research papers J Dummy-atom modelling of stacked and helical IUCr nanostructures from solution scattering data ISSN2052-2525 BIOLOGYjMEDICINE Max Burian and Heinz Amenitsch* InstituteofInorganicChemistry,GrazUniversityofTechnology,Stremayrgasse9/V,Graz8010,Austria.*Correspondence e-mail:amenitsch@tugraz.at Received6February2018 Accepted9April2018 Theavailabilityofdummy-atommodellingprogramstodeterminetheshapeof monodisperse globular particles from small-angle solution scattering data has led to outstanding scientific advances. However, there is no equivalent EditedbyJ.Trewhella,UniversityofSydney, procedure that allows modelling of stacked, seemingly endless structures, such Australia as helical systems. This work presents a bead-modelling algorithm that reconstructsthestructuralmotif ofhelical androd-likesystems.Thealgorithm Keywords:SAXS;stackedstructures;helical isbasedona‘projectionscheme’:byexploitingtherecurrentnatureofstacked structures;shaperetrieval;SasHel;structure systems, such as helices, the full structure is reduced to a single building-block determination;solutionscattering; computationalmodelling;structuralbiology; motif.Thisbuildingblockisfittedbyallowingrandomdummy-atommovements nanoscience. without an underlying grid.The proposed method isverified using avariety of analytical models, and examples are presented of successful shape reconstruc- Supportinginformation:thisarticlehas tion from experimental data sets. To make the algorithm available to the supportinginformationatwww.iucrj.org scientific community, it is implemented in a graphical computer program that encourages user interaction during the fitting process and also includes an option for shape reconstruction ofglobular particles. 1. Introduction Small-angle X-ray scattering (SAXS) is an established technique to study the structural aspects, such as the size and shape,ofmolecularsystemsinsolution(Lietal.,2016).Asthis structural information is not directly apparent from the recorded scattering intensity, one requires a fitting process that generally relies on an underlying mathematical model (Pedersen,1997).Whileavarietyofsuchanalyticalmodelsare available in the literature (Pedersen, 1997), each of them is boundtoagivenshape.Thefittingprocessofanexperimental datasetthereforerequirespreviousknowledgeofthesample suchthattheappropriatemodelcanbechosen.Dummy-atom (DA) modelling, which describes the particle shape as a variable bead assembly, bypasses this issue as the fitting processisnolongerconstrainedtoasinglegeometry(Chacoń etal.,1998;Waltheretal.,2000;Svergunetal.,2001;Franke& Svergun, 2009; Koutsioubas et al., 2016). Consequently, no a priori knowledge regarding the solute’s shape is necessary. This advancement has helped to make SAXS an attractive technique to characterize stable molecules in solution, far beyond the community of scatteringexperts (Yang,2014). However,singlemoleculesinsolutionarenotalwaysstable. In fact, it is in their nature to interact and aggregate, often resulting in highly organized hierarchical systems stretching over several orders of magnitude (Palmer & Stupp, 2008; Wasielewski, 2009; Busseron et al., 2013; Praetorius & Dietz, 2017). Helical and chiral superstructures are a common yet spectacularclassofsuchsystems,e.g.thelengthofthehuman genome DNA double helix exceeds several centimetres (Venter et al., 2001) while the forces that give rise to the 390 https://doi.org/10.1107/S2052252518005493 IUCrJ (2018).5,390–401 research papers Figure1 Theprojectionscheme,visualizedusingtheexampleofasingle-strandedhelix.Forsuchseeminglyendlessgeometries,thereisabuildingblock(upper left-handcorner)thatisrecurrentalongthezaxis(seefullhelixontheright).ForafullbodyconsistingofMstackedbuildingblocks,onefindsdistinct structuralmotifs,suchasthebuildingblockitself,neighbouringbuildingblocks,single-spacedbuildingblocksandsoonupto(M(cid:3)1)-spacedbuilding blocks.Thescatteringintensityofthefullgeometrycanhencebecalculatedbysummingthecontributionsofthesestructuralmotifs,scaledbytheir recurrence.ThisbypassesthedemandforanactualDAmodelrepresentingthefullstructure,aswesolelyevaluateprojectionsofthebuildingblock insteadofactualstackedduplicates,hencethetermprojectionscheme. characteristic helical motif act on the molecular (nanometre) seemingly infinite nature of e.g. helical systems onto a single level (Dobbs, 2007). Amyloid fibrillation, i.e. the aggregation building-blockunit,representedbydummyatoms(DA).This ofproteinsintoinsoluble,oftenhelical,fibrils,hasbeenlinked building block is altered by random DA movements while to critical diseases such as type 2 diabetes or Alzheimer’s simultaneously fitting the corresponding scattering curve (Dobson,2003;vonBergenetal.,2000)aswellastounwanted againsttheexperimentalone.Theproposedmethodisverified drug degradation (Morozova-Roche & Malisauskas, 2007; using simulated data sets of various one-dimensional struc- Vestergaard et al., 2007). Consequently, a comprehensive turesandwesubsequentlyapplyittoexperimentallyobtained understanding of these systems requires structural character- SAXSdata.Thefullalgorithm isimplementedinacomputer ization at the (supra-)molecular scale. Small-angle X-ray program(SasHel),whichalsoincludesanoption forglobular diffraction of aligned fibres [e.g. the famous studies of the geometries. The reduction of the system’s complexity by structureofDNA(Wilkinsetal.,1953;Watson&Crick,1953)] symmetricprojectionandthefastcodeimplementationresult is a powerful technique for this purpose. However, the in a toolkit that allows a full shape retrieval from scattering evaluationofsolutionscatteringdatafromrandomlyoriented dataintheorderof3–90minonastandardworkstation(see helicalstructuresfacestwochallenges:(i)onlyafewanalytical Appendix A),depending on the level ofdetail. models are available from which structural information, such aspitchandtwist,canberetrieved(Schmidt,1970;Pringle& Schmidt,1971;Hamley,2008;Teixeiraetal.,2010);and(ii)the 2. Projection scheme endless nature of helices does not allow DA modelling using current programs (Volkov & Svergun, 2003; Gingras et al., Dummy-atom modelling is based on the idea that a given 2008). particleshapeisrepresentedbyanassemblyofNsmallbeads In this work, we present a bead-modelling algorithm to ofequalscatteringlengthdensity,calleddummyatoms(DA). determine the structural motif of monodisperse systems, AccordingtotheDebyeformula(Debye,1915),thescattering highly elongated in one dimension, in solution from SAXS intensityofthisassemblycanthenbecalculatedfromthebead data. Weuse symmetrical boundary conditions to project the assembly as 391 IUCrJ (2018).5,390–401 BurianandAmenitsch (cid:2) Dummy-atommodelling research papers (cid:2) (cid:3) IðqÞ¼f ðqÞ2XN XN sin qrij ; ð1Þ Fig.1(NBB=400,M=15),thisincreasesthecalculationspeed DA qr approximately seven fold. i¼1 j¼1 ij Thenotionofreducingagivenmodeltoitsstructuralmotifs where the norm of the scattering vector is denoted as q = raisesacriticalpointregardingtherequirednumberofstacks 4(cid:2)sin((cid:3))/(cid:4) (2(cid:3) is the scattering angle and (cid:4) is the radiation M necessary to represent a seemingly endless structure. This wavelength)andf (q)representstheDAformfactor(given number defines the length of the entire projected model L, DA inthisworkbyaGaussiansphereapproximation;Koutsioubas simplyviaL=MH .Accordingtoscatteringtheory,rod-like BB & Pe´rez, 2013; Svergun et al., 1995; Grudinin et al., 2017; structurespresentacharacteristicq(cid:3)1power-lawbehaviourin Fraseretal.,1978).ADAdiameterof0.2nmwasusedforall the intermediate Porod regime (Glatter & Kratky, 1982), reconstructions. As this formalism considers the distances whereas the transition from this Porod regime (q(cid:3)1) to the betweenallpossibleDApairsr =|r (cid:3)r|insidetheassembly, adjacentlower-angleGuinierregime(q(cid:3)0)occursataspecific ij j i its mathematical complexity is OðN2Þ. Hence, the computa- scatteringvectorq dependingonthelengthoftherod,which 1 tional effort increases drastically for large N. Adequate canbeestimatedusingq =181/2/Linthesimplifiedframework 1 modelling of seemingly infinite rod-like geometries requires of the Hammouda model (Hammouda, 2010). Thus, if the very large numbers of DAs (N > 10000), making the Debye experimental data present a q(cid:3)1power law at even the smal- formula appearinadequate in its standardnotation. lestaccessiblescatteringangleq ,thelengthofthestructure min Rod-likestructures,such ashelices,often possess acertain must be larger than L > 181/2/q . Similarly, we use this min structural motif, a building block, which recurs along the relationtodeterminetheminimumnumberofstacksrequired, elongation direction. Hence, only this building block is of according to M > 2 + 181/2/(H q ) (with the additional BB min interest for shape reconstruction. We define a building block term of+ 2to avoid truncationeffects). ofN dummyatomswithitselongationdirectionparallelto BB the z axis (see Fig. 1). The full rod-like structure is then 3. Determination of the stacking distance reproduced by duplicating the building block M times along thezdirection,withastacking distancecorresponding tothe The projection scheme is a faster alternative for calculating buildingblock’sheightH (seeFig.1).Theevaluationofthis the scattering intensity of a stacked structure compared with BB stackedmodelbymeansoftheDebyeformula[equation(1)] the standard Debye formula. As is apparent from equation now includes redundant terms, as distinct motifs inside the (2),theformalismrequiresanadditionalinputparameter,the structurearenowrecurrent.Forinstance,thesinglebuilding- stacking distance between the building blocks H . In the BB block motif would be evaluated at each repetition along z, following, we present a pathway to determine this using the such that the same computation is performed a total of M pairdistance distribution function (PDDF). times. It is therefore sufficient to calculate the scattering ThePDDFcorrespondstothesurface-weightedprobability intensity of the building block only once and scale it by the to find two points separated by a distance r inside a particle numberofstacks.Thesameprincipleappliestointer-building- (Glatter&Kratky,1982).Itisthusahistogramofalldistances blockcontributions.Forexample,asthemotifofneighbouring that appear inside a given particle, weighted by the corre- building blocks can be found (M (cid:3) 1) times in the full struc- spondingelectrondensities.InthecaseofDAassemblies,the ture,wecanevaluatethismotifonceand,again,multiplyitby PDDF can be also interpreted as the volume-weighted pair the corresponding scalar. correlation function p(r). Inmathematicalterms,theDebyeformulacanbeadjusted An analytically available example of this definition yields toneglectthese redundancies,resulting in thecase ofstacked spheresusingtheformalismbasedonthe workofGlatter(1980),fromwhichaseriesofconclusionscan IðqÞ¼fDAðqÞ2(MXNBB XNBB sinq(cid:2)(cid:4)(cid:4)qr(cid:4)(cid:4)r(cid:3)j(cid:3)rr(cid:4)(cid:4)i(cid:4)(cid:4)(cid:3) bsyestdermawonf(tseene Fstiagc.kSe1dinsphtheeressu).ppEovritdinegntliyn,foarsmathtieonsyfsotrema i¼1 j¼1 j i þ2MX(cid:3)1ðM(cid:3)kÞXNBB XNBB sin(cid:5)(cid:4)q(cid:2)(cid:4)(cid:4)(cid:2)rjþk(cid:4)HBBe(cid:3)z(cid:3)(cid:3)r(cid:4)i(cid:4)(cid:4)(cid:6)); cpoenaktasi.nTshteefinrsstppheeareks(bthlueectorarrceesipnoFnidgi.nSg1)PrDelDatFesptroetsheentmseteann q(cid:4) r þk(cid:4)H e (cid:3)r(cid:4) shape of all the spheres involved – it is hence an intra- k¼1 i¼1 j¼1 j BB z i building-blockPDDF.Thefollowingninepeaks(redtracesin ð2Þ Fig. S1) are caused by the repetitive nature of the system, as wheree denotestheunitvectoralongthezaxis(elongation/ for each possible sphere-to-sphere distance a new peak is z stacking direction). This formalismreducesthecalculationof found – they are hence inter-building-block PDDFs. These the scattering intensity to the sum of structural motifs found inter-building-block PDDFs, in particular the distances inside the stacked model (see scheme in Fig. 1). It further between them, therefore hold information on the stacking bypasses the demand for a DA model to represent the full distance between the building blocks. structure,asweevaluateonlyprojectionsofthebuildingblock In order to determine the stacking distance of a structure instead of all stacked duplicates, hence the term ‘projection fromagivenPDDF,itwouldbeintuitivetomeasurethepeak scheme’. Its benefit, compared with the standard Debye distances. However, the exact peak shapes and therefore formula, is a reduction in the mathematical complexity from positionsaredistorteddueto(i)thelinearhigh-rdecayinthe O½ðMN Þ2(cid:5)toOðMN2 Þ.Inthecaseoftheexampleshownin PDDF(seedashedlinesinFigs.S1andS2)and(ii)theoverlap BB BB 392 BurianandAmenitsch (cid:2) Dummy-atommodelling IUCrJ (2018).5,390–401 research papers of neighbouring peak contributions (Glatter, 1980; Feigin & needs to be minimized, where (cid:6) (q ) denotes the experi- exp m Svergun,1987).Adirectmeasurementofthepeakpositionsin mental error and N the number ofdata points. exp the PDDF might therefore result in misdetermination of the In DA modelling, the principle idea behind this minimiza- stackingdistance.Astableapproachtocircumventthisissueis tionprocedureisstraightforward:agiveninitialconfiguration to calculate the derivative of the PDDF (dPDDF). This is gradually altered until the best agreement between the numerically fast and easy operation suppresses the above- corresponding scattering curves is found. Here, we randomly mentioned decay distortion (see dPDDF in Fig. S1). The fillaninitialbuilding-blockvolumewithDAsthataresetfree resulting dPDDF can then be fitted by e.g. a damped sinu- at the start of the fitting procedure (500–1500 DAs per soidal function in which the period is directly related to the building block are suggested, depending on the experimental mean stacking distance (see Fig. S1 and Section S1 in the resolution – see Section S2). In order to optimize the target supporting information). function (cid:5)2, various metaheuristic methods exist such as Amorecomplexexampleillustratingthenameddistortion SimulatedAnnealing(Kirkpatricketal.,1983)ortheGenetic effects is the case of a torus (a ring with a circular cross Algorithm (Mitchell, 1996). In our case, we employ a meta- section; Kawaguchi, 2001). Such a torus presents two char- heuristic fitting procedure (Boussaı¨d et al., 2013; Gogna & acteristic dimensions: the ring–centre diameter and the ring Tayal, 2013) with an antifragile implementation (Taleb & thickness.Asaresult,thePDDFofsuchasingletorusexhibits Douady, 2013) and the algorithm improves (cid:5)2 by randomly twodistinctpeakscorrelatingtothesestructuralfeatures(see moving single DAs. In contrast with current DA modelling Fig.S2;themodelscatteringdata,includinganartificialerror programs,theDAsdonotmoveonanunderlyinggrid.Asthe band,werecalculatedaccordingtoAppendixA).Considering magnitudeofthemovementisscaledbyatemperaturefactor, now the case of multiple stacked rings (Kornmueller et al., we force the system to freeze eventually in a given condition 2015),foreachringthatisaddedontopoftheother(s)wefind andthe algorithm toconverge. anewpeakinthecorrespondingPDDF(seeFigs.S2andS3). DA modelling faces the general problem of uniqueness: as However, in this case, the radial size of the tori (diameter modelsconsistof>103DAs,theinformationcontentgivenby 20nm) was selected to be larger than the stacking distance the scattering data is highly overdetermined by a given between them (7nm), resulting in an increased distortion of configuration.Fittingofthescatteringdatawithoutrestraints the inter-building-block peak positions [see distortion canthereforeleadtophysicallyunfeasibleresults,inparticular phenomenon(ii)above,aswellasFig.S2].AsshowninFig.S2, with regard to the model homogeneity and compactness. the determination of the stacking distance by fitting of the Currentalgorithmscontrolandoptimizethesetwoproperties dPDDF yields astable result. during the fitting process by means of a ‘looseness penalty’ Weemploythisapproachtodeterminethestackingdistance regularization term as a quantitative measure of the DAs’ from all repetitive structures presented in this work. The local vicinity: by counting and maximizing the number of advantage of this choice is that we avoid the use of a contacting neighbours of each DA, a compact and homo- complementary characterization technique, as the underlying geneous configuration is achieved (Svergun, 1999; Franke & PDDFcanbedeterminedfromagivendatasetusingavariety Svergun,2009;Koutsioubasetal.,2016).Weadaptthisproven ofavailablesoftwarepackages.Amoredetaileddiscussionof technique to our grid-free algorithm in two ways. First, we thelimitationsandpossiblecausesofmisdeterminationcanbe introducetheparameter d ,denotingthedistancebetween N12 found in Sections S1.2and S1.3 and Figs.S1–S6. agivenDAandits12(close-packedlimit)nearestneighbours. Similar to the looseness penalty, d quantifies the local N12 vicinityofeachDA,actingasahomogeneityclassifierforthe algorithmtodecideifagivenDApositionisacceptedornot. 4. Fitting algorithm Second, we apply the idea of a regularization term and According to the presented projection scheme, we reduce a introduce a ‘radial compactness’ parameter RC(X) into the seeminglyinfinitestructuretoabuilding-blockmotiftogether minimizationprocedure,keepingtheDAclosetothe(radial) with two boundary conditions,the stacking distance H and BB centre ofmass of the configuration. the number of necessary stacks M. As these two values are As the choice of a grid-free DA algorithm is a departure determined directly from either the PDDF or the scattering from current implementations, we face challenges for which data, the modelling occurs within the single building-block no readily available solutions exist. In particular, we must volume. We thus represent the building-block motif by a introduce new concepts, (i) to obtain a hard-contact limit for dummy-atom configuration X such that the scattering inten- neighbouring DAs, (ii) to allow model scalability over sity I (q ) (where q denotes the q value of the measured calc m m different length scales and (iii) to generate a random move- data points) can be calculated using equation (2). To find a mentdependingonthecurrentannealingtemperature.Inthe configuration that best fits the experimental scattering inten- following subsections we discuss these topics in more detail, sity I (q ), the relation exp m startingwiththeintroductionofthescalingparameterD and X the random-movement generator, and then moving to the 1 XNexp"I ðq Þ(cid:3)I ðq Þ#2 mathematical definitions of d and RC(X) regarding (cid:5)2 ¼ exp m calc m ; ð3Þ N12 N (cid:6) ðq Þ homogeneity and compactness. A general overview of these exp m exp m conceptsandtheirparametersisgiveninTable1.Attheend 393 IUCrJ (2018).5,390–401 BurianandAmenitsch (cid:2) Dummy-atommodelling research papers Table1 Main concepts of the proposed grid-free fitting algorithm, the corresponding mathematical parameters, their function and their implementation comparedwithcurrentsolutionsinfixed-gridDAmodellingprograms. Fordetailsofcurrentfixed-gridmethods,seeSvergun(1999),Franke&Svergun(2009)andKoutsioubas&Pe´rez(2013). Concept Fixed-gridmethods Parameter Function Implementation Scalability Gridsize D Scalingparameter d ,RC(X),randommovement X N12 generator Homogeneity Fixedgrid/loosenesspenalty d AvoidDAclustering/disconnected Movementclassifier/forcedrecon- N12 DAs figuration Compactness Loosenesspenalty/limitedsearch RC(X)/D KeepDAscloseto(radial)centre Regularizationterm/forcedrecon- X,crit volume ofmass/avoidmodelexplosion figuration of this section, we further address the topic of model Alongthezaxisweintroduceacontinuitycondition(seethe uniqueness and repeatability and give a conclusive overview limitedbuilding-blockheightinFig.1)suchthatDAsleaving ofthe algorithm’simplementationin the programSasHel. the building block in the vertical direction are re-projected back inside the building-blockvolume. 4.1. Scalability – scaling parameter D X Allowing DAs to move randomly without an underlying 4.2. Random-movement generator grid brings an advantage in terms of the model volume and As we pursue the concept of grid-free random DA move- therefore the radial size. In a fixed grid system, the final DA mentstofindanoptimalconfiguration,werequirearandom- densityispredefinedbythegridresolution,whichisrelatedto movement generator that depends on a series of parameters, theresolutionofthescatteringpatterninreciprocalspace.In including (i) the dimensions of the current DA configuration the case of reconstruction of a highly elongated structure, (H and D ), (ii) the current temperature T of the fitting BB X radial expansion of the DA configuration during the fitting process(1>T>0)and(iii)thehelicalchiralityofthesample process is only possible by the addition of new DAs, hence (helicalfieldbias(cid:7)).Inevery numericaliterationthroughout increasing the numerical overhead (N ’ D2 for radial DA X thefittingprocess,achosenDAismovedalongtheunitaxes growth).Radialcompression,ontheotherhand,islimitedby e , e and e by the random vector r (T), where x y z rand the grid resolution, such that at a certain point the grid type (e.g. hexagonal packing of the DAs) may impose artificial 0 h (cid:7) (cid:8)i 1 D T Rand þ(cid:7)Tcos 2(cid:2)zi e shtotirgubhceltyumeroalorlenfgecaoattaeurdrseesstt.rhuTachntuistrheiess,ewpxaphreetrriciemutlehanerltygarlitldrimureeitsfoaolsruattihoreenasicsoanlsiekabeollyef rrandðTÞ¼BB@DXXThRandxyþ(cid:7)Tsin(cid:7)H2H(cid:2)BBzBBi(cid:8)ieyxCCA: ð5Þ (cid:8)H TRand e BB z z computationaloverhead(N <10000)mustbemaintained. DA Our grid-free approach does not present such limitations: as Here, a random number generator Rand returns any value thenumberofDAsperbuildingblock(mainlydeterminedby between(cid:3)1 2hd i that are As an ultimate radial boundary, similar to the search- N12 N12,X outside the mean radial distance h|r|i to move towards the volume diameter in DAMMIN (Svergun, 1999), we apply a X centreofmassoftheclosestfractionofDAs.Thisavoidsfree criticaldiameterD =2D suchthatsingleDAsmovingfar X,crit X floatingofasingleDAandthereforeforcestheDAstoremain away from the RCOM are repeatedly forced into a more inacompactconfiguration.Adetaileddescriptionofhowthe compact configuration (see Section S3 and Fig. S27 for a d parameterisimplementedinthefittingalgorithmcanbe detailed description of how the radial compactness is imple- N12 found in Section S3 and Fig.S27. mented in the algorithm). 395 IUCrJ (2018).5,390–401 BurianandAmenitsch (cid:2) Dummy-atommodelling research papers 4.5. Model resolution and uniqueness starting configuration according to a user-defined diameter, the building-block stacking distance and the number of DAs The most obvious visual side effect of using random per building block. Once started, the fitting algorithm under- movementsisrelatedtotheoutersurfaceofthefittedmodel. goesN iterations:startingfromatemperatureT ,thesystem For example, if DAs can only move on an artificial grid, the k 0 coolsdownasdefinedbythequenchingcoefficientq (0

Dummy-atom modelling of stacked and helical nanostructures from solution scattering data

Similar Dummy-atom modelling of stacked and helical nanostructures from solution scattering data