Extraction of Robust Voids and Pockets in Proteins

Raghavendra Sridharamurthy1 Harish Doraiswamy2 Siddharth Patel3 Raghavan Varadarajan3

Vijay Natarajan1,4

1Department of Computer Science and Automation, Indian Institute of Science, Bangalore2Polytechnic Institute of New York University, USA

3Molecular Biophysics Unit, Indian Institute of Science, Bangalore4Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore

IISC-CSA-TR-2013-3http://www.csa.iisc.ernet.in/TR/2013/3/

Computer Science and AutomationIndian Institute of Science, India

May 2013

This report is an extended version of the EuroVis 2013 short paper titled “Extraction of robust voids and pockets in proteins”

Extraction of Robust Voids and Pockets in Proteins

Raghavendra Sridharamurthy∗ Harish Doraiswamy† Siddharth Patel‡

Raghavan Varadarajan§ Vijay Natarajan¶

Abstract

Voids and pockets in a protein refer to empty spaces thatare enclosed by the protein molecule. Existing meth-ods to compute, measure, and visualize the voids andpockets in a protein molecule are sensitive to inaccu-racies in the empirically determined atomic radii. Thispaper presents a topological framework that enables ro-bust computation and visualization of these structures.Given a fixed set of atoms, voids and pockets are rep-resented as subsets of the weighted Delaunay triangu-lation of atom centers. A novel notion of (ε, π)-stablevoids helps identify voids that are stable even after per-turbing the atom radii by a small value. An efficientmethod is described to compute these stable voids for agiven input pair of values (ε, π).

1 Introduction

Protein molecules have a well packed structure, yet theycontain cavities. A cavity refers to both voids (withoutopenings) and pockets (with openings). These cavitiesplay a key role in determining the stability and functionof proteins.

Several methods have been proposed to locate suchcavities in protein molecules. We focus our attentionon geometric methods. Edelsbrunner et al. [6, 8] andLiang et al. [16, 17] propose a definition that is based

∗Department of Computer Science and Automation, Indian Insti-tute of Science, Bangalore. raghavendra@csa.iisc.ernet.in†Polytechnic Institute of New York University, USA.‡Molecular Biophysics Unit, Indian Institute of Science, Banga-

lore.§Molecular Biophysics Unit, Indian Institute of Science, Banga-

lore.¶Author for TR correspondence. Department of Computer Sci-

ence and Automation, Supercomputer Education and Research Cen-tre, Indian Institute of Science, Bangalore. vijayn@csa.iisc.ernet.in

on the theory of alpha shapes and discrete flows in De-launay triangulations. Kim et al. [11, 13] propose adefinition of cavities based on an alternate represen-tation of a set of atoms called beta shapes that faith-fully captures proximity. Tools based on the above ap-proach are available and widely used [2, 14, 12]. Tilland Ullmann [24] employ a graph theoretic algorithm toidentify cavities and compute their volume. Parulek etal. [21] use graph based methods on the implicit repre-sentation of molecular surfaces to identify pockets andpotential binding sites. Varadarajan et al. [1] employa Monte Carlo procedure to position water moleculestogether with a Voronoi region-based method to locateempty space. They discuss the importance of accurateidentification of cavities for the study of protein struc-ture and stability. Novel Voronoi diagram-based tech-niques for the extraction and visualization of cavitieshave also been developed from the viewpoint of study-ing and interactively exploring access paths to activesites [23, 22, 19, 18]. Krone et al. [15] present a visu-alization tool for interactive exploration of protein cav-ities in dynamic data.

1.1 Motivation and Problem StatementThe input used for calculations in previous work comefrom x-ray crystallography data or other lower resolu-tion data. Previous cavity detection methods are sen-sitive to inaccuracies that are inherent in the crystallo-graphic measurements. While the measurements mayguarantee high resolution, it is important to note thateven small inaccuracies may cause a significant differ-ence in the reported number of cavities. The inaccu-racies may also arise due to some fundamental limi-tations such as the notion of radii of atoms, which isdetermined empirically. For example, as illustrated inFigure 1, presence of such inaccuracies may result in acavity detection method to report two distinct but largecavities in place of one or report very small volume cav-ities. Figure 2 illustrates the problem as it occurs in a

1

Figure 1: Left: Two cavities that are apparently verynear to each other may be a single cavity. Right: A verysmall cavity may be reported whereas no such cavitymay exist.

Figure 2: (a) Two voids that appear very near to eachother in a lyzozyme protein (PDB ID: 200l). The solidsurface represents the voids while the wireframe repre-sents the molecule. (b) Zoomed-in view of the voidswithout the molecular skin surface. (c) The two voidsmay be a single void.

lyzosyme protein.We aim to develop an interactive method to compute

robust cavities in proteins. We achieve this by enablingthe user to reduce, if not completely eliminate, the inac-curacies mentioned earlier. We define what robustnesswould mean in this context, why such a notion is impor-tant and also demonstrate the robustness of the method.As an auxiliary task, we visualize the protein moleculesand their cavities in an interactive manner and computetheir properties.

From the biologist’s point of view, obtaining a stableprotein is the starting point of many applications, fromin-vitro studies of binding and interactions, to using theprotein as an antigen or vaccine. Whereas surface pock-ets often form part of the active site of enzymes or inter-acting sites for other proteins, internal voids are oftenrelevant structurally as features that affect the overallstability of the protein. It is established that filling upinternal voids improves the packing of the protein thusincreasing stability. In this respect, detecting and vi-sualizing structurally robust cavities inside the proteininforms the biologist on which mutations to perform toimprove internal packing and get a stable protein.

Our software while detecting robust cavities and cal-culating volumes, also provides an interactive frame-work that the biologist can use, along with their ownknowledge and discretion, to decide which cavities af-

fect their proteins and which mutations would be re-quired for their specific application.

1.2 ResultsThe main results described in this paper are in the con-text of a novel definition and method for computing ro-bust and stable voids in proteins. We employ a sim-ple and succinct structure called the alpha complex torepresent protein molecules. The alpha complex is asimplicial complex that can be stored as a filtration, aseries of simplicial complexes Ki with Ki ⊂ Ki+1.With the aim of computing a set of voids that are stablewith respect to small perturbations in the atom radii,we develop a method that modifies the radii of a selectset of atoms symbolically by systematically processingand modifying the filtration. We show that this modifi-cation results in controlled changes in the number andproperties of voids and does not violate key propertiesof the filtration. The method also supports the elimina-tion of very small or insignificant voids as measured bythe notion of topological persistence [9]. We also de-velop software to visualize the stable cavities togetherwith the molecule, and to calculate cavity volume andsurface areas.

2 BackgroundIn this section, we briefly introduce the topologicalbackground required to define and represent the struc-ture of biomolecules [20, 5, 4].Simplicial Complex. A k-simplex σ is the convex hullof k + 1 affinely independent points. A vertex, edge,triangle, and tetrahedron are k-simplices of dimension0 − 3. A simplex τ is a face of σ, τ ≤ σ, if it is theconvex hull of a non-empty subset of the k+1 points. Asimplicial complex K is used to represent a topologicalspace and is a finite collection of simplices such that(a) σ ∈ K and τ ≤ σ implies τ ∈ K, and (b) σ1, σ2 ∈K implies σ1

⋂σ2 is either empty or a face of both

σ1 and σ2. A subcomplex of K is a simplicial complexL ⊆ K.Voronoi diagram and Delaunay triangulation. LetS ⊆ Rd be a finite set of points. The Voronoi cell Vp, ofa point p ∈ S, is the set of points in Rd whose Eu-clidean distance to p is smaller than or equal to anyother point in S. The collection of Voronoi cells ofall points in S partitions Rd, and is called the Voronoidiagram (Figure 3(a)). The Delaunay triangulation D

2

(a) (b)

Figure 3: (a) Voronoi diagram ofa point set in R2. (b) The De-launay complex is the dual of theVoronoi diagram.

(a) (b)

Figure 4: (a) Intersection of the weightedVoronoi diagram and the union of balls.(b) The dual complex is the dual of this par-tition of the union of balls that captures theincidence relationship.

(a) (b)

Figure 5: (a) Void and(b) Pocket in a collectionof 2D balls.

of S is the dual of the Voronoi diagram and partitionsthe convex hull of S, see Figure 3(b). The above def-initions can be extended to a set of balls or weightedpoints by choosing an appropriate measure of distancebetween a weighted point p and a point in Rd. Thepower distance between p and a point x ∈ Rd is equalto πp (x) = ‖x − p‖2 − wp, where wp is the weight ofp.Alpha Complex. Molecules are often represented us-ing a space-filling model such as a union of balls. Theweighted Voronoi diagram may be extended to repre-sent the contribution from each atom to the union ofballs. Consider an atom p. Define Bp as an open ballhaving the radius of the atom p. Let Vp be the weightedVoronoi cell corresponding to p, where the weight isequal to the square of the atom radius. The contribu-tion from each atom p is equal to Bp ∩ Vp, the inter-section between the ball corresponding to the atom andthe weighted Voronoi cell of p. The corresponding dualstructure is a subcomplex of the weighted Delaunay tri-angulation and called the dual complex, see Figure 4.

Edelsbrunner et al. [7, 10, 3] consider a growth modelwhere the ball radii grow, and track the changes in thedual complex. The growth parameter, α, corresponds toa radius

√r2p + α

2 for a ball centered at p with radius

rp. The weight of the pointw(p) increases tow(p)+α2.Note that α = 0 corresponds to no growth. The dualcomplex corresponding to a set of balls after they aregrown by α is called the alpha complex.

Given a simplicial complex K, a finite sequence,∅ = K0,K1, . . . ,Km = K, of subcomplexes of Kis a filtration if K0 ⊂ K1 ⊂ · · · ⊂ Km. The rankof a subcomplex refers to its position in the filtration.The set of alpha complexes obtained by varying α from−∞ to ∞ is a filtration of the Delaunay triangulation.In particular, we consider the filtration that is generatedby inserting the simplices one at a time and ties brokenbased on the dimension of the simplex.Voids and Pockets. Let the alpha complex K repre-

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Figure 6: A filtration generated by inserting simplicesin a particular order. The arrival time and the behaviorof the simplices are also shown in the boxes.

sent a molecule at a given value α and D be the De-launay complex of the weighted point set. A cavity isa maximally connected component of the complementD −K. Voids and pockets are cavities that are, respec-tively, bounded and not bounded by the union of balls.Figure 5 illustrates a void and a pocket in 2D.

Topological persistence. The alpha complex helpsrepresent and track voids via the growth process. Avoid is represented by a 2-cycle, which is a collectionof 2-simplices whose boundary is empty. A 2-cycleis created when the last triangle in the collection is in-serted and it is destroyed by the insertion of a tetrahe-dron. Topological persistence of a k-cycle measures itslifetime (k ≥ 0) in a filtration [9]. It is equal to thedifference between the ranks of alpha complexes in thefiltration where the cycle is created and destroyed. If a2-cycle doesn’t have a destroyer then its persistence isequal to∞. Given a filtration, the persistence of cyclescan be computed efficiently [9]. The insertion of everysimplex either creates a cycle or destroys a lower di-

3

mensional cycle. The persistence value associated witha simplex is equal to the topological persistence of thecorresponding cycle.

Figure 6 illustrates creation and destruction of cyclesin a filtration of a small simplicial complex. The num-ber in each box denotes the timestamp. The simplexthat is added is denoted in the adjacent box along with apositive or a negative sign which respectively signifiescreation or deletion of a cycle.

3 Robust Voids and their Compu-tation

We introduce a notion of robust voids based on twoparameters, one local and another global. The localparameter is referred to as stability and the global pa-rameter is specified by topological persistence. In or-der to simplify the description, we assume that thevoids are computed for the α-complex correspondingto α = 0. However, the definitions, methods, and sub-sequent analysis are valid for all values of α.

3.1 ε-stable and π-persistent voidsConsider the interval [−ε, ε] of α values, where ε ≥ 0.A void is called an ε-stable void if it remains a singleconnected void within all α-complexes for α values inthe range [−ε, ε]. In other words, using the lifetime ter-minology, the void is born, possibly split into multiplecomponents, and destroyed at α-values that lie strictlyoutside of this interval. A void is π-persistent if itstopological persistence is greater than π i.e., the voidsize measured in terms of its lifetime is greater than π.Combining the two notions of robustness, we call a voidto be (ε, π)-stable if it is both ε-stable and π-persistent.

The above definitions help measure the stability ofthe voids when the radii are perturbed by a smallvalue. The local parameter considers perturbationwithin a small interval centered at the α-value of inter-est whereas the global parameter measures the size ofthe void in terms of its lifetime in the filtration. Voids ofinterest may often not be stable with respect to both no-tions. For example, a large sized void (π-persistent forsome large π) may be born within the interval [−ε, ε].However, note that a small perturbation in the radii ofatoms that line the surface of the void could result in anearlier birth time, hence making the void to be ε-stable.We aim to extract all voids that are either stable as is orcan be made stable via a small perturbation.

3.2 Computing (ε, π)-voidsThe location of the atoms that constitute a proteinmolecule together with their van der Waals radii is ob-tained from the protein data bank in pdb format. Givenε and π, we compute the set of (ε, π)-stable voids asfollows.

1. Compute the weighted Delaunay triangulation ofthe input [14]. The atom centers form the set ofpoints that are weighted using their van der Waalsradii.

2. Build the alpha shape spectrum [3], which is a fil-tration of the weighted Delaunay triangulation.

3. Modify the filtration based on the value of ε.4. Compute the set of (ε, π)-stable voids by identi-

fying all voids [16] of the modified filtration atα = 0, and retaining only those voids that havepersistence at least π.

Modifying the filtration. The filtration of theweighted Delaunay triangulation as defined by theα-values provides an explicit representation of thebirth/death times of each void and the evolution duringits lifetime. We propose to alter the birth/death timesof voids by modifying the filtration instead of directlymodifying radii of atoms that line the surface of thevoid. While the latter approach follows directly fromthe definition, it is cumbersome and computationallyinefficient. For example, varying the radii without ex-plicit control may lead to changes in the triangulationand the alpha complex. These changes need to be ex-plicitly tracked, else they may lead to inconsistenciesbetween the alpha complex that represents the moleculeand the space-fill model. Resolving such inconsisten-cies would necessitate the re-computation of all repre-sentations. On the other hand, the former approach issimpler and computationally efficient.Delayed simplex insertion. One or more simplices areinserted to obtain a rank i+1 simplicial complex from arank i simplicial complex in the filtration. Higher rankscorrespond to higher values of α. The topology of voidsmay change when the simplices are inserted. In partic-ular, the insertion of a triangle may either not affect anyvoid, split a void into two, or create a new void. Onthe other hand, the insertion of a tetrahedron always de-stroys a void. These topology changes may therefore beavoided by delaying the insertion of the simplices thatchange the topology of voids. LetKj be the alpha com-plex corresponding to α = ε. Consider the set of sim-plices, Σ, inserted into the filtration for values of α in

4

the range [−ε, ε]. Let Σt ⊂ Σ be the set of the trianglesthat split a void, and ΣT ⊂ Σ be the set of tetrahedra.We delay the insertion of simplices σi in Σt and ΣTsuch that σi /∈ Kj but σi ∈ Kl, where Kj ⊂ Kl ⊂ D.Simplicial complexes in the filtration of the weightedDelaunay triangulation and the order of simplices thatare inserted to generate the filtration satisfy several con-tainment and incidence properties. These propertiesshould be satisfied for the modified filtration as well.Towards this, we propose a conservative but computa-tionally efficient approach to modify the filtration:

1. Move all tetrahedra in ΣT to the end of the filtra-tion. All such tetrahedra are present in D but notin any Ki ⊂ D.

2. For each triangle in Σt, find its incident tetrahedraτ1, τ2.

3. Delay the insertion of the triangle and the twotetrahedra, τ1 and τ2, to the end of the filtration.

(a)

(b)

Figure 7: 2D illustration of simplex insertion causing avoid to split. (a). Voids occur near to each other and theedge that splits the single void into two. (b). The twovoids merge into one if the simplex insertion is delayed.

To illustrate the modification in 2D let us consider avoid split into two as shown in Figure 7(a). Assumethat the highlighted edge (triangle in 3D) becomes partof the alpha complex in the interval [−ε, ε]. Further italso satisfies the criterion that it bounds two differenttunnels (voids in 3D). So it becomes a candidate fordelayed insertion. We move the edge to the end of thefiltration, which means that it does not belong to thealpha complex as shown in Figure 7(b). Also the radii ofatoms centered at the end points of the edge (triangle in3D) are changed accordingly in relation to the ε value.

Selective modification of the radii of a specific set ofatoms is hence achieved in a controlled manner.Implication of the modified filtration. Consider thedelayed insertion of a triangle (edge in 2D) to avoid thesplit of a void into two as illustrated in Figure 7. Thedelay corresponds to shrinking the atoms centered at thevertices of the triangle. However, note that the radii arenot yet modified. We optionally modify the radii laterfor further analysis of the void. A triangle that createsa void is left untouched and the corresponding void isalso declared to be ε-stable. The triangle insertion maybe advanced to ensure that the void is created outsidethe interval. This corresponds to a small increase in theradii of the atoms centered at the vertices of the trian-gle. We choose not to explicitly advance the triangleinsertion because it does not affect the results for smallvalues of ε. After the filtration is modified as describedabove, we recompute the voids from the alpha complex.

We compute the persistence of the ε-stable voids ob-tained and retain only those having persistence greaterthan π. This pruned set of voids are (ε, π)-stable. Notethat the persistence is computed with respect to the orig-inal filtration. The above notion of stability can be ex-tended to pockets as well.

3.3 Implementation details

The filtration obtained from the alpha shape spectrum(Step 2) is stored as a list. The index of each sim-plex in this list represents the rank of that simplex. Foreach triangle, we additionally store the indices of its co-face tetrahedra. For a given ε, we first compute theranks k1 and k2 of the alpha shape at α = −ε andα = ε, respectively, and the set of voids present in theinterval, α ∈ [−ε, ε]. If the addition of a triangle σk,k1 ≤ k ≤ k2, splits a void, σk is added to the set Σt.The coface tetrahedra τ1 and τ2, of σk is added to ΣT .All tetrahedra having their rank in the range [k1, k2] areadded to ΣT . The set of triangles and tetrahedra in Σtand ΣT , respectively, are sorted in the increasing or-der of their ranks. Instead of explicitly moving thesesimplices to the end of the filtration, we perform an im-plicit move. These simplicies are marked as invalid inthe list containing the original filtration. The new filtra-tion is obtained by traversing the original filtration, ig-noring the invalid simplicies, followed by traversing Σtand ΣT . An advantage of using this approach is that,when the value of ε is changed, it is easy to revert to theoriginal filtration and recompute the new filtration.

5

3.4 Analysis

Let m be the number of simplices in the Delaunay tri-angulation of the input protein having n atoms, m =O(n2). Computing the set of voids takes O(mα(m))time using the union-find data structure. Here, α is theinverse Ackermann function. Given ε, Σt and ΣT iscomputed in O(m) time using a sequential search overthe filtration. Identifying the set of tetrahedra incidenton triangles in Σt, and moving all the simplices to theend of the filtration takes O(m) time. Thus the timerequired to modify the filtration is O(mα(m)).

3.5 Stability of pockets

The notion of stability defined for voids can be extendedto pockets as well. A pocket may be destroyed in thefiltration by the addition of a triangle. By destroyingthe pocket, this triangle creates a void, or a void-pocketpair. In our current implementation, the voids that arecreated by the triangle are given preference over pocketsthat are destroyed in the (−ε, ε) range. We choose to dothis because a pocket with a very narrow opening (sizeless than ε) will not allow any ion to pass into it, andhence functions more as a void. Therefore, the only sta-ble pockets that are identified correspond to those thatare destroyed or split into a void-pocket pair outside ofthe (−ε, ε) range. However, in case a pocket is pre-ferred over a stable void, the algorithm can be easilymodified to delay the insertion of the triangle that de-stroys the pocket, along with its coface tetrahedra.

4 ResultsGiven the input protein, our software RobustVoids firstcomputes the alpha complex of the input. The user canvisualize the protein either as a skin surface, or as avolume mesh for different α-values. The set of (ε, π)-stable cavities are computed using the values of ε andπ specified by the user. The software also reports thecavity volume and surface area.Visualization of stable cavities. Figure 8(a) showsprotein 2CI2, which has 3 voids. Using a value ofε = 0.3 and π = 0.01 results in two (0.3, 0.01)-stablevoids, see Figure 8(b). Note that a value of ε = 0.3is equivalent to an increase / decrease of the radius ofan atom by at most 0.33Å, which is within the tolerated0.5Å used by the biologists. Modifying the filtrationand computing the stable voids for this protein takes

Figure 8: Visualization of voids of the protein 2CI2.The values ε = 0.3 and π = 0.01 was used to computethe set of stable voids. (a) Voids in the volume com-puted at α = 0. Number of voids = 3. (b) The set of(0.3, 0.01)-stable voids. Number of (0.3, 0.01)-stablevoids = 2. (c) Two nearby voids in the protein renderedin solid and the skin surface of the molecule renderedin wireframe mode. (d) The merged stable void shownalong with the skin surface of the molecule.

0.1 s.The protein 4HHB has a total of of 72 voids, shown

in Figure 9(a). Using a value of ε = 1.0 and furtherremoval of voids having persistence π < 0.01 resultsin a total of 70 (1.0, 0.01)-stable voids, see Figure 9(b).Figure 9(c) shows two nearby voids in the protein whichmerge to form a single stable void, see Figure 9(d).Note that a value of ε = 1.0 is equivalent to an increase/ decrease of the radius of an atom by at most 0.33Å,which is within the tolerated 0.5Å used by the biolo-gists. Modifying the filtration and computing the stablevoids for this protein takes 62 secs. Figure 10 showsthe visualization of the (ε, π)-stable voids for the pro-tein 4B87. Modifying the filtration and computing thestable voids for this protein takes 33 secs.

The implementation for now uses a naive approach ofexplicitly computing the voids/pockets while construct-ing the set of simplices which needs to be moved andthus the running time is large. This can be reducedby tracking the creators of voids/pockets as describedin [9].Properties of stable cavities. Figures 11 and 12plots the number and volume of (ε, π)-stable voids forvarious values of ε. Note that increasing the value ofε implies that voids from a larger α range are consid-ered. This could potentially increase the number ofε-stable voids. However, such voids usually have lowpersistence and are therefore not (ε, π)-stable. We use

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Figure 11: Graphs showing the variation of the number of voids with varying ε. Note that there is an increase inε-stable voids as we consider a larger interval but (ε, π) voids are less than or equal to original number of voids.

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Figure 12: Graphs showing the variation of the total volume of voids with varying ε.

Figure 9: Visualization of voids of the protein 4HHB.The values ε = 1.0 and π = 0.01 was used to computethe set of stable voids. (a) Voids in the protein com-puted at α = 0. Number of voids = 72. (b) The setof (1, 0.01)-stable voids. Number of (1.0, 0.01)-stablevoids = 70. (c) Two nearby voids in the protein. (d)These two voids merge together resulting in a singlestable void.

Figure 10: Visualization of voids of the protein 4B87.The values ε = 1.0 and π = 0.01 was used to computethe set of stable voids. (a) Voids in the protein com-puted at α = 0. Number of voids = 47. (b) The set of(1.0, 0.01)-stable voids. Number of (1.0, 0.01)-stablevoids = 44. (c) Two of the nearby voids in the protein.(d) These voids merge together resulting in a single sta-ble void.

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Figure 14: Plot of the computed and actual volumesof the artificial voids that were generated using mutantmodels.

a constant value of π = 0.01 in all experiments. Thetotal volume of all stable voids increases marginally(< 1%) with increasing ε. The merging of two nearbyvoids into a single stable void does not effect the totalvolume. However, volumes of individual voids couldchange drastically. The volume of the stable void inFigure 8(b) is approximately equal to sum of the vol-ume of the unmerged voids. The volumes are verifiedagainst known results [1].Robustness of (ε, π)-stable cavities. Figure 13 plotsthe number of (ε, π)-stable voids and π-persistent voidsfor various values of α for ε = 1.0 in case of 4HHBand 4B87 and ε = 0.3 in case of 2CI2. The number of(ε, π)-stable voids remain constant over an interval asthe method considers all the creation/destruction eventshappening in the interval [−ε, ε] for the computation.But the number of π-persistent voids vary as they aresensitive to creation/destruction of voids at a particularα value.Validation of computed cavity volumes. There isusually a variation in the volumes of cavities computedusing various methods [1]. This variation may arise dueto the different models used for computing the volumes.Therefore, we perform an additional normalization ofthe computed volumes using model mutants [1] to elim-inate such variations. We used 13 different model mu-tants to create a set of artificial voids. We use the result-ing volumes to compute the linear normalization func-tion, see Figure 14. The volumes computed using ourcomputation is normalized as follows:

V olume = 2.54× ComputedV olume+ 60.77.

In order to verify the correctness of the volumes com-puted by our software RobustVoids, we compare themto the volumes computed using MCCavity [1], see Fig-

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Figure 15: Comparison of normalised volumes com-puted using RobustVoids and MC Cavity.

ure 15. Note that we use normalised volumes in thisgraph.

5 Conclusions

We have defined a novel notion of robust cavities that isinsensitive to the perturbation of the atomic radii. Ro-bust cavities are computed via a controlled modificationof the filtration that represents the molecule and its cav-ities. Identifying robust cavities is important so that thebiologist only targets these cavities in tedious mutation-based experiments.

The method addresses the inaccuracies in the mea-surements of the radii by selectively varying the radiifor a specific set of atoms. But the positional uncer-tainties which arise due to the motion of the moleculesis not addressed. We have been able to present somevisual evidence that slight perturbations in the radii re-sults in a larger cavity instead of smaller cavities. Thevalue of ε is lower than the typical experimental errorin crystallographic measurements. We plan to furtherinvestigate the relationship between the perturbation inthe atom radii corresponding to the delayed simplex in-sertion and the structural and functional properties ofthe protein. Future work also includes generalizing theframework to use empirically determined intervals ofradii for each atom type and addressing the issue of bi-ological implications of the method.

Acknowledgements

This work was supported by a grant from De-partment of Science and Technology, India(SR/S3/EECE/0086/2012).

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π-persistent voids(ε,π)–stable voids

(a) Protein 4HHB

−0.4 −0.2 0 0.2 0.4⍺

0

1

2

3

num

ber o

f voi

ds

π-persistent voids(ε,π)–stable voids

(b) Protein 2CI2

−1 −0.5 0 0.5 1⍺

40

45

50

55

60

num

ber o

f voi

ds π-persistent voids(ε,π)–stable voids

(c) Protein 4B87

Figure 13: Graphs showing the variation of the total volume of voids for constant ε and varying α.

References

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