protein structure comparison and contact maps

Protein structure comparison andcontact maps

A ProteinProtein is a complex molecule with a primary, linear structure (a sequence of aminoacids) and a3-Dimensional structure (the protein fold).

Protein STRUCTURE determines its FUNCTION

For instance, the Drug Design problemcalls for constructing peptides with a 3Dshape complementary to a protein, so asto dock onto it.

Motivation:Structure Alignment is Important for:

- Discovery of Protein Function (shape determines function)

- Search in 3D data bases

- Protein Classification and Evolutionary Studies

- Assessment of Fold Prediction quality (e.g. CASP)-…..

Problem: Align two 3D protein structures

Contact Maps

CONTACT MAPSUnfolded protein

Unfolded protein

Folded protein = contacts

CONTACT MAPS

Unfolded protein


Contact map = graph

CONTACT MAPS

Unfolded protein


Contact map = graph

OBJECTIVE: align 3d folds of proteins = align contact maps

CONTACT MAPS

Contact Maps are related to fold: Similar folds similar contact maps

We studied the problem of determining contact map similarityWe studied the problem of determining contact map similarity

Contact Maps are related to fold: Similar folds similar contact maps

We studied the problem of determining contact map similarityWe studied the problem of determining contact map similarity

In the period 2001-2004 ------------------------------

-I.P. formulation via Branch & Cut (RECOMB)

-Use of Compact Optimization instead of separation (AIRO)

-Lagrangian Relaxation (RECOMB)

(Pubblications: RECOMB proceedings, AIRO proceedings, OR Letters, Journal of Comp. Bio., 4OR)

The Contact Map AlignmentProblem

Non-crossing Alignments

Protein 1

Protein 2

non-crossing map of residues in protein 1 and protein 2

The value of an alignment

Value = 3


Value = 3


We want to maximize the value


NP-Hard (Goldman, Istrail, Papadimitriou, 1999)

Integer Programming Formulation

(5th RECOMB conference)

The use of Integer Linear Programming

Integer Programming Formulation

• Model a difficult problem by 0-1 variables, linear objective function and linear constraints

• Can find optimal solution by branch and bound

• Bound comes from LP relaxation (polynomial)

• Bound can be used to access quality of any feasible sol

CONTACT-CONTACT VARS

yef for e and f contacts

f

yef

RESIDUE-RESIDUE VARS

xij for i and j residuesyef

i

j

xij

(i) 0-1 VARIABLES(i) 0-1 VARIABLES

e

maximize ef yef

(ii) OBJECTIVE(ii) OBJECTIVE

over all feasible x and y

(iii) CONSTRAINTS (FEASIBILITY)(iii) CONSTRAINTS (FEASIBILITY)

y(ip)(jq) <= xij and y(ip)(jq) <= xpq

non-crossing

i

j

i’

j’

xij + xi’j’ <= 1

i

j

p

q

activation

Non-crossing clique Constraints

Variables x define a graph Gx:

• A node for each line• An edge between each pair of crossing lines

i

j

i’

j’

ij

i’j’

Variables x define a graph Gx:

• An independent set corresponds to a noncrossing alignment• Gx has nice proprieties (it’s a perfect graph)• It’s easy (poly) to find large independent sets in Gx

• A node for each line• An edge between each pair of crossing lines

i

j

i’

j’

ij

i’j’

Clique Constraints

Non-crossing constraints can be extended to

CLIQUE CONSTRAINTS

xij <= 1[i,j] in M

For all sets M of mutually incompatible (i.e. crossing) lines

All clique constraints satisfied imply a strong bound!

Clique Constraints

Maximal cliques in Gx

Structure of Maximal cliques in Gx

1. Pick two subsets of same size


2. Connect them in a zig-zag fashion


3. Throw in all lines included in a zig or a zag


The result is a maximal clique in Gx

Separation of Clique Inequalities

Separation of Clique InequalitiesPROBLEM

There exist exponentially many such cliques (O(22n) inequalities).How do we add them ?

PROBLEM


SOLUTION

We don’t add them in the original LP, but only when needed at runtime. Not all of them will be needed, so we are fine as long as…

PROBLEM


SOLUTION


SEPARATION

…we can generate in polynomial time a clique inequality when needed,i.e., when violated by the current LP solution x*

x*ij > 1[i,j] in M

PROBLEM


SOLUTION


SEPARATION

…we can generate in polynomial time a clique inequality when needed,i.e., when violated by the current LP solution x*

x*ij > 1[i,j] in M

THEOREM

We can find the most violated clique inequality in time O(n2)

n2

1n11 2

2

i

u


Create n1 x n2 grid

n2

1n11 2

2

i

u

Orient all edges and give weights

x*iu

x*iu


Create n1 x n2 grid

Create n1 x n2 gridOrient all edges and give weightsThere is violated clique iff longest A,B path has length > 1

A=(1,n2)

B=(n1,1)


.25 .20 .30

0

.35 0 .15

.20 0

• The method which adds violated inequalities by separation is called BRANCH-and-CUT

• The method can get stuck in long runs of cut additions each of which “cuts very little”

• There is an alternative to this, called COMPACT OPTIMIZATION

Detour:

COMPACT OPTIMIZATIONCOMPACT OPTIMIZATION

vs vs

BRANCH and CUTBRANCH and CUT

Polynomial number of “simple” constraints

LP Separation Paradigm

Exponential number of “combinatorial” constraints

TSP, O(n) Degree constraints

TSP, O(2^n)Subtour Elimination


SEPARATION ALGORITHM:Look for a (the most) violated constraint

Cast as an OPTIMIZATION problem

A violated combinatorial constraint exists

+ some of the combinatorial ones

Add inequalities


END YESYES NONO






Add inequalities


END YESYES NONO

E.g. SHORTEST PATH, MAX FLOW,E.g. SHORTEST PATH, MAX FLOW, BIP MATCHING...BIP MATCHING...

A violated combinatorial constraint exists






Add inequalities


END YESYES NONO A violated combinatorial

constraint exists

...IT’S AN LP ITSELF !!...IT’S AN LP ITSELF !!


Polynomial number of “simple” constraints+ some of the combinatorial ones

Add inequalities

END YESYES NONO A violated combinatorial

constraint exists

Polynomial number of constraintsPolynomial number of constraintsfor the separation problem as an LPfor the separation problem as an LP


Polynomial number of “simple” constraints+ some of the combinatorial ones

Add inequalities

END YESYES NONO


Polynomial number of constraintsPolynomial number of constraintsto force that no violated combinatorialto force that no violated combinatorialconstraint existsconstraint exists




Polynomial number of constraintsPolynomial number of constraintsto force that no violated combinatorialto force that no violated combinatorialconstraint existsconstraint exists

Org variables x

Org variables x+ new variables y

Variables x, y


ThTh: Optimization iff Separation: Optimization iff Separation (Grotschel, Lovasz, Schrijver, 1981)

Compact OptimizationCompact Optimization: solve an LP with an exponential n. of: solve an LP with an exponential n. ofinequalities by lifting to a space with a polynomial n. ofinequalities by lifting to a space with a polynomial n. ofinequalities, solving, and projecting backinequalities, solving, and projecting back

ThTh: Compact Optimization iff Compact Separation: Compact Optimization iff Compact Separation (Carr, Lancia, 2002)

Somewhat known (Maculan, Pulleyblank) Somewhat known (Maculan, Pulleyblank) rediscovered (Carr, Lancia, 00)rediscovered (Carr, Lancia, 00)

the application to contact maps comparison

21

maxEf

efEe

y

iuij

vuji xy )(

),)(,(

1221 ),(,),(, EvuViEvuVi

ivij

vuij xy )(

),)(,(


1

ijQij

x for all cliques Q of mutually intersecting alignment lines

A=(1,n2)

B=(n1,1)

Separation of cliques: there is a Q such that x*(Q) > 1if and only if the longest A-B path in this grid is > 1

ijx*

ijx*

2 i n1

1

j

Note: Longest path on GRID can be cast as LP...

Vars: zp = longest up to p.Constraints: zp >= zq + len(q,p) for arc (q,p)

21

maxEf

efEe

y

iuij

vuji xy )(

),)(,(


ivij

vuij xy )(

),)(,(


1

ijQij


21

maxEf

efEe

y

iuij

vuji xy )(

),)(,(


ivij

vuij xy )(

),)(,(


1

ijQij


jiijji zxz ,1,

jiijji zxz ,,1

21, VjVi

02,1 nz

21, VjVi

11,1nz

1, jiz

jiz ,1 jiz ,

ijx

ijx

What about Heuristics?Genetic algorithms

Genetic Algorithm Overview• A Population of candidate solutions that

evolve (improve) over time

• Recombination creates new candidate solutions viacrossover and mutation

Populationat time t

Populationat time t+1

Recombinationoperators

Evaluationfunction

Blue Parent

Offspring

Red Parent

Crossover• Crossover selects pieces from both parents and

creates two offspring solutions– Select a set of edges in one parent to copy to the child

– Copy as many edges as possible from the other parent

– Add random edges to fill any remaining space

These edges conflict with existingedges and are not copied





Crossover• Crossover selects pieces from both parents and creates two

offspring solutions– Select a set of edges in one parent to copy to the child– Copy as many edges as possible from the other parent– Add random edges to fill any remaining space

Mutation• Mutation introduces small changes to

existing solutions by shifting edge endpoints


existing solutions by shifting edge endpoints– Select a set of endpoints to shift


existing solutions by shifting edge endpoints– Select a set of endpoints to shift

This edge “fell off” theend of the contact map

and is removed


existing solutions by shifting edge endpoints– Select a set of endpoints to shift– Randomly add new edges

Computational Results

Compact Optimization vs SeparationCompact Optimization vs Separation

INSTANCE SEPARATION COMPACT OPTIMIZATION

PROT1 PROT2 n m cols rows nLPs time cols rows nLPs time speedup

1b3c 1svf 92 101 4359 6085 1944 28072 6474 10225 1 207 136x1nmg 1svf 92 112 4722 6932 2371 38877 6837 11072 1 314 124x1svf 2b3c 92 101 4359 6085 2118 26996 6474 10225 1 232 116x1bw5 1svf 96 91 4209 5393 1776 14010 6504 9889 1 136 103x1bct 1hlh 105 104 5354 7196 1477 18159 8104 12593 1 186 98x1bw5 1joy 100 115 5805 8055 2434 62426 8304 12955 1 663 94x1svf 1szt 97 101 4584 6349 1118 9744 6924 10934 1 142 69x1svf 2new 93 91 4074 5624 1349 10147 6234 9853 1 149 68x1joy 1svf 94 90 4086 5667 1174 6719 6291 9985 1 115 58x1f22 1svf 93 88 3975 5423 827 4937 6135 9652 1 110 45x1hlh 2new 98 120 5996 10658 1126 28941 8396 13112 1 829 35x1qr9 1svf 100 105 4851 6738 454 3328 7326 11590 1 116 29x1mdy 1svf 100 89 4323 5545 558 2382 6798 10397 1 83 29x1bhb 1svf 97 104 4683 6352 442 2860 7023 10937 1 165 17x1bct 1svf 100 75 3861 4662 316 739 6336 9514 1 54 14x1tn9 1bmr 109 191 12068 15432 328 73449 15036 21164 1 5185 14x1sfc 1svf 101 86 4269 5773 173 513 6789 10714 1 49 10x1svf 1wdc 99 82 4047 5318 170 427 6477 10081 1 61 7x1fza 1fzb 145 194 14664 21142 144 7811 19920 31511 1 1471 5x1ehj 1f22 100 116 5851 8346 65 284 8347 13240 1 96 3x


• 269 proteins– 64 to 72 residues– 80 to 140 contacts

• Selected 597 pairs of proteins out of 36046 possible– roughly as many similar pairs as dissimilar

pairs

Optimality Gap0 1 2 3 4 5 > 5

Number ofInstances

42 48 72 71 76 95 193

Average/MaxNum. Residues

66.4/69 66.8/72 66.7/71 67.0/72 67.0/71 66.8/72 66.8/72

Average/MaxNum. Contacts

61.1/92 56.3/89 57.3/93 59.7/95 61.5/88 64.7/89 71.4/133

Num. GA Best 38 44 63 61 64 74 155Num. LS1 Best 25 20 35 31 33 35 82Num. LS2 Best 5 0 0 1 5 12 53

Skolnick Clustering Test

Skolnick ResultsSkolnick Results• Four Families

1 Flavodoxin-like fold Che-Y related

2 Plastocyanin

3 TIM Barrel

4 Ferratin

• alpha-beta

• 8 structures

• up to 124 residues

• 15-30% sequence similarity

• < 3Å RMSD

• Four Families1 Flavodoxin-like fold Che-Y related

2 Plastocyanin

3 TIM Barrel

4 Ferratin

• beta

• 8 structures



• < 2Å RMSD

Skolnick ResultsSkolnick Results


2 Plastocyanin

3 TIM Barrel

4 Ferratin

• alpha-beta

• 11 structures



• < 2Å RMSD



2 Plastocyanin

3 TIM Barrel

4 Ferratin

• alpha

• 6 structures



• < 4Å RMSD


Family Style Residues Seq. Sim. RMSD Proteins1 alpha-beta 124 15-30% < 3A 1b00, 1dbw, 1nat, 1ntr,

1qmp, 1rnl, 3cah, 4tmy2 beta 99 35-90% < 2A 1baw, 1byo, 1kdi, 1nin,

1pla, 3b3i, 2pcy, 2plt3 alpha-beta 250 30-90% < 2A 1amk, 1aw2, 1b9b, 1btm,

1hti, 1tmh, 1tre, 1tri,1ydv, 3ypi, 8tim

4 170 7-70% < 4A 1b71, 1bcf, 1dps, 1fha,1ier, 1rcd


2 Plastocyanin

3 TIM Barrel

4 Ferratin


ClusteringClustering

Define score(P1, P2) as

0 <= # shared contacts

Min # of contacts of P1,P2

<= 1

Put P1, P2 in same family if score(P1, P2) >= threshold

Clustering validation

We got some known families from biologists, PDB.

Experiment: Take a family F of proteins and align them against each other and against the remaining.

We got some known families from biologists, PDB.

0.05 MISMATCH0.1 MISMATCH0.15 MISMATCH0.2 MISMATCH0.25 MISMATCH0.3 MISMATCH0.35 MATCH…… ……1.0 MATCH

score proteins were…

Experiment: Take a family F of proteins and align them against each other and against the remaining.

TYPICAL BEHAVIOUR


• Performance– 528 alignments– 1.3% false negative– 0.0% false positive


The Lagrangian The Lagrangian Relaxation Approach Relaxation Approach

(RECOMB)(RECOMB)

- RECOMB-1 main meritsRECOMB-1 main merits:

showing optimality is feasible for a rigorous and well defined accepted similarity measure

providing a way to obtain bounds to optimal value

- RECOMB-1 main drawbacks:RECOMB-1 main drawbacks:

works only for small proteins (60 residues, 90 contacts)

can be slow and involved: relies on LP

The RECOMB-2 approach:The RECOMB-2 approach:

Can solve larger instanceslarger instances

EasierEasier to implement (no LP)

FasterFaster (no LP)

Provides good heuristicgood heuristic solutions

Provides boundsbounds from optimum

LP Lagrangian

60 resid., 80 contacts

150 resid., 250 contacts

100 res., 200 cont. 1000 res., 2000 cont.

2 hrs 5 min

Bound type

< 10% in < 1 hr

Max proved optimal

Max B&B root time

OLD NEW

Side by side

Integer Quadratic Programming Formulation

0-1 VARIABLES: For each possible line l =(i,j), a variable xl (i and j residues)

i

jxl

(i) x>=0, integer

Node to Node Variables


i

j

xl(i) x>=0, integer



i

j

xl

CONSTRAINTS: The lines must not cross

(i) x>=0, integer



i

j

xl


xl <= 1l in C

(i) x>=0, integer

for each clique C(ii)


Node to Node Variables0-1 VARIABLES: For each possible line l =(i,j), a variable xl (i and j residues)

i

j

xl


xl <= 1l in C

(i) x>=0, integer


OBJECTIVE: Pick pairs of lines aligning residues in contact


i

j

xl


xl <= 1l in C

(i) x>=0, integer



l m



i

j

xl


xl <= 1l in C

(i) x>=0, integer



l mb(l,m) = 1

define constant coefficients b



i

j

xl


xl <= 1l in C

(i) x>=0, integer



l m b(l,m) = 0



i

j

xl


xl <= 1l in C

(i) x>=0, integer



b(l,m) xl xmis 1 iff alignment linesm and l give a sharng



i

j

xl


xl <= 1l in C

(i) x>=0, integer



(iii) max m b(l, m) xl xm


l


i

j

xl


xl <= 1l in C

(i) x>=0, integer




(iii) max m b(l, m) xl xml

The main idea

We are looking for a noncrossing matching max a quadratic function

A = max i j bij xi xj

s.t. x is a noncrossing matching

THE MAIN IDEA




THE MAIN IDEA

Hard!Hard! What if the function was linear?




B = max i b’i xi


THE MAIN IDEA



3 2 1 2



B = max i b’i xi


THE MAIN IDEA



52 1



B = max i b’i xi


THE MAIN IDEA






B = max i b’i xi

s.t. x is a noncrossing matching3 1 4 1

THE MAIN IDEA





B = max i b’i xi


Easy!Easy! It’s same as sequence alignment problem. DP, O( n1 x n2 )

a t c t c g

c g t c

THE MAIN IDEA





B = max i b’i xi



a t c t c g

c g t c

The idea is then to find b’ such that A B

THE MAIN IDEA





B = max i b’i xi



a t c t c g

c g t c

The idea is then to find b’ such that A B In fact, it is always A B

THE MAIN IDEA

This b’ is found by Lagrangian Relaxation of a formulationof the model.

L.R. consists in removing constraints and put them in the objectivefunction, weighted by some penalties

It’s a successful technique for large optimization problems

We skip the technical details.

Lagrangian RelaxationLagrangian Relaxation

The approach yields GOOD FEASIBLE HEURISTIC The approach yields GOOD FEASIBLE HEURISTIC SOLUTIONS (best than previous methods), i.e. the SOLUTIONS (best than previous methods), i.e. the noncrossing matching of the lagrangian subproblemsnoncrossing matching of the lagrangian subproblems

Hence,we get LOWER BOUNDS –as well as UPPER BOUNDS- to the optimum

Lagrangian Relaxation


• Branch-and-Bound Results• 269 proteins

– 70 -100 residues– 80 to 140 contacts

• Picked 597 (REC’01) and 10,000 new pairs of proteins out of 36046 possible (would have taken up to few months with old method, took a weekend on PC)

RECOMB 01

Optimality Gap0 1 2 3 4 5 > 5

Number ofInstances

42 48 72 71 76 95 193

Average/MaxNum. Residues

66.4/69 66.8/72 66.7/71 67.0/72 67.0/71 66.8/72 66.8/72

Average/MaxNum. Contacts

61.1/92 56.3/89 57.3/93 59.7/95 61.5/88 64.7/89 71.4/133

Num. GA Best 38 44 63 61 64 74 155Num. LS1 Best 25 20 35 31 33 35 82Num. LS2 Best 5 0 0 1 5 12 53

Lagrangian

We pushed our algorithm to its limit by optimally aligning very large proteins which are known to be very similar.

For instance, we optimally aligned a protein of 891 residues and 1944 contacts to a protein with 887 residues and 1937 contacts

Further use of Lagrangian

Skolnick Clustering Test

Family Style Residues Seq. Sim. RMSD Proteins1 alpha-beta 124 15-30% < 3A 1b00, 1dbw, 1nat, 1ntr,

1qmp, 1rnl, 3cah, 4tmy2 beta 99 35-90% < 2A 1baw, 1byo, 1kdi, 1nin,

1pla, 3b3i, 2pcy, 2plt3 alpha-beta 250 30-90% < 2A 1amk, 1aw2, 1b9b, 1btm,

1hti, 1tmh, 1tre, 1tri,1ydv, 3ypi, 8tim

4 170 7-70% < 4A 1b71, 1bcf, 1dps, 1fha,1ier, 1rcd

• Four Families1 Flavodoxin-like fold Che-Y related2 Plastocyanin3 TIM Barrel4 Ferratin

- Fix similarity level , define Pi and Pj in same familiy iff score(Pi, Pj) >=


INVESTIGATING THE CONTACT THRESHOLD

At low threshold there are no contacts

At high threshold, all pairs are in contact

Hence, interesting contact maps (that highlight similarity) lie somewhere in between

We performed experiment: vary contact threshold and similaritylevel and align, until can retrieve the clusters

Clustering a 0-1 matrix of similarity amounts to find a block-diagonal structure (done by TSP)

ijx

We found best results for threshold =7-8 angstrom and similarityabout 65%

We find something like this

A server for contact map alignment

Possible practical project:

User inputs PDB proteins and threshold and retrieves alignment

Also, can compute contact maps from PDB files

Joint work with

-Sorin Istrail (Caltech)-Bob Carr (Sandia Natl Labs)-Brian Walenz (Celera genomics)-Alberto Caprara (University of Bologna)

Multiple contact map alignment

Research project:

protein structure comparison and contact maps

Documents

protein fold

gxstructure of maximal

mapsa protein

protein structure comparison

linear structure

alignmentthe value

noncrossing map of residues

linear constraints