The use of connected masks for reconstructing the single particle image from X-ray diffraction data

Transcription

1 Mathematical Biology and Bioinformatic V. 10. Suppl. P. t1 t19. doi: / t1 The tranlation of original article Lunin V.Y., Lunina N.L., Petrova T.E. Matematichekaya biologiya i bioinformatika V P doi: / ======================= MATHEMATICAL MODELING ====================== УДК: The ue of connected mak for recontructing the ingle particle image from X-ray diffraction data Lunin V.Y., Lunina N.L., Petrova T.E. Intitute of Mathematical Problem of Biology, Ruian Academy of Science, Puhchino, Mocow Region, , Ruia Abtract. The problem of recontructing the image of a ingle macromolecular object from X-ray diffraction data can be formulated a a problem of the recontruction of the 3D electron denity ditribution from the magnitude of it Fourier tranform. Thi problem can be reduced to a erie of tandard X-ray crytallography tak, namely, the recovery of a periodic function from it Fourier coefficient (tructure factor) magnitude, which are determined in an X-ray diffraction experiment. In thi work, a new approach to the olution of thee tak i uggeted which i baed on the ue of connected binary mak a an approximation of the required electron denity ditribution. The approach include the random generation of a great number of connected mak, the election of the mak that are in agreement with an experimental and a priori information about the object, and the alignment and the averaging of the phae et of the tructure factor that correpond to the elected mak. The averaged phae value together with the experimentally determined magnitude are ued for the calculation of the Fourier ynthei, which i applied for the viualization of the object under tudy. The approach can be ued in tudie of both ingle particle and crytalline pecie; however, it hold the greatet promie for invetigation of ingle object. The reult of teting the approach are preented. Key word: X-ray crytallography, phae problem, XFEL, cattering by a ingle particle. INTRODUCTION X-ray diffraction i the main method of determining the atomic tructure of biological macromolecule and their complexe. However, a eriou retriction of the method i that a ample for the X-ray experiment ha to be prepared a a monocrytal. Thi requirement i dictated by the fact that the intenitie of the wave cattered by a ingle molecule are many order of magnitude le than the intenity of the incident X-ray beam, which trongly complicate their regitration. The interference of wave cattered by a et of molecule that are in the node of the regular periodic grid lead, for ome particular direction, to the multiple amplification of econdary wave, which make their regitration poible. The development of new powerful ource of X-ray, uch a X-ray free electron laer, allow one in the nearet future to talk about X-ray diffraction experiment with ingle big molecule or complexe of molecule [1, 2]. Thi poibility make timely the development of approache to the determination of the 3D tructure of biological macromolecule baed on X- ray diffraction data obtained in the experiment with a ingle molecule. In the preent work, one poible approach i uggeted. The main tacle for the tructure determination from X-ray diffraction data i the phae problem. The experiment enable one to determine directly only the magnitude of the complex function, which i the Fourier tranform of the function that decribe the patial

2 LUNIN et al. ditribution of electron in the object under tudy (the function of the ditribution of the electron denity). The olution of the phae problem (calculation of the phae value of the Fourier tranform) make it poible to retore the electron denity ditribution by calculating the invere Fourier tranform uually named a the Fourier ynthei of the electron denity. It hould be noted that the olution of the phae problem in the cae of a ingle particle i a theoretically impler tak than in the cae of a crytal. During X-ray diffraction through a crytal, one can meaure the intenity of cattered wave only for a dicrete et of pecific direction (Bragg reflection), while, in a ucceful experiment with a ingle molecule, one can obtain the information about the intenity of cattering in all direction. The accuracy of the information obtained by calculating the ynthei of the electron denity depend on the number of reflection (in cae of a crytal) or the ize of the region of cattering angle (in cae of a ingle molecule) included in the calculation. Thi accuracy i characterized by the reolution value and depend on the limiting value of the cattering angle, for which it wa poible to regiter the intenity of cattered wave. A common practice in crytallography i a tep-by-tep progre in the determination of the electron denity ditribution from low to high reolution. It thi work, we dicu approache to the olution of the phae problem at the initial tage of the tudy, namely, in the zone of low and medium reolution (mall angle cattering). The information obtained allow one to determine the general contour of the molecule and i a tarting point for the further ue of the method for increaing the reolution. The author of the preent work uggeted a et of approache to the olution of the phae problem in the low reolution zone in the tudie of crytal tructure [3, 4]. Here, we attempt to extend thee approache [5] into the cae of the cattering by a ingle molecule. A it will be hown below, the tak of the recovery of the electron denity ditribution in cae of the cattering by a ingle molecule can be conidered a a erie of uual crytallographic tak relevant to imaginary crytal containing the molecule under tudy buried in a large volume of a olvent in the unit cell. An increae in the portion of the olvent in the imaginary cell potentially facilitate the olution of the phae problem but lead to a ignificant increae in the computing time. The method i baed on the ue of the compactne and connectivity feature of the region of high denity value of the electron denity in biological macromolecule. Earlier, the Monte Carlo type procedure wa uggeted [5], which conit of a random generation of phae et and the calculation of the correponding Fourier ynthee, the election of thoe phae et that provide the connectivity of the region of high electron denity value in Fourier ynthei, the alignment and averaging of elected phae et. Here, we invetigate an alternative approach to the ue of connectivity feature. In the new procedure, jut hypothetically connected region are randomly generated. For the further work, thoe region are elected, for which the magnitude of the Fourier tranform reproduce well the value obtained in the experiment. The phae et of the Fourier tranform correponding to the elected region are further aligned and averaged, which lead to the required olution of the phae problem. Thi approach demand much le computing calculation and provide a higher quality of the reulting phae et compared with the earlier ued approach [5]. 1. FUNDAMENTALS OF THE METHOD 1.1. Theoretical bai of the X-ray diffraction analyi A principal cheme of an X-ray diffraction experiment i hown in Fig. 1. An object under tudy i placed in an X-ray beam, and the intenitie of ariing econdary wave diverging in all direction are regitered. During the experiment, the object i rotated, which make it poible to obtain at the output a et of two-dimenional roentgenogram correponding to different orientation of the object relative to the initial beam. In the kinematic theory of diffraction, a primary beam i conidered a a plane monochromatic electromagnetic wave, and the interaction of electron only with thi primary wave i conidered. By the action of t2

3 THE USE OF CONNECTED MASKS FOR RECONSTRUCTINQ THE SINGLE PARTICLE IMAGE FROM X-RAY DIFFRACTION DATA thi incident wave, electron of the object begin to ocillate and become the ource of new pherical wave. Thee wave are ummed to form cattered wave. A complex magnitude of a cattered wave E differ from the magnitude of the primary wave E 0 by two multiplier: ( ) 0 E =F E. (1) The multiplier i equal to the portion of the energy flow of the initial wave that come with the cattered wave to the detector upon the cattering by only one electron. Thi portion i 12 very mall (it can be etimated to be ~ 10 ), which jut preent the main problem during F i called the tructure the regitration of the cattered radiation. The complex multiplier ( ) factor and i determined by the electron denity ditribution in the object that catter. Fig. 1. Scheme of an X-ray diffraction experiment. The value of the tructure factor depend on the direction of the cattering (more exactly, on the vector σ σ 0 =, (2) which i called the vector of cattering) and i related through the Fourier tranform to the electron denity ditribution ( r ) in the object under tudy: ( ) ( ) exp 2 (, ) F = r i r dv r. (3) The intenity of the cattered wave regitered in the experiment i proportional to the quare of the tructure factor magnitude, i.e., depend on the ditribution of electron in the object under tudy. Thi allow one to formulate the problem of determination of the ditribution I. ( r ) from the et of experimentally determined intenitie ( ) Theoretically, the electron denity ditribution can be recovered from tructure factor value through the invere Fourier tranform ( ) ( ) exp 2 (, ) r = F i r dv, (4) however, there are two tacle. Firt, an X-ray experiment enable one to obtain the value of only tructure factor magnitude and doe not allow one to meaure phae value. Second, it follow from formula (2) that the experiment allow one to obtain the value of tructure factor magnitude only for the vector that atify the retriction 2. Therefore, even with the known phae of the tructure factor, the invere Fourier tranform enable one to t3

4 LUNIN et al. obtain, intead of the exact ditribution of electron denity ( r ), only the Fourier ynthei of a finite reolution d ( ) ( ) exp 2 i(, ) r = F r dv. (5) 1 d In thi cae, a poible reolution d cannot be le than half of the wavelength 2. Let the object under tudy be a crytal; that i, there i a great number of identical imilarly orientated molecule placed in the node of the crytal lattice built on the bai a, b, c. A hift of the molecule poition in the pace by vector u lead to an appearance of an additional phae multiplier exp 2 i(, ) u in it tructure factor. Suppoe that vector u run through all node of lattice, and additional condition for vector are atified: ( ) ( ) ( ), a = h,, b = k,, c = l, where h, k, l are integer. (6) In thi cae, all additional multiplier in the tructure factor that correpond to ingle molecule will be equal to 1; and a multiple amplification of the cattered wave due to the ummation of identical contribution from a great number of molecule in the crytal will happen. The intenity of thi wave become big enough to be regitered in the experiment. If condition (6) are not atified, the phae of additional complex multiplier are almot random, and the tructure factor correponding to ingle molecule give zero a a reult of ummation. In thi cae, the intenity of the cattered wave become to be too mall to be detected experimentally. Condition (6) are called the diffraction condition (Laue-Bragg- Wulff), and the correponding cattered wave are called the reflection. Below, we will talk about the magnitude and phae of the tructure factor correponding to thee or other reflection bearing in mind tructure factor involved in equation (1) for thee wave. In the cae of a crytal, the formula for the calculation of tructure factor and electron denity have the form: ( ) ( ) exp 2 ( ) F = r i,r dvr, (7) V ( ) ( ) exp 2 i( ) r = F,r, (8) where V i a parallelepiped built on vector a, b, c (the unit cell of the crytal), and V i the volume of the unit cell. The ummation in (8) occur over all node of the integer grid * * * built on the vector of the bai a, b, c, which i conjugated to the bai a, b, c : * = ha + kb + lc where h, k, l are integer. (9) Note that thee are exactly the ame cattering vector for which the diffraction condition (6) are atified, that i, whoe tructure factor magnitude can be meaured in the experiment Scattering by a ingle particle. Reduction to crytallographic problem Let u go back to the cae of cattering by a ingle molecule. Let 0 ( ) p electron denity ditribution in a ingle molecule, and F ( ) factor calculated according to (3). Let u chooe an arbitrary bai,, r decribe the are the correponding tructure a b c in the pace. Let the vector be choen big enough for the molecule to be poitioned completely in the t4

5 THE USE OF CONNECTED MASKS FOR RECONSTRUCTINQ THE SINGLE PARTICLE IMAGE FROM X-RAY DIFFRACTION DATA parallelepiped built on thee vector. Among all tructure factor F( ) ubet correponding to the node of the integer grid t5, we will elect a abc = ha + kb + lc, where h, k, l are integer, (10) * * * where a, b, c i the bai conjugated to the bai a, b, c. Conider, finally, an imaginary crytal with the unit cell parameter a, b, c inide which there i one molecule with electron denity ditribution ( ) 0 r inide the molecule and the denity outide the boundarie of the molecule equal to zero. Structure factor for thi crytal have the form: p ( ) = ( ) exp 2 i(, ) dv = ( ) exp 2 i(, ) dv = ( ) cryt F 0 r r r 0 r r r F. (11) Vabc Thu, the etting of the Fourier tranform for a ingle particle on an integer grid abc i equivalent to the etting of tructure factor for an imaginary crytal tructure in which the particle i in the crytal unit cell with the parameter a, b, c. And, correpondingly, the problem of the recovery of electron denity from the magnitude of the Fourier tranform p F can be reformulated a the problem of the recovery of the electron denity ( ) cryt ditribution in the imaginary cell from tructure factor magnitude ( ) F :. abc Becaue of thi, a vat majority of method of the olution of the phae problem in protein crytallography could be applied to the determination of the tructure of a ingle particle. It hould be noted that, with thi reduction of the problem to the problem of the macromolecular crytallography, we ue only a part of the potentially available information, i.e., only the Fourier tranform magnitude taken in the node of the grid (10). Both the bai e1, e2, e 3 and the length a, b, c of the unit cell can be elected by a variety of way. The data on the cattering by a ingle particle contain much more information than a uual crytallographic problem; and thi give more poibilitie for the olution of the phae problem. Although, the formulation of the problem uggeted in the previou ection i formally imilar to the ordinary crytallographic problem, it ha an eential difference. With large value of abc,, parameter, we obtain a unit cell in which only a mall part i occupied by an unknown electron denity, and denity value in the greater part of the unit cell are known (equal to zero). Thi make the problem of the tructure determination overdetermined [6] and make poible it olution. A ignificant tacle for the olution i that we do not know in which particular point the denity i equal to zero. If thi information (a mak of the molecule region) i obtained by ome way, different iteration procedure can be applied for the olution of the phae problem [6 8]. In thi paper, we dicu the way to obtain thi mak. It hould be mentioned that, by increaing the cell parameter, we involve a great deal of experimental information (a greater quantity of reflection) in the work and deal with a greater number of tructure factor of the imaginary crytal (with the reolution being the ame). On the other ide, thi increae lead to an increae in the portion of the olvent in the unit cell, which increae the information redundancy and can facilitate the olution of the phae problem Phae problem. Ab initio method of the olution One of the central problem of the crytal tructure determination i a lo in the experiment of half of the neceary information. For the recovery of the electron denity uing (8), both the magnitude and phae of tructure factor are neceary; an experiment allow one to obtain only value of the magnitude. The lack of the information ha to be compenated for by ome other information about the object. A a rule, either the

6 LUNIN et al. experimental data obtained in additional experiment (with iomorphou derivative or at different wavelength of X-ray) or the known tructure of a homologou protein are ued a an additional information. A pecial group of method are o-called direct or ab initio method in which additional general information not relevant to a particular object under invetigation i ued. An example of thi information i the connectivity of the region of high electron denity value [4, 9]. For uing thi property, a Monte-Carlo type procedure wa developed, and the complex of the program GENNEM wa created in which thi procedure i implemented. The procedure conit of a few tep. In the firt tep, a hypothetical et of poible value of tructure factor phae i randomly generated. The phae with experimental tructure factor magnitude are ued to calculate the Fourier ynthei of the electron denity. From the Fourier ynthei, a region of high denity value i determined (a cut-off level i a parameter of the method), and the number of the connectivity component of thi region i determined. If the number of thee component i no greater than the given number, the phae et generated i conidered a admiible and i elected for further analyi. The generation of the phae et i repeated until the number of elected admiible phae et reache a given number. At the econd tep, the phae et elected are aligned [10, 11]. The neceity of the alignment i dictated by the fact that the tructure factor magnitude do not fix the choice of the origin. The electron denity ditribution of the initial object and that of the ame object but hifted by ome vector have identical et of tructure factor magnitude (but different et of phae). Therefore, with the random generation of the phae et, a ituation may arie that two phae et generated would lead to the ame (or very imilar) image of the molecule but formally different phae value. Thi difference can be eliminated by a hift of the econd image (and correponding tranformation of the phae). After the alignment of all phae et, the phae value for each tructure factor are averaged, and an indicator of the diperion of the phae value in different et (a figure of merit) i calculated. The mean value and the figure of merit (a correcting multiplier) are ued to calculate Fourier erie (8), which i a current approximation to the olution of the problem of determining the electron denity in the object. The approximation found can be ued for the modification of the firt tep of the procedure. Random phae value can be then be generated with allowance for the information about their mot probable value. The procedure can be repeated everal time, with varying individual probability ditribution upon the generation of the value of tructure factor phae until the attainment of convergence. An additional criterion for the election within thi procedure can be retriction upon the molecule ize; thu, it i required that the region of high value wa within the parallelepiped of the given ize Uing mak of the molecule region for the olution of the phae problem In cae of low and middle reolution, the main information extracted from the electron r i uually a mak of the molecule region (characteritic function). denity ditribution ( ) Thi mak i uually defined a a region of the highet value on the Fourier ynthei of the electron denity (referred below to a the region of high value, RHV). With a given cut-off level crit, we define a mak of the region a a binary function 1, if mak ( r) = 0, if t6 ( r) ( r) crit crit. (12) An alternative way to define a RHV i to preet the deired ize of the high value region and chooe the cut-off level crit o that function (12) would ditinguih the region of the deired ize. The deired volume of the region can be pecified by different way. Thi can be, for example, the abolute volume of the region in the three-dimenional pace (in Å3) or the number of grid node if the region i built on ome grid in the unit cell. Thi can be alo the

7 THE USE OF CONNECTED MASKS FOR RECONSTRUCTINQ THE SINGLE PARTICLE IMAGE FROM X-RAY DIFFRACTION DATA relative volume that i the ratio of the region volume to the volume of the unit cell. In ome cae, it i convenient to characterize the ize of the region by a volume that correpond to a particular tructure unit. For thi purpoe, we will ue a pecific volume, which i calculated a an average volume per one amino acid reidue of the protein tructure being tudied. At the initial tep of the tructure invetigation, an object of the earch can be not the electron denity ditribution itelf but a binary mak that decribe bet the molecule. Moreover, the property of the binarity per e can be ued a an additional information about the object under tudy for the recontruction of phae value [12]. In thi work, we ugget a new approach to the ab initio tructure determination of biological molecule at low and middle reolution. The approach can be applied in cae of both crytalline ample and a ingle molecule, but it computing advantage become particularly ignificant in tudie of ingle molecule. The object of the earch i a mak of the molecule region given at ome grid in the unit ell b, 0 m N, 0 n N, 0 p N. (13) Here, x y z mnp x y z N, N, N are the number of grid node on the ide of the unit cell. At the firt tep, connected et of point coniting of a given number of point are randomly generated. To each et of generated point, the binary function mak 1 for ( r) = 0 for t7 r r i contructed. From thi function, the magnitude and phae of tructure factor are calculated. If the calculated magnitude are cloe enough to the experimental value (for example, have a correlation above the given value), then the correponding phae et i conidered a admiible and i kept for further work. The generation i repeated until the neceary number of admiible phae et i elected. The next two tep of the work, the alignment and averaging, are accomplihed a decribed above in ection 1.3. Upon the election of generated mak, additional requirement of the election can be impoed. Thee requirement are formulated in term of the characteritic of mak; for example, the extenion of the mak in different direction i limited or the required radiu of the inertia of mak i given. At the beginning of the work with the tructure, the addition of node to the mak being generated can occur with an equal probability for all grid node. With the advance of the work, the Fourier ynthei obtained after the averaging of elected phae et can be converted into the ditribution of probabilitie for the preence in the mak of particular grid node. After thi, the generation of the mak at the next circle can be performed with allowance for thi probability ditribution Tet object 2. RESULTS The atomic model of the trimer of the membrane protein AcrB [13] coniting of 3129 amino acid reidue (23811 non-hydrogen atom) wa ued. The overall view of the trimer i hown in Fig. 2. Two tet et of complex tructure factor correponding to two variant of an imaginary crytal tructure were calculated. The AcrB molecule wa aumed to be placed in the rectangular unit cell with dimenion 120, 120, and 150. Å in the firt cae (the mall unit cell) and 180, 180, 225 Å in the econd cae (the extended unit cell). In both cae, the pace group wa aumed to be P1 (the abence of crytallographic ymmetry). The firt cae reemble an ordinary tak of the crytal tructure determination in the preence of a large volume of the olvent (about 75%). The econd cae correpond to an imaginary crytal (14)

8 LUNIN et al. (with an abnormally large volume of the olvent) conidered in the ingle particle X-ray tudy. The tructure factor magnitude are conidered a known experimentally determined F value and referred below a experimental value. The calculated phae were ( ) exact aumed to be exact phae value ( ) During tet, they were conidered a unknown. and were ued only for the control of the reult. Fig. 2. Overall tructure of the AcrB protein. Three monomer compriing the molecule are hown by different color. The boundarie of a mall and an extended unit cell are hown in red and blue. It hould be noted that the AcrB molecule ha a ymmetry axi of the third order. The ymmetry wa not taken into account either in the choice of the unit cell or during the retrieval of the value of tructure factor phae. Therefore, the manifetation of thi ymmetry on molecule image obtained erved a a confirmation of the correctne of the olution. The ab initio determination of the tructure factor phae wa performed in a reolution zone of 25 Å. For the control, different indicator were calculated alo in the zone of lower reolution of 40 and 30 Å. The ditribution of the number of the tructure factor in the reolution zone i hown in Table 1. Table 1. Ditribution of the number of tructure factor in the reolution zone Dimenion of the unit cell [Å] Reolution zone (d min) [Å] Number of tructure factor Small unit cell 120, 120, Extended unit cell 180, 180, Control criteria The choice of the mot optimal criteria for the comparion of the et of the phae, or tructure factor magnitude, or Fourier ynthee of the electron denity i a ubject of the eparate dicuion in protein crytallography. In thi paper, we ue two of the mot popular criteria. For the comparion of tructure factor magnitude calculated from a binary mak and F, we ue the non-centric correlation coefficient: their experimental value ( ) CM = F calc ( ) F ( ) calc ( F ( ) ) F ( ) ( ) 2 2. (15) t8

9 THE USE OF CONNECTED MASKS FOR RECONSTRUCTINQ THE SINGLE PARTICLE IMAGE FROM X-RAY DIFFRACTION DATA The criterion for the comparion of two phae et (for example, ab initio olution and the exact phae) i defined in two tep. At the firt tep, we define a formal correlation coefficient of two phae et a: FCP( 1 ( ), 2 ( ) ) = FCP( 1 ( r), 2 ( r) ) = 2 ( ) ( ) ( ( )) ( 1( ) 2( )) 1 2 dv F co r r r = ( ( )) ( ( )) ( F ( )) 1 r dvr 2 r dvr Here, 1 ( r ) and 2 ( ) F ( ) and the phae et being compared. (The term ( 0). (16) r are the Fourier ynthee calculated with experimental magnitude F i omitted in thee calculation.) It hould be mentioned that thi criterion i pecific for a particular X-ray experiment becaue it i calculated uing the experimental tructure factor magnitude. The above formal comparion of phae et i not alway acceptable in the tak of the recontruction of phae value from the tructure factor magnitude. Thi i becaue the tructure factor magnitude do not fix the origin of the coordinate. To be more precie, two function, ( ) 1 = r and = ( r ) tructure factor magnitude but different phae value 2 u, which differ by a hift at the vector u, have the ame ( ) ( ) ( ) 2 = 1 + 2, u. (17) Thee two function provide one and the ame image of the tructure and have to be conidered, in the frame of the tructure determination tak, a equivalent. During the random generation of mak or phae, a cae i poible that the image being generated (or Fourier ynthei calculated from phae being generated) differ by only a hift or become very imilar after the application of the properly elected hift. In the frame of our tak, thee phae et have to be conidered a cloe. Therefore, a the econd tep in the determination of the level of cloene, we introduce the alignment, i.e. the election of a vector u that would provide a maximal value of the formal criterion (16): CP ( 1 ( ), 2 ( ) ) max FCP( 1 ( r u), 2 ( r) ) u 2 ( F ( ) ) co( 1( ) + 2 (, u) 2( ) ) = = max. u ( F ( ) ) In the pace group P1, the maximization i performed over all vector of the hift u. A et of poible vector of the hift can be limited in the cae of the preence of crytallographic ymmetry [11]. One more tranformation of the function of the electron denity ditribution that doe not lead to a change in the tructure factor magnitude i the tranition to an enantiomer; the r have the ame et of the tructure factor magnitude. At low and function ( r ) and ( ) medium reolution, the choice of a correct enantiomer i uually impoible. Therefore, in the procedure of the alignment, an additional poibility of the change of the enantiomer i introduced in cae that uch a change i not limited due to the ymmetry (for example, when one work with the group P1): CP ( 1 ( ), 2 ( ) ) = max max FCP 1 ( ), 2 ( ) =1 2 ( ) (18) r u r. (19) u t9

10 LUNIN et al Averaging of the phae et. Weighted Fourier ynthei In ab initio phae determination procedure dicued in thi work, the reult of the firt tep i a population of the phae et choen. The implet proceing procedure of the phae et i their averaging, which i applied to the phae et previouly aligned relative to each other [4]. To be more precie, the averaging i applied to the Fourier ynthee calculated F and different phae et. Thi lead to uing experimental tructure factor magnitude ( ) a weighted Fourier ynthei which can be calculated a 1 bet ( r) = m( ) F ( ) exp i ( ) exp i2 (, ) V r. (20) bet For each tructure factor, the bet phae ( ) determination (the figure of merit) are determined: t10 and the indicator of the reliability of it M bet 1 m( ) exp i ( ) = exp i j ( ). (21) M j = 1 It i eaily een that m ( ) characterize the catter of phae value in different et relative to bet the averaged phae value ( ) : M 1 m. (22) ( j ) bet ( ) = co ( ) ( ) M j = 1 When uing the weighted Fourier ynthee (20), we will alo apply, along with the phae correlation coefficient, the weighted correlation coefficient CP w, which i the reult of the optimization over all poible phae hift CP w ( 1 ( ), 2 ( ) ) = max max FCPw 1 ( ), 2 ( ) =1 u ( ) r u r, (23) where intead of the formal correlation coefficient (16), the weighted formal correlation coefficient i ued: FCP w ( 1( ) 2( ) ) 2 ( ) co( 1( ) 2( )) ( ) F ( ) m, =. ( F ( ) ) m( ) F ( ) ( ) The averaging of random phae et. The efficiency of ab initio procedure At low reolution, the correlation coefficient CP a a criterion of ucce hould be ued with caution. The point i that the et of the tructure factor magnitude for biological macromolecule often have anomalouly large value for a relatively mall number of reflection in the low reolution zone (Fig. 3). Thi i why the image on low reolution map can be determined motly by a mall number of thee trong reflection. In thi cae, high CP value can indicate a conenu of phae value only for a mall number of dominant reflection. In addition, it i neceary to take into account that when random and exact phae value are compared, then the intuitively expected zero value of the correlation coefficient can appear only when the formal criterion FCP i ued. In cae of the preliminary alignment of the phae et being compared, the value of the correlation coefficient can be much higher. The ditribution of the value of the CP correlation coefficient calculated for randomly generated phae et are hown on Fig. 4, and the characteritic of thee ditribution are given in Table 2. (24)

11 THE USE OF CONNECTED MASKS FOR RECONSTRUCTINQ THE SINGLE PARTICLE IMAGE FROM X-RAY DIFFRACTION DATA The averaging of the preliminary aligned phae et lead to greater value of the correlation coefficient CP w (Table 3, 4). Thu, for a tet object in the reolution zone of 40 Å, the correlation coefficient value of 0.7 can be obtained imply by the averaging of randomly generated phae et. Thi bae value ha to be taken into account when evaluating the ucce of any procedure of the olution of the phae problem. The procedure can be conidered ucceful only if it allow one to obtain higher value of the correlation coefficient than thoe allowed by the averaging of random phae et F Å 30Å 25Å 20Å E E E E E-03 Fig. 3. Structure factor magnitude for AcrB (the mall cell cae). For each reflection, the tructure factor magnitude i hown a a function of 2. p 70A 40A 25A CP Fig. 4. Empirical ditribution of the correlation coefficient CP of exact and randomly generated phae of the tructure factor in different reolution zone (AcrB, the mall cell cae). Table 2. Value of the correlation coefficient CP for randomly generated and exact phae value (AcrB, the mall cell cae) Reolution Number of CP d min [Å] reflection min max ave igma ave+3igma t11

12 LUNIN et al. Table 3. Value of the correlation coefficient CP w for exact phae and phae obtained by the averaging of 100 randomly generated phae et (AcrB, the mall cell cae, five independent tet a e) Variant Reolution a b c d min [Å] CP w d e Table 4. Value of the correlation coefficient CP w for exact phae and phae obtained by the averaging of 100 randomly generated phae et (AcrB, the extended cell, five tet a-e) Variant Reolution a b c d min [Å] CP w d e A comparion of Table 3 and 4 how that the quality of averaged random phae decreae with an increae in the cell dimenion of the imaginary crytal Generation of random phae et. The criterion of finitene for the region of high electron denity value. Retriction of the number of connected component Here, the tet were arranged in the following way. In each tet, a deired volume VHDR of the region of high electron denity value wa et. A great number (up to hundred million) of random phae wa generated. For every phae et, the Fourier ynthei wa calculated uing thee phae and the et of the experimental tructure factor magnitude. A region of high value of the given volume VHDR wa elected, and it wa checked whether thi region can be placed entirely in the crytal cell without croing the border. If thi condition wa atified, the checked phae et wa conidered a admiible and wa elected for further analyi. The generation of phae et wa continued until 100 admiible phae et were elected. Then, thee 100 phae et were aligned and averaged. The accuracy of the phae et thu obtained i given in Table 5 and 6. For each preet value of volume VHDR, the tet wa repeated five time with different tarting contant for the generator of random number. It i neceary to point out that at big VHDR value, the calculation procedure required coniderable proceor time (about a full day). Table 5. Accuracy of the phae value obtained by averaging the randomly generated phae et leading to a finite region of high value on the Fourier ynthei. The phae correlation CP w value calculated in different reolution zone (40, 30, 25 Å) are given for five independent tet (the mall cell cae) CP w*100 40/30/25Å Volume of high denity region: pecific [Å 3 per reidue]/relative/number of point /65/63 78/72/70 76/71/69 81/75/73 84/78/75 87/81/78 74/69/67 76/72/69 81/75/73 82/77/74 82/77/74 75/70/67 76/71/68 79/74/71 80/75/72 86/80/77 86/80/77 74/69/66 78/73/70 80/75/72 83/78/75 86/80/77 75/70/68 75/68/66 81/76/75 83/78/75 85/79/75 86/80/ t12

13 THE USE OF CONNECTED MASKS FOR RECONSTRUCTINQ THE SINGLE PARTICLE IMAGE FROM X-RAY DIFFRACTION DATA Table 6. Accuracy of the phae value obtained by averaging the randomly generated phae et leading to a finite region of high value on the Fourier ynthei. The phae correlation CP w value calculated in different reolution zone (40, 30, 25 Å) are given for five independent tet (the extended cell cae) CP w*100 40/30/25Å Volume of high denity region: pecific [Å 3 per reidue]/relative/number of point t /51/49 55/53/51 50/48/46 62/59/57 65/62/60 73/69/67 75/71/69 57/54/53 55/52/50 54/51/50 64/61/60 66/63/61 72/68/66 74/70/68 53/51/49 65/62/60 66/63/61 67/64/62 67/64/62 72/69/66 73/70/67 56/53/52 56/54/52 63/60/58 65/62/60 67/64/62 70/67/65 73/69/67 49/47/46 51/49/48 61/58/56 61/58/56 63/59/58 69/66/64 73/69/67 An analyi of Table 3 and 4, a well a Table 5 and 6, allow one to make following concluion. Firt, the procedure decribed in thi ection make it poible to obtain a higher quality of averaged phae than in cae of phae obtained by averaging randomly generated phae without any preliminary election. Therefore, thi procedure can be conidered a a working ab initio procedure of the determination of the phae. Second, an increae in the cell volume with the retention of the pecific volume of the region of high value having a phyical ene lead to a coniderable decreae in the quality of the phae et obtained. In other word, the procedure conidered eem to be more uited for the work with real crytal than with imaginary crytal in tudie of ingle particle. In the election of generated phae et, the attempt to impoe additional retriction on the number of connectivity component in the region of high value led only to an inconiderable increae in the quality of averaged phae et Generation of random connected mak. Averaging without election The goal of thi erie of tet wa to examine how ueful, in term of the phae problem olution, the retriction uch a the connectivity and binarity of the region of high value may be. The tet wa performed in the following way. In the unit cell, a uniform grid with a tep approximately equal to the third of the nominal reolution of 25 Å (i.e., about 8.3 Å) wa introduced. Each tet wa performed for a given volume VHDR of the region of high value on the Fourier ynthei of the electron denity, with the number of the grid node inide the mak Nmak correponding to thi volume (Table 7, 8). A hundred of random connected mak coniting of Nmak node were generated. Each mak wa ued for the calculation of the tructure factor magnitude and phae. Thee 100 phae et were aligned and averaged. The accuracy of the phae value thu obtained i given in Table 7, 8. For each given value of the volume VHDR, the tet wa repeated five time with different tart contant of the generator of random number. Table 7. Reult of the averaging of phae et calculated from randomly generated connected mak. The phae correlation CP w value calculated in different reolution zone (40, 30, 25 Å) are given for five independent tet (the mall cell cae) CP w*100 40/30/25Å The volume of the mak: pecific [Å 3 per reidue]/relative/number of point /57/55 70/65/63 77/73/70 77/72/69 85/80/77 85/80/77 85/80/77 64/60/57 70/65/63 75/71/68 81/77/74 83/78/75 85/79/77 85/80/77 66/61/58 72/67/65 78/74/71 80/75/72 84/80/77 87/82/79 84/79/76 63/59/57 72/68/66 78/73/71 80/76/73 82/77/74 86/81/78 87/81/78 62/57/55 71/65/63 75/70/68 81/76/73 87/82/79 83/78/75 86/80/77

14 LUNIN et al. Table 8. Reult of the averaging of phae et calculated from randomly generated connected mak. The phae correlation CP w value calculated in different reolution zone (40, 30, 25 Å) are given for five independent tet (the extended cell cae) CP w*100 40/30/25Å The volume of the mak: pecific [Å 3 per reidue]/relative/number of point /57/55 70/65/63 77/73/70 77/72/69 85/80/77 85/80/77 85/80/77 64/60/57 70/65/63 75/71/68 81/77/74 83/78/75 85/79/77 85/80/77 66/61/58 72/67/65 78/74/71 80/75/72 84/80/77 87/82/79 84/79/76 63/59/57 72/68/66 78/73/71 80/76/73 82/77/74 86/81/78 87/81/78 62/57/55 71/65/63 75/70/68 81/76/73 87/82/79 83/78/75 86/80/77 An analyi of the table allow one to make the following concluion. Firt, the procedure decribed above make it poible to obtain phae of higher quality than the quality of averaged randomly generated phae et ued without any preliminary election. Thu, thi procedure can be conidered a a working ab initio procedure for phae determination. Second, a comparion of Table 5 and 7 how that, in the cae of a mall cell, the quality of the phae et obtained a a reult of the averaging i approximately the ame in both cae, when phae are randomly generated and the election of the variant i baed on the demand of the finitene of HDR, and when connected mak are generated and the phae calculated from the mak are directly averaged. However, the computational expene with the econd approach are coniderably lower. Third, in the cae of the averaging of phae obtained from connected mak, the extending of unit cell dimenion lead to a coniderable increae in the quality of the olution of the phae problem. Thu, the new approach implement the advantage when working with a ingle particle, which are attained due to the poibility of uing more experimental information (a greater number of tructure factor magnitude, which are included in the calculation when one work with a large unit cell of an imaginary crytal) Generation of random connected mak. Selection baed on the correlation of tructure factor magnitude In thi erie of tet, an additional requirement to randomly generated mak wa included, namely the correpondence of the tructure factor magnitude calculated from the mak to the experimental one. The tet wa performed in the following way. In the unit cell, a uniform grid with a tep approximately equal to the third of the nominal reolution of 25 Å (i. e., about 8.3 Å) wa introduced. Each tet wa carried with two parameter preet: the volume VHDR of the region of high value on the Fourier ynthei of the electron denity and the required accuracy of the correpondence of tructure factor magnitude calculated from the mak to the experiment. A great number of random connected mak coniting of Nmak node were generated. Each generated mak wa ued to calculate tructure factor magnitude and phae. For the calculated tructure factor magnitude, the coefficient of their correlation with experimental value (15) wa calculated. If thi coefficient wa higher than a given threhold value, the generated phae et wa conidered a acceptable and wa kept for further analyi. The generation wa continued until 100 acceptable phae et were elected. The elected phae et were aligned and averaged. t14

15 THE USE OF CONNECTED MASKS FOR RECONSTRUCTINQ THE SINGLE PARTICLE IMAGE FROM X-RAY DIFFRACTION DATA Table 9. Reult of the averaging of phae et calculated from randomly generated connected mak having a given level of correlation (25) of tructure factor magnitude. The phae correlation CP w value calculated in different reolution zone (40, 30, 25 Å) are given for five independent tet (the mall cell cae). The bet reult i highlighted CP w* /30/25 Å Correlation of magnitude z crit (25 Å) Mak volume: pecific [Å 3 per reidue]/relative/number of point /57/54 69/64/62 72/67/64 80/75/72 87/82/79 88/83/80 87/82/ /57/55 68/63/61 73/68/66 83/78/75 88/83/80 90/84/81 89/83/ /55/53 65/60/57 73/68/66 81/76/73 88/83/80 89/83/80 89/83/ /55/53 62/58/56 71/66/64 81/76/73 88/83/79 83/78/74 Table 10. Reult of the averaging of phae et calculated from randomly generated connected mak having a given level of correlation (25) of tructure factor magnitude. The phae correlation CP w value calculated in different reolution zone (40, 30, 25 Å) are given for five independent tet (the extended cell cae). The bet reult are highlighted CP w* 100% 40/30/25 Å Correlation of magnitude z crit (25 Å) Mak volume: pecific [Å 3 per a.a.]/relative/number of point /69/67 78/74/71 86/82/79 89/85/83 91/86/84 91/87/85 90/86/ /68/66 77/73/71 85/81/79 90/86/84 91/87/85 92/88/85 91/87/ /68/66 77/73/70 88/84/82 91/87/84 92/88/85 92/88/86 90/86/ /69/67 81/77/74 89/85/83 92/88/85 94/90/87 94/90/87 93/88/86 a b c d Fig. 5. AcrB model and the region of high electron denity value on unweighted Fourier ynthei calculated with experimental value of tructure factor magnitude and ab initio determined phae: a reolution 25 Å, cut-off level 3σ; b reolution 40 Å, cut-off level 3σ, one of the monomer of the model i not hown; c reolution 40 Å, cut-off level 2σ; d reolution 40 Å, cut-off level 3σ. t15

16 LUNIN et al. The accuracy of the phae value obtained in thi way i given in Table 9 and 10. In thee table, the rigidity of the election of acceptable mak i given in term of the normalized z CM core, which wa determined in the following way. A a preliminary tep, a great number (in our tet, 10000) of random connected mak of the given ize were generated. For each mak, the coefficient of the correlation between the calculated and the exact tructure factor magnitude wa calculated. For value of CM, the mean value CM and the mean tandard deviation CM were calculated. Then, when the mak generation wa performed with the election, the calculated CM value for each et of tructure factor magnitude wa tranformed to a normalized value z CM CM CM =, (25) which jut wa compared with the given threhold value z crit. The table how that the introduction of the election according to the correpondence of the calculated tructure factor magnitude to the experiment ignificantly improve the quality of the reulting phae compared with that attained in the averaging procedure without election. Fig. 5 how the image of the object obtained from the Fourier ynthei calculated with experimental tructure factor magnitude and the bet phae found. ADDITIONAL NOTES AND CONCLUSIONS The teting of the new approach to the ab initio olution of the phae problem of the biological crytallography howed it applicability in both traditional tudie of crytal pecie and invetigation of ingle biological macromolecular object. The approach i baed on the Monte-Carlo type procedure [2, 3], which conit of a few tep. At the firt tep, an object of the earch i binary mak erving a an approximation of the region of high value of the electron denity function in the object under tudy. For thi purpoe, a great number of connected binary mak are randomly generated, and for each of them, the tructure factor magnitude and phae are calculated. The mak (and the phae correponding to it) i conidered a admiible if the level of the correpondence of the tructure factor magnitude calculated from the mak and the experimental value exceed a given limit. The generation i performed until the preet number of admiible phae et are obtained. A criterion of the cloene of the tructure factor magnitude may be the coefficient of the magnitude correlation, although other criteria may be ued (for example, the tatitical likelihood [16]). Along with the criterion of the cloene of tructure factor, the criteria of another type may be ued at the tep of the election of mak, which are related to their expected parameter or are baed on the reult of other experiment (e.g., mall angle X-ray diffraction or neutron cattering). Random mak can be generated either on aumption of the equal probability of all grid node in the unit cell or with the ue of a priori preference etablihed at previou tep of the work. In cae of large unit cell value and rigid retriction on the degree of correpondence of the mak to experimental data, the generation of the required number of admiible mak can demand a ignificant amount of the computer time. However, the natural parallelim of the election proce allow one to ignificantly decreae time expene in the work on computer with parallel architecture. At the econd tep, the alignment and, in the implet cae, the averaging of the phae et are performed. A more accurate procedure involve uing the method of the cluter analyi, which allow one to ditinguih, among elected variant, compact cluter for further averaging inide thee cluter. It hould be noted that the averaging and cluter analyi are performed uing pecial metric relevant to a pecific X-ray experiment. The phae value CM t16

17 THE USE OF CONNECTED MASKS FOR RECONSTRUCTINQ THE SINGLE PARTICLE IMAGE FROM X-RAY DIFFRACTION DATA obtained a a reult of the averaging together with tructure factor magnitude determined experimentally are ued to build Fourier ynthei. 100 rand CP w *100 rp 50 rp rp 100 rp rp 150 rm rm 75 rm rm 125 rm 150 rm rm 200 rm rm 150 z=2 rm 175 z= rm 175 z=2 rm 175 z=1.5 Å 3 /re rm 200 z= Fig. 6. Phae correlation CP w*100 in the reolution zone of 40 Å for averaged phae et with the different ize of the region of high value: rand random phae without election (ection 2.4); rp random phae with election baed on the finitene property of the region of high value (ection 3.5); rm random mak without election (ection 2.6); rmz random mak with election baed on the magnitude correlation at different election level (ection 2.7). The pecific volume of the region of high value i hown on the X-axi. For all condition, the reult of five independent experiment in the mall cell are hown. CP w * rp rp 75 rp 100 rp rp 150 rp 175 rp 200 rp rm 75 rm 100 rm 125 rm rm 175 rm 200 rm 225 rm 150 z= rm 150 z=2 rm 175 z=1.5 rm 175 z=2 rm 175 z= rm 200 z=1.5 rm 150 z=2.5 Å 3 /re rm 200 z=2 rm 200 z= Fig. 7. Phae correlation CP w*100 in the reolution zone of 40 Å for averaged phae et with the different ize of the region of high value: rand random phae without election (ection 2.4); rp random phae with election baed on the finitene property of the region of high value (ection 3.5); rm random mak without election (ection 2.6); rmz random mak with election baed on the magnitude correlation at different election level (ection 2.7). The pecific volume of the region of high value i hown on the X-axi. For all condition, the reult of five independent experiment in the extended cell are hown. t17

18 LUNIN et al. The method wa teted in two regime. A mall unit cell modeled a crytal ample with a large content of the olvent in the cell. An extended unit cell modeled a reduced tak of tructure determination uing X-ray diffracting from a ingle particle. The reult of the teting allowed one to find optimal parameter for the procedure and howed that the procedure doe make it poible to advance in the determination of the phae of the tructure factor. The accuracy of the value found increae with increaing ize of the unit cell, which make thi method epecially promiing for the invetigation of ingle particle. In the cae of the traditional crytallographic tak, a comparion of the new method with the one uggeted earlier howed a compatible efficiency of the olution of the phae problem, with the computing efficiency of the new approach being higher. At the ame time, the new method lead to a much better olution of the phae problem when one ha to do with cell containing artificially increaed portion of the olvent. Fig. 6 and 7 illutrate the accuracy of determining the phae of tructure factor with the ue of different approache and parameter of the method. The work wa upported by the Ruian Foundation for Baic Reearch (grant No ). REFERENCES 1. Krupyankii Yu.F., Balabaev N.K., Petrova T.E., Sinityn D.O., Gryzlova E.V., Terehkina K.B., Abdulnayrov E.G., Stepanov A.S., Lunin V.Y., Grum-Grzhimailo A.N.. Femtoecond X-ray free-electron laer: A new tool for tudying nanocrytal and ingle macromolecule. Ruian Journal of Phyical Chemitry B V. 8. P Sinityn D.O., Lunin V.Y., Grum-Grzhimailo A.N., Gryzlova E.V., Balabaev N.K., Lunina N.L., Petrova T.E., Terehkina K.B., Abdulnayrov E.G., Stepanov A.S., Krupyankii Yu.F. New poibilitie of X-ray nanocrytallography of biological macromolecule baed on X-ray free-electron laer. Ruian Journal of Phyical Chemitry B V. 8. P Lunin V.Y., Lunina N.L., Petrova T.E., Skovoroda T.P., Urzhumtev A.G., Podjarny A.D. Low-reolution ab initio phaing: problem and advance. Acta Crytallographica Section D: Biological Crytallography V. 56. P Lunin V.Y., Urzhumtev A.G., Podjarny A. Ab initio phaing of low-reolution Fourier ynthee. In: International Table for Crytallography. Volume F. Crytallography of Biological Macromolecule. Second Edition. Ed. Arnold E., Himmel D.M., Romann M.G. John Wiley & Son, P Lunin V.Y., Lunina N.L., Urzhumtev A.G. Connectivity propertie of high-denity region and ab initio phaing at low reolution. Acta Crytallographica Section D: Biological Crytallography V. 56. P Bricogne G. Method and program for direct-pace exploitation of geometric redundancie. Acta Crytallographica Section A: Foundation of Crytallography V. 32. P Fienup J.R. Recontruction of an object from the modulu of it Fourier tranform. Optic Letter V P Eler V. Solution of the crytallographic phae problem by iterated projection. Acta Crytallographica Section A: Foundation of Crytallography V. 59. P Baker D., Krukowki A.E., Agard D.A. Uniquene and the ab initio phae problem in macromolecular crytallography. Acta Crytallographica Section D: Biological Crytallography V. 49. P Lunin V.Y., Urzhumtev A.G., Skovoroda T.P. Direct low-reolution phaing from electron-denity hitogram in protein crytallography. Acta Crytallographica Section A: Foundation of Crytallography V. 46. P t18