hybridization networks using a SAT-solver Vladimir Ulyantsev and - - PowerPoint PPT Presentation

Constructing parsimonious hybridization networks using a SAT-solver

Vladimir Ulyantsev and Mikhail Melnik, presented by Alexey Sergushichev

AlCoB 2015, Mexico

Phylogenetic tree

• Binary tree with set of taxa as leaves
• Can be defined for a particular gene

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Hybridization network

• Directed acyclic graph with

a single root

• Reticulation nodes:

in-degree=2, out-degree=1

• Regular nodes:

in-degree=1, out-degree=2

• Leaves: taxa

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Displaying a tree

• Select direction a reticulation nodes
• Collapse simple paths

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Hybridization network problem

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• Find a hybridization network for a set of

phylogenetic trees T1, T2, .. Tt with the minimal number of reticulation nodes

• Is NP-complete even for t=2

Most parsimonious network

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For two trees:

• CASS (heuristic)
• MURPAR (heurisic)

For multiple trees:

• PIRNCH (heuristic)
• PIRNC (exact)

Existing solutions

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• Fix hybridization number k
• Make Boolean formula f so that f ∈ SAT iff

there is a hybridization network for k

• Check satisfiability with a SAT-solver
• Find minimal k with satisfiable formula
• Restore the network

Reduction to SAT

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• Boolean formula f in CNF form:

𝑔(𝑤1, 𝑤2, … ) = 𝑤1 ∨ ¬𝑤2 ∨ . . . ∧ … ∧ . . .

• Whether values for 𝑤1, 𝑤2, … exist that makes f

true

• Can be seen as conjunction of multiple

constraints

• Constraints can be of the form

𝑤1 ∧ ¬𝑤2 ∧ . . . → 𝑤3

SAT

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• 2n+ 2k - 1 nodes

– [1, n] — leaves (L) – [n+1, 2n + k - 1] — regular nodes (V) – [2n+k, 2n+2k-1] — reticulation nodes (R)

Network structure

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• 𝑚𝑤,𝑣 and 𝑠

𝑤,𝑣 — u is a left (right) child of v for v

in V

• 𝑞𝑤,𝑣— u is parent of v for v in L+V
• 𝑞𝑚, 𝑞𝑠 and c — parent child relations for

reticulation nodes

• 𝑃(𝑜2) variables

Network structure variables

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• Nodes have only one left child, right child,

parent

• u is child of v → v is parent of u
• u is parent of v → v is left of right child of u
• 𝑃(𝑜3) constraints

Network consistency constraints

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Network consistency constraints: Actual clauses

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• For a tree T
• Choice of a parent for reticulation nodes
• Variables for correspondence between

network and tree nodes

• Collapsing non-branching paths

– Whether particular nodes were removed or not – Parent relations after collapsing

• 𝑃(𝑢𝑜2) variables

Displaying structure

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• All T nodes are uniquely mapped to network

nodes

• Parent relations in the tree uniquely

correspond to the network structure after selecting directions at reticulation points and collapsing paths

• Parent relations in the network are consistent
• 𝑃(𝑢𝑜3) constraints

Displaying consistency constraints

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Displaying consistency constraints: Actual clauses (1)

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Displaying consistency constraints: Actual clauses (2)

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All clauses

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Additional optimizations

• Splitting into independent problems
• Symmetry breaking

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• 57 grasses dataset by Group G.P.W. et al
• CryptoMiniSAT solver
• 1000 s time limit
• Comparison with PIRNs

Experiments

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• 57 grasses datasets by Group G.P.W. et al

Grass Phylogeny Working Group

• CryptoMiniSAT solver
• 1000s time limit
• Comparison with PIRNs

Experiments

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Results

• Exact solution (out of 57)

– PhyloSAT: 36 – PIRNC: 29

• Non-exact

– PhyloSAT: 48 (40 optimal) – PIRNCH: 43 (36 optimal)

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Results for k >= 6

hybridization number (time in seconds)

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• Different SAT-solvers
• Improving reduction
• Using upper and lower bounds on k
• Searching for all minimal solutions

Future work

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Conclusions

• Constructing parsimonious hybridization

networks can be approached with reducing to SAT

• This approach outperforms known exact

solver and compares well with heuristic solver

• Solving bigger instances is still challenging

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The End

https://github.com/ctlab/PhyloSAT Vladimir Ulyantsev (ulyntsev@rain.ifmo.ru)

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