Cross-Validation of Data in SAXS and Cryo-EM

Transcription

1 2015 IEEE Intenational Confeence on Bioinfomatics and Biomedicine (BTBM) Coss-Validation of Data in SAXS and Cyo-EM Bijan Afsai Jin Seob Kim Gegoy S. Chiikjian Cente fo Imaging Science Johns Hopkins Univesity Baltimoe, Mayland Depatment of Mechanical Engineeing Johns Hopkins Univesity Baltimoe, Mayland Depatment of Mechanical Engineeing Johns Hopkins Univesity Baltimoe, Mayland Abstact-Cyo-Electon Micoscopy (EM) and Small Angle X-ay Scatteing (SAXS) ae two diffeent data acquisition modalities often used to glean infomation about the stuctue of lage biomolecula complexes in thei native states. A SAXS expeiment is geneally consideed fast and easy but unveiling the stuctue at vey low esolution, wheeas a cyo-em expeiment needs moe extensive pepaation and post-acquisition computation to yield a 3D density map at highe esolution. In cetain applications, one may need to veify if the data acquied in the SAXS and cyo-em expeiments coespond to the same stuctue (e.g., pio to econstucting the 3D density map in EM). In this pape, a simple and fast method is poposed to veify the compatibility of the SAXS and EM expeiments. The method is based on aveaging the 2D coelation of EM images and the Abel tansfom of the SAXS data. The esults ae veified on simulations of confomational states of lage biomolecula complexes. I. INTRODUCTION Cyo-Electon Micoscopy (EM) [1] and Small Angle X-ay Scatteing (SAXS) [2]-[4] ae two popula -but vey diffeentdata acquisition modalities used to glean infomation about the stuctue of lage biomolecula complexes in thei native states. Both methods diffe fom cystallogaphy in that no cystallization is needed (at the expense of lowe esolution). The advantage is that cystallization in many cases is eithe a vey lengthy pocess o may destoy biologically impotant featues. A SAXS expeiment is geneally consideed as fast and easy, but unveiling the stuctue at vey low esolution, wheeas a cyo-em expeiment needs moe pepaation and post-acquisition computation to yield a 3D density map at a highe esolution. The low esolution in SAXS can be oughly attibuted the fact that, the SAXS data is the spheical aveage of the scatteing patten of the complex unde study. In cyo EM, simila to standad tomogaphy, one obtains numeous 2D pojections of a 3D complex, but in contast to tomogaphy the pojections ae at andom (unknown) diections; moeove, the pojections ae extemely noisy. As a esult, the 3D volume econstuction in cyo-em involves complicated postpocessing and still the esolution is not vey high. In ode to investigate the elationship between the function and the shape of a biomolecula complex, one may want to combine these two expeimental data modalities (o fuse them). In such applications, one may fist need to veify if the data acquied in the SAXS and cyo-em expeiments coespond to the same stuctue (e.g., pio to econstucting the 3D density map). In this pape, we intoduce a simple yet effective method that enables fast veification of the compatibility of data collected in SAXS and Cyo-EM expeiments. Roughly speaking, we elate the plana coelations of EM images to the SAXS data. To the best of ou knowledge, ou wok is the fist attempt in establishing such a elation and the deivations in Section III-A ae new. The main benefit of ou appoach is that it enables the validation without the need fo aligning and classification of the EM images o 3D econstuction of the volume, which ae complicated steps [1]. The tanslation-invaiance popety of the coelation function is the key enabling facto to achieve this. The combination and validation of SAXS and EM data has been appeaed in the liteatue (e.g., [5], [6]). Howeve, such methods ae mostly based on image pocessing techniques o visual veification. Coelation functions in the context of EM data have been used befoe [7], but it became clea that fo econstuction of 3D density map the coelation function might not be adequate [8]. Hee, we pove that the coelation functions of EM images afte aveaging can be elated to the SAXS data via the Abel tansfom. This pape is oganized as follows: In Section II we eview the mathematical modeling the EM and SAXS data. In Section III the elation between the SAXS and EM data is established and an algoithm fo the validation is expessed. In Section IV we pefom some simulations to suppot ou appoach, and Section V concludes the pape. The focus of this pape is deiving the basic mathematical methodology and a plausibility study with simulated data. Applications with actual SAXS and EM data will appea late. II. THE EM AND SAXS DATA AND THEIR RELATION A. The Cyo-EM Data The eade is efeed to [1] fo detailed physical and mathematical analysis of cyo-em modeling and econstuction. Hee, we poceed with a vey basic mathematical model. We model the 3D atomic density of a lage biomolecula complex of inteest as a unifom density map x(), whee X : IR 3 -+ IR>o and (x,y,z)t E IR 3 [1]. Although any unifom density function IR 3 -+ IR ::o will wok hee, we specifically use x() as the chaacteistic function, which is defined as [9], [lo] x() { if EB if f- B /15/$ IEEE 1224

2 whee B denotes a biomolecula complex viewed as a solid body. With the chaacteistic function, a numbe of geometic quantities can be computed. Fo example, the volume of the complex body is computed as V(B) lir3 x()d 1d whee d dxdydz is the usual integation measue fo IR 3. In a cyo-em expeiment a fozen sample containing the complex is imaged. Inside the sample instances of the complex appea at andom positions and oientations. This can be modeled by andom igid body motion 9 (R, t) E SE(3), whee R E SO(3) is the otation component and t E IR 3 is the tanslation component; SE(3) and SO(3) denote, espectively, the Lie goups of igid body motions and otations in IR 3. A copy of the complex unde andom otation and tanslation 9 (R, t) E SE(3) can be modeled as 1 B Xg() X(g-l. ), (1) whee denotes the usual action of SE(3) on IR 3 defined as 9. R + t. In the pocess of imaging, essentially evey copy of the complex is imaged, and this is modeled by the pojection along the z axis in a global fame: (Xg()) P X (x, y) 1 X(g-l. )dz (2) Thus an EM image is the pojection of a andomly tanslated and oiented copy of the complex. In eality, the EM images ae highly noisy to the extent that signal-to-noise atio (SNR) of 1/100 (i.e., the noise enegy 100 times the signal enegy) is quite COlmnon. Moeove, othe effects such as the contast tansfe function of the micoscope futhe deteioate the images. Hee, we ignoe such effects. The standad econstuction of the 3D volume x() fom the 2D noisy pojections {X (x,y)}g is a complicated and lengthy pocess [1]. We show in Section III that cetain infomation that elates to the SAXS expeiment can be obtained fom the EM images with elatively simple opeations and low computational load. B. The SAXS Data Befoe pogessing futhe, we have to mention that the SAXS and EM data ae geneated based on diffeent physical pinciples and atomic popeties. Howeve, one expects that the infomation elevant to the geomety of a lage complex would not be much affected by this diffeence. Thus we postulate that both SAXS and EM expeiments yield infomation about the complex modeled by a unifom density X. Having this assumption in mind, we fomulate the mathematics of SAXS data. The data collected in a SAXS expeiment can be elated to the (spatial) coelation function of x() (also known as the Patteson function [4]), which is defined as Cx() x() * X( -) lir3 x(')x(' - )d', (3) whee * denotes the convolution opeation. Thus the coelation function Cx () is the convolution of X( ) with its eflection acoss the ongin. The coelation function fo a function defined on IR and IR2 is defined similaly. In the fequency domain we have Cx(w) Ix(wW, whee i(w) denotes the Fouie tansfom of f() and w E IR 3 is the spatial fequency vecto. Let us wite u, whee u E 2 ( 2 being the unit sphee in IR 3 ) and Illl E R So we wite Cx() Cx(u). Then the SAXS expeiments gives a pofile which (in spatial domain) can be expessed as (see [3] fo details): The meaning of this equation is that the SAXS data '/x () is the spheical aveage of the coelation function Cx(). Note that the pai distance distibution function px(), which is one of two key quantities in SAXS expeiments, can be witten as (see [4] fo details) The souce of equation (4) is that the copies of the complex x() ae andomly (unifomly) diected in the liquid sample. This is, in fact, elated to ou appoach in elating the EM data and the SAXS data. Note that we make a distinction between oientation, which is coded by a otation matix R E SO(3), and diection which is coded by a diection vecto u E 2. III. RELATION BETWEEN SAXS AND EM DATA A. Relating the SAXS and EM Data via the Abel Tansfom Let us denote the tanslated vesion of x() (i.e., x( - o» as Xo ' whee o is the tanslation vecto. We stat by noting that Cx() CXo (), i.e., the coelation function is invaiant unde tanslations. This holds both in the case of ID o 2D coelations. Thus the plana (2D) coelation of an EM image X (x, y) (see (2) fo its definition) only depends on R E SO(3), the otation pat of 9 E SE(3). Theefoe, we wite Cx (x,y) Cx (x,y) X (x,y) * X (-x, -y) (5) 1 x (x',y')x (x' - x,y' - y)dx'dy' x',y' whee X is defined by setting 9 (R,O) in (2). We also note that pojection opeation COlmnutes with convolution and coelation opeations. This follows fom the Fouie slice theoem [11]. Theefoe, using the tanslation-invaiance popety and the commutativity popety we can wite: (CXg())P Cxl (x,y) X (x,y) * X (-x, -y), (6) whee P denotes the pojection opeation as defined in (2). Next, we assume that the R component of 9 is unifomly distibuted on SO(3), i.e., the EM images ae coming fom unifomly oiented copies of the complex. This assumption is impotant in ou deivations. Howeve, we stess that it is known that, in pactice, the unifomity assumption may be violated, as thee ae the so-called pefeed oientations that most copies of the complex assume in the fozen sample [1, ch. 3]. The existence of pefeed oientations is a quite complicated (4) 1225

3 phenomenon; thus hee we etain the unifomity assumption and leave studying the effect of pefeed oientations and the extent to which the unifomity assumption can be elaxed to a late wok. Now, unde the unifomity assumption, by aveaging all the plana coelations acoss all oientations R we get the ciculaly symmetic function: ER{ Cx)'z (x, y)} 1'xP (J x2 + y2) 1'xP (), (7) whee, hee, with some abuse of notation, J x2 + y2 and ER denotes expectation (aveage) with espect to the andom vaiable R. By the commutativity popety we have 1'xp() ER{Cx)'z(x,y)} (ER{CxR()})P, (8) that is, 1'xP () is the aveage of the coelation of XR acoss all (andom) oientations R of the complex. In the above, we also used the fact the mathemtical expectation and pojection opeations commute. The next step is to elate the above aveaged plana coelation to the SAXS data 1'x() in (4). Notice that 1'x() is the aveage of CXR acoss the coset 2 SO(3)/ SO(2). But unde the unifomity assumption 1'x () is not diffeent fom the aveage of CXR() ove SO(3), namely, ER{CxR()}. Thus it follows that, unde the unifomity assumption, the EM aveaged coelation 1'xP () in (8) is equal to the pojection of the SAXS data 1'x(), when viewed as a 3D spheically symmetic function. This pojection can be expessed in tems of the Abel tansfom [12, Ch. 9]. The Abel tansfom of a one dimensional function f : [0,00) -7 IR is defined as Abel(f)() 21 f(')' d'. (9) v,2-2 This elation is easy to pove using Pythagoas's theoem and a simple change of vaiable. Fom ou discussion it follows that: 1 1'x(')' 1'xp() Abe1bx)() 2 d,, (10) v,2-2 which means that the aveage of the coelations of EM images equals the Abel tansfom of the SAXS data. In pactice, we expect this equality to hold up to a scale, due to the fact the souces fo EM imaging and SAXS expeiment have diffeent amplitudes. Moeove, we ae tacitly assuming that the SAXS and EM data ae based on the same physical popeties of the complex. In eality, howeve, the EM images ae fomed based on the scatteing of electon beams by, pimaily, the nuclei of the atoms, wheeas the SAXS data is fomed based on the diffaction of X-ay beams by electons in the atoms. B. Evaluating the Abel Tansfom: Accuacy Issues Ou appoach equies evaluating the Abel tansfom of the SAXS data 1'x(). In the Abel tansfom, the integand is singula at ' i- O. This can cause some eo in evaluating the tansfom, especially when the SAXS data 1'x () is known with only finite adial esolution o noise is high. Note that fom (10) fo 0, the singulaity is emoved, and Abelbx)(O) 2 fo 1'x(')d'. In geneal, thee ae two appoaches in evaluating singula integals: eliminating the singulaity with a change of vaiable and ignoing the singulaity [13]. The fist method, essentially equies knowing the integand with infinite accuacy, which is impactical in ou application. Howeve, the method of ignoing-the-singulaity is pactical, and as its name suggests equies no special povision. The caveat, howeve, is that the pesence of a singulaity advesely affects the ate of convegence in tems of the integation step-size [14]. We elaboate on this issue (following [14]). Let f : (0, 1] -7 IR be continuous on (0, 1]. Assume that f can be witten as f() CXg() with ex > -1, g() being a function whose deivative on [0, Ii exists, is continuous and integable in absolute value, i.e., fo If Id < 00. Notice that unde these conditions the integal fo f ( )d exists. Such a singulaity is called an algebaic singulaity [14]. In the case of the Abel tansfom we have ex -, as fo most densities Cx() and, hence, 1'x() ae smooth. Now, let us take the simple ectangle integation ule In(f) L 1 f(;), the step-size being h. Note that, in this appoximation, we ae ignoing the singulaity at 0 (e.g., instead of witing L - 1 f(;), which obviously cannot be calculated). The appoximation eo is En I f0 1 f()d - In(f)l. It can be shown that En n 0 with the ate ( ) Hcx [14]. This means that fo the Abel tansfom the ate of convegence will be )n. It is inteesting also to mention that, even moe complicated integation methods such as the midpoint, tapezoid, o quadatue ules have the same ate of convegence [14], although the actual eo fo a given step-size might be diffeent. To put this ate of convegence in pespective, note that if f is continuous on [0,1] (no singulaity, ex 0), then the ate would be. Also ecall that in the case of a function with smooth second ode deivative on [0,1], the convegence ate of the tapezoid and the midpoint ules impove to,&. Finally, we stess that the appoximation eo always exists (due to the discetization in computing the integal), howeve, in this case, due to the singulaity it is wose compaed with the case of a nonsingula integand. Next, we pefom a numeical expeiment in the case of the SAXS data of a spheical object. Conside a spheical object of adius R and with unifom density, i.e., {I if xs() xs() 0 Illl( ) R, othewise. Fo such a body the coelation function is given by CXs() Cxs() { l 7f(4R+)(2R- )2 if 2 Illl( ) 2R, othewise. (11) (12) This fomula is deived using the fomula fo the intesection volume of two sphees. Due the spheical symmety of Cxs ' we have 1'xs () Cxs (). We use the ectangle integation method with vaious step-sizes to evaluate the Abel tansfom of Cxs () with R 100. Figue 1 shows the esults fo stepsizes h 1,1/2,1/4,1/8,1/16,1/32,1/64. Also the exact 1226

4 cuve (calculated using Matlab's adaptive step-size integation method) is shown. Clealy the slow ate of impovement in the accuacy matches the theoy explained above. Abel tansfom of CXs (R100), ectangle integation method 10 ' h1/2 h1i4 ----h1/8 - - h1i h h1/64 --exact Fig. 1: The effect of integation step-size in evaluating the Abel tansfom of Cxs () (12). In ou applications, we cannot have vey small step-sizes because usually the SAXS data is available on a finite gid and additionally the values at the gid nodes can be noisy. Notice that an intepetation of the above eo analysis is that if we have I'x () at a esolution of we get its Abel tansfom with an eo of ode In. Howeve, this also means that any eo in I'x() may be amplified in Abelbx)(). Thus the net effect the singulaity is that we have to accept some modeate eo in evaluating the Abel tansfom of I'() and hence in the matching between the EM pofile I'xP () and the SAXS pofile Abelbx)() (equation (10». This may hinde ou ability to distinguish between complexes that have vey simila SAXS data I'x(). C. Removing the Noise in EM Plana Coelations The noise in an EM image can be extemely stong and we need to devise methods to mitigate its effect. Let us denote the noisy vesion of the image X (x, y) by X (x, y). We assume an additive noise model (which is a easonable assumption in the EM imaging mechanism). Thus we can wite x (x, y) X (x, y) + v(x, y), (13) whee v(x, y) is the noise. If we assume that the noise is spatially white with zeo mean and vaiance O', and futhe that it is uncoelated with the image X, then we can see that if x y 0, othewise. (14) We have deived this equation unde the egodicity assumption meaning that statistical and spatial aveages ae equal, which in pactice holds to a good extent. Note that (14) means that if we know o can estimate the noise vaiance O', then we can emove the effect of noise fom the coelations Cx ' In pactice, we may be able to estimate O' fom pats of the image, whee most likely the actual pojection image of the complex is not pesent (e.g., the cones of the image). IV. EXPERIMENTS An impotant poblem in stuctual biology is to detemine the confomational states of a lage biomolecula complex (e.g., open vs. closed). Both SAXS and EM expeiments can be pefomed fo this pupose, and in some cases one may want to veify if the samples in the SAXS and EM expeiments contain the complex at the same confomational state. We pefom two simulated expeiments on a synthetic model and one on a simulated complex with ligand-binding domains. A. Confomational States: The Two-Body Model We simulate diffeent confomational states of a lage biomolecula complex by modeling the complex with two solid ellipsoids of the same size and vaying the (unknown) angle e between the pincipal axes of the two ellipsoids. We denote the complex at confomation associated with angle e by Xe. The goal is to decide whethe the EM and SAXS expeiments wee conducted on the complex at the same confomation in the espective samples, and also to examine the angula esolution that can be achieved. Befoe poceeding futhe, we stess that although ou mathematical models wee in spatially continuous setting, in the expeiments we move to a discete spatial setting. The data is geneated as follows. The axes lengths of each ellipsoid ae a 21.3, b c 6. 7 A. A volume V of size N x N x N (N 129) voxels is geneated that contains the complex Xe with confomational angle e. The complex Xe itself is modeled by a solid (unifom) body as explained ealie in Section II. To simulate the EM expeiment, the complex is otated (inside V) aound the cente of V by a andomly (unifomly) geneated otation matix R. Then a pojection along the z-axis is computed, which esults in an N x N image. The top two images in Figue 2 show andom pojections of confomational angles of e 30 and e 45. Next samples of zeo mean white Gaussian noise of vaiance O' is added to the images as shown in the second ow of Figue 2. In the figue, the signal-to-noise atio (SNR) is /0' Fo each image the SNR is defined as the aveage enegy of the pojection of the complex to the enegy of the noise in that image; and O' is detemined accodingly (the enegy of noise is N O' ). In this expeiment F 100 andom otation matices ae unifomly geneated on SO(3) and coesponding to each otation matix K 10 noisy samples ae geneated. Thus a total of 1000 noisy images ae geneated. Next, the spatial coelation of each plana image is found. Notice that this esults in a coelation image of size 2N - 1. To simulate the noise-emoval step descibed in Section III-C, a patch of size W x W pixels, whee W 35, at a cone of each image is selected and the vaiance of noise estimated in that patch. Then the vaiance of the noise is subtacted fom the coelation at (0,0) (see (14»; and all such denoised coelations ae aveaged. The esult is an almost ciculaly symmetic image. Howeve, we futhe sylmnetize it by cicula aveaging to get an appoximation of I' o (). The cicula aveaging is pefomed at athe high angula esolution. Since the images ae discete spatially, fo 1227

5 cicula aveaging one needs to pefom spatial intepolation. We also used intepolation to incease the esolution in the adial vaiable to H 0. 5A. This impoves visualization and smoothness of the esults. To simulate the SAXS data geneation we simply find the 3D coelation of the volume V and pefom a spheical aveaging (combined with spatial intepolation) to get an appoximation of 'Yxo() (see (4)). We obtain 'Yxo() with esolution of H 0. 5A. Finally, to estimate the SAXS pofile Abelhxo)(), we use the ectangle integation ule. The bottom left plot in Figue 2 shows the gaphs of the estimated EM pofiles 'Y o () and SAXS pofiles Abe1hxo)() fo confomational angles of e 30 and e 45. It can be seen that the coesponding SAXS and EM pofiles ae simila fo modeate to lage values of. Howeve, fo small values of the SAXS pofiles fo both 30 and 45 ae becoming close to each othe and deviate fom the EM pofiles. The fact that the SAXS pofiles fo small ae vey simila is expected, and eflects the fact that SAXS data can give low-esolution infomation about a stuctue. In fact, theoetically 'Y X 300 (0) 'Y X 450 (0) and 'Y X 300 () ;::::; 'Y X 450 () fo small since in both confomations the total volume is the same and the individual bodies ae the same. Thus effectively Abelhx300 )() ;::::; Abelhx450 )() fo small. The obseved discepancy between the EM and SAXS pofiles can be attibuted to both noise and the integation eo in evaluating the Abel tansfom. In the bottom ight plot of Figue 2 we have plotted the nomalized vesion of the EM and SAXS pofiles, whee each pofile integates to 1. Specifically, we define the nomalized pofiles as p,. n( ) 'Y o() Ab In( ) Abelhx)() 'Yx o oo p, e 'Yx. Jo 'Yxo()d Jo oo Abe1hx)()d (15) In pactice this nomalization is vey useful, pimaily because the EM and SAXS pofiles ae, at best, popotional to each othe and may diffe in thei scaling significantly since they come fom vey diffeent expeimental souces. Next we pefom a simila expeiment to estimate the smallest esolvable angle. Fo that we define a notion of distance between the SAXS and EM pofiles. Based on the fogoing discussion, the diffeence at small may be vey lage but with vey little elation to the confomational angles. Thus we choose a value o and we compae the pofiles fo ;::: o. We choose a simple L 2 -based distance as: d("vp,n IXOI' Abeln ("V IX02 )) 100 I'Y () - Abelnhxo2 )()12d. TO (16) Of couse, in pactice a discetized vesion of this will be used. In this expeiment, we chose the SNR much lowe at 160' At this extemely low SNR, no vestige of the (pojection of the) complex is visible. We geneate confomations with angles e E {30, 35,..., 70 }. Table I shows the enties of a distance matix, whee each enty is the distance defined in (16) between SAXS and EM pofiles at espective indicated angles. All the distances ae scaled by a fixed numbe fo ease of pesentation. We have chosen o ;::::; 20A in (16). As it can be seen, in some cases (undelined) the EM and SAXS pofiles of angles that diffe by 5 can be close than those of the actual (coect) angles. Howeve, fo diffeences lage than o equal to 10 such a confusion is not obseved. Ou expeiments show that in a lage numbe of tials confusion fo angles lage than equal to 10 is vey ae. Thus, the angula esolution is oughly ±5 with vey high cetainty. EM pojection fo two-ellipsoid model, EM pojection fo two-ellipsoid model, (}4So Noisy vesion of the above Image, SNR 1/10 Noisy vesion of the above Image, SNR 1/ EM and SAXS pofiles fo two-ellipsoid x10 ' model, 030 ', 45 ', SNR1/10 -EM pofile,930 "... SAXS pofile, 030 ' - -EM pofile,84s o ---SAXS pofile,d--4s " Nomalized EM and SAXS pofiles fo two ellipsoid model, 030 ', 45 ', SNR1/10 -EM pofile,o30 "... SAXS pofile, EM pofile,o4s o ---SAXS Ofile,94S o Fig. 2: The top ow shows andom EM pojections of the two body complex Xe at two confomational angles e 30 and e 45 (see Section IV-A fo moe details). The middle ow shows the same images contaminated by noise (SNR 1/10). The bottom gaphs show the estimated coesponding SAXS and EM pofiles (the left gaph shows un-nomalized pofiles and the ight one shows the nomalized pofiles-see (15)). B. Glutamate Recepto Ligand-Binding Domain Confomations Hee, we pefom a athe simila expeiment while we geneate the data fom Potein Data Bank (PDB) [15]. Specifically, we conside the ligand-binding domain (LBD) of glutamate ecepto, which is known to play a cucial ole in human bain activities such as memoy fomation and leaning pocess [16]. We conside thee confomational states of LBD: 'apo' o unliganded state (PDB enty: IFTO), antagonist-bound state (PDB enty: IFTL), and patial agonist-bound state (PDB enty: IFTK). It is expeimentally shown that the confomation of 'apo' state is moe simila to the antagonist-bound state than 1228

6 SAXS OA lao OA u u OAI OAI 65 u u TABLE I: A distance matix fo the two-body model expeiment in Section IV-A. Each enty shows the distance (16) between the SAXS and EM pofiles at the coesponding bodyangles. All the distance values ae nomalized by a common facto fo ease of pesentation. The undelined en ties show the cases whee the distances between the SAXS and EM pofiles of two diffeent angles give a distance smalle than the coect angle FTL.pdb Un-nomalized pofiles -SAXS_1FTl -SAXS_1FTK - - EM lftl - - B,(::lFTK o -10 1FTK.pdb Nomalized pofiles (By aea unde cuve) to the agonist-bound one [17]. As we will see shotly, the same conclusion can be dawn fom ou simulation esults. In ou simulation, all atom coodinates wee consideed. We simulate the effect of each atom by a 3A x 3A x 3A cube of constant intensity centeed at each atomic coodinate pesent in the coesponding PDB file of each complex. The est of simulation is as in the pevious example with volume size N 129, SNR1/100, numbe of andom otations F 100, numbe of images pe oientation K 10, and patch size W 35 fo denoising. In the fist expeiment we compae the SAXS and EM pofiles of complexes IFTL and IFTK. The top panels in Figue 3 show the atomic configuation of these two complexes. The left bottom gaph shows the unnomalized SAXS and EM pofiles fo both. The lage jump at 0 is due to eo in estimating the noise vaiance. As it can be seen the pofiles of distinct confomations ae easily sepaable despite the fact that the EM and SAXS of each complex diffe slightly (due to noise and the singulaity effect). Hee, howeve, this diffeence fo smalle values of is significant, pesumably because the total volume of IFTL and IFTK is substantially diffeent. In the next expeiment depicted in Figue 4 we compae the pofiles of IFTO and IFTL. As it can be seen the two confomations ae quite simila and thei coesponding pofiles also become indistinguishable, which confoms with the ealie mentioned fact that these two confomational states ae close. Note that in the cuent simulations, we used unifomly distibuted andom otations. In eality, howeve, it might be the case that thee ae pefeed oientations of the molecules in the sample gid [I, ch. 3]. In ode to test that, we tied non-unifom otations, such as the case whee thee ae finite fixed numbe of otational diections. Ou peliminay and limited investigation shows that esults such as those in Fig. 3 and 4 wee not influenced significantly by non-unifom otations. In ou futue wok, we will ty to undestand how these esults will be affected due to the pesence of pefeed oientations and the violation of the non-unifomity assumption (see III-A and [1, ch. 3]), togethe with applying ou method to lage and moe complex macomolecules (A) Fig. 3: Compaing SAXS and EM pofiles (Abelbx)() and 1'xP (), espectively) and thei nomalized vesions fo IFTL and IFTK, see Section IV-B fo details FTL.pdb Un-nomalized pofiles 0.08 I I 0.06 I 1FTO.pdb 60 Nomalized pofiles (By aea unde cuve) 0.04 " " (A) -SAXS_1FTL -SAXS_1FTO - - EM lftl - - EMlFTO Fig_ 4: Compaing SAXS and EM pofiles (Abe1bx)() and 1'xP (), espectively) and thei nomalized vesions fo IFTL and IFTO, see Section IV-B fo details_ V. CONCLUSIONS We have pesented a simple and fast appoach to check the compatibility of SAXS and cyo-em data_ By noting that the SAXS data is indeed deived fom the coelation function (i.e., the Patteson function, which is the self-convolution of the density function), we poposed a method based on the coelation functions both in SAXS and EM. In this pape, 1229

7 we deived the basic mathematical methodology. In paticula, we used the fact that aveaging the coelation functions of EM images (pio to 3D econstuction) is equivalent to the Abel tansfom of the SAXS data. Then we investigated the plausibility with simulated data and the actual PDB data. We also discussed limitations due to the singulaity in the Abel tansfom in SAXS and the noise effect in EM images. An impotant question is the effect of pefeed oientations in ou esults and simulations, which is the subject of ou futue eseach. The cuent study is a fist step towad the fusion of two diffeent expeimental modalities, which in tun will enhance the undestanding of the elationship between the function and the shape of lage bimolecula complexes. [17] D. R. Madden, N. Amstong, D. Svegun, J. Peez, and P. Vachette, "Solution X-ay scatteing evidence fo agonist- and antagonist-induced modulation of cleft closue in a glutamate ecepto ligand-binding domain," J. BioI. Chem., vol. 280, pp , ACKNOWLEDGMENT This wok has been suppoted by the National Institute of Geneal Medical Sciences of the NIH unde awad numbe ROIGMl The authos thank the anonymous eviewes fo insightful COlmnents. REFERENCES [1] J. Fank, "Thee-dimensional electon micoscopy of macomolecula assemblies: Visualization of biological molecules in thei native," [2] D. I. Svegun and M. H. Koch, "Small-angle scatteing studies of biological macomolecules in solution," Repots on Pogess in Physics, vol. 66. no. 10, p [3] L. Feigin, D. I. Svegun, and G. W. Taylo. Stuctue analysis by smallangle X-ay and neuton scatteing. Spinge, [4] D. I. Svegun, M. H. Koch, P. A. Timmins, and R. P. May, Small Angle X-Ray and Neuton Scatteing fom Solutions of Biological Macomolecules. Oxfod Univesity Pess, [5] B. Vestegaad, S. Sanyal, M. Roessle, L. Moa, R. H. Buckingham, J. S. Kastup, M. Gajhede, D. I. Svegun, and M. n. Ehenbeg, "The SAXS solution stuctue of RFI diffes fom its cystal stuctue and is simila to its ibosome bound cyo-em stuctue," Molecula Cell, vol. 20, no. 6, pp , [6] P. Bon, E. Giudice, J.-P. Rolland, R. M. Buey, P. Babie, J. F. Diaz, Y. Peyot, D. Thomas, and C. Gamie, "Apo-Hsp90 coexists in two open confomational states in solution." Biology of the celli unde the auspices of the Euopean Cell Biology Oganization, vol. 100, no. 7, pp , [7] M. Schatz and M. Van Heel, "Invaiant classification of molecula views in electon micogaphs," Ultamicoscopy, vol. 32, no. 3, pp , [8] M. van Heel, M. Schatz, and E. Olova, "Coelation functions evisited," Ultamicoscopy, vol. 46, no. 1, pp , [9] H. Dong and G. S. Chiikjian, "A computational model fo data acquisition in SAXS," in BDB '14 Poceeding of the 5th ACM Confeence on Bioinfomatics, Computational Biology, and Health Infomatics, 2014, pp [l0] H. Dong, J. S. Kim, and G. S. Chiikjian, "Computational analysis of SAXS data acquisition," 1. Comput. Bioi., vol. 22, pp , [11] F. Nattee, The mathematics of computeized tomogaphy. Siam, 1986, vol. 32. [l2] R. Bacewell, Fouie analysis and imaging. Spinge Science & Business Media, [13] P. J. Davis and P. Rabinowitz, Methods of numeical integation. Academic Pess, [14] M. EI-Tom, "On ignoing the singulaity in appoximate integation," SIAM Jounal on Numeical Analysis, vol. 8, no. 2, pp , [15] H. Beman, J. Westbook, Z. Feng, G. Gilliland, T. Bhat, H. Weissig, I. Shindyalov, and P. Boune, "The Potein Data Bank," Nucleic Acids Res., vol. 28, no. 1, pp , [16] R. Dingledine, K. Boges, D. Bowie, and S. F. Taynelis, "The glutamate ecepto ion channels," Phamacol. Rev., vol. 51, pp. 7-61,