Phylogenetic networks: recent questions and results or: constructing a level2 phylogenetic network f

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Phylogenetic networks recent questions and
results (or constructing a level-2 phylogenetic
network from a dense set of input triplets in
polynomial time)

• Leo van Iersel1, Judith Keijsper1, Steven Kelk2,
  Leen Stougie12
• (1) Technische Universiteit Eindhoven (TU/e)
• (2) Centrum voor Wiskunde en Informatica (CWI),
  Amsterdam
• Email S.M.Kelk_at_cwi.nl
• Web http//homepages.cwi.nl/kelk

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Part 1 Context
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Phylogenetic tree reconstruction
Phylogenetic tree reconstruction is essentially
the science of efficiently inferring and
constructing plausible evolutionary trees when we
only have limited input data about the species
concerned At the intersection of biology,
bioinformatics, computer science and
mathematics.
Orangutan
Gorilla
Chimpanzee
Human
(This tree borrowed from a presentation by Tandy
Warnow)
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Dominant methods in phylogenetic reconstruction

• Character-based methods
• Maximum Parsimony ( Minimum Steiner Tree)
• Maximum Likelihood
• Bayesian methods (Markov Chain Monte Carlo)
• Distance-based methods
• Neighbour Joining
• UPGMA
• Quartet/triplet-based methods

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Triplet-based methods (1)

• Quartet-based methods used for constructing
  unrooted evolutionary trees no root ( most
  distant ancestor) and edges have no direction
  (e.g. edge between species X and Y does not say
  whether X evolved into Y, or vice-versa.)
• Triplet-based methods are used for constructing
  rooted evolutionary trees there is a root and
  edges are directed.
• The central idea build a single, big
  evolutionary tree for a set L of species by
  combining smaller evolutionary trees on subsets
  of L such that the big tree respects the
  structure of the smaller trees.
• In triplet-based methods, the small input trees
  are always defined on size-3 subsets of the
  species set L (and are called rooted triplets.)

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Triplet-based methods (2)

• For example. Suppose I want to reconstruct a
  plausible evolution for the species set
  W,X,Y,Z.
• I am given a set of rooted triplets zwx, yxw,
  xyz, wzy. (Note zwx wzx.)

solution
algorithm
w
z
x
y
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Triplet-based methods (2)

• For example. Suppose I want to reconstruct a
  plausible evolution for the species set
  W,X,Y,Z.
• I am given a set of rooted triplets zwx, yxw,
  xyz, wzy. (Note zwx wzx.)

solution
z
w
x
algorithm
w
z
x
y
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Triplet-based methods (2)

• For example. Suppose I want to reconstruct a
  plausible evolution for the species set
  W,X,Y,Z.
• I am given a set of rooted triplets zwx, yxw,
  xyz, wzy. (Note zwx wzx.)

solution
z
w
x
algorithm
x
y
z
w
z
x
y
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From trees to networks

• The algorithm of Aho et al. (1981) can be used
  to construct trees from rooted triplets.
• Butwhat if the algorithm fails? Why might the
  algorithm fail?
• Possible reason 1 The underlying evolution is
  tree-like, but the input triplets contain errors.
• Possible reason 2 The triplets are correct, but
  the underlying evolution is not tree-like.
  Biological phenomena such as hybridization,
  horizontal gene transfer, recombination and gene
  duplication can lead to evolutionary scenarios
  that are not tree-like!
• Response try and construct not phylogenetic
  trees, but phylogenetic networks

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From trees to networks (2)

• For example, suppose the input is xyz, xzy.

z
y
x
(Note that there are cases when, even if there is
at most one triplet per 3 species, a tree is not
possible)
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From trees to networks (2)

• For example, suppose the input is xyz, xzy.

x
y
z
z
y
x
(Note that there are cases when, even if there is
at most one triplet per 3 species, a tree is not
possible)
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From trees to networks (2)

• For example, suppose the input is xyz, xzy.

z
y
x
z
y
x
(Note that there are cases when, even if there is
at most one triplet per 3 species, a tree is not
possible)
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Level-k phylogenetic networks
root (only one!)
A level-k phylogenetic network is a rooted,
directed acyclic graph where every biconnected
component (in the underlying undirected graph)
contains at most k recombination vertices. This
network here is a very simple example of a
level-1 network. In a level-1 network, the
cycles are vertex-disjoint, hence the
alternative name galled tree.
split-vertex
z
y
x
leaf- vertex
recombination-vertex
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Level-k phylogenetic networks
root (only one!)
A level-k phylogenetic network is a rooted,
directed acyclic graph where every biconnected
component (in the underlying undirected graph)
contains at most k recombination vertices. This
network here is a very simple example of a
level-1 network. In a level-1 network, the
cycles are vertex-disjoint, hence the
alternative name galled tree.
split-vertex
z
y
x
leaf- vertex
recombination-vertex
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What Jansson Sung ( Nguyen) did

• A set of input triplets is dense iff, for every
  subset of 3 species, there is at least one
  triplet corresponding to those 3 species.
• A dense set of input triplets for n species
  contains thus O(n3) triplets.
• Jansson Sung (2006) showed the following

Given a dense set of triplets T for a set L of
species, it is possible to determine in
polynomial-time whether a level-1 phylogenetic
network N exists such that all the triplets in T
are consistent with N. (And if so, to construct
such a network.)

• They later showed, together with Nguyen, how to
  do this in time linear in T. They also showed
  that, in the non-dense case, the problem is
  NP-hard.
• But what about level-2 networks, and higher?

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Main result Given a dense set of triplets T for
a set L of species, it is possible to determine
in time O(T3) whether a level-2 phylogenetic
network N exists such that all the triplets in T
are consistent with N. (And if so, to construct
such a network.)
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Part 2 The algorithm
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Algorithm, high-level idea

• The algorithm is conceptually (fairly) simple,
  but the proof of correctness and the technical
  details are rather complex.
• The high-level idea is as follows
• PARTITION the set of leaves (i.e. species) into
  a correct partition P
• INDUCE a new set of triplets T where every
  block of the partition P becomes a single leaf (a
  kind of meta-leaf if you like)
• SOLVE a simpler version of the problem for T to
  get a network N
• RECURSE inside each leaf of N

• Step 3 is the critical part of the algorithm. It
  brings together two issues
• (a) why is it sufficient to only solve a simpler
  version of the problem?
• (b) how do we solve this simpler version of the
  problem?

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Definition inducing new triplet sets from
partitions of the leaf set

• Suppose I have a partition P P1, P2, , Pt
  of the leaf set L.
• Suppose I have a dense set of triplets T on the
  leaf set L.
• Let T be a new triplet set on leaf set q1,
  q2,, qt defined as follows
• qiqjqk is in T if and only if i?j?k and there
  exists a triplet xyz in T such that x is in Pi,
  y is in Pj and z is in Pk
• Then we say that T is the triplet set induced
  by the partition P of L.
• Critically if T is dense, then T is also
  dense.
• In some sense this can be perceived as a
  coarsening of the input set.

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Definition simple level-2 networks
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An example of a simple level-2 network
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Definition SN-set

• Jansson Sung introduced the idea of the SN-set.
• SN-sets are special subsets of the leaves L, and
  are defined with respect to triplet sets.
• All sets containing just a single leaf, are
  SN-sets.
• More generally, an SN-set is any subset of leaves
  obtained by taking the closure of the following
  operation on some subset S of the leaves L

z
x
y
some subset S of the leaves
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Definition SN-set

• Jansson Sung introduced the idea of the SN-set.
• SN-sets are special subsets of the leaves L, and
  are defined with respect to triplet sets.
• All sets containing just a single leaf, are
  SN-sets.
• More generally, an SN-set is any subset of leaves
  obtained by taking the closure of the following
  operation on some subset S of the leaves L

In other words, if there is some pair of leaves
x,z in the set S such that xyz is a triplet and
y is not in the set S, add y to S, and repeat
until no more leaves can be added. An SN-set is
any set that can be constructed this way.
z
x
y
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Definition maximal SN-set

• The SN-set that is equal to the total leaf set L,
  is called the trivial SN-set.
• An SN-set that is non-trivial, and is not a
  strict subset of any other non-trivial SN-set, is
  called a maximal SN-set.
• Jansson and Sung proved that the set of maximal
  SN-sets partition the leaf set L. So no two
  maximal SN-sets overlap, and they completely
  cover the set of input leaves.
• It is polynomial-time solveable to find all the
  SN-sets, and all the maximal SN-sets.
• Jansson Sung solved the level-1 problem by
  observing that they could treat the maximal
  SN-sets like meta-leaves, thus reducing the
  problem to recursively solving the problem on the
  triplets induced by the maximal SN-sets.
• Our idea is similar, but SN-sets in level-2
  networks are (unfortunately) rather more complex
  creatures than in level-1 networks.

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Definition (highest) cut-edges

• In a phylogenetic network N, a cut-edge (x,y) is
  an edge whose removal disconnects the
  (underlying) graph.
• A cut-edge (x,y) is said to be a trivial cut
  edge iff y is a leaf.
• A cut-edge (x,y) is said to be highest iff there
  is no cut-edge (p,q) such that there is a
  directed path from q to x in N.

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So each maximal SN-set can be expressed as the
union of the leaves reachable by one or more
highest cut-edges.
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A first attempt at reducing the problem to simple
level-2 networks

• Now, suppose we have a dense set of triplets T
  and there exists a level-2 network N such that
  all the triplets in T are consistent with N. (Of
  course we dont know what N is yet)
• Suppose we construct a partition P of L as
  follows. The blocks of P are the sets of leaves
  reachable from highest cut-edges in N. (Each
  maximal SN-set of N thus corresponds to one or
  more blocks in P.)
• Let T be the new set of triplets induced by the
  partition P. In other words, if we collapse the
  set of leaves below highest cut-edges into
  meta-leaves, T is the new set of triplets we
  get. (Nice property the maximal SN-sets of T
  are in 11 correlation with the maximal SN-sets
  of T.)
• Critical fact 1 the only level-2 networks where
  all cut-edges are trivial, are simple level-2
  networks.
• Critical fact 2 there exists some simple
  level-2 network N such that the triplets in T
  are consistent with N. Furthermore, if we find
  such an N, and then recursively construct
  networks within each meta-leaf, we obtain a
  network consistent with T!

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But.thats a non-deterministic argument

• So, it looks like we can indeed reduce the
  problem in some sense to finding simple
  level-2 networks.
• But that analysis was based on knowing where the
  highest cut-edges are in a hypothetical solution
  N. And we dont know Nthis is precisely what
  were looking for!
• We can, however, compute the maximal SN-sets of
  the input triplet set T.
• We need to be able to say something more about
  how maximal SN-sets of T relate to highest
  cut-edges in hypothetical solutions. Then we can
  base the recursion on maximal SN-sets, instead of
  highest cut-edges.

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Central Theorem (simplified). Suppose there is a
dense triplet set T consistent with some simple
level-2 network N. Then there exists a level-2
network N (not necessarily simple) such that,
with the exception of perhaps one maximal SN-set
with respect to T, every maximal SN-set appears
below a single cut-edge in N. The remaining,
odd-one-out maximal SN-set (if it exists) will
be equal to the union of leaves below two
cut-edges.
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Observe how SN-set C,G,F has been pushed
below a single cut-edge.
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An existence argument

• If some solution N exists for T, then a simple
  level-2 solution N exists for T (induced by the
  highest cut-edges of N) where the maximal SN-sets
  of T are tightly correlated with the maximal
  SN-sets of T. Finding N gives the starting point
  for a solution to T.
• But by the Central Theorem, all (except maybe
  one) of the maximal SN-sets of N can be pushed
  below highest cut-edges to give a solution N
  for T.
• If we re-expand all the meta-leaves of N, we
  obtain a new solution N for T. Crucially, all
  (except maybe one) of the maximal SN-sets of T
  will be beneath single cut-edges in N. The
  odd-one-out will be beneath two cut-edges.
• So if we substitute N as N in the first step,
  we come to the following conclusion
• We can find a solution for T by finding a simple
  level-2 solution for the set of triplets induced
  by the maximal SN-sets of T, and recursing. We
  need to correctly guess the odd-one-out maximal
  SN-set, however, and split that into two
  meta-leaves. Fortunately we can just try
  splitting each maximal SN-set in turn.

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subnetwork below highest cut-edge
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whole maximal SN-set is now below a cut-edge!
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Finding simple level-2 networks

• So we know that, if we analyse the maximal
  SN-sets carefully, and construct an appropriate
  new set of triplets, we can recursively reduce
  the entire problem to finding simple level-2
  networks.
• But how do we algorithmically construct a simple
  level-2 network that is consistent with a given
  dense set of triplets?

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g
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Conclusions open problems

• So we know how to efficiently construct level-2
  networks from dense triplet sets. Whats next?
• Applicability how useful is it?
• Initial implementation programming and
  fine-tuning
• Improving running time in the spirit of the
  SN-tree of JSN
• Complexity what about level-3 and higher?
• Bounds worst-case, best-case scenarios
• Building all networks
• Properties of output networks as function of
  input
• Different triplet restrictions
• Confidence how good are the solutions?
• Exponential-time exact algorithms for NP-hard
  problems

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Conclusions open problems

• So we know how to efficiently construct level-2
  networks from dense triplet sets. Whats next?
• Applicability how useful is it?
• Initial implementation programming and
  fine-tuning
• Improving running time in the spirit of the
  SN-tree of JSN
• Complexity what about level-3 and higher?
• Bounds worst-case, best-case scenarios
• Building all networks
• Properties of output networks as function of
  input
• Different triplet restrictions
• Confidence how good are the solutions?
• Exponential-time exact algorithms for NP-hard
  problems

Thank you for your attention!