Emergence of Preferred Structures in a Simple Model of Protein Folding
Author(s): Hao Li, Robert Helling, Chao Tang and Ned Wingreen
Source: Science, New Series, Vol. 273, No. 5275 (Aug. 2, 1996), pp. 666-669
Published by: American Association for the Advancement of Science
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found in cells much closer to the articular
surfacethan normal.We concludethat Ihh,
producedby differentiatingchondrocytes,
delays the differentiationof growth plate
chondrocytesby stimulatingthe synthesis
of perichondrialPTHrP,whichthen actson
the PTH/PTHrPreceptoron chondrocytes.
REFERENCESAND NOTES
1. J. Potts Jr. et al., in Endocrinology, L. DeGroot, Ed.
(Saunders, Philadelphia,1995), vol. 2, pp. 920-966.
2. A. E. Broadus and A. F. Stewart, in The Parathyroids.
Basic and ClinicalConcepts, J. P. Bilezikian,M. A.
Levine, R. Marcus, Eds. (Raven, New York, 1994),
pp. 259-294.
3. H. JOppneret a/., Science 254, 1024 (1991).
4. A. B. Abou-Samra et al., Proc. Natl. Acad. Sci.
U.S.A. 89, 2732 (1992).
5. P. Urena et al., Endocrinology 133, 617 (1993).
6. J. Tian, M. Smorgorzewski, L. Kedes, S. G. Massry,
Am. J. Nephrol. 13, 210 (1993).
7. M. Karperienet al., Mech. Dev. 47, 29 (1994).
8. K. Lee, J. D. Deeds, G. V. Segre, Endocrinology 136,
453 (1995).
9. J. J. Orloffet a/., Am. J. Physiol. 262, E599 (1992).
10. N. Inomata, M. Akiyama, N. Kubota, H. JOppner,
Endocrinology136, 4732 (1995).
11. S. Fukayama, A. H. Tashjian Jr., J. N. Davis, J. C.
Chisholm, Proc. Natl. Acad. Sci. U.S.A. 92, 10182
(1995).
12. T. B. Usdin, C. Gruber, T. I. Bonner, J. Biol. Chem.
270,15455 (1995).
13. A. van de Stolpe et aL, J. Cell BioL 120, 235 (1993).
14. A. C. Karapliset al., Genes Dev. 8, 277 (1994).
15. N. Amizuka, H. Warshawsky, J. E. Henderson, D.
Goltzman, A. C. Karaplis, J. Cell BioL 126, 1611
(1994).
16. A. Vortkamp, K. Lee, B. Lanske, G. V. Segre, H. M.
Kronenberg, C. Tabin, Science 273, XXX(1996).
17. M. R. Capecchi, ibid. 244,1288 (1989).
18. Two genomic clones (X-8 and X-15) were isolated
from a X-DASH11129/SvJ mouse livergenomic library
with the
(1 X 107 plaque-forming units per milliliter)
coding sequence of the rat PTH/PTHrPreceptor as a
probe (4, 27). Forconstruction of the targetingvector,
a 4.5-kb Bam HIfragment containing exon El of the
PTH/PTHrPreceptor gene was used as the 5' flanking region and subcloned by blunt end ligationinto a
similarlytreated Xho Isite in the pPNT plasmid (14). A
5.6-kb Eco RIfragmentcontainingthe 3' untranslated
region of the PTH/PTHrPreceptor gene, cloned into
the Eco RIpolylinkersite of pPNT, was used as the 3'
flankingregion. The resultingtargeting vector was linearized at the unique Not Isite in the pPNT backbone
for electroporation of ES cells. Double selection was
carriedout, and resistant ES clones were analyzed by
Southern (DNA)blot analyses with an externalintronic
0.9-kb Sac l-Xho Ifragment at the 5' end of the gene.
Positive clones (12%) were injected into recipient
blastocysts to generate chimeras as described (28).
Progeny of two independent clones yielded the identical phenotype. All animal experimentation followed
institutionalguidelines.
19. B. Lanske et al., data not shown.
20. Twenty-five embryos from E6.5 to E9.5 were examined histologically and had normal appearing
Reichert's membrane. Five E9.5 embryos were examined by in situ hybridizationfor PTH/PTHrPreceptor mRNA (29). The hybridizationsignal was strong in
four embryos but was completely absent from one, a
presumed PTH/PTHrP (-/-) fetus. Immunohistochemical staining was carriedout on paraffinsections
by standard techniques. Hybridizationwas done with
a rabbitantibody to a-laminin(dilution,1/300; Gibco).
To increase the sensitivity, we used a second antibody specific forthe firstone (swine antibodyto rabbit
immunoglobulinG complex; DAKO).Visualizationof
this complex was obtained by hybridizationwith the
ABComplex (horseradish peroxidase; DAKO).
21. M. Karperien,P. Lanser, S. W. de Laat, J. Boonstra,
L. H. K. Defize, Int. J. Dev. Biol., in press.
666
22. F. Beck, J. Tucci, P. V. Senior, J. Reprod. Fertil.99,
343 (1993).
23. M. McLeod, Teratology 22, 299 (1980).
24. Fresh tissues were obtained by cesarean sections
from fetuses derived from heterozygous interbreeding at day 18.5 of gestation. Fetuses were fixed in
10% formalin-phosphate-buffered saline (PBS) (pH
7.2) and subsequently decalcified in neutral 40%
EDTA.Various parts of the skeleton were embedded
in paraffin, and 5- to 10-,um-thick sections were
stained with hematoxylin and eosin.
25. E. Schipani, K. Kruse, H. JOppner, Science 268, 98
(1995).
26. Hind limbs of El 6.5 fetuses were severed at midfemur, stripped of skin, and then placed on a filter
paper (pore size, 0.8 pum)on a wire mesh in a Falcon
organ culture dish, and 1 ml of BGJb medium (Gibco
BRL) was added. The hind limbs, which lay at the
air-fluidinterface, were cultured at 37?C in a humidified atmosphere of 95% air-5%C02 withdailychanges of medium. The hind limbs were treated with either
M human PTHrP (1-34), recombinant murine
10-7
Shh (5 ,ug/ml) (16), or vehicle (BGJb medium-0.1 %
27.
28.
29.
30.
bovine serum albumin)alone from day 2 for 4 days.
The samples were never exposed to serum. At the
termination of culture, the hind limbs were fixed in
10% formalin-PBS, paraffin-embedded, and cut in
serial sections.
X. F. Kong et aL, Biochem. Biophys. Res. Commun.
200, 1290 (1994).
A. Bradley, in Teratocarcinomas and Embryonic
Stem Cells:A PracticalApproach, E. Robertson, Ed.
(IRL,Washington, DC, 1987), pp. 1 13-152.
Complementary 35S- labeled RNA probes were transcribed from the rat PTH/PTHrPreceptor cDNA (4)
and from the human lhh cDNA (16). Insitu hybridization was done as described (8).
We thank C. Tabinforvaluablecollaborationand helpful reading of the manuscript, T. Doetschmann for
reagents, E. Samson for technical assistance with
histology, and M. Mannstadt for help and technical
expertise. Supported by National Institutesof Health
grants DK47038 and DK47237. B.L. was supported
in part by a fellowship of the Max-Kade Foundation.
10 April1996; accepted 20 June 1996
Emergence of Preferred Structures in a
Simple Model of Protein Folding
Hao Li, Robert Helling,*Chao Tang,t Ned Wingreen
Protein structures in nature often exhibit a high degree of regularity (for example, secondary structure and tertiary symmetries) that is absent from random compact conformations. With the use of a simple lattice model of protein folding, it was demonstrated
that structural regularities are related to high "designability" and evolutionary stability.
The designability of each compact structure is measured by the number of sequences
that can design the structure-that is, sequences that possess the structure as their
nondegenerate ground state. Compact structures differ markedly in terms of their designability; highly designable structures emerge with a number of associated sequences
much larger than the average. These highly designable structures possess "proteinlike"
secondary structure and even tertiary symmetries. In addition, they are thermodynamically more stable than other structures. These results suggest that protein structures are
selected in nature because they are readily designed and stable against mutations, and
that such a selection simultaneously leads to thermodynamic stability.
Natural proteinsfold into specificcompact
structuresdespite the huge numberof possible configurations(1). For most singledomainproteins,the informationcoded in
the amino acid sequence is sufficient to
determinethe three-dimensional(3D) folded structure,which is the minimumfreeenergy structure (2). Protein sequences
must undergo selection so that they fold
into unique 3D structures.Becausefolding
mapssequencesto structures,it is relevant
to ask whetherselection principlesalso apply to structuresthat have evolved in nature.Proteinstructuresoften exhibit a high
degree of regularity-for example,secondarystructuressuch as a helices and P3sheets
and tertiary symmetries-that is absent
NEC Research Institute, 4 Independence Way, Princeton, NJ 08540, USA.
*Present address: Second Institutefor Theoretical Physics, DESY/Universityof Hamburg, Hamburg, Germany.
t To whom correspondence should be addressed. E-mail:
tang@research.nj.nec.com
SCIENCE * VOL. 273
*
from randomcompact structures.What is
Does nature
the originof these regularities?
select special structuresfor design?What
are the underlyingprinciplesthat govern
the selection of structures?
Here we describeresultsfrom a simple
model of proteinfolding that suggestsome
answersto these questions.We focuson the
propertiesof each individualcompactstructure by determining the sequences that
have the given structureas their nondegenerategroundstate. We show that the number of sequences (Ns) associatedwith a
given structure(S) differsfromstructureto
structure and that preferred structures
emerge with NS values much largerthan
the average.These preferredstructuresare
with secondarystructuresand
"proteinlike,"
symmetries, and are thermodynamically
more stable than other structures.
Our resultsare derivedfrom a minimal
model of proteinfolding,which we believe
capturesthe essential components of the
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51}11
REPORTS
problem.In this model, a protein is represented by a self-avoidingchain of beads
placedon a discretelattice, with two types
of beads used to mimic polar (P) and hydrophobic(H) aminoacids(3). A sequence
is specifiedby a choice of monomertype at
each position on the chain, (tuj, where ai
could be either H or P, and i is a monomer
index. A structureis specifiedby a set of
coordinatesfor all the monomers, riJ.The
energyof a sequencefoldedinto a particular
structureis given by short-rangecontact
interactions
H = XE,,,tA(ri-r)
(1)
whereA(ri- r) = 1 if rEandr1areadjoining
lattice sites but i and j are not adjacentin
position along the sequence,and A(ri -r)
= 0 otherwise.Dependingon the types of
monomersin contact, the interactionenergy will be EHH, EHP, or EPP,corresponding
to H-H, H-P, or P-P contacts, respectively
(Fig. 1).
This simplemodelhas somejustification
in nature.The majordrivingforce for protein folding is the hydrophobicforce (4).
The tendencyof aminoacidsto avoidwater
drivesproteinsto fold into a compactshape
with a hydrophobiccore, and sucha forceis
effectivelydescribedby a short-rangecontact interaction.Although20 differentamino acids exist in nature,quantitativeanaldatarevealsthat they
ysisof natural-protein
fall into two distinct groups (H and P)
accordingto their affinities for water (5).
Experimentalevidence also indicates that
certainproteinscan be designedby specification of only this HP pattem of the sequence (6).
We chose the interactionparametersin
Eq. 1 to satisfythe followingphysicalconstraints:(i) compactshapeshave lowerenergiesthan any noncompactshapes;(ii) H
monomersare buriedas much as possible,
which is expressedby the relation EPP>
EHP > EHH, which lowers the energy of
configurationsin which H residuesarehidden fromwater;and (iii) differenttypes of
monomerstend to segregate,which is expressedby 2EHP> Epp+ EHH.Conditions
(ii) and (iii) werederivedfromour analysis
of the real-proteindata contained in the
matrix of interresidue
Miyazawa-Jernigan
contact energiesbetween differenttypesof
amino acids (5).
We have studied the model on a 3D
cubic latticeand on a 2D squarelattice.We
focus on the "designability"
of each compact structure.Specifically, we count the
numberof sequences(Ns) that have a given
compact structure (S) as their unique
groundstate, which requiresidentification
of the minimum-energy compact conformations of each sequence. Because all compact
with AA
Fig. 1. Structures
the largestvalueof Ns
for3D 3 by 3 by 3 (A)
and2D 6 by6 (B)systems. Eachsequence
representsone of the
Ns possible sequences. Blackbeads indicate H residues;gray
beads indicateP residues. Two beads are
consideredto be in
contact if they are
nearestneighborsbutarenotconnectedbythe backbone.
structureshave the same total numberof
contacts,we can freelyshift and rescalethe
interactionenergies,leaving only one free
parameter.Throughoutthis paper,we chose
EHH- -2.3, EHP= -1, and E = 0,
which satisfyconditions(ii) and (iii) above.
The resultsare insensitiveto the value of
EHH as long as both these conditions are
satisfied.
For the 3D case, we analyzeda chain
composedof 27 monomers.We considered
all the structuresthat forma compact3 by
3 by 3 cube.Therearea total of 51,704 such
structuresunrelated by rotational, reflection, or reverse-labelingsymmetries.For a
given sequence,the groundstatestructureis
determinedby calculationof the energiesof
all compactstructures.We completelyenumeratedthe groundstatesof all 227 possible
sequences,showing that 4.75% of the sequences have unique groundstates. As a
result of this complete enumeration,we
obtained all possible sequences that "design"a given structure that is, that have
that structureas their uniquegroundstate.
Thus,Ns is a measureof the designabilityof
a given structure,and this informationis
availablefor all compactstructures.
Compactstructureswere found to differ
markedlyin terms of their designability.
Therearestructuresthat can be designedby
a largenumberof sequences,and there are
structuresthat can be designedby
"poor"'
only a few or even no sequences.Forexample, the top structurecan be designedby
3794 different sequences (Ns = 3794),
whereasthere are 4256 structuresfor which
N= = 0. The numberof structureswith a
given NS value decreases monotonically
(with small fluctuations)as NS increases
(Fig. 2A). Forstructuresthat contributeto
the longtail of the distribution,
Ns >> Ns
61.72, whereNs is the averagenumber.We
refer to these structuresas "highlydesignable." The distribution differs markedly
from the Poisson distributionthat would
result if the compact structureswere statistically equivalent. For a Poisson distribution with NS = 61.72, the probabilityof
finding even one structurewith Ns > 120
is 1.76
x
10-6.
SCIENCE * VOL.273
*
Highlydesignablestructuresexhibit certain secondarystructuresthat are absent
fromrandomcompactstructures.We examined the compact structureswith the 10
largestNs values and found that all have
parallel running lines folded in a regular
manner(Fig. 1A). These structurescontain
eight or nine strands(three amino acids in
a row), whereasthe averagestructurehas
only 5.4 strands.
To ensure that the above resultswere
not artifactsof small size (the 3 by 3 by 3
cube), we also performedsystematicstudies
of size dependencein two dimensions(the
study of largerstructuresin three dimensions is not practicalbecauseof the limitations of computingpower).We studiedsystems of sizes 4 by 4, 5 by 5, 6 by 5, and 6 by
6 on a 2D squarelattice. Forsystemsof sizes
6 by 5 and 6 by 6, a randomsamplingof
sequenceswas performed(7). Comparison
of systemsof differentsizes requiresappropriaterescalingof the axes. We chose bin
sizesfor NS to be proportionalto Ns, and
rescaledthe numberof structuresby a factor
proportionalto the total numberof structures.For the 6 by 5 and 6 by 6 cases, we
ensuredthat the randomsampling of sequencesproduceda reliabledistributionby
doublingthe numberof sequencesuntil a
fixed distributionwas achieved.
The systems of different sizes in two
dimensionsall showedthe samequalitative
behavioras that apparentin three dimensions. In each instance, there are highly
designablestructuresthat standout. Forthe
6 by 5 and 6 by 6 systems,for which the
total numbersof structuresare sufficiently
large to producesmooth distributions,the
two distributionsshowedvirtuallyidentical
shapes(Fig. 2B). The tail of the 2D distribution could be fitted by an exponential
function (Fig. 2B, inset). In contrast,the
falloff of the tail in the 3D distributionis
slightlyless than exponential.
Similarlyto the 3D case, the highly designablestructuresin two dimensionsalso
exhibit secondarystructures.In the 2D 6 by
6 system, as the surface-to-interiorratio
approachesthat of real proteins,the highly
designablestructuresoften have bundlesof
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887
pleats and long strands,reminiscentof a
helices and a strandsin real proteins;in
addition, some of the highly designable
structureshavetertiarysymmetries(Fig.1B).
The highlydesignablestructuresare, on
average, thermodynamicallymore stable
than other structures.The stability of a
structurecan be characterizedby the average energygap (8s), averagedover the NS
sequencesthat design the structure.For a
given sequence,the energygap (8s) is defined as the minimumenergy requiredto
change the ground-statestructureto a differentcompactstructure.Forthe 3D 3 by 3
by 3 structures,there is a markedcorrelation between NS and &s(Fig. 3). Highly
designable structureshave average gaps
much largerthan those of structureswith
smallNS values,andthere is a suddenjump
in 8s) for structureswith NS 1400. The
numberof structureswith largegaps is 60.
The abruptjump in &sis somewhatunexpected comparedwith the smoothdistribution of Ns, Such an abrupttransitionpro-
vides a meansof differentiatingthe special,
highly designablestructuresfrom the ordinary ones. According to this distinction,
highly designablestructuresconstituteonly
a smallfraction(0.12%)of all the compact
structures.
The fact that highly designablestructures are more stable than other structures
can be understoodqualitativelyby considering a particularsequenceassociatedwith
a highlydesignablestructure,S. A mutation
of the sequencemay change the energyof
the structureS as well as those of the competingstructures.If the gap is large,it is less
probablethat the energiesof the competing
structureswill shift below that of the struc*tureS. Thus, the structureS is likely to
remainas the groundstate of the mutant.
Therefore,a largegap is likely to correlate
with a largenumberof sequencesthat design the structure.
An importantapproachin studyingreal
protein structuresis to assessmutationeffects and homologoussequences(sequences
relatedby a common ancestor)(8). In our
simple model, we referto the NS different
sequencesthat designthe samestructureas
"homologous."Analysis of mutation patternsof the homologoussequencesforhighly designablestructuresrevealedphenomena similarto those observedin realproteins.
For example, sequenceswith no apparent
similarities(with differenttypes of monomer at more than half of the sites) can
design the same structure. Furthermore,
some sites arehighly mutable,whereasothers are highly conserved. The conserved
sitesfora given structurearegenerallythose
sites with the smallestor largestnumberof
sides exposedto water.Figure4 shows the
probability,Ppof findinga P monomerat a
particularsite, calculatedfor the structure
with the largestNS for the 3D 3 by 3 by 3
and the 2D 6 by 6 systems(Fig. 1). Forthe
3D case, there are sites that are perfectly
conservedwith Pp = 0 and Pp = 1.
The mutationpatternfor the 2D 6 by 6
system shows additional characteristics:
1.5
0.~~~~~~~~~~~E
103
>-
1
co
0
o
0
oo
ai 0.5 _
101
0
1000
200000
0
4000
Fig. 3. Averageenergy gap of 3D 3 by 3 by 3
structures
plottedagainstNS of the structures.
0.2
co
103
ll
10
1
I
I III
l
100
1 0001__
I
I I1
0
A
10
lo
1
ooC
1--02r
1.0
1.0
B
!
v
0.8
I'
I\
3 by by 3
of
3ooe
finin
Fig. 3. Avroageility
ga210
2
30
5100
fof the structureswthth
strutiulre ploted
calcuainteds
n~~~~~~~~~~~~~~~~~~
102 _un
0.2
101
~ ~ ~~C
102
0.6 NS fo 3
1 MDc
5
0.4
1.0
100
0
101
I
II
50
100
150
B
ffnigaPmnmra
patclrste0
tucuewt6h
acltd
o h
100
Ns
by 6(B
3f byi3d(A)
aPmndo2Dr
larges N frorb3Di3iby
Fig. 2. (A)The number of structures with a given N8 for a 3D 3 by 3 by 3 system. (B) The number of
structures with a given Ns for 2D 6 by 5 (A) and 6 by 6 (El)systems. (Inset)Same data in a semi-log plot.
668
~
30
Seuec
Fi.84 rbblt
200
I
10
~
5 20
10
SCIENCE * VOL. 273
*
systems.
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All use subject to JSTOR Terms and Conditions
-REPORTS
The pleated regions have alternatingarrangements(with period 4) of H and P
types, the region of long folded strandsis
essentiallyall H type, and the region connectingthese substructures
(similarto turns
in proteins)containspredominantlyP type
monomers.
Mutationpatternscan also be characterized by calculatingthe entropyof the homologous sequencesfor a given structure
from
Entropy=
Isites-
Ppln(Pp)
+ (Pp - 1)ln(1 - PP)]
(2)
Given only the knowledgeof entropy,an
estimate of the numberof all possiblehomologous sequences can be made, Nest
=
exp (entropy), assumingthat mutation of
each site is independent. We calculated
NeSt for all the structuresand comparedit
with the exact value Ns, For the highly
designable structures, Nest is a good order-
of-magnitudeestimatefor Ns, Forexample,
for the top structurein the 3D 3 by 3 by 3
3.5. However, for the less
case, NestNsdesignable structures, Nest greatly overesti-
matesNs, The largedeviationbeginsat NS
1400, at the boundarybetweenlarge-gap
and small-gapstructures.Thus, for highly
designable structures,the mutations are
roughly independent, whereas they are
highly correlatedfor other structures.
Although our resultsfor the 3D system
werederivedfor smallstructures(3 by 3 by
3), we believe similarresultshold for much
largerstructures.Evidence to this effect is
providedby recentstudiesof designin larger structuresby Yue and Dill (9) with a
similarmodel.In a few examplesstudiedby
these researchers,
they foundthat sequences
with a smallground-statedegeneracycorresponded.to structureswith certainproteinlike secondarystructuresand tertiarysymmetries.In light of our results,we believe
that such proteinlike structuresare the
highly designablestructureswith largeNS
values.This interpretationis differentfrom
Location.
that of Yue and Dill, who suggestthat minimaldegeneracyis sufficientto produceproteinlike secondarystructuresand tertiary
symmetries.Our resultsshow that it is possible to find sequencesthat uniquelydesign
even "poor"structures.It is the requirement
that many sequences design a particular
structurethat leadsto proteinlikesecondary
structuresand tertiarysymmetries.
Although the detailed structuresof real
proteinsare determinedby manyfactorsfor example, hydrogen bonding and the
shapes of the amino acids-our results
fromthe simplemodel suggestthat there is
a principleof design and evolutionarystability that shouldplay a crucialrole in the
selection of protein structures;that is, real
protein structuresmust be highly designable and mutable. Becausehighly designable structuresare also morestable, such a
selection principle solves the problem of
thermodynamic stability simultaneously.
Froman evolutionarypoint of view, highly designablestructuresare more likely to
have been chosen through randomselection of sequences in the primordialage,
and they are stable against mutations.
Our proposed principle of selection
based on designability and mutability
shouldhave importantcorollariesin prediction and design of protein structure.If, in
fact, nature selects only highly designable
structures,then structurepredictionalgorithmsshould limit the searchof the conformationalspace to these special structures, which might constitute only a tiny
fraction of the total number of possible
structures.Indeed,certainstructuralmotifs
occur frequentlyin the protein structure
data bank (10), and it has been estimated
that only -1000 possiblefolds exist in the
structuresof naturalproteins(11). A relatively successfulalgorithmfor structureprediction has been developedwith the use of
the templates from known protein structures(12). Our studylends theoreticalsupport to such an approach.Furtherimprovement dependson finding practicalways to
Location.
New
identifyhighly designablestructures.
An importantquestionconcernsthe kinetic accessibilityof these structures,and
whetherthereareotherselectionprinciples
imposedby kinetics.Studieshave examined
structureselectionbasedon foldingkinetics
(13). It is likely that ouLrhighly designable
structuresalso fold more readilybecauseof
the large gap in their excitation spectrum
(14). We have performedsuccessfulpreliminary folding simulationsfor some highly
designablestructures(15). A more systematic study of kinetics, including ordinary
structures,is underway.
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372, 631 (1994).
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(1994).
15. R. Melin,H. Li,N. Wingreen, C. Tang, in preparation.
16. We thank W. Bialek, M. Hecht, A. Libchaber, G.
McLendon, and Y. Meirfor helpful discussions.
19 March 1996; accepted 4 June 1996
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