Implicit Representation of Molecular Surfaces

Transcription

1 Imliit Reresentation of Moleular Surfaes Julius Parulek Ivan Viola Deartment of Informatis, University of Bergen Deartment of Informatis, University of Bergen. (a) (b) () (d) (e) (f) (g) (h) Figure : Ray-asting based visualization of eight roteins reresented by the solvent exluded surfaes (SES). Surfaes are defined by imliit funtions omuted loally and without erforming any reomutation stes. The roteins ontain 3023 (a), 2872 (b), 2820 (), 3346 (d), 2530 (e), 2555 (f), 78 (g) and 4744 (h). Small disontinuities are aused by the reision arameter that aets a lose viinity of the 0 iso-level of the imliit funtion. A BSTRACT Moleular surfaes are an established tool to analyze and to study the evolution and interation of moleules. One of the most advaned reresentations of moleular surfaes is alled the solvent exluded surfae. We resent a novel and a simle method for reresenting the solvent exluded surfaes (SES). Our method requires no reomutation and therefore allows us to vary SES arameters outright. We utilize the theory of imliit surfaes and their CSG oerations to omose the imliit funtion reresenting the moleular surfae loally. This funtion returns a minimal distane to the SES reresentation. Additionally, negative values of the imliit funtion determine that the oint lies outside SES whereas ositive ones that the oint lies inside. We desribe how to build this imliit funtion omosed of three tyes of athes onstituting the SES reresentation. Finally, we roose a method to visualize the iso-surfae of the imliit funtion by means of ray-asting and the set of rendering arameters affeting the overall erformane. From the biomoleular oint of view, the struture of moleules is given by three-dimensional organization of atoms. While hemial struture of the atoms is often quite stable, the overall shae, generated by fore fields of atoms, an vary raidly. Therefore, to understand the shae in its full omlexity, virtual omuter models of moleules are instrumental. A seifi tye of moleules are roteins that atively artiiate in various ell roesses. Essentially, they are long-hained maromoleules that onsist of standard amino aids. These amino aids an ontain between 0 and 25 atoms eah, where tyially the moleular hain an ontain u to a few thousand of amino aids. The set of atoms that is ommon for every amino aid is alled the bakbone. The set of atoms onneted to the bakbone is alled the side hain and is unique for every amino aid. Many roteins are enzymes that need to be triggered by other biomoleules alled ligands. Suh roteins an, for instane, transort ligands to different arts of the ell or even between the ells. In the latter ase, these form so-alled hannels. In order to disover whih ligands an bind to a given rotein, one needs to ommuniate the rotein shae aordingly. A ommon aroah to reresent rotein surfae to disover the ligand binding site, is to utilize a solvent moleule, e.g. the water moleule. The reason is that the ligand an ratially aess all the laes that are reahable by the solvent itself. Therefore, one of the ossible strutural reresentations of the moleules is defined via the solvent whih an reveal ossible binding sites. In moleular visualization, many different methods of reresenting moleules have been roosed [3]. One an use different strutural reresentations like sae fill, balls-and-stiks, liorie, bakbone, and ribbon. These an be shown through distint illustrative Keywords: Visualization of Moleular Surfaes, Geometrial Modeling, Imliit Surfaes, Bioinformatis Visualization Index Terms: J.3 [Comuter Aliations]: Life and Medial Sienes Biology and Genetis I.3.4 [COMPUTER GRAPHICS]: Comutational Geometry and Objet Modeling Boundary reresentations, Curve, surfae, solid, and objet reresentations Julius.Parulek@uib.no ivan.viola@uib.no IEEE Paifi Visualisation Symosium February - 2 Marh, Songdo, Korea /2/$ IEEE I NTRODUCTION 27

2 tehniques, e.g. artoon reresentation, or by means of emhasizing the deth eretion like halos and ambient olusion [28]. One of the most emloyed advaned reresentations of moleular surfaes is alled the solvent exluded surfae (SES), whih was first roosed by Rihards [2]. So far, SES beame a very well established reresentation of the iso-surfae aroximating the otential field of atoms while taking into onsideration the solvent moleule. The main utilization of SES reresentation is studying the moleular dynamis in order to disover ligand binding sites and their evolution over the moleular dynami simulation. It is imortant to mention that one an aly more omlex, hysially-lausible models, whih inlude eletrostati otential fields (EPF). The major advantage of using SES reresentation with reset to EPF-based reresentation is its relatively fast omutation mainly when dealing with large moleular dynami simulations (MDS) ontaining several thousands of animation frames. The EPF models are omutationally very extensive and are usually rasterized via artial differential equations on a regular grid. Although, when suh reresentation is reomuted, one an ombine SES and EPF reresentations into a merged visualization. The SES reresentation is obtained by rolling a solvent moleule (aroximated by a shere of radius R) on the so-alled solventaessible surfae (SAS). The SAS is reresented as the boundary of extended van der Waals sheres by radius R. The resulting SES then deomoses into three tyes of surfae athes: onvex sherial athes, onave sherial athes (sherial triangles) and toroidal athes (Fig. 2). Figure 2: An examle of a moleular surfae. The solvent exluded surfae (union of full irles and their toroidal interonnetions) is formed by rolling a ball (blue irle) on van der Waals sheres while the enter lies on the boundary of solvent aessible surfae (union of dashed irles). The SES reresentation deits a single residual sequene of roteinase 3 (Pr3) etide ontaining 360 atoms. The SES surfae an be be desribed by three tyes of athes, the onvex sherial one (green), the onave sherial one (blue) and the toroidal one (red). The main drawbak when using SES reresentation in visual exloration is the lak of interativity, sine the geometri SES reresentation requires usually substantial reomution [24, 30]. Here, to delimit the atom seletion, artiiating in the SES reresentation, or just to adjust the solvent radius R, requires rereating the entire surfae. The essential arameter settings interation during the visual exloration of a maromoleule an not be erformed in realtime. Moreover, to reresent the surfae for time-varying rotein simulations the heavy load is ut to the reomutation ste even further. Essentially, one needs to to reroess several time-stes in advane. Therefore, biologists often refer to limit the bindingsite analysis to the artiular key frames of the simulation. With the reent advane of omutational hardware aabilities, larger and larger dynami rotein sequenes are being rodued, whih learly demands more flexible SES reresentation that would allow for an instant surfae reresentation without having done any reomutational stes. In this aer, we roose a novel aroah to reresent SES surfae without reomutation. Our new reresentation is based on theory of imliit surfaes, namely its generalization alled funtional reresentation [8]. We roose a rendering aroah to ahieve interative visualization by introduing several renderingquality arameters affeting the overall erformane. The instantaneous SES reresentation allows us to rovide interativity neessary for rotein analysis. Partiularly, we aim at roviding biologists with the ossibility of erforming hanges of solvent radius R and seeking atany timeintervals ofthesimulation. 2 RELATED WORK We relate our work to three areas. Firstly, we introdue tehniques of moleular visualization. Seondly, we desribe imliit modeling tehniques. Finally, we resent real-time rendering tehniques of imliits. 2. Visualization and Reresentation of Moleular Surfae There are several tyes of moleular surfae reresentations introdued in the literature. One of the basi reresentations is when the rotein atoms are deited as sheres, where the radius orresonds to the van der Waals fore (vdw surfae) [3]. The extension of this surfae by a solvent radius is alled the solvent aessible surfae (SAS). The most widely used reresentation is denoted as solvent exluded surfae (SES) [2, 6]. In 992, Edelsbrunner and Müke introdued α-shae reresentation and in 2007, Ryu et al. extended it to the β-shae. In 999, Edelsbrunner [4] also roosed a new reresentation of moleules alled the skin-surfae. In our work, we fous on reresenting and visualizing the SES reresentation, beause it is a standard tehnique and was requested by our biology ooeration artners for binding-site exloration. There has been a lot of effort ut into generation of SES in the literature. In 983, Connolly [3] roosed an analytial desrition of the SES reresentation. More than ten years later, Sanner et al. [24] resented the redued surfae algorithm for onstrution of SES reresentation. In the same year, Totrov and Abygyan introdued the ontour-buildu algorithm [30] to form the SES reresentation. Reently, Lindow et al. [4] roosed a seeding u and arallelization of ontour-buildu algorithm. The visualization is then erformed by a ray-asting method. In 2009, Krone et al. [] introdued the utilization of the redued surfae method for diret ray-asting of moleular surfaes. Here, all tehniques exloit reomutation stes neessary to render the final surfae. Triangularization methods of SES are oular as well, with the most reent ones being roosed in 2009 by Ryu et al. [22]. Additionally, SES an be deomosed into a set of quadris, whih an be effiiently rendered on the GPU (Setion 2.3). In 20, Krone et al. [2] introdued an aroximation of the SES reresentation via quadrati olynomial kernels. This work was aimed at the traking of rotein avities. Here, nevertheless, the resulting iso-surfae only imitates the exat SES reresentation. In our work, we utilize Protein Data Bank (PDB) file format, whih stores the rotein information (e.g., atom tyes, residual sequenes). The trajetories of the atoms are stored in the DCD file format that is used as a standard in the Visual Moleular Dynamis (VMD) tool [9]. The file an ontain tens of thousands of rotein trajetories. The other oular moleular viewers are for instane PyMOL [25], MetaMol [2] and QuteMol [28] to name a few. So far, none of these tools an visualize dynami trajetories and/or would allow for interative visual rotein exlorations. 28

3 To the best of our knowledge, there are no aroahes that would allow fordiretvisualization ofses reresentation without any reomutation. We ross this ga by introduing a new aroah that omutes the SES instantly on the fly during ray-asting. Contrary to the revious aroahes, we emloy a theory of imliit surfaes, where the geometry of a single atom is desribed by its distane based imliit funtion. 2.2 Imliit Modeling Imliit surfaes (imliits) rovide a way to easily model omlex dynamially hanging geometri objets. Moreover, they naturally enable the modeling of smooth, songe like objets in a onvenient way. The set of tehniques, known today as imliit modeling, was used for the first time by Blinn []. Pasko et al. generalized the reresentation of imliits, by ombination of the different forms of imliit models [8], and denoted it as funtion reresentation (Fre). The inequality () desribes an imliit solid (objet): f () 0, () where =(x,x 2,x 3 ) E 3. Funtion f is alled an imliit surfae funtion (imliit funtion), whih lassifies the sae into two half-saes f () > 0and f () < 0. Comlex objets an be reated from simle ones via Construtive Solid Geometry (CSG) oerations. The basi set-theoreti oerations an be defined using the min and max oerators: union( f, f 2 ) = f f 2 = max( f, f 2 ) intersetion( f, f 2 ) = f f 2 = min( f, f 2 ) subtration( f, f 2 ) = f \ f 2 = min( f, f 2 ). Analytial exressions that aroximate these oerators were roosed by Rii [20]. The other analytial definitions of the settheoreti oerations are known as R-funtions [8]. Nevertheless, the analytial versions do not reserve the distane harateristis as well as min/max oerators do [5]. Therefore, in our work we utilize the basi min and max oerators (2) sine they are fast to omute and reserve distane harateristis of the imliit funtion better. However, they are disontinuous where f = f 2.This has been studied by Fayolle [5]; whih we took as an insiration for our solution. 2.3 Real-time rendering of imliit objets In order to visualize models based on imliits, one an onvert them to a mesh reresentation rior to rendering them as a set of ath rimitives [6]. However, when dealing with omlex models and shaes, suh as for instane the moleular surfaes, one would need to generate millions of triangles to erform a fully detailed surfae reresentation. Therefore, in reent years, authors turn to diret visualization tehniques, whih is being reresented by ray-asting methods. Sine imlits enomass different forms of geometrial models, the atual ray-asting method is roosed aording to the tye of imliits we are dealing with. For examle, the ray-asting of algebrai surfaes is overed extensively by omuter grahis literature, whih goes bak several deades. Hanrahan introdued ray-asting of algebrai imliit models u to the fourth order [7]. Later work addressed ray-asting of large number of quadris on GPU aimed at moleular visualization as well [29, 9, 5]. Later on, Sigg et al. [26] introdued very fast GPU rendering of sheres, ellisoids and ylinders fousing mainly at moleular rendering. Nevertheless, these aers assume that the inut set of quadris is already formed, whih is not the ase with our instant imliit reresentation. In 992, Hart introdued a more robust aroah for ray-asting of distane based imliit surfaes alled shere traing [8]. Sine (2) in our work, we are dealing with distane based imliit funtions also, we adoted Hart s tehnique, beause it is easy to imlement and erforms very well on distane based surfaes. The reent work addresses ray-asting of general imliit surfaes on GPU using interval arithmetis [0] and via so-alled adative marhing oint method [27]. However, their methods assume that the funtion is defined globally, whereas our funtion is omuted loally only. 3 METHOD OVERVIEW Our algorithm omutes the imliit funtion reresenting SES on the fly during ray-asting. This funtion returns the minimal distane from a samle oint (given on a ray) to the surfae. To omute the funtion, the following roedures are erformed. The k losest atoms to the oint are retrieved in the asending order. Aording to the number of atoms k the funtion f SES is generated. Here, all ossible airs ( k 2) of atoms are tested, whether there might be an intersetion between their solvent extended isosurfaes. If the test is ositive, the intersetion oint that is the losest one to is estimated and stored. Then all ossible trilets ( k) 3 are heked aording to the retrieved air-wise intersetion oints and neighboring atoms. Again, if the test is ositive, the intersetion oint of the trilet is estimated and stored. To evaluate f SES all the stored intersetion oints then generate the solvent shere funtion. Afterwards, using omuted f SES, the ray is advaned to a next oint, by the funtion value. When the value of f SES is lose to 0 (to the iso-surfae), a deth and a normal at the oint are stored. The imlementation of our ray-asting method is erformed and arallelized using CUDA, where the ray-traversal and the funtion evaluation is done er every image ixel. Finally, aording to the stored deth values and the omuted normals, we erform the shading omutation and enhane the deth eretion by means of sreen sae ambient olusion [7]. 4 SES REPRESENTATION Let us first introdue the geometrial basis allowing us to reresent the surfae using imliit modeling tehniques. The goal is to define an imliit funtion desribing the surfae in a ertain 3D neighborhood of a given oint, in order to evaluate f SES (). The imliit funtion f SES fulfils the following roerties: for the given 3D oint the funtion f SES () will return the minimal distane to SES reresentation. f SES () > 0for inside the objet, f SES () < 0for outside the objet and f SES ()=0 means the lies on the boundary (iso-surfae). funtion f SES is C ontinuous in a neighborhood of the surfae, i.e. gradient f SES is ontinuous in this neighborhood. Let us assume that the set of atoms is defined as C = {(,r ),...,( n,r n )}) with the solvent radius defined by R. We define a set of imliit funtions (imliits) defined as F = { f, f 2,..., f n }, where eah f i () =r i i reresents an atom i with the orresonding van der Waals radius r i. Note that funtion f delimits the sae into the interior and the boundary ( f 0) and the exterior ( f < 0) half-sae. We also define an extended set of imliits, where shere atoms are enlarged by the solvent radius R, asg = {g = f + R,g 2 = f 2 + R,...,g n = f n + R}. Sine f i and g i are distane funtions the gradient of both is defined via f i ()= g i ()= i i. Additionally, we define a set of atom enters S that are loser to a oint than 2R, S = { i f i () 2R i C}. The set S C delimits the oint set C to atoms that might artiiate in 29

4 generating the surfae (Fig. 3a), aording to the oint that is a arameter of the imliit funtion. The exression of set S is derived diretly from the detetion of two inident atoms whih holds if i j < 2R + r i + r j (Fig. 3b). (a) 2R Figure 3: Forming the oint set S. a) Out of all the oints, the set S ontains those whih belong to the area of influene of the oint (full irles). b) The oint lies above the R distane from the atom 2, but it might still artiiate in forming the final iso-surfae. After forming set S the resulting funtion an be exressed as follows: f SES ()= f SAS () (R x ), (3) x fsas (0) where f SAS is defined as follows: 2R (b) R to both iso-surfaes of g and g 2,i.e. = g () g () and 2 = g 2 () g 2 () (Fig. 5b). Now we define a rediate B 2 g ( 2 ) 0 g 2 ( ) 0, (6) whih reresents whether oint lies within the the toriodal setion. Furthermore we define a distane based imliit funtion desribing the surfae inside the toroidal setion. In order to do so, we exloit the fat that the toroidal iso-surfae an be defined as minimal distane of to the oints belonging to the intersetion irle of g and g 2, I = {x g (x) =g 2 (x) =0}. However, omutation of all the oints would be very imratial. Therefore, we omute only the losest oint x 2 I to diretly x 2 = argmin x I x. (7) One we have this oint x 2, we define the resulting imliit funtion f S ()= f 2, onneting and 2, as follows: { R x 2 if B f 2 ()= 2 (8) f () f 2 () otherwise, where f () f 2 () is required to merge smoothly the toroidal ath with the both sheres. Figure 4 demonstrates the resulting imliit funtion iso-lines of the analogous senario in 2D. Note that the resulting funtion orretly omutes distane values u to the extent of R from the SES iso-surfae. S f SAS ()= g i (), (4) i= where the S i= g i = max{g,...,g S }. The term fsas (0) reresents all iso-surfae oints of the funtion f SAS, on whih the union of the solvent sheres is erformed. This union is subsequently subtrated from the f SAS. The formula (3), however; does not have a losed form solution. Therefore, our solution, reroduing Eq. 3, is based on building, in a iee-wise fashion, the SES imliit funtion aording to a given oint and the size of the set S, S = k. We denote the resulting imliit funtion as f SES ()= f S (). In the following setions, we desribe the onstrution of f S aording to S. We introdue four essential ases, i.e. when S ontains no atom, one atom, two atoms, and three or more atoms. The first two ases an be solved easily. When there is no atom inluded in S, the imliit funtion is defined as n f SES ()= f i (). (5) i= In ratie, sine there is no atom in 2R neighborhood of the oint, we an set the f SES () = 2R, in order not to evaluate all the oints. In a ase when there is only a single atom loser to than 2R, S = { }, the resulting f S () is defined solely by the atom, f S ()= f (). 4. Toroidal imliit funtion In a ase when there are two atoms loser to than 2R, S = 2, the resulting f S () desribes a toroidal ath that blends smoothly with both atom sheres. Let us assume that S = {, 2 }. We firstly detet whether oint lies in the toroidal setion of both extended sheres, g and g 2 (Fig. 5a, the blue triangle). The toroidal setion lis the toroidal iso-surfae within the ossible extent. In order to assess whether oint lies inside this area, we firstly generate two oints and 2 that are rojetions of the oint Figure 4: A 2D version of the toroidal imliit funtion with rendered iso-levels. The funtion returns orret distane values from the surfae u to R distane from the surfae (arrows). To evaluate the intersetion oint x 2 of g and g 2 we exloit Newton s iterative formula. The natural solution would be to utilize an analytial aroah to solve the intersetion of two sheres; nevertheless we aim at roviding a more robust aroah that ould ossibly handle other tyes of atom reresentation than sheres. For examle, in a ase when two atoms are very lose to eah other (overlaing) they an be aroximated by an imliit tube, whih an our quite frequently. Additionally we estimate the losest intersetion oint only, whih is a rerequisite for Eq. 8. Therefore, we aroximate both extended funtions g and g 2 at an unknown oint x,whereg (x)=g 2 (x)=0, by means of their first order Taylor exansion at oints g () and g 2 (): 0 = g (x) g () (x ) g () 0 = g 2 (x) g 2 () (x ) g 2 (), where (x ) exresses the vetor leading to the desired oint x. Additionally, it is required that the vetor (x ) lies in a lane (9) 220

5 erendiular to both gradients g () and g 2 () (Fig. 5): ( g () g 2 ()) (x )=0. (0) Using Eqs (6,9,0), we obtain a system of three linear equations where we would like to exress an unknown oint x from: g () g () g 2 () =(x ) g 2 () () 0 g () g 2 () (a) x-2 g 2 (b) g2 By denoting the matrix of gradients as M() for the oint, and the left side vetor of the equation by v() T, the solution an be obtained by x = v() T M() + (2) Please note that when B 2, the oint lies in the lose viinity of oint x. However, it is still an aroximation, and it is neessary to iterate through system () numerous times. Eq. 2 hanges then to: x i+ = v(x i ) T M(x i ) + x i, (3) where x 0 = is the initial guess. The formula (3) has been already utilized to omute the intersetion of two imliit funtions in [5], whih was aimed at generating distane aroximation of boolean oerations on funtional reresented objets. The system (3) stos when the oint x lies in lose viinity (ε N ) of both iso-surfaes, g (x) ε N g 2 (x) ε N. Here, we an delimit the number of the maximal number of stes, sine the oint lies lose to x. For instane we evaluated an average number of stes required for x 2 evaluation between two sheres, for ε N = 0.000, to 3 7stes. In a ase when a self-intersetion ours i.e the ondition desribed by Ryu et al. [23] (Fig. 5e), we benefit from the fat that oint x 2 always lies losest to oint (7). Therefore, selfintersetion is takled imliitly, whih is an another reason for referring Newton s iterative method over the analytial root finding. In Figure 5d, we showase the final toroidal iso-surfae for two atoms, where we also demonstrate the self-intersetion senario (Fig. 5f). In the following, the intersetion oint x retrieved via () between g i and g j is denoted as x i j. 4.2 Sherial triangle imliit funtion In a ase when there are three or more oints loser to than 2R, S 3, the resulting f S () desribes a sherial triangle. Here, the resulting funtion reate sherial triangle ath that blends smoothly with three toroidal athes. For simliity, let us assume that S ontains only 3 oints, S = {, 2, 3 }. In order to disover whether oint lies in the sherial triangle setion (liing the sherial triangle), we will exloit the intersetion oints omuted in the revious ase. Firstly, we extend the rediates B 2 (6) to B 2 to enomass not only toroidal setion but also an additional region that might be enomassed by the area of influene of the third oint 3 (Fig. 6a). Additionally, it must be fulfilled that intersetion oint x 2, obtained via (), lies within the oosite extended imliit funtion g 3 (Fig. 6b). Therefore, the new rediate B i j is defined as follows: B 2 g () R g 2 () R g 3 (x 2 ), (4) where g 3 (x 2 ) is the extended funtion of the third oint evaluated at the intersetion oint of g i and g j. Now, in order to determine whether oint lies inside the sherial triangle setion, we defined rediate B 2 3 as follows: B 2 3 B 2 B 3 B 2 3. (5) g g 2 () x-2 (e) Figure 5: Forming the toroidal ath for two atoms. a) The toroidal ath (red) lies only in the area marked by the blue triangle. b) In order to determine that the given oint belongs to this area, we erform () rojetion of B to iso-surfaes g ( )=0 and g 2 ( 2 )=0 and (2) evaluation of both funtions for oosite oints. ) In a ase that the oint lies inside the toroidal setion, we retrieve the intersetion oint x 2 of g and g 2 using the fat that the vetor (x 2 ) lies in the lane erendiular to both gradients g () and g 2 (). d)the visualization of the toroidal ath. The atoms are lose enough to get the ontinuous iso-surfae. e) Self-intersetion an our when the solvent is thiker than the height of the atual toroidal setion. f) The visualization of the self-interseted toroidal ath. Visual omarison of forming a onave sherial ath using original rediates B and their udated version B is shown in Figure 6 and 6d. Furthermore, in order to seify the sherial imliit funtion defined by, 2 and 3, we loate an intersetion oint fulfilling g (x) =g 2 (x) =g 3 (x) =0. Aordingly, we define the resulting imliit funtion f S ()= f 2 3, onneting, 2 and 3,asfollows: f 2 3 ()= { R x 2 3 if B 2 3 f 2 () f 3 () f 2 3 () otherwise, (6) where the seond branh is required to smoothly onnet the sherial triangle with the all three toroidal athes. In order to loate the oint x, we utilize Newton s iterative method of root finding of all three funtions, where the system () hanges to the following: g () g 2 () g 3 () =(x ) g () g 2 () g 3 (). (d) (f) (7) In order to solve (7) we utilize again the iterative formula resented in (3). The system (7) stos when the oint x lies in lose viinity (ε N ) of all three iso-surfaes, g (x) ε N g 2 (x) ε N g 2 (x) ε N. The definition of f S () for all the oints in S is straightforward: f S ()= (i, j,k) ( S 3 ) f i j k (). (8) 22

6 Similarly to the revious ase, when a self-intersetion ours; we utilize the fat that oint x 2 3 lies losest to oint and the self-intersetion is takled imliitly (Fig. 6e). We showased the examle omosed of 2 atoms in Figure 6f. x -2 3 (a) x-2 (b) () (d) (e) (f) Figure 6: The formation of the sherial triangle ath. a) Evaluation of the rediate B 2. The oint x 2, reresenting the intersetion of g and g 2, must lie inside the area of influene of g 3. Therefore the area (the blue ellise) for x 2 estimation is extended to all the oints that fulfil g () R g 2 () R. b) One the intersetion oint x 2 is omuted, it is evaluated against the third extended funtion g 3 (x 2 ) 0 (Eq. 4). ) When using the original rediates artifats an aear aused by the fat that the intersetion oint x i j lies too far away from the oint k. d) The examle of the orreted rediate B, where the raks disaeared. e) The self-intersetion issue is solved imliitly using the roerty of the oint x 2 3.f)Anexamle of the SES, defined by (8), omosed of 2 atoms. 5 IMPLEMENTATION Our imlementation of the visualization ieline is erformed using modern GP-GPU tehniques, namely CUDA and GLSL. On CUDA, we erform the ray-asting of the whole sene and store the iso-surfae oints and their normals. Afterwards, using GLSL shaders we erform an image based enhanement by means of the sreen sae ambient olusion [7]. The arallelization is done on the ray basis, where a single ray is generated for every ixel. 5. Data Struture The ruial art of our method is to retrieve the set S that reresents all neighboring oints that might artiiate in forming the funtion. There have been several effiient GPU strutures roosed to retrive k-losest oints to a given one [33]. Nevertheless, we utilize a simle and straightforward aroah that is based on an uniform satial subdivision. This has has been already utilized by the broader moleular visualization ommunity [, 4]. The atoms are sorted into ubi voxels with a lateral length of 2radius max + 2R max,whereradius max reresents maximum (van der Waals) radius of all inluded atoms and R max reresents maximal allowed radius of the solvent. We set the maximal allowed radius to 2, whih means that values of R an be interatively varied from 0 to 2. The sorting of the atoms into the grid is done in O(n), whih also reresents the only and the neessary reomutation ste in our ieline. Then in order to build the set S it is required to visit 3x3x3 neighboring voxels for a given oint. Thus, for a given time-ste, we need to send to GPU only the atom enters and their radii, and the voxels with Ids of atoms. 5.2 Rendering Parameters The rendering roess deends on several arameters, whih affet the overall rendering erformane and the reision ratio. With this reset these arameters an be taken as level of detail arameters. In the general ase, the size of the set S, might ontain an unlimited number of atoms, whih would make it imossible to store them in CUDA memory, even with the version 4.0 that allows to alloate dynami arrays. Fortunately, Varshney et al. has exerimentally shown that a tyial atom in a rotein has aroximately 45 neighboring atoms within the radius of a water moleule [32]. Therefore, we limit the maximum size of S to 45 = S. However, suh a onfiguration would still rodue ( 45) 3 = 490 trilets to evaluate in Eq. (8). Aordingly, we introdue arameter k max whih seifies the maximum number of atoms being stored when building the set S, with the maximal limit being 45. With reset to this limitation, we have to hoose whih atoms are retrieved. Therefore, all the atoms in S are stored in the inreasing order, i.e. S = {, 2,..., k f () f 2 ()... f kmax () 2R}. Sine we are ordering only k max number of items, the time omlexity of the sorting is O(). We evaluated the quality of the resulting surfae aording to different k max values. Here we notied that for k max > 0, we obtained only small surfae imrovements (Fig. 7). On the other hand, when rendering avities in details, the number of S-inluded atoms has to be inreased sine all the atoms along the avity irumferene (and for whih f i () 2R) might artiiate to the final surfae. Figure 7: Comarison of the SES visual quality when having set the arameter k max = 5 (left), k max = 0 (middle) and k max = 5 (right). Note that there are only negligible imrovements between the last two. The resulting imliit funtion defines the imliit surfae at oints f SES ()=0, whih an ause a erformane bottlenek during the evaluation of the exat oints laying on the iso-surfae. Therefore, we determine whether the oint lies on the iso-surfae if and only if f SES () < ε,whereε reresents the minimal allowed roximity to the surfae. Here, the exat ε value is seified on the fly; either to inrease erformane or to imrove the surfae quality. When evaluating both systems () and (7), we seify the ε N and the maximum allowed number of iterations L available for intersetion oint estimations, i.e x L = v(x L ) T M(x L ) +x L. Sine the intersetion is always evaluated when oint is lose to the intersetion oints themselves, we set L = Ray-asting As was mentioned before, the arallelization is done on the ray basis. Here we omute the ray entry (t min )andexit(t max ) arameters within the bounding box of the urrently bounded sene. For simliity, we erform only the first hit traversal. A generated ray is roessed in a ste-wise fashion until the t max is reahed or we hit the iso-surfae. Sine the set S is ordered inreasingly from,we erform both toroidal and sherial triangle imliit omutations only when S > 0andg () 0. This reflets the fat that must lie at least in one area of influene of all the oints in S. Therefore, the funtion f SES () returns a signed distane value reresenting the losest roximity to the SES iso-surfae, although only when being loser to the surfae than R. Using this fat, the rayasting 222

7 roedure inrement the ray arameter t by f, sine the funtion has negative values outside, i.e. t = t f (Fig. 8). The notieable defiieny is that it is neessary to erform many stes when being lose to the surfae. -f Figure 8: Ray-asting a simlified sene. We erform stes of the size of f SES, in order not to ski the existing iso-surfae. 5.4 Funtion evaluation One of the disadvantages of imliit funtion evaluation is the time omlexity when evaluating the funtion for all the oints in S.In general ase, where the oints have arbitrary toology, the evaluation of sherial triangle funtion would lead to O( S 3 ) omlexity. Fortunately the number of atoms artiiating generation of imliit funtion is uer bounded. With reset to this restrition we store the required ombinations of ( k ( 2) and k 3), used by Eq. (8) into linear arrays into the onstant CUDA memory. Comutations of rediates, intersetions and funtion values is ahieved in O(), sine we are iterating over systems (,7) in the uer bounded number of omutational stes. While evaluating, the only real bottlenek is being traversal of neighboring oints in order to onstrut the set S. This is done in linear time O(n). To otimize the rayasting, we ahe the intersetion oints while rogressing along the ray. 5.5 Performane We evaluated the erformane of the new rendering method on the sequene of eight roteins (Fig. ); murine oronin- (3023 atoms, a), g-atin (2872 atoms, b), otassium hannel (2820 atoms, ), roteinase 3 (3346 atoms, d), immunoglobin (2530 atoms, e) roliferati ell nulear antigen (2555 atoms, f), bateriorhodosin (78 atoms, f) and tubulin (4744 atoms, h). We were varying all resented arameters affeting the erformane (k max, ε, ε N ). The erformane measurements were done on a workstation equied with two (2 GHz) roessors and 2.0 GB RAM and with the GPU, NVIDIA GeFore GTX 480. It is imortant to mention here that we do not aim at outerforming any existing rendering tehniques [, 4] regarding FPS, sine we omute the iso-surfae on the fly. On the other hand, our aim here is to define SES reresentation in a simle way and to rovide users with an interative resonse for hanging any rotein or solvent arameters. Furthermore, to study moleular dynamis using visualization of arbitrary setions of the sequene without the need of any reomutation stes. Therefore, the erformane (FPS) is more about the fat that suh a system an rovide an interative resonse when, for instane, inluded in the existing rotein exloration environment. The level of interativity is deendent on the three aforementioned arameters, k max,ε and ε N. We observed that the rendering erformane is affeted signifiantly by the arameter k max.forinstane, by setting ε = 0.0, ε N = 0.005, R =.4 and by inreasing inrementally k max from 5 to 20, we get the following FPS: k max a b d e f g h Nevertheless, by setting ε < and setting k max > 20 we get < FPS for all the roteins. Therefore, this onfiguration an be used to rovide high-quality images but a less interative resonse. To guarantee onstant framerates (above 5FPS), we imlemented a rogressive refinement strategy that automatially redues these arameters,e.g.; ε = 0.2, ε N = and k max = 5, when erforming visual exloration (zooming, rotation) or seeking for different time-stes. When a user does not interat with the sene, these arameters are automatially inreased to the higher level to get the finer details. We demonstrated the sequene of two of resented roteins (roteindanf,wherewesetk max = 0 as a onstant) to the bioinformatis sientist, where we ahieved satisfatory interativity. Here, the feedbak, whih we have reeived was that visualization rovides a suffiient amount of details for an overview on the rotein surfae with reset to the avity exloration. Additionally, the rovided interativity was highly areiated. 6 ERROR ESTIMATION AND LIMITATIONS The visual surfae reision is diretly affet by the minimal allowed roximity, ε, to the iso-surfae during the ray-asting. The arameters k max and ε N affet the funtion evaluation error. When a given oint x on the ray r fulfils f (x) ε, the oint x is onsidered to lie on the iso-surfae, whih atually holds only if ray r intersets the surfae in a oint x,i.e. f (x )=0. Sine the funtion f is C ontinuous, via the mean value theorem in several variables, there exist suh a arameter t [0, ] that f (x) f (x )= f ( ( t)x +tx ) (x x ). (9) When denoting oint =( t)x +tx and sine f (x )=0, we get the errore deendent on the following equation f () (x x )= f (x) ε. (20) Using Eq. (20) we an identify the ase, when funtion values vary raidly and gradient magnitude an beome very high, then distane between x and x needs to be signifiantly smaller in order to fulfil Eq. (20). In our aroah, this reates the artifats sine the arameter ε is taken as a onstant during ray-asting. Instead, as a future work, it needs to be adjusted aording to funtion omlexity in the neighborhood of x. We have disussed the orresonding visualization artifats with the bioinformatial seialist. Sine the SES reresentation is utilized in avity disoveries redominantly, whih are reognized only when a ball of radius R fits inside an oening, the artifats were onsidered insignifiant when setting ε R. As a onsequene, the major limitation of utilizing our imliit reresentation for visualization uroses is that it does not have to be a 00% SES reresentation. Here, the differene to existing aroahes lies in the fat that while these aroahes generate a set of quadris, our imliit funtion is defined globally and therefore we have to rely on the numerial aroah while traking the isosurfae. Additionally, even one seifies k max = 45, it an theoretially haen that the number of neighboring atoms an beome higher (with the reset to MD simulations). Nevertheless, this is not the tyial ase, sine we have notied only little imrovements when setting k max > 0. Another limitation is deendent more on the utilized grahis hardware, for whih GPU an stall when evaluating too many oints in set S. 223

8 7 UTILITY POTENTIAL Our imliit reresentation is tailored for the urose of deteting, analyzing and visualizing avities; i.e tunnels or hannels. Tunnels are essentially athways leading from a avity to a rotein ore. Channels lead through the whole rotein struture having avities on both ends. The study of how these athways evolve is highly imortant and required for drug design. Usually, to disover a hannel, it is neessary to find the avity and then trak the avity during the rotein simulation, to find out whether it reahes the surfae. Contrary to revious aroahes [, 4], for a given oint x our SES imliit funtion (Setion 4) omutes the losest distane to the surfae, the gradient at x, and lassifies x aording to whether it lies inside or outside of the rotein atoms. By means of numerial methods, we an omute the volume area of the avity using oint-to-objet lassifiation. Another utilization is to trak down the avity automatially using the distane roerty of the imliit funtion and using its gradient. We an delimit the avity detetion only to laes, where a solvent will fit in. In other words, we loate oints where f SES < R, i.e. oints that are above the distane of solvent radius R. 8 CONCLUSION We resented a new method of reresenting SES by means of funtional reresentation of objets. The imliit funtion omutes SES reresentation loally. For a given oint the funtion returns the orret minimal distane to the surfae, although only u to the distane of the solvent radius R. Additionally, the funtion is ositive inside the SES objet and negative outside. The main ontribution is that we reresent the moleular surfae without reomutation, thus enabling instantaneous visual analysis. Moreover, we roosed a simle ray-asting method whih is used to render even large roteins at reasonable interativity. We introdued a set of arameters that affet the overall rendering erformane. These an be adjusted in aordane with the rendering erformane versus a reision ratio. Sine the work forms the first stes in a new diretion of SES reresentation, we have erformed only an informal evaluation. We showed the imlementation of our method to seialists where we demonstrated it on a sequene of the two resented roteins. The ossibility of hanging the solvent radius or even to ik u atoms in related views, was found to be highly interesting and romising. ACKNOWLEDGEMENTS We give thanks to Nathalie Reuter for roviding the moleular dynamis simulation datasets and the neessary feedbak, and to Armin Pobitzer for final touhes with mathematial formulas. REFERENCES [] J. Blinn. A generalization of algebrai surfae drawing. ACM Transations on Grahis, : , 982. [2] M. Chavent, B. Levy, and B. Maigret. MetaMol: high-quality visualization of moleular skin surfae. 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