Freezing and folding behavior in simple off-lattice heteropolymers

0.49 MB·English

by  J. E. Magee

Freezing andfoldingbehavior
insimpleoff-latticeheteropolymers
J. E. Magee,1 J. Warwicker,2 and L. Lue3
1Department of Chemical Engineering
2Department of Biomolecular Sciences
3Department of Chemical Engineering,
UMIST, PO Box 88, Manchester, M60 1QD, UK
(Dated:January11,2010)
We have performed parallel tempering Monte Carlo simulations using a simple continuum heteropolymer
modelforproteins. Alltenheteropolymersequenceswhichwehavestudiedhaveshownfirst-ordertransitions
at low temperature to ordered states dominated by single chain conformations. These results are in contrast
4
withthetheoreticalpredictionsoftherandomenergymodelforheteropolymers,fromwhichwewouldexpect
0
0 continuoustransitionstoglassybehavioratlowtemperatures.
2

n
I. INTRODUCTION
a
J

6 Manybiologicalheteropolymers(inparticularproteins)canadoptasingle,sequencedependentconformation(thefold,orna-
1 tivestate)underphysiologicalconditions.Thisspecificityallowsgroupsofatomswithinsuchamoleculetobeplacedaccurately
inspace,allowingthevastrangeofprecisebiochemicalfunctionswithinlivingorganisms.Uponalteringtheenvironmentaround

] theprotein(e.g.pH,solventquality,temperature,etc.),themoleculecanbemadetolosethissingleconformation(i.e.unfoldor
t
f denature),adoptingafluctuating,disorderedstate. Thisfoldingtransitionhasbeenshowntobebothreversibleandfirst-order
o
formanyproteins[1],andisstillpoorlyunderstood.
s
. The modern theoretical viewpoint on the thermodynamics of protein folding (recently reviewed by Pande et al. [2]) uses
t
a conceptsfromthetheoryofspinglasses. ThisapproachwasintroducedbyBryngelsonandWolynes[3],whousedtherandom-
m
energymodel(REM)[4]todescribethelowtemperaturebehaviorofheteropolymers. Thissimplemodelassumestheenergies
- ofthepossibleconformationsofaheteropolymertobeindependentanduncorrelated,andpredictsasecondordertransition(no
d
latentheat)toaglassystatedominatedbyalowenergyconformation. ThemorecomplicatedtheoryofShakhnovitchetal.[5]
n
usesanothertrickfromthespinglassrepertoire,thereplicaapproach,toarriveatthesameconclusions.
o
c The low energy conformation occupied in the glassy state is not necessarily the native state. Depending upon details of
[ conditions, sequenceandsample history, the heteropolymermay well become“misfolded”; thatis, lockedin to a non-native,

lowenergyconformation. Indeed,mostrandomlyselectedsequencesshouldhaveseveralclosely-spacedconformationsatthe
1
low end of their energy spectrum, and will not reliably or reproducibly fold into only one of these. Some annealing of the
v
1 heteropolymersequence(i.e.designwork)shouldbeneededto“drivedown”theenergyofachosen(target)conformationwell
9 belowtheclosely-spacedregionoftheenergyspectrum,preventingsamplingofcompetingconformationsandallowingrobust
2 folding[2]. Simulations[2, 6]andnumericalstudies[7]usinglatticeheteropolymermodelssupportthesepredictionsandthe
1 approximationsbehindthemodels.
0
However,therandomenergyapproximationisessentiallymeanfieldinnature,andcannotaccuratelytreatsystemswithper-
4
sistent correlations [8]. Further, lattice models artificially constrain the configurations available to a polymer. While lattice
0
homopolymerresultshavebeeninstructiveathightemperatures[9,10],theydonotpredictthelowtemperaturefreezingtran-
/
at sition seenin thesimulationsofZhouetal. [11, 12], whereanisolated homopolymerfreezesinto a single, apparentlyunique
m crystallinestateviaafirst-ordertransition.
We have performedMonte Carlo simulationsusing ten randomsequence realizations of a simple off-lattice heteropolymer
-
d modeltotestthesepredictionsincontinuumsystems. Wereportthat,ratherthanexhibitingglassybehavioratlowtemperature,
n alltensequencesstudiedpassthroughafirst-ordertransitiontoanorderedlowenergystate,dominatedbyasingleconformation,
o
withoutneedforsequencedesignwork.
c
:
v
Xi II. MODEL
r
a Our heteropolymersare modeledas freely-jointedchainsofN hard-spheremonomersof diameters , each assigneda type;
we denote the type of monomer i by s. The model is a simple extension of that used by Zhou et al. [11, 12] and is shown
i
schematicallyinFig.1. Bondedmonomersinteractviathepotential
¥ r < (1−d )s
u (r)= 0 (1−d )s < r < (1+d )s (1)
i,i+1
¥ (1+d )s < r

The parameter d is a bond-lengthflexibility parameter; this allows for conversionto molecular dynamics simulation and has
beenshowntomakenophenomenologicaldifferencetohomopolymerfoldingresults[11,12]. Weused =0.1,allowingaten
percentvariationinbondlength.
Ifmonomersiand jarenotdirectlybonded(thatis,|i−j|>1),thentheyinteractviaasquare-wellpotential:
¥ r < s
uij(r)= es0isj lss << rr < ls (2)

wherer denotesthe distancebetweenmonomersiand j, ls isthesquarewellwidth,ande is aninteractionmatrix giving
ss′
square-wellinteractionstrengths(square-welldepths)betweenthemonomertypes.Inthiswork,weuseamodifiedhydrophobic-
polar,ormHPmatrix[36]. Thisisgivenby:
−1 −1
e mHP=e ′ 2 (3)
−1 0
(cid:18) 2 (cid:19)
wheree ′isthecharacteristicenergyscale.Withthismodel,monomerswiths =0arehydrophobic-like(H-type),preferringtype-
i
symmetric interactions, while type s =1 monomersare polar-like (P-type), preferringtype-asymmetricinteractions. Similar
i
interactionmatricesarewidelyusedinon-latticeproteinfoldingsimulationsandnumericalwork[13,14,15]andarebelieved
to be good approximations to real intraprotein interactions [16]. In the dense, globular heteropolymer state, we expect this
interactionmatrix to lead to configurationswith a core of hydrophobicmonomerssurroundedby polarmonomers, as seen in
globularproteins.
Weusereducedunitsscaledbythehard-sphereradiuss andtheenergyscalee ′,suchthat
r∗ = r/s
E∗ = E/e ′ (4)
T∗ = k T/e ′
B
Polypeptidesoflengthgreaterthanapproximately30aminoacidscanformstablefoldeddomains(e.g.the35aminoacidWW
domain[17]). WehavechosentostudyN=64lengthheteropolymers;thislengthisshortenoughforsimulationinreasonable
time,whilelongenoughthatequivalentrealproteinscanfold. WehavestudiedtenrandomsequencesinteractingviathemHP
interaction matrix, each with a 1:1 H:P composition ratio. If we consider P-type monomersto map onto charged and polar
amino acid residues, and H-type monomers to map to all other amino acid residues, this composition ratio is similar to that
foundinrealproteins[18].
III. SIMULATIONMETHOD
Monte Carlo simulations were performed for a single polymer in isolation. As such, we have not used periodic boundary
conditions.Wehaveimplementedcrankshaft,pivotandcontinuumconfigurationalbias(CCB)polymermoves[19].Bondlength
fluctuation is allowed both in the CCB moveimplementation, as well as throughstandard Monte Carlo particle displacement
moves performedon the monomers. CCB moves regrow end segments of up to four monomers. Moves are performed with
relativeprobabilitiesintheratioN:(1/2):(1/4):(1/4)forparticledisplacement,crankshaft,CCBandpivotmovesrespectively.
Initialequilibrationrunslast3×109MonteCarlomoveattempts;datacollectionrunslast1×109moveattempts.
TheMonteCarlo simulationsalso useparalleltempering(also knownasreplicaexchange)[20]. SeveralMonteCarlo sim-
ulationsofthesamesystematdifferingtemperaturesareruninparallel; thesesimulationsattempttoexchangeconfigurations
at regular intervals. This helps to maintain the ergodicity of the simulations; an individual system which has become stuck
at low temperature in a metastable “trap” may be promoted to higher temperatures, where it can explore a wider range of
phasespace,aidingitsequilibration. Ourparalleltemperingsetupusestenindividualsimulationsacrossthetemperaturerange
T∗=0.15...1.5,distributedsuchthattheratiobetweenconsecutivetemperaturesisconstant[21]. Configurationswapmoves
areattemptedevery1000MonteCarlomoveattempts.
The simulations sample the configurational energy E, the radius of gyration r , the two-particle density r (2)(r), and the
g
instantaneouscontactmapcforthesystemevery105MonteCarlomoveattempts. Thetwo-particledensityisdefinedas
r (2)(r)= 1 (cid:229) (cid:229) d (r-r ) (5)
ij
N
* i j6=i +
wherer isthedistancebetweenmonomersiand j[22].
ij
The instantaneouscontactmap c reflectsa heteropolymerconformation;it is an N×N matrix, with each elementc equal
ij
tooneifmonomersiand j arewithineachotherssquarewell,andzerootherwise. Theinstantaneouscontactmapistherefore
givenby:
0 r <ls
c = ij (6)
ij 1 r >ls
ij
(cid:26)
Theaveragecontactmap c givesusinformationonthestructureadoptedbythesystematgiventemperature. Wecanuse
ij
ittocalculateanormalizedthermalaverageoverlap:
(cid:10) (cid:11)
(cid:229) 2
c
ij
hQni= i6=(cid:229) j(cid:10) (cid:11) (7)
c
ij
i6=j
(cid:10) (cid:11)
This normalized overlap parameter can take values between zero and one. Values close to one indicate that the system is
dominated by only a very small number of very similar conformations; small values indicate that the system is exploring a
varietyofdifferentconformations.
Energyandradiusofgyrationdatafromtheindividualsimulationswithinaparalleltemperingrunaretiedtogetherusingself
consistentmultiplehistogramextrapolation[23]. Thisallowsthe histogramsofenergyE andotherobservables{A}recorded
fromtheindividualsimulationsto be combinedto estimate W (E,{A}), the densityofstates ofthe system. Fromthis, we can
interpolatethe behaviorof the system across, and to a certain extentbeyond,the simulated temperatureand parameterrange.
Weusetheextrapolateddistributionatgiventemperaturetocalculatetheconstantvolumespecificheatc∗,givenby
v
E∗2 −hE∗i2
c∗= (8)
v T∗2
(cid:10) (cid:11)
Histogramextrapolationmethodsrelyongoodsamplingacrosstherangeofinterestbytheindividualsimulations. Assuch,
simulationsaroundfirst-ordertransitionswithasignificantfreeenergybarrierbetweenphasescanbeproblematic.Ifsimulations
on either side of the transition temperaturesample onlythe phase which is moststable at that temperature,but no simulation
actually observes the system passing back and forth between phases, the extrapolation process can “blur out” the transition,
leavingasignatureonlyinanincreaseintheerrorestimateandtheresponsefunctions(e.g.specificheat)acrosstheregionof
thetransition.Worsestill,ifthesystemexhibitsstrongmetastability,thiscanintroducealargesystematicerrorintoanyestimate
forthepositionofthetransition.
Toavoidthisproblem,we haveusedmulticanonicalsampling[24]to directlyconfirmordenyphasecoexistencewherewe
suspecttheexistenceofafirst-ordertransition. Wehaveusedaniterativemulticanonicalscheme[25]. RepeatedbiasedMonte
Carlosimulationsrefineourestimateforthedensityofstatesofthesystem, withtheaimofsamplingevenlyinenergyacross
thesuspectedrangeofcoexistence.Thebiascanthenberemovedfromtheresultsofthesesimulations,andtherefinedestimate
forthedensityofstatescanbesplicedbackintothewidersimulationresults. Theindividualsimulationsusedinthisprocedure
areperformedusingthedatacollectionrunparametersstatedabove.
IV. SIMULATIONRESULTS
Plots of E∗, c∗ and r∗ for our sequences, interpolatedacross the studied temperature range, are shown in Fig. 2. The phe-
v g
nomenologydoes notvary considerablywith sequence. Above a temperatureof T∗ ≈0.9, the heteropolymerexists in a dis-
orderedcoilstate -a highenergystructurethatrapidlyincreasesitsradiusofgyrationanddecreasesits heatcapacity(energy
fluctuations)withtemperature. Between T∗ ≈0.3andT∗ ≈0.9, wesee theexpectedlowtemperaturecollapsedglobulestate
with a small radius of gyration (order of 0.05s ) and with relatively high heat capacity (order of 2.0k ). Around T∗ ≈0.26,
B
however, we see sharp spikes in the specific heat, associated with sudden drops in both conformationalenergyand radius of
gyrationwithdecreasingtemperature,suggestiveoffirst-ordertransitions.Wehaveusedmulticanonicalsamplingtostudythese
apparenttransitions;forallsequences,wehavefoundtemperaturesatwhichdoublepeakedprobabilitydistributionsoccurfor
conformationalenergy,indicatingtwo-phasecoexistence. TheseprobabilitydistributionsareshowninFig.3(a). Thesequence
dependenceoftheprobabilitydistributionsissmall.
In Fig. 3 we plot coexistence temperatures for the sequences studied against the energy difference D E∗/T∗ between the
coexisting phases. The quantitiesare clustered around T∗ ≈0.27 and D E/k T ≈56, but otherwise appear uncorrelated. For
B
comparison,thetypicallatentheatD H ofthefoldingtransitionforaglobularproteinisoforderof100k T [26].
B
A. Structuralcharacterofthelowenergyphase
We presenttwo-particledensitiesr (2) data, calculatedusingEq.(5), for sequence4 in Fig. 4; these plotsare typicalof the
resultsforalltensequencesstudied. Thesedatawerecollectedbeforethemulticanonicalalgorithmwasused,andmayexhibit
somedegreeofmetastability. Wefirstconsiderr (2) (r∗),thehydrophobic-hydrophobictwo-particledensities,inFig.4(a). We
HH
see thatbelow the transitiontemperature,thefirst threeneighborshells areclear anddistinct, separatedby regionswith near-
zero hydrophobicparticledensity. The nextthreeneighborshells are much moresparsely populated(limited bythe diameter
of the collapsed state), but remain distinct, separated by sharp minima. These data show a low energy phase structure with
localtranslationalorderbetweenhydrophobicmonomers. Forr (2)(r∗)abovethetransition,thefirstthreeneighborshellsare
HH
still clear up to T∗ =0.7, but not distinctly separated. The next three neighbor shells have blurred together into one single
peak. Above the transition temperature, the hydrophobicmonomers have no translational order, as expected in a disordered
globularphase.Weshowtypicallow-energyphaseandglobularphaseconfigurationsinFig.5;theorientationofthelow-energy
configuration has been chosen to clearly show the hydrophobicmonomersarranged in hexagonalcrystal planes, whereas no
crystalstructureexistsintheglobularconfiguration.
Figures4(b)and(c)showhydrophobic-polarandpolar-polartwo-particledensitiesr (2)(r∗)andr (2)(r∗),respectively. Nei-
HP PP
thershowanysignoftranslationalorder,eitheraboveorbelowthetransition. Thepolarmonomersremaininadisorderedstate
acrossthetemperaturerangestudied.
WeshowthethermalaveragecontactmapsinFig.6. Notethatthesedataarecollectedfromequilibratedparalleltempering
runsbeforethemulticanonicalalgorithmisused,andassuchexhibitmetastability. Atlowtemperatures(belowT∗≈0.32),we
seethatthecontactmapsaredominatedbyverypersistentcontacts,suggestingthatthelowtemperaturestructureisdominated
byveryfewconfigurations.Contactmapscollectedathighertemperaturesshowlargelowpersistenceregions,asexpectedfora
disorderedglobularstate.
Toquantifythis,plotsofthenormalizedoverlapparameter,calculatedfromthesecontactmaps,areshowninFig.7. Aswith
thecontactmaps,theseresultsexhibitmetastabilityaroundthetransition. Forallsequences,weseealargeincreaseoncooling
in hQ i from values of hQ i≈0.3 in the disordered globule phase to values of hQ i≈0.85 in the low energy phase. This
n n n
confirmsthatthelowenergyphaseisdominatedbysmallfluctuationsaroundasingleconformation.
V. DISCUSSIONANDCONCLUSIONS
Ourresultsindicatethat,forallofthetenrandomheteropolymersequenceswhichwehavestudied,thereexistsalowtemper-
aturetransitionbetweenthefamiliarcollapsedliquid-likeglobulephaseandalowenergyphase. Thesetransitionsdonothave
theglassycharacterexpectedfromreplicatheory,showinginsteaddouble-peakedenergydistributionfunctionswithsubstantial
associated latent heat. In each case, the low energy phase is more compact than the disordered globule, having a lower r ,
g
exhibitslocaltranslationalorderandcrystalplanes,asshownbythetwo-particledensityfunctionr (2)(similartothelowenergy
phasesseeninoff-latticehomopolymersimulations[11,12]),andisnotglassy,beingdominatedbyasingleconformationofthe
polymerbackbone,asshownbythenormalizedthermalaverageoverlapparameterhQ i. Inanalogytothemorefamiliarphase
n
behaviorofbulksystems,wedescribethetransitionasafreezingtransition,andthelow-energystateasafrozenstate. Wenote
theobviousparallelswithproteinfoldingtransitions, wherepolypeptideheteropolymersundergoa first-orderphasetransition
toacollapsedsingleconformation,exhibitingrepeatingstructuralunits.
Thelargecomputationaleffortassociatedwithsimulatinghigh-density,lowtemperaturecontinuumpolymerstateshaslimited
thenumberofsequenceswe havebeenabletostudy. Nevertheless,wecanusetheNewcombe-Wilsonscoremethod(method
3 in Ref. [27]) to estimate with 95% confidence that at least 72.25% of 64 length 1:1 HP composition ratio sequences will
demonstrateanequivalentfirst-orderfreezingtransitiontoanon-glassysingleconformationstate.
Wesuggestthatthereasonthatweseedifferentbehaviorfromequivalentlatticemodels,whichundergolowtemperatureglass
transitions,isamatterofsymmetries.Asevidencedbyther (2)data(seeFig.4),thedisorderedglobularstateforoursequences
has no translational order, with hydrophobic monomers able to occupy any position, within the bounds of bond lengths and
hard-sphereoverlap.Passingbelowthefreezingtemperature,thefrozenstatesofoursequencesgaincrystalline-likeorder[37];
hydrophobicmonomers must occupy lattice points, or points very near them. The spatial symmetry of the system has been
brokeninamannerwhich,inthreedimensions,requiresafirst-orderphasetransition.Inthelatticemodel,theapproximationis
thatspatialsymmetryisalreadybrokenforthesystematalltemperatures,andthereisnopossibilityofsuchanorder-disorder
transition. Further,thesymmetryofthelatticeisfixedinadvance;ifthefrozenstate oftheoff-latticesystemisonadifferent
lattice,orifsomedistortionofthelatticeisnecessaryforstability,thelatticemodelcannotaccuratelyportraytheorderedfrozen
state. Withcooling,thelatticesystemwillremaininadisorderedstate,eventuallyformingaglass.
Thecrystallinenatureofthefrozenstatesseenheredonotresembletheorderedstructuresseeninbiologicalheteropolymers
(i.e., proteins); we see no signature of large scale helices or b -sheets in the contact maps. We believe this to be due to the
completeflexibilityofthechainsinthemodel. Ithasbeensuggested[28]thathelicesandb -sheetlike“saddle”structuresare
optimalshapes for a stiff stringlike object, in the same manner thatthe ubiquitousface centeredcubic crystal structure is the
optimalarrangementforsetsofspheres.Wearecurrentlyinvestigatingthismatterforbothhomo-andheteropolymericsystems.
Itcanbeseenfromthecontactmaps(seeFig.6)thatthefrozenconfigurationsarestronglysequencedependent.However,the
thermodynamicbehavioroftheheteropolymers(Fig.2)hasverylittle(thoughstillstatisticallysignificant)sequencedependence.
We believe that the thermodynamicbehavior, in the HP model at least, has stronger composition dependence than sequence
dependence. Our1:1H:Pcompositionratio(withitsfreezingtemperaturesbunchedaroundT∗≈0.27)liesbetweenthelimits
ofthepurehydrophobichomopolymer(withafreezingtemperatureofT∗=0.33[11]),andthepurepolarhomopolymerwith
itseffectivelyhard-spheremonomers,whichshouldnotfreezeinisolation.
Itisgenerallysuggestedthatthefractionofheteropolymericsequencescapableoffreezingtouniqueconformationsshould
be small, around10−9 [29]. Dill and coworkersestimate the chanceof finding a proteinsequence which will fold to a given
approximatestructureasbeingoforder10−10 [15,30,31]. Recentexperimentalwork,usingalargerandom-sequenceprotein
librarytofindsequenceswhichbindtoATP,estimatesthefractionofproteinsequenceswhichare“functional”asaround10−11
[32]. For our system, we have estimated that over 70% of 64-length 1:1 composition ratio HP sequences freeze to unique
conformations. Similar over-estimation is also seen for lattice models; the impressive direct enumeration performed by Li
et al. [14] on an HP lattice model suggests that as many as 4.75% of all 27-length HP sequences have unique ground state
configurations.
Webelievesuchover-estimationtobeduetoalackofsourcesoffrustrationinourmodel,ascomparedtorealproteinsystems.
Ourmodelhasonlytwomonomertypes,nosizepolydispersityorsidechains,completelyflexiblebonds,andnoorientational
dependencein its interaction potential. Any of these factors could significantly lower the numberof model sequenceswhich
exhibitfolding-likebehavior.
In summary, we have demonstratedthat a simple continuumheteropolymermodelcan exhibit a first-order transition to an
orderedlowtemperaturestatedominatedbyasingleconformation,withoutglassybehavior.Thisbehaviorissimilartothatseen
inproteinfolding.Itwouldbeofinteresttoseewhetherextensionstothismodelyieldfurtherprotein-likecharacter,forexample
secondarystructure,orinsteadintroducefrustrationswhichpreventtheobservedbehaviors.
Acknowledgments
ThisworkissupportedbytheBBSRC(grantreferenceB17005).
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Figures
Figure1: Schematicdrawingofmolecularmodel;asystemofsphericalmonomers(filledcircles),diameters .Monomersarejoinedtogether
inanon-branchinglinearsequencewithbondsoflengths (1−d )<l<s (1+d ).Non-bondedmonomersinteractiftheyarewithindistance
ls ofeachother(shownwithdottedcircles);inthefigure,monomersi+1andi+2areinteracting.
(a) -0.5
-1
-1.5
N
*E/ -2
-2.5
-3
-3.5
20
(b)
15
*v10
c
5
0
(c)
0.064
0.062 0.25
0.2
*rg 0.06 0.15
0.1
0.058
0.05
0.056 0.15 0.5 1 1.5
0.15 0.5 1 1.5
*
T
Figure2: (a)Energyperunitmonomer,(b)heatcapacityand(c)radiusofgyrationasafunctionoftemperatureforthetensequencesstudied.
Thescaleofplot(c)hasbeenchosentoshowthelowtemperaturebehavior;theinsetshowstherg behavioroverthefulltemperaturerange
studied.CurvesarecalculatedatT∗intervalsof0.01usingW E,rg calculatedfromparalleltemperingrunsasdescribedinthetext.Estimated
errorsaresmallerthansymbolsizeforradiusofgyrationandenergydata.
(cid:0) (cid:1)
(a) (b)
70
1.4
65
1.2
1 60
x
e
o
N) 0.8 *Tc
*E/ >/ 55
p( *E
0.6
D<
50
0.4
45
0.2
0 40
-3.2 -3 -2.8 -2.6 0.25 0.255 0.26 0.265 0.27 0.275 0.28
* *
E /N T
coex
Figure3: (a)Coexistenceenergydistributions p(E∗/N)forthesequencesstudied. (b)CoexistencetemperatureT∗ plottedagainstenergy
coex
differenceD E/kBT betweenthecoexistingphases.Colordenotessequenceinbothgraphs.