On the Simplicity of Evolution Algorithms for Phylogenetic Networks

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On the Simplicity of Evolution Algorithms for
Phylogenetic Networks
Leo van Iersel and Steven Kelk
Part of this research has been funded by the
Dutch BSIK/BRICKS project AFM2.
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• Combine a set of small trees (triplets) into a
  single network that is as simple as possible

algorithm
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Phylogenetic Trees
Root (common ancestor)
Time
Split vertices (ancestors)
Leaves (species)
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Phylogenetic Networks
Root
Split vertex
Reticulation vertex
Leaves
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Phylogenetic Networks
Root

• Reticulation can model
• Recombination
• Hybridization
• Horizontal gene transfer
• Ambiguity

Split vertex
Reticulation vertex
Leaves
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How Simple is a Network?

• Number of reticulations total number of
  reticulation vertices (indegree two vertices)
• Level maximum number of reticulation vertices in
  a biconnected component

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Example level-2 network with 4 reticulations
blue biconnected component red reticulation
vertex
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New Algorithms

• Given dense set of triplets
• Construct level-1 network with a minimum number
  of reticulation vertices
• Given dense set of triplets
• Construct level-2 network with a minimum number
  of reticulation vertices

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Triplets
abc
bca
acb

• A triplet set is dense if for each combination of
  three leaves it contains at least one of the
  three possible triplets

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Triplet Consistency
A triplet abc is consistent with a network if
this network contains a subdivision of abc
abc
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Triplet Consistency
Also triplet bca is consistent with this network
because it also contains a subdivision of bca
bca
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Previous results

• Aho et al. (1981)
• Constructing (phylogenetic) trees from triplets
  in polynomial time
• Jansson, Nguyen and Sung (2004)
• Constructing level-1 networks from dense triplet
  sets in polynomial time
• Van Iersel, Keijsper, Kelk, Stougie, Hagen and
  Boekhout (2008)
• Constructing level-2 networks from dense triplet
  sets in polynomial time

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New Algorithms

• Given dense set of triplets
• Construct level-1 network consistent with the
  input triplets that contains a minimum number of
  reticulation vertices
• Given dense set of triplets
• Construct level-2 network consistent with the
  input triplets that contains a minimum number of
  reticulation vertices

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Old algorithm
New algorithm
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SN-sets

• Definition. Subset S of the leaves is an SN-set
  if there is no triplet xyz withx?S and y,z?S
• Lemma Jansson and Sung.Any two SN-sets are
  either disjoint or one is included in the other

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Observation. A set of leaves below a cut-arc is
an SN-set
Because then xyz with x?S and y,z?S is not
consistent with the network
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Algorithm for Level-1

• For each SN-set S, from small to large
• Construct a network consisting of a root
  connected to two optimal networks NS1, NS2 for
  included SN-sets S1, S2
• And all possible networks consisting of a cycle
  connected to at least three optimal networks for
  included SN-sets
• The optimal solution NS is the network with the
  minimum number of reticulation vertices over all
  constructed networks

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Case 1 root is not in a cycle
S1 and S2 are the maximal SN-sets
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Case 2 root is in a cycle
Which SN-sets are S1, S2, S3, S4, S5?
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The Maximal SN-sets
Some maximal SN-sets may be divided below a path
ending in a reticulation vertex
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• Guess X
• Find maximal SN-sets that do not contain X

• There can still be a maximal SN-set below a path,
  but

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• There exists a different solution where this
  SN-set is below a single cut-arc
• This solution has the same number of reticulations

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To conclude

• We know how to find the sets S1,,S5
• A single cycle can be constructed with the
  algorithm by Jansson, Nguyen and Sung
• Optimal networks NS1,,NS5 have been computed in
  earlier iterations

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Real network
Constructed network
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Conclusion and Discussion

• We can construct level-1 and level-2 phylogenetic
  networks with a minimum number of reticulations
  in polynomial time
• Input triplet set has to be dense
• There must exist a network consistent with all
  input triplets
• Level can be at most two

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Thank you