Comparative analysis of protein structure using multiscale additive functionals

0.22 MB·English

by  Marconi Soares Barbosa

Comparative analysis of protein structure using multiscale additive functionals
Marconi Soares Barbosa1, Rinaldo Wander Montalva˜o2, Tom Blundell2 and Luciano da Fontoura Costa1
1Institute of Physics at S˜ao Carlos, University of S˜ao Paulo,
S˜ao Carlos, SP, PO Box 369, 13560-970, Phone [phone],
FAX [phone], Brazil, [email protected] and
2Department of Biochemistry, University of Cambridge, 80 Tennis Court Road, Cambridge,
CB2 1GA, Phone [phone], FAX [phone] United Kingdom
(Dated: February 9, 2008)
This work reports a new methodology aimed at describing characteristics of protein structural
7 shapes, and suggests a framework in which to resolve or classify automatically such structures
0 into known families. This new approach to protein structure characterization is based on elements
0 of integral geometry using biologically relevant measurements of shape and considering them on
2 a multi-scale representation which align the proposed methodology to the recently reported tube
n picture of a protein structure as a minimal representation model. The method has been applied
a withgood resultstoasubsetofproteinstructuresknowntobeespecially challengingtorevertinto
J families, confirming thepotential of theproposed method for accurate structure classification.

4
PACSnumbers:
1

]
M I. INTRODUCTION II. ADDITIVE SHAPE FUNCTIONALS
Q
Evolutionhasproducedahugenumberofproteinfam- We start by describing the mathematical aspects of
.
o ilies and super-families whose members possess similar the adopted procedure. The Minkowski functionals of
bi sequences and three-dimensional structures. Restraints a body K in the plane are proportional to the famil-
- on evolutionary divergence are mainly related to the iar geometric quantities of area A(K), perimeter U(K)
q protein function, and therefore selective pressure tends and the connectivity or Euler number χ(K). The usual
[
to operate on the three-dimensional structure [1]. The definition of the connectivity from algebraic topology in

1 HOMSTRAD [2] is an example of a database of pro- two dimensions is the difference between the number of
v tein structures organized into homologous families. As connected nc components and the number of holes nh,
9 a consequence of the global proteomic effort, the num- χ(K) = nc−nh. In the Euclidean space, there are two
1 ber of known structures is growing at an impressive rate kinds of holes to consider. First, we have the pure hole,
0
and has passed the total of 39000 structures. It is re- acompletelyclosedregionofwhite voxelssurroundedby
1
0 markable progress but, on the other hand, it also in- black voxels. Second, the tunnels. The Euler charac-
7 troduces an overwhelming amount of data to be man- teristic is consequently given as χ(K) = nc −nt +nh,
0 ually classified on those databases. With more than 400 where nt is the number of tunnels and nh is the number
o/ structures solved every month, the challenge for auto- of pure holes. There is an additional geometric quan-
i matic protein structural comparison and classification is tity to consider in the three-dimensional space, namely
b greaterthanever. Mostoftheproteincomparisonmeth- the mean curvature or breadth B(K). By exploring the
-
q odsdependmainlyuponstructuralalignmentandRMSD additivityoftheMinkowskifunctionals,theirdetermina-
: measures, and therefore are not completely reliable [3]. tionreducestocountingthemultiplicityofbasicbuilding
v
While RMSD is a good measure of structure similarity blocks that disjointly compose the object. For example
i
X for almost identical proteins, it cannot be used to judge a voxel can be decomposed as a disjointed set of 8 ver-
r dissimilarity since it violates the triangle inequality. It tices, 12 edges, 6 faces and one open cube. The same
a means that any system based on RMSD alone is unable process can be applied to any object in a lattice. For a
to clusterstructures and,consequentlyincapable ofclas- three-dimensional space, which is our interest regarding
sifying them into families. In addition, the reliance on protein structures, see [5, 6], we have
sequence alignments introduces a drawback because it is
virtuallyimpossible to avoiderrorsduring the alignment V(P)=n3, S(P)=−6n3+2n2, (1)
construction. 2B(P)=3n3−2n2+n1, χ(P)=−n3+n2−n1+n0,
In this paper we investigate the potential of an al-
gorithm adapted to automatically classify proteins into Wheren3 isthe numberofinteriorcubes, n2 is thenum-
HOMSTRAD families. This algorithm is based on con- ber of open faces, n1 is the number of sides and n0 is
cepts of Integral Geometry [4], know as Morphological the number of vertices. So, the procedure to calculate
Image Analysis (MIA), which has been recently applied Minkowski functionals of a pattern P can be reduced to
toaseriesofproblemsduetoitssimplicity indesignand counting the number of elementary bodies of each type
implementation. FieldsasdiverseasNeuroscience[5]and that compose a voxel (cubes, faces, edges and vertexes)
MaterialsSciences[6]havebenefitedfromthisapproach. belonging to P.
2
III. PROTEIN STRUCTURE, TUBE PICTURE
AND MULTISCALE SIGNATURES
2.5e+05
Theproteinstructureinourapproachisdefinedessen-
tially by the geometrical/topological nature of its back- 2e+05
bone. All α-carbon atom coordinates are identified from
a.pdb fileandaninterpolationschemeisusedtoconnect 1.5e+05
neighboring atoms by a straight path. This design pro- Area
cedure attaches a variable resolution to the method, as 1e+05
the highly refined atomic scale data has to be truncated
during the process. 50000
In our analysis the calculation of the Minkowski func-
tionals are incorporated into a multiscale framework. In 2 4 6 8 10 12
Dilation radius
suchascheme,allfourquantitiesarecalculatedasafunc-
tion of a control parameter as some transformation is
made on the structure of interest. In this paper we con-
sider this transformation to be the process of exact dila-
tions and the controlparameterthe dilation radius. Our
choice is particularly suited as the exact dilation proce-
dure naturally fits itself in what has been described as 30000
thetubepicture forproteinstructureanalysis[7],amini-
mthaeliisnttbriicoaptheysaiscpaelcrtesasoofntinhgeogfetohmeeptrroyt/etionpmoloodgyel.arWehaicle- Perimeter20000
counted for at each spatial scale by the Minkowski func-
tionals,thespacesurroundingthebackboneisprobedby 10000
performingthedilationofthestructureandthisinforma-
tion is condensed in what we call henceforth multiscale
2 4 6 8 10 12
signatures. The behaviorofsuchsignatures,particularly Dilation radius
the topologicallyrelated ones,can be discontinuous. For
examplethe processofdilationmaychangeabruptlythe
number of pure holes or tunnels at particular scales and
these facts are registered for all scales in the multiscale
signature for the connectivity or Euler number (charac-
teristic).
2000
IV. RESULTS AND DISCUSSION Mean breadth1000
Figure 1 shows all considered functionals signatures
0
for a set of 71 proteins which were chosen specifically
because of their similarities. The range of scales shown
inthesegraphsencompassestheinitialstructureandthe 2 4 6 8 10 12
Dilation radius
final filled volume without holes and tunnels (χ = 1).
Therearebothsimilaritiesandstrikingdifferenceswhose
subtleties, until now, have been handled only by more
complex algorithms.
ForeachofthosesignaturesinFigure1weselectthree 50
featuresinanattempt togloballycharacterizethe struc-
0
ture and, by doing so, minimize the amount of data
nteiveoaenlduaeladst.efodFrotfhruettuhsreteacsnlidagasnsraidfitucdraeetvsioiaontfbioAansr,eedaitosannindMteiPngerkraoilmw,seaktneidrfu,tnwhcee- Euler Characteristic-1-0500
scale at which the integral of the curve reaches half of
-150
the total value. For the signatures of the Connectivity
and of the Mean Breadth we measured the standard de-
-200
viation, the integral of the curve and the monotonicity
2 4 6 8 10 12
index given by i = (i +i +i )/i where i are the Dilation radius
s d p s s,d,p
counts for each time the curve increase, decrease or stay
FIG. 1: Multiscale signatures associated with the four
Minkowski functionals in the Euclidean space.
3
A B C D E Error Posterior.Error above global measures and quantifies the classification
potential of the proposed framework. Such a discrimi-
A (asp) 13 0 0 0 0 0.0000000 0.0000477
nantanalyisprojectsthe measurementsin sucha wayas
B (ghf13) 0 11 0 0 0 0.0000000 0.0000025
to optimize their separation, expressed in terms of high
C (ghf22) 0 0 12 0 1 0.0769231 0.1051906
interclass and low intraclass dispresions. It is remark-
D (kinase) 0 0 0 16 0 0.0000000 0.0247150 able that, although the structures were specially chosen
E (phoslip) 0 0 0 0 18 0.0000000 -0.0104105 to make a reduction into families difficult, this approach
Overall 0.0140845 0.0221997 managedtoperfectlyclassifyfouroutofthefivefamilies.
AmistakewasmadeinclassC,wereitmisclassified1out
of 13 structures. It is worthwhile to note that although
TABLE I: The result of a classical discriminant analysis for exhibiting different foldings, alpha plus beta in the class
the12 features extracted from the multiscale signatures. C and all alpha in the class E, their average length and
topological properties in general are quite similar. Fig-
ure 2 shows a two-dimensional section of the complete
constant. Table I shows the numeric results obtained featurespacedefinedbymeasuresfromthemeanbreadth
andconnectivityonly. Itprovidesamoreeconomicaldis-
criminating clustering, albeit with overlaps.
V. CONCLUSION
3500 In this paper we have accessed the potential of the
multi-scaleMinkowskifunctionalsforproteinmorpholog-
ical characterization and structural analysis. We found
mean breadth curve3000 toathofdpaaitosnlttaoihnlgyeiccssteiassl,feufatenasoctfstuuisrobtenrssaut.lacsFntouatrirraeeatselplkdobntbuoeytnwtontinahtleeloyfrahesmasuuviitlleytesdhoiofgtbohsttlatryhuinsicseitmdukriiflneaosdrr,
Integral of 2500 ABC ntharmouelgyhtaheclgalsysciocsaylldhisycdrrimolainsaenfatmanilayly2s2i,styhieelcdlaedssfifiucllaytiaocn-
D curate results. These results are comparable with the
E
2000 bestapproachsofar[9],whichusesconsiderablymorepa-
rametersandis basedonacomplexconcept. Inaddition
2000 2500 3000 to the classification result, it is important to emphasize
Perimeter standard deviation the simplicity ofthe algorithmandthe clearrelationship
between the quantities used for the characterizationand
familiargeometrical,topologicalandbiologicalconcepts.
FIG. 2: A scatter plot derived from the mean breadth and This direct relation to familiar measurements, combined
theperimeter alone leads to a discriminative feature space.
with the simplicity for implementing the MIA approach,
suggeststhatthis kindofanalysisisaparticularlyuseful
by classicaldiscriminant analysis[8] basedon the twelve tool for classifying the shape of protein structures.
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