Computing close bounds on the minimum number of

recombinationsDan Gusfield UCD

Y. Song, Y. F. Wu, D. Gusfield (ISMB2005)

D. Gusfield, D. Hickerson (Dis. Appl. Math 2005)

D. Gusfield, V. Bansal (Recomb 2005)

Reconstructing the Evolution of Binary Bio-Sequences (SNPs)

• Perfect Phylogeny (tree) model

• Phylogenetic Networks (DAG) with recombination (ARG)

• Efficiently Computed Lower and Upper Bounds on the number of recombinations needed.

• Decomposition Theorem and open question.

Geneological or Phylogenetic Networks

• The major biological motivation comes from genetics and attempts to reconstruct the history of recombination in populations.

• Some of the algorithmic and mathematical results also have phylogenetic applications, for example in hybrid speciation, lateral gene transfer.

• Critical distinction is whether there is a fixed linear ordering of the sites (some results need it and some do not).

The Perfect Phylogeny Model for binary sequences

00000

1

2

4

3

510100

1000001011

00010

01010

12345sitesAncestral sequence

Extant sequences at the leaves

Site mutations on edgesThe tree derives the set M:1010010000010110101000010

Classic NASC: Arrange the sequences in a matrix. Then (with no duplicate columns), the sequences can be generated on a unique perfect phylogeny if and only if no two columns (sites) contain all four pairs:

0,0 and 0,1 and 1,0 and 1,1

This is the 4-Gamete Test

When can a set of sequences be derived on a perfect phylogeny

with the all-0 root?

A richer model

00000

1

2

4

3

510100

1000001011

00010

01010

12345101001000001011010100001010101 added

pair 4, 5 fails the threegamete-test. The sites 4, 5``conflict”.

Real sequence histories often involve recombination.

10100 01011

5

10101

The first 4 sites come from P (Prefix) and the sitesfrom 5 onward come from S (Suffix).

P S

Sequence Recombination

A recombination of P and S at recombination point 5.

Single crossover recombination

called ``crossing over” in genetics

Network with Recombination

00000

1

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3

510100

1000001011

00010

01010

12345101001000001011010100001010101 new

10101

The previous tree with onerecombination event now derivesall the sequences.

5

P

S

Multiple Crossover Recombination

4-crossovers

2-crossovers = ``gene conversion”

A Phylogenetic Network

00000

52

3

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4S

p

PS

1

4

a:00010

b:10010

c:00100

10010

01100

d:10100

e:01100

00101

01101

f:01101

g:00101

00100

00010

Minimizing recombinations

• Any set M of sequences can be generated by a phylogenetic network with enough recombinations, and one mutation per site. This is not interesting or useful.

• However, the number of (observable) recombinations is small in realistic sets of sequences. ``Observable” depends on n and m relative to the number of recombinations.

• Two algorithmic problems: given a set of sequences M, find a phylogenetic network generating M, minimizing the number of recombinations . Find a network generating M that has some biologically-motivated structural properties.

Minimization is NP-hard

The problem of finding a phylogenetic network that creates a given set of sequences M, and minimizes the number of recombinations, is NP-hard. (Wang et al 2000) (Semple 2004)

A super-exponential-time method computes the exact min (Song and Hein). Works only for a small number of sequences.

Wang et al. explored the problem of finding a phylogenetic network where the recombination cycles are required to be node disjoint, if possible.

They gave a sufficient but not a necessary condition to recognize cases when this is possible. O(nm + n^4) time. A full solution is

in Gusfield, Eddhu, Langley O(nm + n^3) time (also root-unknown case solved later in the same time bound).

Practical and Accurate Lower Bound Computation

• Unconstrained recombination.• Want efficient computation of lower bounds on

the minimum number of recombinations needed.

Many methods: HK, Haplotype, History, Connected Comps, Composite Interval, Gall-Interval Composite Method, Subsets, Recmin, Composite ILP, HapBound (-S,-M). Some require a linear order, and others are more general.

Incompatible Sites

A pair of sites (columns) of M that fail the

4-gametes test are said to be incompatible.

A site that is not in such a pair is compatible.

0 0 0 1 01 0 0 1 00 0 1 0 01 0 1 0 00 1 1 0 00 1 1 0 10 0 1 0 1

1 2 3 4 5abcdefg

1 3

4

2 5

Two nodes are connected iff the pairof sites are incompatible, i.e, fail the 4-gamete test.

Incompatibility Graph

M

The grandfather of all lower bounds - HK 1985

• Arrange the nodes of the incompatibility graph on the line in order that the sites appear in the sequence. This bound requires a linear order.

• The HK bound is the minimum number of vertical lines needed to cut every edge in the incompatibility graph. Weak bound, but widely used - not only to bound the number of recombinations, but also to suggest their locations.

Justification for HK

If two sites are incompatible, there must have been some recombination where the crossover point is between the two sites.

1 2 3 4 5

HK Lower Bound

1 2 3 4 5

HK Lower Bound = 1

More general view of HK

Given a set of intervals on the line, and for each interval I, a number N(I), define the composite problem: Find the minimum number of vertical lines so that every interval I intersects at least N(I) of the vertical lines.

In HK, each incompatibility defines an interval I where N(I) = 1.

The composite problem is easy to solve by a left-to-right myopicplacement of vertical lines: Sort the intervals by right end-point;Process the intervals left to right in that order; when the rightendpoint of an interval I is reached, place there (if needed) additional vertical so that N(I) lines intersect I.

This general approach is called the Composite Method(Simon Myers 2002).

If each N(I) is a ``local” lower bound on the number ofrecombinations needed in interval I, then the solution tothe composite problem is a valid lower bound for thefull sequences. The resulting bound is called the compositebound given the local bounds.

Example: New bound

The number of non-trivial connected components in the incompatibility graph is a lower bound on the needed number of recombinations (Gusfield, Hickerson; Bafna, Bansal).

CC Bound = 2 in the prior example.

The CC bound can be arbitrarily larger than the HK bound,but can also be smaller.

When the CC bound is used for the local bound N(I) in each interval I,then the resulting composite bound is provably as large as theHK bound, and is usually larger (in our simulations).

Better composite bounds are obtained by using the``haplotype bound” as the local bound.

Haplotype Bound (Simon Myers)

• Rh = Number of distinct sequences (rows) - Number of distinct sites (columns) -1 (folklore)

• Before computing Rh, remove any site that is compatible with all other sites. A valid lower bound results - generally increases the bound.

• Generally really bad bound, often negative, when used on large intervals, but Very Good when used as local bounds in the Composite Interval Method, and other methods.

Composite Interval Method using RH bounds

Compute Rh separately for each of the C(m,2) intervals of the m sites; let N(I) = Rh(I) be the local lower bound for interval I. Then compute the composite bound using these local bounds.

Polynomial time and gives bounds that often double HK in our simulations.

Composite Subset Method (Myers)

• Let S be subset of sites, and Rh(S) be the haplotype bound for subset S. If the leftmost site in S is L and the rightmost site in S is R, then use Rh(S) as a local bound N(I) for interval I = [S,L].

• Compute Rh(S) on many subsets, and then solve the composite problem to find a composite bound.

RecMin (Myers)

• World Champion Lower Bound Program (available). Often RecMin gives a bound three times as large as HK.

• Computes Rh on subsets of sites, but limits the size and the span of the subsets. Default parameters are s = 6, w = 15 (s = size, w = span).

• Generally, impractical to set s = w = m, so generally one doesn’t know if increasing the parameters would increase the bound. (example: Myers bound of 70 on the LPL data).

Optimal RecMin Bound (ORB)

• The Optimal RecMin Bound is the lower bound that RecMin would produce if both parameters were set to their maximum values (s = w= m).

• In general, RecMin cannot compute (in practical time) the ORB.

• Practical computation of the ORB is our first contribution.

Computing the ORB

• Gross Idea: For each interval I, use ILP to find a subset of sites S that maximizes Rh(S) over all subsets in interval I. Call the result Opt(I).

• Set N(I) = Opt(I), and compute the composite bound using those local bounds.

• The composite bound is the ORB. -- the result one would get by using all 2^m subsets in RecMin, with s = w = m.

We have moved from doing an exponential number of simplecomputations (computing Rh for each subset), to solving a quadratic number of (possibly expensive) ILPs.

Is this a good trade-off in practice? Our experience - very much so!

How to compute Opt(I) by ILPCreate one variable Xi for each row i; one variable Yj for each column j in interval I. All variables are 0/1 variables.

Define S(i,i’) as the set of columns where rows i and i’ have different values. Each column in S(i,i’) is a witness that rows i and i’ differ.

For each pair of rows (i,i’), create the constraint: Xi + Xi’ <= 1 + ∑ [Yj: jin S(i,i’)]

Objective Function: Maximize ∑ Xi - ∑ Yj -1

Alternate way to compute Opt(I) by ILP

Create one variable Yj for each column jin interval I. All variables are 0/1 variables.

S(i,i’) as before.

For each pair of rows (i,i’) create the constraint: 1 <= ∑ [Yj: jin S(i,i’)]

Objective Function: Maximize N - ∑(Yj) -1

Finds the smallest number of columns to distinguish the rows.

First remove any duplicate rows. Let N be the number of rowsremaining.

Two critical tricks

1) Use the second ILP formulation to compute Opt(I). It solves much faster than the first (why?)

2) Reduce the number of intervals where Opt(I) is computed:

If the solution to Opt(I) uses columns thatonly span interval L, then there is noneed to find Opt(I’) in any interval I’ containing L. Each ILP computationdirectly spawns at most 4 other ILPs. Apply this idea recursively.

I

L

With the second trick we need to find Opt(I) for only 0.5% - 35% of all the C(m,2) intervals (empirical result). Surprisingly fast in practice (with either the GNU solver or CPLEX).

Bounds Higher Than the Optimal RecMin Bound

• Often the ORB underestimates the true minimum, e.g. Kreitman’s ADH data: 6 v. 7

• How to derive sharper bound? Idea: In the composite method, check if each local bound N(I) = Opt(I) is tight, and if not, increase N(I) by one.

• Small increases in local bounds can increase the composite bound.

Bounds Sharper Than Optimal RecMin Bound

• A set of sequences M is called self-derivable if there is a network generating M where the sequence at every node (leaf and intermediate) is in M.

• Observation: The haplotype bound for a set of sequences is tight if and only if the sequences are self-derivable.

• So for each interval I where Opt(I) is computed, we check self-derivability of the sequences induced by columns S*, where S* are the columns specified by Opt(I). Increase N(I) by 1 if the sequences are not self-derivable.

Algorithm for Self-Derivability

• Solution is easy when there are no mutations --only recombinations are allowed and one initial pair of sequences is chosen as ``reached”.

• Two reached sequences S1 and S2 can reach a third sequence S3, if S3 can be created by recombining S1 and S2.

• Do BFS to see if all sequences can be reached by successive application of the ``reach operation”.

• Clearly polynomial time and can be optimized with suffix-trees etc. (Kececiouglu, Gusfield)

Self-derivability Test with mutations allowed

• For each site i, construct set MUT(i) of sequence pairs (S1, S2) in M where S1 and S2 differ at only site.

• Try each sequence in M as root (which is the only reached sequence initially).

• For each root, enumerate all ways of choosing exactly one ordered pair of sequences (Sp, Sq) from each MUT(i). Sp is allowed to ``reach” Sq.

• Run the prior self-derivability algorithm with these new permitted reach relations, to test if all sequences in M can be reached. If so, then M is self-derivable, otherwise it is not.

Checking if N(I) should be increased by two

If the set of sequences is not self-derivable,we can test if adding one new sequence makes

it self-derivable.the number of candidate sequences is

polynomial and for each one added to M we check self-derivability. N(I) should be increased by two if none of these sets of sequences is self-derivable.

Program HapBound

• HapBound –S. Checks each Opt(I) subset for self-derivability. Increase N(I) by 1 or 2 if possible. This often beats ORB and is still practical for most datasets.

• HapBound –M. Explicitly examine each interval directly for self-derivability. Increase local bound if possible. Derives lower bound of 7 for Kreitman’s ADH data in this mode.

HapBound vs. RecMin on LPL from Clark et al.

Program Lower Bound Time

RecMin (default) 59 3s

RecMin –s 25 –w 25 75 7944s

RecMin –s 48 –w 48 No result 5 days

HapBound 75 31s

HapBound -S 78 1643s

2 Ghz PC

Example where RecMin has difficulity in Finding Optimal Bound

on a 25 by 376 Data MatrixProgram Bound Time

RecMin default 36 1s

RecMin –s 30 –w 30 42 3m 25s

RecMin –s 35 –w 35 43 24m 2s

RecMin –s 40 –w 40 43 2h 9m 4s

RecMin –s 45 –w 45 43 10h 20m 59s

HapBound 44 2m 59s

HapBound -S 48 39m 30s

Frequency of HapBound –S Bound Sharper Than Optimal

RecMin Boundms param. Rho=1 Rho=5 Rho=10 Rho=20

Theta=1 0.0% 0.4% 0.5% 1.5%

Theta=5 0.7% 4.0% 10.4% 27.0%

Theta=10 1.4% 9.2% 17.8% 40.4%

Theta=20 1.4% 10.5% 27.8% 45.4%

For every ms parameters, 1000 data sets are used.

Computing Upper Bounds

• The method is an adaptation of the ``history” lower bound (Myers).

• A non-informative column is one with fewer than two 0’s or fewer than two 1’s.

Single History Computation

1) Set W = 02) Collapse identical rows together, and remove

non-informative columns. Repeat until neither is possible.

3) Let A be the data at this point. If A is empty, stop, else remove some row r from A, and set W = W + 1. Go to step 2).

Note that the choice of r is arbitrary in Step 3), so the resulting W can vary.

History Lower Bound

Theorem (Myers) Let W* be the minimum W obtained from all possible single history computations.

Then W* is a valid lower bound on the number of recombinations needed.

Naïve time: theta(n!) (RecMin), but can bereduced to theta(2^n) (Bafna, Bansal).

Converting the History Lower Bound to an Upper Bound

• Given a set of rows A and a single row r, define w(r | A - r) as the minimum number of recombinations needed to create r from A-r (well defined in our application).

• w(r | A-r) can be computed in linear time by a greedy-type algorithm.

Upper Bound Computation

1) Set W = 02) Collapse identical rows together, and remove non-

informative columns. Repeat until neither is possible.3) Let A be the data at this point. If A is empty, stop, else

remove some row r from A, and set W = W + W(r | A-r). Go to step 2).

Note that the choice of r is arbitrary in Step 3), so the resulting W can vary.

This is the Single History Computation, with a change instep 3).

Note, even a single execution of the upper bound computation gives a valid upper bound, and a way to construct a network. This is in contrast to the History Bound which requires finding the minimum W over all histories.

However, we also would like to find the lowest upper bound possible with this approach. This can be done in O(2^n) time by DP. In practice, we can use branch and bound to find the lowest upper bound, but we have also found that branching on the best local choice, or randomizing gives good results.

Branch and Bound

(Branching) In Step 3) choose r to minimize w(r | A-r) + L(A-r), where L(A-r) is some

fast lower bound on the number of recombinations needed for the set A-r. Even HK is good for this purpose.

(Bounding) Let C be the min for an full solution found so far; If W + L(A) >= C, then backtrack.

Kreitman’s 1983 ADH Data

• 11 sequences, 43 segregating sites

• Both HapBound and SHRUB took only a fraction of a second to analyze this data.

• Both produced 7 for the number of detected recombination events

Therefore, independently of all other methods, our lower and upper bound methods together imply that 7 is the minimum number of recombination events.

A minimal ARG for Kreitman’s data

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

SHRUB produces code that can be input to an open source program to display the constructed ARG

Papers and Software

wwwcsif.cs.ucdavis.edu/~gusfield/

A general result and open question

Recombination Cycles

• In a Phylogenetic Network, with a recombination node x, if we trace two paths backwards from x, then the paths will eventually meet.

• The cycle specified by those two paths is called a ``recombination cycle”.

Blobbed-trees: generalizing galled-trees

• In a phylogenetic network a maximal set of intersecting cycles is called a blob.

• Contracting each blob results in a directed, rooted tree, otherwise one of the “blobs” was not maximal.

• So every phylogenetic network can be viewed as a directed tree of blobs - a blobbed-tree.

The blobs are the non-tree-like parts of the network.

Ugly tanglednetwork insidethe blob.

Every network is a tree of blobs. How do the tree partsand the blobs relate?

How can we exploitthis relationship?

The Decomposition Theorem (Recomb 2005)

For any set of sequences M, there is a blobbed-tree T(M) that derives M, where each blob contains all and only the sites in one non-trivial connected component of the incompatibility graph. The compatible sites can always be put on edges outside of any blob. Moreover, the tree part of T(M) is unique. And it is easy to find the tree part.

This is weaker than the result for galled-trees: itreplaces “must” with “can”.

The main open question

• Is there always a blobbed-tree where each blob has all and only the sites of a single connected component of the conflict graph, which minimizes the number of recombinations over all possible phylogenetic networks for M?

• If so, then we can optimize on each component separately, and find lower bounds on each component separately and then add those bounds to get a valid lower bound for M.