Close Lower and Upper Bounds for
the Minimum
Reticulate Network of Multiple
Phylogenetic Trees
Yufeng Wu
Dept. of Computer Science & Engineering
University of Connecticut, USA
ISMB 2010
1
Reticulate Networks
Gene trees: phylogenetic
trees from gene sequences
- Assume: Binary and rooted
- Different topologies at different
genes
1:
2:
3:
4:
Gene B
AGC
TGT
AAC
AGT
TB
TA
Reticulate evolution:
one explanation
- Hybrid speciation,
horizontal gene transfer
1:
2:
3:
4:
Gene A
TCG
TCA
CGG
CCG
1
2
3
4
1
3
2
4
Reticulate network:
A directed acyclic graph
displaying each of the
gene trees
Keep two
red edges
Reticulation event(s): nodes
with in-degree two or more
1
Keep two
black edges
2
3
4
2
The Minimum Reticulation Problem
Given: a set of K gene trees G.
NP complete: even for K=2
Current approaches:
Problem: reconstruct reticulate
networks with Rmin(G), the
minimum number, reticulation
events displaying each gene tree.
T1
1
• exact methods for K=2 case (see
Semple, et al)
• impose topological constraints (e.g.
galled networks, see Huson, et al.) or
work on small-scale topologies
T3
T2
2
3
1
4
2
1
4
1
2
4
3
For simplicity: a reticulation
node with exactly two incoming
edges (our approach allows
more general case)
N
2 reticulation
events. Minimum!
3
2
3
4
3
Close Lower and Upper Bounds for Minimum
Reticulation of Multiple Gene Trees
Key idea: developing novel lower and upper bounds for
Rmin(G): G is the set of K gene trees.
RH(G) < Rmin(G)
< SIT(G)
RH(G): Lower bound
Rmin(G): Minimum
Novel: first non-trivial bound
Challenging for K  3
SIT(G): Upper Bound
Works for any K
Bounds provides range of Rmin(G)
If RH(G)=SIT(G), then Rmin(G) = RH(G) = SIT(G)
4
Pairwise Distance
Pairwise distances forT1, T2 and T3
T1
Pairwise reticulation distance
of T1 and T2: d(T1,T2), the
1
1
minimum reticulation in any
reticulate network for T1 and T2
1
T2
T3
Rmin(T1,T2,T3)  max(1,1,1) = 1
Question: can Rmin(T1,T2,T3) = 1?
1
T3
T2
T1
2
3
4
1
2
3
1
2
4
3
?
Choosing same reticulate
edge  same gene trees
Rmin(T1, T2 and T3)  2!
4
v
Imaginary network
with one reticulation
node
T’
T
Display Vector
1
Tree T is displayed in a
network
2
3
4
1
3
2
VT: 0 1
Each tree has a display vector
1
4
VT’: 1 0
0 1 0 1
v1
v2
2
3
4
VT : Display vector of T, how T is displayed in the network
• one bit per reticulation node  length of display vector = number of
reticulation nodes in the network
• value 0/1: at each reticulation node, which edge (the 0-edge or 1-edge) is kept
for T?
Intuition: display vectors can not be too similar
Lemma: D(VT1,VT2)  d(T1,T2) for any network displaying T1 an T2.
D(VT1,VT2): Hamming distance of VT1 and VT2.
d(T1,T2): pairwise reticulation distance of T1 and T2
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T1
The RH Lower Bound
Key: if R reticulation events
possible, then exist K length R
display vectors, satisfying the
distance constraints: Hamming
distance D(VT,VT’)  d(T,T’)
• Analogy: Selecting K points on R
dimensional binary hypercube s.t. the
points can not be too close
• If such K points do not exist, then
we must need at least R+1 reticulation
events.
RH lower bound: the smallest R
s.t. K points can be selected on
R-dimensional hypercube
satisfying the distance
constraints.
3
2
T2
2
T3
Question: can Rmin(T1,T2,T3) = 3?
T3
?
T2
T1
Rmin(T1, T2, T3)  4!
No polynomial time algorithm is known for
general HPP problem.
We use integer linear programming to solve it.
Closed-form formula of RH bound for K=3.
Upper Bound
Problem: how to reconstruct a network for T1, T2, …, TK using small
(may not be minimum) number, U, of reticulation events?
• U: an upper bound
Key idea: sequentially insert gene trees Ti into a growing network N
• Each step inserts a tree into N.
• New reticulation events are needed to display Ti in N.
• Minimize the new reticulation events at each step.
T1
1
2
3
4
1
N
1
T3
T2
2
3
4
1
2
N
2
3
4
1
4
3
N
2
3
4
1
2
3
4
SIT Upper Bound: Stepwise Insertion of Trees
Insertion of tree into a network: given a reticulate network N and a
gene tree T, grow N by adding the minimum number of reticulation
events to make T displayed in N
• NP complete
• Practical computation using integer linear programming
SIT bound
• Try all ordering of T1, T2, …, TK
• For each ordering, insert each tree Ti and compute the number
of reticulation events needed for inserting each Ti. Obtain a
network for each order.
• SIT bound = the smallest reticulation events in these networks.
• Heuristics when K is large or trees are large and different
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Simulation
PIRN: a downloadable open-source software tool
• Implemented in C++ and GNU GLPK (and CPLEX)
Generation of Simulation Data: a two-stage approach
• Simulate a reticulate network N backwards in time for n species
• Randomly select K trees embedded in N.
Evaluation Creteria:
• How often exact minimum is found when lower and upper
bounds match?
• The gap between the lower and upper bounds
• Average running time (see paper)
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Performance of PIRN: % of Datasets
Optimal Solution Found
% LB=UB
Horizontal axis: number of
taxa
Vertical axis: % of datasets
lower = upper bounds
K: number of gene trees
r: level of reticulation
Average over 100 datasets
Number of taxa
Lower and upper
bounds often match for
many data
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Performance of PIRN: Gap of Bounds
Gap
Horizontal axis: number of taxa
Vertical axis: gap between
lower and upper bounds
K: number of gene trees
Gap between the lower
and upper bounds is often
small for many data
Number of taxa
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Reticulate Network for Five Poaceae Trees
ndhF
phyB
RH bound: 11
SIT bound: 13
rbcL
rpoC2
ITS
13
Reticulate Network for Five Poaceae Trees
ITS
SIT bound: 13 reticulation events used in the network
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Acknowledgement
• More information available at:
http://www.engr.uconn.edu/~ywu
• Research supported by National
Science Foundation
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