, T}, the node will be given the set {A, C, T}
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CSE 5290, Fall 2011

124. Fitch Algorithm (cont.)
2) Assign labels to each vertex, traversing the tree from root to leaves
Assign root arbitrarily from its set of letters
For all other vertices, if its parent’s label is in its set of letters, assign it its parent’s label
Else, choose an arbitrary letter from its set as its label
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125. Fitch Algorithm (cont.)
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126. Fitch vs. Sankoff Both have an O(nk) runtime
Are they actually different?
Let’s compare …
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127. Fitch
As seen previously:
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128. Comparison of Fitch and Sankoff
As seen earlier, the scoring matrix for the Fitch algorithm is merely:
So let’s do the same problem using Sankoff algorithm and this scoring matrix A T G C 1
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129. Sankoff
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130. Sankoff vs. Fitch
The Sankoff algorithm gives the same set of optimal labels as the Fitch algorithm
For Sankoff algorithm, character t is optimal for vertex v if st(v) = min1<i<ksi(v)
Denote the set of optimal letters at vertex v as S(v)
If S(left child) and S(right child) overlap, S(parent) is the intersection
Else it’s the union of S(left child) and S(right child)
This is also the Fitch recurrence
The two algorithms are identical
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131. Large Parsimony Problem
Input: An n x m matrix M describing n species, each represented by an m-character string
Output: A tree T with n leaves labeled by the n rows of matrix M, and a labeling of the internal vertices such that the parsimony score is minimized over all possible trees and all possible labelings of internal vertices
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132. Large Parsimony Problem (cont.)
Possible search space is huge, especially as n increases
(2n – 3)!! possible rooted trees
(2n – 5)!! possible unrooted trees
Problem is NP-complete
Exhaustive search only possible w/ small n(< 10)
Hence, branch and bound or heuristics used
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133. Nearest Neighbor Interchange A Greedy Algorithm
A Branch Swapping algorithm
Only evaluates a subset of all possible trees
Defines a neighbor of a tree as one reachable by a nearest neighbor interchange
A rearrangement of the four subtrees defined by one internal edge
Only three different rearrangements per edge
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134. Nearest Neighbor Interchange (cont.)
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135. Nearest Neighbor Interchange (cont.)
Start with an arbitrary tree and check its neighbors
Move to a neighbor if it provides the best improvement in parsimony score
No way of knowing if the result is the most parsimonious tree
Could be stuck in local optimum
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136. Nearest Neighbor Interchange
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137. Subtree Pruning and Regrafting Another Branch Swapping Algorithm
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138. Tree Bisection and Reconnection Another Branch Swapping Algorithm
Most extensive swapping routine
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139. Homoplasy
Given:
1: CAGCAGCAG
2: CAGCAGCAG
3: CAGCAGCAGCAG
4: CAGCAGCAG
5: CAGCAGCAG
6: CAGCAGCAG
7: CAGCAGCAGCAG
Most would group 1, 2, 4, 5, and 6 as having evolved from a common ancestor, with a single mutation leading to the presence of 3 and 7
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140. Homoplasy But what if this was the real tree? 11/24/2018
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141. Homoplasy
6 evolved separately from 4 and 5, but parsimony would group 4, 5, and 6 together as having evolved from a common ancestor
Homoplasy: Independent (or parallel) evolution of same/similar characters
Parsimony results minimize homoplasy, so if homoplasy is common, parsimony may give wrong results
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142. Contradicting Characters
An evolutionary tree is more likely to be correct when it is supported by multiple characters, as seen below
Human
Lizard
MAMMALIA
Hair
Single bone in lower jaw
Lactation
etc.
Frog
Dog
Note: In this case, tails are homoplastic
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143. Problems with Parsimony
Important to keep in mind that reliance on purely one method for phylogenetic analysis provides incomplete picture
When different methods (parsimony, distance-based, etc.) all give same result, more likely that the result is correct
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144. Phylogenetic Analysis of HIV Virus
Lafayette, Louisiana, 1994 – A woman claimed her ex-lover (who was a physician) injected her with HIV+ blood
Records show the physician had drawn blood from an HIV+ patient that day
But how to prove the blood from that HIV+ patient ended up in the woman?
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145. HIV Transmission
HIV has a high mutation rate, which can be used to trace paths of transmission
Two people who got the virus from two different people will have very different HIV sequences
Three different tree reconstruction methods (including parsimony) were used to track changes in two genes in HIV (gp120 and RT)
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146. HIV Transmission
Took multiple samples from the patient, the woman, and controls (non-related HIV+ people)
In every reconstruction, the woman’s sequences were found to be evolved from the patient’s sequences, indicating a close relationship between the two
Nesting of the victim’s sequences within the patient sequence indicated the direction of transmission was from patient to victim
This was the first time phylogenetic analysis was used in a court case as evidence (Metzker, et. al., 2002)
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CSE 5290, Fall 2011