Presentazione Oratore

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Presentazione Oratore

• Romeo.Rizzi_at_dimi.uniud.it
• cel phone [phone]
• appassionato di Ottimizzazione Combinatorica,
  algoritmi,
• operante in Biologia Computazionale
• Ricercatore Operativo ed Informatico

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More reliable protein NMR peak assignment via
improved 2-interval scheduling
Zhi-Zhong Chen Department of Mathematical
Sciences Tokyo Denki University Joint work with
T. Jiang, G.-H. Lin, R. Rizzi, J. Wen, D. Xu,
and Y. Xu.
http//rnc.r.dendai.ac.jp/chen/papers/pnmr.pdf
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Agenda

• Introduction
• Problem Definition
• Problem Formulation
• Old and new results
• Algorithms the greedy filtering technique
• Algorithms an approximation algorithm
• Open problems
• Biography

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Agenda

• Introduction
• Problem Definition
• Problem Formulation
• Old and new results
• Algorithms the greedy filtering technique
• Algorithms an approximation algorithm
• Open problems
• Biography

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Introduction

• Nuclear Magnetic Resonance (NMR)

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Introduction

• Nuclear Magnetic Resonance (NMR)
• Use the strong magnetic wave to align nuclei
  (isotopes).
• When this spin transition occurs, the nuclei are
  said to be in resonance with the applied
  radiation.

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NMR measurement

• Chemical Shift
• ppm
• Electrons in the molecule have small magnetic
  fields
• When applied string magnetic field, electrons
  tend to oppose the applied field.
• NMR Spectrum

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NMR spectroscopy

• Study the physical, chemical, and biological
  properties.
• Problem
• Identified sequence.
• Unknown (complete) structure.
• Known basic structure.
• Unknown the structure corresponding to AAs.

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Procedure to determine protein structure using NMR

• Three steps
• Data generation
• Involves corresponding resonance peaks to AA and
  forming spin system.
• Data interpretation
• Involves matching spin system to amino acids
  providing inter and intra AA distance and angle.
• NMR Structure calculation
• Involves structure determination using Molecular
  Dynamics (MD) Energy Minimization (EM).

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Agenda

• Introduction
• Problem Definition
• Problem Formulation
• Old and new results
• Algorithms the greedy filtering technique
• Algorithms an approximation algorithm
• Open problems
• Biography

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A Problem of Data Interpretation

• Steps for doing this data interpretation
• Map resonance peaks from different NMR spectra to
  same residue.
• Identify adjacency relationship.
• Assign the segments to the protein sequence.

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Peak Assignment

• Assignment procedure need to address two crucial
  information
• Different AA types have different distribution of
  spin system.
• The adjacency info, scalar coupling, between spin
  systems are obtained by identifying their common
  resonance frequencies.

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The NMR spectral peak assignment Problem
FID signal
1-D NMR spectral
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The NMR spectral peak assignment Problem
2-D spectral
1-D spectral
An isolated spin system
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The NMR spectral peak assignment Problem
correct assignment
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The NMR spectral peak assignment Problem
A protein a sequence of amino acids
Given
ACVSANDLLEDAVNFGAWEKLSDNWWOQSWERDDSFDMSDFAQLL
Segments of spin systems generated from the
protein
Output A maximum-likelihood assignment of
the segments to contiguous amino acids.
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Agenda

• Introduction
• Problem Definition
• Problem formulation
• Old and new results
• Algorithms the greedy filtering technique
• Algorithms an approximation algorithm
• Open problems
• Biography

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The interval scheduling problem
Given
A single machine M,
?job j ?time unit u p( j, u) the profit of
starting
executing Job j at time unit u
Output A scheduling that maximizes the total
profit of executed jobs.
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The k-interval scheduling problem
The special case of the interval scheduling
problem where each job requires at most k time
units.
The 1-interval scheduling problem is exactly the
maximum weight bipartite matching problem.
Chen et al., 2002 ?k?2 The k-interval
scheduling problem is APX-hard, even if the
processing time of each job is 1 or 2 and the
profit of executing a job is proportional to the
processing time of the job.
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Relationship between the two types of problems
The interval scheduling problem

The NMR spectral peak assignment problem
?k
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Agenda

• Introduction
• Problem Definition
• Problem Formulation
• Old and new results
• Algorithms the greedy filtering technique
• Algorithms an approximation algorithm
• Open problems
• Biography

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Best known approximation algorithms
Bar-Noy et al., 2001 The interval scheduling
problem has a poly-time 2-approximation
algorithm.
Open question Is there a poly-time
r-approximation algorithm for the interval
scheduling problem with r lt 2 ?
Chen et al., 2002 There is a 1.7778-approximatio
n algorithm for the special case of the
2-interval scheduling problem where profits are
proportional to length of jobs.
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New results

• A poly-time (13 / 7)-approximation algorithm for
  
• the 2-interval scheduling problem.

• A new efficient heuristic (namely,
• Greedy filtering the above algorithm)
• for the NMR spectral peak assignment problem
• that outperforms the previous best.

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Agenda

• Introduction
• Problem Definition
• Problem Formulation
• Old and new results
• Algorithms the greedy filtering technique
• Algorithms an approximation algorithm
• Open problems
• Biography

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Greedy filtering
ACVSANDLLEDAVNFGAWEKLSDNWWOQSWERDDSFDMSDFAQLLKCVY
TRAKSLDLSKGTE




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Agenda

• Introduction
• Problem Definition
• Problem Formulation
• Old and new results
• Algorithms the greedy filtering technique
• Algorithms an approximation algorithm
• Open problems
• Biography

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The (13 / 7)approximationalgorithm

• Very complicated.

• It consists of four algorithms and
• outputs the best scheduling returned by them.

• The main tool in the design
• maximum weight bipartite matching
• careful manipulation of the input.

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The idea behind Algorithm 1
The given time interval
Partition into basic intervals
Partition 1
a basic interval (3 time units)
only the first and the last may have lt3 time
units
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Algorithm 1constrained scheduling
The given time interval
The given time interval
Partition into basic intervals
Partition 1
The constrained interval scheduling problem for
Partition 1 ? Execute at most one job within
each basic interval. ? Each executed job must
finish within one basic interval. ? Maximize the
total profit of executed jobs.
This constrained problem can be reduced to the
maximum weight bipartite matching problem!
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Algorithm forconstrained scheduling
The given time interval
The given time interval
Partition into basic intervals
Partition 1
Step 1. Construct an edge-weighted
bipartite graph.
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Algorithm forconstrained scheduling
The given time interval
The given time interval
Partition into basic intervals
Partition 1
The given set of jobs
Purple profit 1 Blue profit 2 Red
profit 6 Green profit 3
Step 2. Remove invalid edges.
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Algorithm forconstrained scheduling
The given time interval
The given time interval
Partition into basic intervals
Partition 1
The given set of jobs
Purple profit 1 Blue profit 2 Red
profit 6 Green profit 3
Step 3. Compute a maximum-weight
bipartite matching.
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The idea behind Algorithm 1
The given time interval
The given time interval
Partition into basic intervals
Partitions 1, 2, 3
Three constrained interval scheduling problems
for them.
Obtain three constrained schedulings in poly-time.
The best among the three constrained schedulings
is good!
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The idea behind Algorithm 1
The given time interval
The given time interval
Partition into basic intervals
Partitions 1, 2, 3
The best among the three constrained schedulings
achieves a total profit of at least p1(S) / 3
2p2(S) / 3, where S is a maximum-profit
scheduling, p1(S) (resp., p2(S)) is the total
profit of executed 1-jobs (resp., 2-jobs).
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The idea behind Algorithm 1
Scheduled by S
The given time interval
The given time interval
Job i
Job j
Job k
Partition into basic intervals
Inheriting from S
Partitions 1, 2, 3
Algorithm 1 achieves a better-than-2 ratio if
p2(S) is relatively large compared to p1(S).
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The main idea behind Algorithms 2 3
The given time interval
Job 4
Job 1
Job 2
Job 3
Job 5
The given set of jobs
? Obtain a maximum-profit preemptive scheduling
(i.e., each 2-job may be split into two
1-jobs) in poly-time.
? Then use the preemptive scheduling to obtain
three nonpreemptive schedulings in a novel way.
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Computing an optimalpreemptive scheduling
The given time interval
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Computing an optimalpreemptive scheduling
The given time interval
The given set of jobs
Purple profit 1 Blue profit 2 Red
profit 6 Green profit 3
Step 2. Compute a maximum-weight
matching Mun in the graph.
Blue profit 1 Red profit 3
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Using Mun to classifythe given time units
The given time interval
The given set of jobs
Purple profit 1 Blue profit 2 Red
profit 6 Green profit 3
Blue profit 1 Red profit 3
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Sketch ofAlgorithm 2
The given time interval
The given set of jobs
Purple profit 1 Blue profit 2 Red
profit 6 Green profit 3
Blue profit 1 Red profit 3
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Sketch ofAlgorithm 2
The given time interval
Try to catch those assignments of 1-jobs in S
to isolated or border time units.
The given set of jobs
Purple profit 1 Blue profit 2 Red
profit 6 Green profit 3
For each block , isolate .
Use the idea of Algorithm 1 to compute an
interval scheduling.
Blue profit 1 Red profit 3
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Sketch ofAlgorithm 3
The given time interval
The given set of jobs
Purple profit 1 Blue profit 2 Red
profit 6 Green profit 3
Blue profit 1 Red profit 3
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Sketch ofAlgorithm 3
The given time interval
The given set of jobs
Purple profit 1 Blue profit 2 Red
profit 6 Green profit 3
Partition the given time interval by grouping
each block.
Blue profit 1 Red profit 3
Compute a constrained scheduling s.t. each block
hosts only one job.
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CombiningAlgorithms 2 and 3
The given time interval
Try to catch those assignments of 1-jobs in S
to isolated or border time units.
The given time interval
Try to catch those assignments of 1-jobs in S
to isolated or internal time units.
Algorithm 2 or 3 achieves a better-than-2 ratio,
if the 1-jobs assigned to the isolated time
units by S have a relatively large total profit
compared to p(S).
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Sketch of Algorithm 4
The given time interval
The given set of jobs
Union of odd sets gives a scheduling of some
jobs, and union of even sets gives a scheduling
of the rest.
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CombiningAlgorithms 1 and 4
The given time interval
Algorithm 1 or 4 achieves a better-than-2 ratio,
if the 1-jobs assigned to the isolated time
units by S have a relatively small total profit
compared to p(S).
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Agenda

• Introduction
• Problem Definition
• Problem Formulation
• Old and new results
• Algorithms the greedy filtering technique
• Algorithms an approximation algorithm
• Open problems
• Biography

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Open problems
Is there a poly-time r-approximation
algorithm for the interval scheduling problem
with r lt 2 ?
Better fast heuristics for NMR spectral peak
assignment ?
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Agenda

• Introduction
• Problem Definition
• Problem Formulation
• Old and new results
• Algorithms the greedy filtering technique
• Algorithms an approximation algorithm
• Open problems
• Biography

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References

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