-10 2012 XVIELAVIO ,ValedosVinhedos,RS,BrazilFebruary5 CarlileLavor ,AntonioMucherino,LeoLiberti,NelsonMaculan TheDiscretizableMolecularDistanceGeometryProblem

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Discretizable Molecular Distance Geometry Problem

Carlile Lavor, Antonio Mucherino, Leo Liberti, Nelson Maculan

clavor@ime.unicamp.br

XVI ELAVIO, Vale dos Vinhedos, RS, Brazil

February 5 ^th - 10 ^th 2012

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Protein conformations

Proteins play many vital functions in the bodies of living beings.

They are chains of amino acids. They fold into unique

three-dimensional conformations, which define their functions.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Protein Data Bank (PDB)

It’s a web database of protein conformations ( ˜76000 ).

Two techniques are mainly employed:

1 X-ray diffraction

provides (in general) conformations with a good resolution;

requires the crystallization of the molecule;

2 Nuclear Magnetic Resonance (NMR)

analyzes the molecule in solution (crystallization not required);

only provides information from which the conformation can be obtained.

What does NMR provide?

Distances between some pairs of atoms of the molecule.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Protein Data Bank (PDB)

It’s a web database of protein conformations ( ˜76000 ).

Two techniques are mainly employed:

1 X-ray diffraction

provides (in general) conformations with a good resolution;

requires the crystallization of the molecule;

2 Nuclear Magnetic Resonance (NMR)

analyzes the molecule in solution (crystallization not required);

only provides information from which the conformation can be obtained.

What does NMR provide?

Distances between some pairs of atoms of the molecule.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Molecular Distance Geometry Problem

Let G = (V , E, d) be a weighted undirected graph, where V the set of vertices of G − corresponds to the set of atoms;

E the set of edges of G − corresponds to the set of known distances;

d the weights associated to the edges of G

the numerical value of each weight corresponds to the known distance.

Definition

Find a conformation x = {x 1 , x ₂ , . . . , x _n } such that all the following constraints are satisfied:

||x i − x _j || = d _ij ∀i , j : i 6= j,

where ||x i − x _j || is the computed distance between x _i and x _j , and

d _ij is the generic weight of the graph G.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

PDB and MDGP

The majority of protein conformations on the PDB related to NMR experiments are obtained by solving the corresponding MDGP by the meta-heuristic Simulated Annealing (SA).

Disadvantages of this approach:

SA is one of the less performing meta-heuristic searches;

there is no guarantee to converge to the optimal solution(s);

even though multi-start techniques are used, there is no

hope to identify all optimal solutions.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Discretizable MDGP (DMDGP)

Instances satisfying the following two assumptions can be discretized.

Given a graph G = (V, E, d ) and a total ordering on V , Ass.1 (1, 2, 3) ⊂ V must be a clique and

for each i ∈ V such that i > 3,

{(i − 3, i ), (i − 2, i), (i − 1, i)} ⊂ E ;

Ass.2 for each i ∈ V such that i > 2, the strict triangular inequality d _{i −2,i} < d _i−2,i−1 + d _i−1,i

must hold.

C. Lavor, L. Liberti, N. Maculan, A. Mucherino, The Discretizable Molecular Distance Geometry Problem,

Computational Optimization and Applications, Online First Articles, March 2011.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Discretizing with torsion angles

If the two assumptions hold, the cosine of the torsion angle among each quadruplet of consecutive atoms can be computed.

i − 3

i − 2

i − 1

i

i

θ

θ

d

d

An atomic position can be computed for each of the two values of

the corresponding torsion angle.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Discretizing with three spheres

If the assumptions hold, the two possible positions for the atom x _i can be computed by intersecting three spheres:

S ₋₁ : center: x _i−1 , radius: d _i−1,i ; S ₋₂ : center: x _i−2 , radius: d _i−2,i ; S ₋₃ : center: x _i−3 , radius: d _i−3,i .

The intersection of the spheres S _{− 1} , S _{− 2} and S _{− 3} can be

one point, two points, a circle, empty.

The intersection can be computed by solving two linear systems.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Discretizing with three spheres

If the assumptions hold, the two possible positions for the atom x _i can be computed by intersecting three spheres:

S ₋₁ : center: x _i−1 , radius: d _i−1,i ; S ₋₂ : center: x _i−2 , radius: d _i−2,i ; S ₋₃ : center: x _i−3 , radius: d _i−3,i .

The intersection of the spheres S _{− 1} , S _{− 2} and S _{− 3} can be

one point (probability 0), two points,

a circle (impossible, because of the strict triangular inequalities), empty (impossible, because of protein steric constraints).

The intersection can be computed by solving two linear systems.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Discretizable MDGP (DMDGP)

When these two assumptions are satisfied, the domain of the penalty function can be reduced to a discrete set.

In particular, a binary tree of atomic positions can be defined, and

solutions to the problem can be found by exploring this tree.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Complexity

Definition

S UBSET -S UM . Given nonnegative integers a 1 , . . . , a n , is there a partition into two sets, encoded by s ∈ {−1, +1} ⁿ , such that each subset has the same sum, i.e. P n

i=1 s(i )a i = 0?

By reduction from the Subset-sum problem (which is known to be NP-hard), we can prove the following:

Theorem

The DMDGP is NP-hard.

C. Lavor, L. Liberti, N. Maculan, A. Mucherino, The Discretizable Molecular Distance Geometry Problem,

Computational Optimization and Applications, Online First Articles, March 2011.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Branch & Prune (BP) algorithm

Branches of the binary tree are pruned as soon as at least one of

the known distances is violated. In this way, an exhaustive search

on the remaining branches is not so expensive.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Branch & Prune (BP) algorithm

Branches of the binary tree are pruned as soon as at least one of

the known distances is violated. In this way, an exhaustive search

on the remaining branches is not so expensive.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Branch & Prune (BP) algorithm

Branches of the binary tree are pruned as soon as at least one of

the known distances is violated. In this way, an exhaustive search

on the remaining branches is not so expensive.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Branch & Prune (BP) algorithm

Branches of the binary tree are pruned as soon as at least one of

the known distances is violated. In this way, an exhaustive search

on the remaining branches is not so expensive.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Branch & Prune (BP) algorithm

Branches of the binary tree are pruned as soon as at least one of

the known distances is violated. In this way, an exhaustive search

on the remaining branches is not so expensive.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Branch & Prune (BP) algorithm

Branches of the binary tree are pruned as soon as at least one of

the known distances is violated. In this way, an exhaustive search

on the remaining branches is not so expensive.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The Branch & Prune (BP) algorithm

Branches of the binary tree are pruned as soon as at least one of

the known distances is violated. In this way, an exhaustive search

on the remaining branches is not so expensive.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

MD-jeep

MD-jeep is an implemention of the BP algorithm in C programming language.

Legend of the following Tables:

Name - instance name;

n - instance dimension;

|E| - number of known distances;

#Sol - number of found solutions;

CPU - CPU time;

LDE - Largest Distance Error:

LDE = 1

|E|

X

(i,j)∈E

| ||x i − x _j || − d _ij |

d _ij

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Generation of the instances

We downloaded a subset of proteins from the PDB, and we generated a subset of instances for the DMDGP.

Only backbone atoms N–C α –C are considered, and only distances shorter than 6Å are included in the instance.

Most papers written by the operation research community consider this kind of instances.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Generation of the instances

We downloaded a subset of proteins from the PDB, and we generated a subset of instances for the DMDGP.

Only backbone atoms N–C α –C are considered, and only distances shorter than 6Å are included in the instance.

Most papers written by the operation research community consider this kind of instances.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Other software for distance geometry

DGSOL

by Jorge Moré, Argonne National Laboratory

It is based on the idea of approximating the objective function with a sequence of smoother functions converging to the original objective function.

Class of problems: DGP with interval data.

SDP-based facial reduction method by Nathan Krislock, INRIA, Nice

It is based on the idea of reducing the search space to subsets of faces, where a subproblem is defined and solved independently from the others.

Class of problems: DGP with exact data and anchors.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Some computational experiments

Instance BP-One BP-All SDP-based DGSOL

Name n |E| CPU LDE CPU #Sol CPU LDE CPU LDE

1brv 57 476 0.00 1.54e-14 0.00 1 0.03 1.24e-14 1.48 2.74e-01

1aqr 120 929 0.00 1.86e-09 0.00 2 0.06 2.54e-13 7.77 4.88e-01

2erl 120 1136 0.00 1.33e-14 0.00 1 0.06 2.52e-13 9.38 2.92e-01

1crn 138 1250 0.00 2.24e-13 0.00 1 0.06 2.24e-14 9.47 2.24e-01

1ahl 147 1205 0.00 1.50e-09 0.00 8 0.07 2.41e-14 6.95 1.46e-01

1ptq 150 1263 0.00 2.30e-13 0.00 1 0.08 2.54e-14 9.16 1.21e-01

1brz 159 1394 0.00 3.53e-13 0.00 2 0.07 2.01e-13 11.39 4.66e-01

1hoe 222 1995 0.00 3.18e-13 0.00 1 0.12 1.31e-13 16.83 2.06e-01

1lfb 232 2137 0.00 5.31e-14 0.00 1 0.11 1.86e-14 38.94 2.88e-01

1pht 249 2283 0.00 2.73e-12 0.00 1 0.10 9.54e-14 42.50 2.00e-01

1jk2 270 2574 0.00 2.09e-13 0.00 1 0.15 2.74e-14 86.98 4.05e-01

1f39a 303 2660 0.00 2.68e-12 0.00 1 0.12 3.91e-13 37.24 2.80e-01

1acz 324 3060 0.00 3.15e-12 0.02 4 0.13 3.04e-13 35.97 3.97e-01

1poa 354 3193 0.00 1.36e-13 0.00 1 0.20 2.53e-12 64.03 4.67e-01

1fs3 378 3443 0.00 8.08e-13 0.01 1 0.17 2.27e-13 54.68 2.69e-01

1mbn 459 4599 0.00 1.36e-09 0.00 1 0.22 9.67e-14 124.24 4.46e-01

1rgs 792 7626 0.00 4.22e-13 0.01 1 0.42 1.58e-13 237.93 4.69e-01

1m40 1224 20382 0.02 1.00e-12 5.26 1 0.71 1.08e-12 1142.49 4.89e-01

1bpm 1443 14292 0.02 2.85e-13 0.02 1 0.76 7.73e-13 398.29 5.06e-01

1n4w 1610 16940 0.02 1.19e-12 0.02 1 0.86 5.44e-13 994.51 5.26e-01

1mqq 2032 19564 0.02 4.90e-12 0.06 1 1.22 6.17e-13 451.58 5.40e-01

1rwh 2265 21666 0.02 2.08e-13 0.06 1 1.38 3.01e-12 934.29 5.38e-01

3b34 2790 29188 0.07 1.17e-11 0.07 1 1.68 3.00e-13 940.95 6.47e-01

2e7z 2907 42098 0.08 4.26e-12 0.09 1 1.88 2.88e-13 915.39 6.40e-01

1epw 3861 35028 0.16 3.19e-12 0.25 1 2.31 1.45e-12 2037.86 4.92e-01

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Visualizing the results

Solutions can be printed in text files in PDB format.

Visualization software (such RasMol) are able to read files in this format.

Two views of one of the found solutions for the instance 1mbn .

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Comparison to DGSOL

Comparison between two solutions, one found by BP, one found by DGSOL.

BP: LDE = 1.36e-09, RMSD = 2.87e-07 DGSOL: LDE = 4.46e-01, RMSD = 1.18e+01

RMSD: Root Mean Square Deviation.

One of the solutions found by the BP algorithm is actually the conformation

the protein 1mbn has.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

An artificial backbone of hydrogens

Given the set of hydrogens related to the protein backbone, how can we sort them so that the necessary assumptions are satisfied?

The order is specified through the labels on the edges.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Reconstructing the real backbone



 



 



||a − b 1 || = d a,b

||a − b 2 || = d a,b

||a − b 3 || = d a,b

||a − b 4 || = d a,b

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Reconstructing the real backbone



 



 



||a − b ₁ || = d _a,b

||a − b ₂ || = d _a,b

||a − b ₃ || = d _a,b

||a − b ₄ || = d _a,b

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Reconstructing the real backbone



 



 



||a − b ₁ || = d _a,b

||a − b ₂ || = d _a,b

||a − b ₃ || = d _a,b

||a − b ₄ || = d _a,b

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Reconstructing the real backbone



 



 



||a − b ₁ || = d _a,b

||a − b ₂ || = d _a,b

||a − b ₃ || = d _a,b

||a − b ₄ || = d _a,b

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

A particular ordering for protein backbones

The ordering is defined by the arrows in red.

If this special ordering is used, we are able to discretize real instances containing data from NMR.

C. Lavor, L. Liberti, A. Mucherino, The interval Branch-and-Prune Algorithm for the Discretizable Molecular

Distance Geometry Problem with Inexact Distances, to appear in Journal of Global Optimization, 2011.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

A particular ordering for protein backbones

How to discretize real instances?

All distances required for the discretization (Ass.1) are obtained from the chemical structure of the protein backbones (so they are independent from the instance);

Distances between pairs (i, i + 1) and (i , i + 2) are always exact;

Distances between pairs (i, i + 3) may be represented by intervals:

1 d _i,i+3 is 0: it means that this is a duplicated atom, there is no

branching because it can only take the same position of its previous copy;

2 d _i,i+3 is exact: the standard discretization process is applied, and

hence two possible positions for the current atom are computed;

3 d _i,i+3 is represented by an interval: D sample distances are taken

from the interval and the discretization process is applied for any

chosen sample distance; 2 × D atomic positions are generated.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

One branch of the new tree

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

interval Branch & Prune

0: IBP(i, n, d , nbranches) if (x

is a duplicated atom) then

assign to x

the same coordinates of its previous copy;

IBP(i + 1,n,d ,nbranches);

else

if (d(i − 3, i) is exact) then b = 2;

else

b = nbranches;

end if for (k = 1, b) do

compute the k

atomic position for the i

atom: x

; check the feasibility of the atomic position x

: if (x

is feasible) then

if (i = n) then a solution is found;

else

IBP(i + 1,n,d ,nbranches);

end if else

the current branch is pruned;

end if

end for

end if

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Instance with 3 amino acids

We firstly consider this simple molecule.

Bond lengths and bond angles are constant, torsion angles can

vary into a predefined interval.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Preliminary experiments

layer atom duplicated? branches w/out pruning branches with pruning atom

1 N 1 no 1 1 N 1

2 H 1 no 1 1 H 1

3 H 1 no 1 1 H 1

4 CA 1 no 2 2 CA 1

5 N 1 yes 2 2 N 1

6 HA 1 no 24 18 HA 1

7 CA 1 yes 24 18 CA 1

8 C 1 no 48 36 C 1

9 N 2 no 576 360 N 2

10 CA 2 no 1152 720 CA 2

11 H 2 no 2304 10 H 2

12 N 2 yes 2304 10 N 2

13 CA 2 yes 2304 10 CA 2

14 HA 2 no 27648 70 HA 2

15 C 2 no 55296 140 C 2

16 CA 2 yes 55296 140 CA 2

17 N 3 no 663552 1400 N 3

18 C 2 yes 663552 1400 C 2

19 CA 3 no 1327104 2800 CA 3

20 H 3 no 2654208 4 H 3

21 N 3 yes 2654208 4 N 3

22 CA 3 yes 2654208 4 CA 3

23 HA 3 no 31850496 9 HA 3

24 C 3 no 63700992 18 C 3

25 CA 3 yes 63700992 18 CA 3

26 O 3 no 764411904 52 O 3

27 C 3 yes 764411904 52 C 3

28 O 3 no 9172942848 10 O 3

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The first symmetry

It is possible to prove that:

If the DMDGP problem has a solution, then it has at least another symmetric solution.

Pairs of solutions of the DMDGP are symmetric.

Theorem

Let x : G → R ³ be a solution for the DMDGP, defined by the

torsion angles ω _1,4 , . . . , ω _{n − 3,n} . If we invert the sign of sin ω _{i − 3,i} ,

for i = 4, ..., n, then we obtain a new solution x ⁰ : G → R ³ for the

DMDGP.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

The other symmetries

We recently proved that, for each vertex i belonging to this set B = {v ∈ V :6 ∃(u, w) s.t. u + 3 < v ≤ w}

there is a symmetry on the binary tree.

In other words, for each i ∈ B, there are two symmetric branches on the tree having as root a common position for x i − 1 .

Theorem

With probability 1, the number of solutions of the DMDGP is a

power of 2.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

An instance with 2 symmetries (and 4 solutions)

The set B related to this small instance is {4, 6}. Feasible branches are marked in light yellow.

If one solution is already available, BP can exploit B in order to

guide the search towards the feasible branches of the tree. This is

the idea behind symBP.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Outline

1 The MDGP Introduction The DMDGP The BP algorithm MD-jeep

2 Solving NMR instances First approach Making order

An extension of BP: iBP

Preliminary experiments with NMR instances

3 Symmetries and symBP

Some theoretical results

Computational experiments

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Comparisons between BP and symBP

instance BP symBP

name n |E| #Sol #DDF time #Sol #DDF time

lavor_10 10 29 4 44 0.00 4 2 0.00

lavor_20 20 77 64 86 0.00 64 7 0.00

lavor_30 30 161 64 592 0.01 64 18 0.00

lavor_40 40 194 2048 30720 0.05 2048 19 0.01 lavor_50 50 203 1024 46728 0.07 1024 49 0.01

lavor_60 60 357 256 71352 0.12 256 89 0.01

lavor_70 70 591 16 232 0.00 16 67 0.00

lavor_100 100 605 2 4460924 7.55 2 815010 1.37

lavor_200 200 1844 32 35298 0.09 32 394 0.01

lavor_300 300 2505 4 38364 0.07 4 9378 0.03

Experiments with Lavor instances.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Comparisons between BP and symBP

instance BP symBP

name n |E| |B| #Sol time #Sol time

lavor_400 400 2600 35 10

8 10

2

lavor_500 500 4577 44 10

310 10

4 lavor_600 600 5933 25 10

30 10

7 lavor_700 700 5679 71 10

7050 10

134

All these experiments are stopped when 100,000 solutions are found. The

cardinality of B is so large that neither BP nor symBP are able to enumerate all

solutions.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Comparisons between BP and symBP

instance BP symBP

name n |E| #Sol #DDF time #Sol #DDF time

1m40 1224 20382 2 2488 0.03 2 115 0.02

1bpm 1443 14292 2 2890 0.04 2 519 0.02

1mqq 2032 19564 2 4066 0.08 2 737 0.04

3b34 2790 29188 2 5664 0.15 2 1798 0.10

1epw 3861 35028 2 11974 0.45 2 3269 0.21

Experiments with PDB instances.

DMDGP Carlile Lavor

The MDGP

Introduction The DMDGP The BP algorithm MD-jeep

Solving NMR instances

First approach Making order An extension of BP:

iBP Preliminary experiments with NMR instances

Symmetries and symBP

Some theoretical results Computational experiments

Thanks!

clavor@ime.unicamp.br