1. Algorithmic Problems Related to Sequences and Phylogenetic Trees
Bhaskar DasGupta
Department of Computer Science
University of Illinois at Chicago
Chicago, IL
11/22/2018

2. Substructure Comparison Problems Sequences
Outline
Introduction
Substructure Comparison Problems
Sequences
Nonoverlapping local alignment
Proteins
Transformation Based Distances
Phylogenetic Trees
Why compare?
A few distances
Genomes
Syntenic Distance
Conclusions
11/22/2018

3. Computational Molecular Biology A Computer Scientist’s Participation
Get to know the computational problems
Talk to biologists
State the computational problems as precisely as possible
Investigate computational aspects of the problems
exact solutions
difficult/easy ?
time/space efficient solutions ?
approximate solutions (if exact solution is hard or not time/space efficient)
guaranteed quality of approximation ? (tradeoff with space/time?)
deterministic vs. randomized algorithms
implementation aspects
programming cleverness to reduce space/time
algorithmic engineering techniques to reduce space/time
interaction with the biologists
are the solutions biologically meaningful ?
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4. Few Computer Science Jargons
When we say What we really mean
Maximization/minimization problem Problem in which we maximize/minimize some objective function
Problem is NP-complete/hard Exact solution for large size problem will most likely require too much time
Polynomial-time solution Solvable in reasonable time in a reasonably fast computer
Approximation algorithm An approximate solution computed in reasonable time
with approximation ratio r with an objective function value of a (for maximization/minimization) least (at most r) of the optimum
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5. Substructure Similarity (or, equivalently, Dissimilarity)
a matches to a’ with similarity 10
b matches to b’ with similarity 15
c matches to c’ with similarity 11
total similarity 36
Goal: match disjoint substructures to maximize total similarity
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6. Many short vs. fewer long substructures
Few Complications
Many short vs. fewer long substructures
Measure of similarity between substructures
Examples:
rmsd (root-mean-square distance) between 3D substructures
edit distance between subsequences
syntenic distance between multi-chromosome genomes
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7. Non-overlapping local alignment
Sequences
Non-overlapping local alignment
total similarity 10+15=25
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8. The problem
Input: pairs of fragments, one from each sequence (or, equivalently a
set of rectangles).
the weight of each pair (rectangle) is their similarity measure
Output: a set of pairs (rectangles) such that
no two rectangles overlap on the x-axis
(i.e., matched fragments of the first sequence are disjoint)
no two rectangles overlap on the y-axis
(i.e., matched fragments of the 2nd sequence are disjoint)
total similarity of selected fragment pairs is maximized
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9. not allowed in the input data
Further assumption
We can preprocess input data (rectangles or fragment pairs) to ensure that
for any two rectangles, the projection of one on the y-axis does not enclose that of another
not allowed in the input data
for any two rectangles, the projection of one on the x-axis does not enclose that of another
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10. An optimal solution of total similarity 25
An illustration
Input: A G 15 G 2 C 1 C 10 T A A G C A C C
An optimal solution of total similarity 25
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11. (n = number of rectangles (fragment pairs))
Previous results
(n = number of rectangles (fragment pairs))
Bafna, Narayanan and Ravi (WADS’95)
NP-complete
O(n2) time approximation algorithm with approximation ratio 3.25
converts to a problem of finding maximum-weight independent set in a 5-clawfree graph
gives approximation algorithm for (d+1)-clawfree graphs with approximation ratio of
Halldórsson (SODA’95)
approximation algorithm with approximation ratio of about 2.5 when all weights are one
again uses clawfree graphs
Berman (SWAT’00)
O(n4) time algorithm with approximation ratio of about 2.5
via clawfree graphs again
11/22/2018

12. (Berman, DasGupta and Muthukrishnan, SODA’02)
Our recent results
(Berman, DasGupta and Muthukrishnan, SODA’02)
O(n log n) time approximation algorithm with approximation ratio 3
very simple to implement
uses a 2-phase approach (or, equivalently, the local-ratio technique)
Extensions to d dimensions (d > 2)
Inputs are similarity measures of d fragments, one from each of given d sequences
Motivation: multiple sequence comparison problems
Generalization of our above approach:
O(n d log n) time approximation algorithm with approximation ratio of 2d-1
current best (Bar-Yehuda, Halldórsson, Naor, Shachnai and Shapira, SODA’02):
polynomial time algorithm with approximation ratio 2d
uses repeated linear programming and continuous version of local-ratio techniques
11/22/2018

13. Common substructure between protein structures
(work in progress with Jie Liang and Andrew Binkowski)
Comparison of 2 4-helix bundles that differ by topological rearrangement, ROP and cytochrome b56
Topological cartoons of 1ROP and 256B. Helices are drawn as cylinders and loops as lines. Residue numbers of structurally equivalent segments are indicated on the cylinders.
The alignment is non-sequential.
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14. Few interesting points:
Motivation:
discovering similar substructures from different proteins is essential for recognizing remote evolutionary relationship at the level of protein fragments
Few interesting points:
it is not easy to characterize topological structures such as void, pocket, or tunnel where ligand and other molecules bind.
Current computational tools do not perform very well on discovering similar substructures.
For example:
(a) protein structures are typically represented by distance matrices or contact maps, which record pairwise inter-distances between selected atoms (typically Cα atoms) on the primary sequences
(b) finding common substructures becomes matching submatrices of the two contact maps
(c) Heuristic algorithms have been developed and have proven to be useful. But, they are time consuming (typically O(n6)), and cannot be used for more demanding tasks such as identifying spatial functional motifs
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15. Our approach in work in progress
reduce the problem to various constrained rectangle-packing problems
use combinatorial methods (such as the local-ratio technique) to design approximation algorithms for these problems
Our final goal
identification of the most discriminating geometric and chemical features and their combinations for various proteins
development of a robust method to compute the similarity/dissimilarity of two shape distributions of these features
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16. Transformation based distances
Objects
Transformation rules (with costs) 10 15 12 9
Goal: find distance between two specified objects 10 15 9
cost = = 34 10 12
cost = = 22
distance between
and
is 22
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17. Distances between Phylogenetic trees
Objects:
Evolutionary trees (phylogenies) on n nodes
Transformation Rules:
How to modify trees locally consistent with biological applications?
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18. inferring phylogenies
Why compute distances between phylogenies ?
First motivation
parsimony method
compatibility
method
compare
them
for
similarity
and
discrepancy
input data
maximum-likelihood
method
distance matrix
method
different methods for
inferring phylogenies
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19. Why compute distances between phylogenies ?
Second motivation
To find out information about rare genetic events such as recombination or gene conversion
recombination
gene
conversion
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20. Few distances that we have looked at......
Nearest neighbor interchange (nni) distance
Linear cost subtree transfer distances
Synopsis of our works on these distances
proving that exact solution is NP-hard
providing fast approximate solutions
investigating fixed-parameter tractability
some implementation works .....
11/22/2018

21. order of genes in any chromosome is unknown or ignored
Genomic Distance
Syntenic distance between multi-chromosome genomes
(Ferretti, Nadeau and Sankoff, 1996)
treats genomes at a higher level of abstraction
chromosome
gene 4 9 3 6 10 8 1 5 7 2
order of genes in any chromosome is unknown or ignored
intra-chromosomal events (e.g., reversal, transposition) do not affect chromosomal assignment
inter-chromosomal events are important
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22. Inter-chromosomal events Fission Fusion1 2 3 5 2 1 3 5 4 4 1 4 2 3 5 1 4 2 3 5
(Reciprocal) translocation 5 1 2 3 4 6 7
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23. Syntenic distance between two genomes
minimum number of fission, fusion and translocations necessary to transform one genome to another
Other related problems
finding the median of 3 genomes for the syntenic distance metric
(useful for phylogentic tree inference problem from synteny data)
Synopsis of our work on these problems
showing NP-hardness of exact computation
giving efficient approximation algorithms
exhibiting fixed-parameter tractability
11/22/2018

24. Genome partitioning with applications to DNA microarray chip design
Other problems......
Genome partitioning with applications to DNA microarray chip design
Consensus sequence reconstruction problems
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25. THE END
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