Large Scale Combinatorial Optimization Problems in Bioinformatics

1
Large Scale Combinatorial Optimization Problems
in Bioinformatics

• Nathan Edwards
• Center for Bioinformatics and Computational
  Biology
• University of Maryland, College Park

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Sample Preparation for Peptide Identification
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Single Stage MS
MS
m/z
4
Tandem Mass Spectrometry(MS/MS)
m/z
Precursor selection
m/z
5
Tandem Mass Spectrometry(MS/MS)
Precursor selection collision induced
dissociation (CID)
m/z
MS/MS
m/z
6
Novel Splice Isoform

• Human Jurkat leukemia cell-line
• Lipid-raft extraction protocol, targeting T cells
• von Haller, et al. MCP 2003.
• LIME1 gene
• LCK interacting transmembrane adaptor 1
• LCK gene
• Leukocyte-specific protein tyrosine kinase
• Proto-oncogene
• Chromosomal aberration involving LCK in
  leukemias.
• Multiple significant peptide identifications

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Novel Splice Isoform
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Novel Splice Isoform
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Polymerase Chain Reaction
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PCR
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Applications of k-mer sets

• Peptide Identification
• Set of all human amino-acid 30-mer peptide
  sequences...
• ...that occur at least twice in dbEST
• PCR Primer Design
• Unique 20-mers
• What does it mean to be unique?

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k-mer superstrings

• Completeness
• All of the required k-mers are represented
• Correctness
• No additional k-mers are represented
• Minimize the total representation length
• Correlates with running time

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Shortest superstring problem

• General strings (arbitrary length)
• Completeness only!
• Classical NP-hard problem
• Garey and Johnson
• Approximate within 2.5OPT
• Max-SNP hard
• One of the first algorithmic approaches to genome
  assembly

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de Bruijn Sequences

• de Bruijn sequences represent all words of length
  k from some alphabet A.
• A 0,1, k 3 s 0001110100
• A 0,1, k 4 s 0000111101011001000

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de Bruijn Graph A 0,1, k 4
1
1
0
1
0
1
1
0
1
0
1
0
1
0
0
0
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de Bruijn Sequences Graphs

• de Bruijn graphs (k,A)
• Edges represent length k words from A
• Each node has
• in degree A
• out degree A
• Eulerian tour constructs de Bruijn sequence.

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SBH-graph
ACDEFGI, ACDEFACG, DEFGEFGI
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Compressed SBH-graph
ACDEFGI, ACDEFACG, DEFGEFGI
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Sequence Databases CSBH-graphs

• Original sequences correspond to paths

ACDEFGI, ACDEFACG, DEFGEFGI
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Sequence Databases CSBH-graphs

• All k-mers represented by an edge have the same
  count

1
2
2
1
2
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Correct, Complete Enumeration

• Set of paths that use each edge at least once

ACDEFGEFGI, DEFACG
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Related work

• Chinese Postman Problem
• Edmonds and Johnson, 73
• Undirected graph, weighted edges
• Shortest path that uses all the edges
• Solvable in polynomial time
• Construct minimum weighted matching between nodes
  of odd-degree
• Add matching to graph and find Eulerian path
• Minimize weight of extra edges used

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Correct, Complete, Enumeration

• Chinese postman problem, except
• Directed graph
• Add edges from nodes with surplus in-degree to
  nodes with surplus out-degree
• Fixed cost teleportation option
• Can always start a new sequence
• Find optimal set of additional edges
• Transportation problem / min cost flow instance

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Patching the CSBH-graph

• Use artificial edges to fix unbalanced nodes

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C3 Enumeration
in-out
in-out
Cost k
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C3 Enumeration
in-out
in-out
Cost 0
Cost 0
Cost k
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C2 Enumeration
in-out
in-out
4
10
Shortcut paths
7
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C2 Superstring
in-out
in-out
Cost 0
Cost 0
Cap 1
Cost 0
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Large scale instances

• Millions of nodes and edges
• cSBH-graph instance
• Min cost flow instance
• cSBH-graph instance takes days to construct
• 2 days on 250 CPUs
• Algorithms must be linear in problem size
• Out-of-core Eulerian path algorithm?

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Grid computing

• Heterogeneous machines
• Varying disk/memory/MHz/cores capabilities
• Centralized scheduler
• Jobs started asynchronously
• Other jobs may preempt current job
• Input files may need to be staged
• 250 simultaneous requests for a 3Gb file?
• How to guarantee integrity of input files?
• Problem decomposition may be non-trivial
• Jobs sizes need to fit the least capable machine
• Sometimes need to game the scheduler
• Need to ensure the integrity of job output

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Uniqueness Oracles

• Oracle for uniqueness of 20-mers in the Human
  genome (n3Gb)
• Count occurrences in the genome 0,1,2
• Construct 20-mer superstring for 20-mers with
  count 1
• Construct 20-mer superstring for 20-mers with
  count gt 1
• Easy for exact sequence match O(n)
• Fast automata, indexing, hash tables.

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Inexact sequence match

• Inexact sequence matching O(nmk)
• Errors/Mismatches (k) 1,2,3
• distinct 20-mers (m) O(n)
• Achieve expected linear time using a hybrid
  approach
• Exact search for short chunks of queries
• Expensive alignment only where chunks match
• Large chunks ) Fast, but miss occurrences
• Small chunks ) Slow, find all matches

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Inexact sequence match

• Baeza-Yates Perleberg
• Correct and O(n) for small k
• At least 1 chunk is observed with no error.
• Form of locality sensitive hashing

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Locality Sensitive Hashing

• For each query
• store a (set of) hash(es) in hash-table
• At each position in the genome
• look-up a (set of) hash(es) in hash-table
• if any are in the table, do more expensive check
• Need to weigh
• sensitivity (false negatives) against
• specificity (false positives)
• Our application requires no false negatives

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Random Projection

• Choose T templates of l random care positions

g
t1
t1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 0
0
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Random Projection

• Choose T templates of l random positions

g
t1
t2
t1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 0
0 t2 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0
0 1
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Random Projection

• Choose T templates of l random positions

g
t1
t2
t1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 0
0 t2 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0
0 1
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Random Projection

• Choose T templates of l random positions

g
t1
t2
t1 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 0
0 t2 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0
0 1
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Gapped seed-set design problem

• Given
• mer-size m ( 20 )
• errors k ( 1,2,3)
• cares l ( 10,12,14 )
• Find the smallest set of templates with no false
  negatives.
• Minimize running time.

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Alternative Formulation 1(for k 2)

• Cover the edges of Km with copies of Km-l
• How many triangles to cover K6?
• Some instances of (m,2,m-3) cover each edge
  exactly once
• Steiner triple systems

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Alternative Formulation 2

• Set cover instance
• Ground set all possible placements of the k
  errors (alignments)
• Covering sets all possible placements of the l
  care positions
• For (m20,k2,l10),
• 190 elements, 184,756 sets!
• Greedy approximation algorithm works

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Alternative Formulation 3
Templates
Positions (m)
Remove any kposition nodes, 1 templatehas
degree l.
l
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Alternative formulation 3

• Template t has care at position i xti
• Alignment a is not covered by template t yta
• Alignment a is not covered by any template za

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Alternative Formulation 3

• Polynomial size in terms of number of templates
• Select T in advance and test whether T is
  sufficient.
• Greedily add T templates.
• Apply iteratively to achieve feasible solution
• Extremely weak LP relaxation
• Lots of symmetry!
• Hard to solve useful instances

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Solution for (20,2,10)

• .................... Positions
• t1
• t2
• t3
• t4
• t5
• t6

• Need at least 4 templates, 6 is optimal

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Remember the application!

• We are checking some templates twice!
• We compute the hash(es) at each position in the
  genome
• Any template that is a shift of another will be
  computed at some nearby genomic position!

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Solution for (20,2,10)

• .................... Positions
• t1
• t2
• t3
• t4
• t5
• t6

• Need at most 3 templates...can we do better?

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Alternative formulation 3 with template shift
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Solution for (20,2,10) w/ shift

• .................... Positions
• t1
• t2

• Optimal is 2 templates...

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Alternative Formulation 3 with shift

• Polynomial size in terms of number of templates
• Select T in advance and test whether T is
  sufficient.
• Greedily add T templates.
• Apply iteratively to achieve feasible solution
• Greedy not known to be good!
• Extremely weak LP relaxation
• Much less symmetry
• Hard to solve useful instances

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Conclusions

• Lots of interesting optimization problems in
  bioinformatics
• Usually very large scale
• Need good empirical algorithms/solvers
• Modeling tradeoffs abound
• Speed/Memory/Optimality/Correctness
• Many variants of LSH in different domains
• Resource constrained allocation problems
• ...due to limitations in biotechnologies
• As the technologies scale up, so do the issues