2 Review: Modeling Molecules with Spherical Balls

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1 CS273: Algorithms for Structure Handout # 12 and Motion in Biology Stanford University Thursday, 6 May 2003 Lecture #12: 6 May 2004 Topics: Geometric Models of Molecules II Scribe: Itamar Rosenn 1 Introduction The size and shape of molecules such as proteins have a strong influence in determining their function. In this lecture, we elaborate upon methods discussed in the previous lecture for modeling molecules geometrically, and explore how those methods can be used to accurately quantify the size and shape of the molecule itself. 2 Review: Modeling Molecules with Spherical Balls Space-Filling Diagrams In biology, space-filling diagrams are used to model the space occupied by a molecule. Each atom is modeled as a spherical ball; thus, since a molecule is a collection of atoms, it is modeled accordingly as a union of balls. When modeling molecules using spherical balls, we consider modeling the surface in one of three ways: The van der Walls surface is the surface of what is covered by the atoms, using the van der Walls radius of each atom. The solvent-accessible surface is generated by rolling a spherical probe around the van der Waals surface to reflect the accessibility of the molecule to a solvent. The molecular surface consists of the solvent-accessible surface offset inwards to remove areas touched by the probe surface, resulting in a smoother model than the original van der Waals surface, with less extreme crevices. Power Diagrams and Delaunay Triangulations A power diagram divides the space of our molecular model into several regions, each containing a unique atom of the molecule. For a given atom, its region in the power diagram consists of the space that is closer in power distance to that atom than to any other atom. We can use this notion of a power diagram to decompose our model into information about its atoms.

2 2 CS273: Handout # 12 Figure 1: (A) van der Waals surface, (B) solvent-accessible surface, and (C) molecular surface More formally: In three-dimensional space, let x y be the Euclidean distance between two points x and y, and let b i (z i, r i ) be a spherical ball whose center is z i and radius is r i. Thus, we define b i (z i, r i ) as {x R 3 : x z i r i }, the set of points that are at least as close to z i as the radius r i. The union of a set B of balls is B = {x R 3 : x b B}. The complement of this union, R 3 B, consists of an unbounded region that corresponds to the area outside of the molecule, and zero or more bounded regions that are cavities of B, which correspond to bounded empty spaces inside the molecule. The complement of the space occupied by the molecule will be addressed further in our discussion of pockets and voids in section 6. Define the power distance of a point x R 3 from a ball b i = b(z i, r i ) as Π i (x) = z i x 2 r 2 i. Note that the power distance between balls b i = (z i, r i ) and b j = (z j, r j ) is Π(ij) = z i z j 2 r 2 i r 2 j. The power diagram of a molecule is a collection of power cells for each of the spherical balls, where the power cell of a ball b i B is defined as the set of points at least as close to b i as to any other ball in B: V i = {x R 3 : Π i (x) Π j (x), j B} The dual of the power diagram of a molecule is its Delaunay T riangulation, which expresses the connectivity of our union of balls. This structure is formed by connecting the centers of each pair of atom-balls whose cells are adjacent in the power diagram. The convex hull of the atom centers consists of the outer boundary of the Delaunay Triangulation.

3 CS273: Handout # 12 3 Figure 2: (a) a molecular model segmented into a power diagram, (b) the corresponding Delaunay Triangulation, and (c) the alpha shape and corresponding dual complex (see section 4). 3 Simplicial Complexes To understand how we can use the notions of a power diagram and a Delaunay Triangulation to extract useful information about the actual molecule, we first need to understand a handful of basic topological concepts. In topology, a vertex is referred to as a 0-simplex, an edge as a 1-simplex, a triangle as a 2-simplex, and a tetrahedron as a 3-simplex, where the integer in each term signifies the dimension of the element. The boundary of a simplex consists of other, lower-dimension simplices that we call the faces or subsimplices of the simplex. Simplicial complexes are objects that are collections of simplices. Their construction adheres to the following two rules: (i) For every simplex in the construction, its faces are also part of the construction. (ii) The intersection of any two simplices is either a face of both simplices, or empty. 4 Alpha Shapes We have already seen an example of a simplicial complex: the Delaunay Triangulation of our molecular model. This complex consists of many vertices, edges, triangles, and tetrahedra. However, we need an organized way of building up these components. The natural

4 4 CS273: Handout # 12 Figure 3: (A) 0-simplex, 1-simplex, 2-simplex, and 3-simplex; (B) a legal complex of simplices; (C) intersection patterns of simplices that are not allowed in a complex. way to arrange these subsimplices of the Delaunay Triangulation is in a sequence using a ball growth model, in which the atom centers are expanded into balls of increasing radii. When two, three, or four atom balls have grown large enough so they collide, an edge, a triangle, or a tetrahedron spanning their centers is added to the complex, respectively. When an element is introduced into the complex in this manner, it is marked with the time point at which it has been introduced. This process of growing the atom balls and building the Delaunay complex in a chronological manner is known as filtration. We let the time t range from to, and grow the weight (i.e. squared radius) of each ball b i to (ri 2 + t) at time t. Since the power diagram of our model remains the same at all times, the dual complexes that arise through the filtration process must be subcomplexes of our entire Delaunay complex. Also, since as each ball grows, it covers an increasingly larger portion of its region in the power diagram, the dual complexes that result from filtration only increase over time. Instead of explicitly using a time parameter t, we use α = t. The motivation behind this convention is that if we set the initial radius of a ball to zero, the radius of the ball at time t is α. Let B α be our collection of balls and K α the corresponding complex at time t = α 2. K α is referred to as the α-complex, and its dual, the current shape of our molecule, is referred to as the α-shape of B. For small enough α, all radii are imaginary,

5 CS273: Handout # 12 5 so our union of balls and its corresponding dual complex are both empty. When α grows large enough, meaning enough time has passed for the balls to grow sufficiently large, the corresponding α-complex is equivalent to the Delaunay complex of our model. Thus, the process of filtration yields a sequence of complexes that contains the empty complex at one endpoint, the full Delaunay complex D at the other endpoint, and along the way reflects the shape of our model at increasingly higher levels of resolution: K α K β D for every < α 2 β 2 < where our filtration sequence is the ordered set: { = K 0, K 1,..., K m = D}. Note that D contains only finitely many simplices; thus there are only finitely many subcomplexes of D that appear during the filtration. Figure 4: (A) The alpha complex for small alpha consists mostly of low-dimension simplices. (B) The alpha complex for medium alpha features more extensive simplicial complexes. (C) The alpha complex for large alpha is connected, and contains previously built complexes along with new simplices in its complex. 5 Computing Metric Properties of a Molecule The problem of computing metric properties of a molecule such as surface area and volume is difficult because the spherical balls representing the atoms overlap due to chemical bonds, van der Waals contacts, and solvent contacts. As a result of these intersections, the area or volume of the molecule cannot be computed simply as a sum of the area or volume of the individual atoms. To account for intersections in computing metric properties, we can use the principle of inclusion-exclusion: when two atoms overlap, we subtract the metric value of the overlap from the sum of the metric values of the individual atoms. When three atoms overlap, we first subtract the pairwise overlaps, and then add the triple overlaps. This process

6 6 CS273: Handout # 12 continues when there are four, five, or more atoms that intersect. To compute our desired metric property using inclusion-exclusion, it is sufficient to use only the intersections represented as elements of our dual complex, which we can construct using the process of filtration explained above. Restricting our formula to these terms yields a polynomial-size computation. Note that without this restriction afforded by the dual complex, all possible intersections would have to be considered, resulting in an exponential-size computation. Figure 5: By inclusion-exclusion, area of union = A + B + C + D AB AC AD BC BD CD + ABC + ABD + ACD + BCD ABCD. Using only the simplices in the Delaunay Triangulation, this simplifies to A + B + C + D AB AC AD BC CD + ABC + ACD. Note that there is an equivalence among the cancelled terms: BD = BCD + ABD ABCD. 6 Pockets, Voids, and Topological Persistence As mentioned earlier, our motivation in quantifying the size and shape a molecule is that these properties affect how the molecule interacts with other molecules, which in turn helps to specify its function. Sometimes, the site of this interaction is a concavity of the molecule, known as a pocket, which is surrounded on most sides by atoms of the molecule. For example, if the molecule is a protein, a solvent can bind to the molecule at a pocket by gaining access to the molecule through the mouth, which is the area of pocket not enclosed by atoms. A void is a cavity that is bound on all sides by atoms of the molecule, and is therefore inaccessible to other molecules. During filtration, as the atom-balls swell, more and more balls intersect each other. Some of these intersections close the mouths of pockets in the model, first turning these

7 CS273: Handout # 12 7 pockets into voids and then subsuming all the empty space inside the voids, until no pockets or voids are left in the model. Indeed, the Delaunay Triangulation, which is the final complex generated by our filtration, by definition contains only unbound empty space outside the convex hull of the molecule. Thus, we can identify and analyze pockets through the process of filtration by noting which concavities of one alpha shape become voids in some later alpha shape and are eventually subsumed entirely. More precisely, pockets of a certain alpha shape are identified and measured using a discrete flow relation defined on the corresponding alpha complex. Discrete flow is defined on simplices of the Delaunay complex that have not yet appeared on the alpha complex; for example, in the two-dimensional case, we consider triangles of the Delaunay Triangulation that are empty in our current alpha complex, meaning that they have not yet appeared (compare figure 2b with 2c). According to the flow relation, an obtuse empty triangle flows to the triangle that borders it on the edge opposite the obtuse vertex, whereas an acute empty triangle is a sink that collects flow from all its neighboring empty triangles. A series of empty triangles that flows to a sink define a pocket, while a series of empty triangles that flow to infinity define the exterior of the molecule. The actual size of a pocket is computed by summing the size of empty triangles that define the pocket and then subtracting the fraction of atom space that appears in those triangles. Figure 6: (a) A pocket formed by five empty Delaunay triangles. Obtuse triangles flow into the sink. The top of triangle 1 forms the mouth of the pocket. (b) This structure is not identified as a pocket because the empty triangles flow to infinity rather than to a sink; this concavity does not get closed into a void at any point of the filtration. We may regard filtration as an evolutionary growth process in which topological features such as pockets and voids are created and later destroyed (see figure 7). The

8 8 CS273: Handout # 12 lifetime, or persistence, of each feature is an interval whose boundaries are the alpha value corresponding to the complex at which the feature first appears and the alpha value corresponding the complex at which it disappears. Very short-lived topological attributes may be considered data noise and removed, while attributes that persist longer are kept as meaningful topological features. Figure 7: Emergence and disappearance of topological features of the alpha shape through filtration. References [1] H. Edelsbrunner and E.P. Mucke, Three-dimensional alpha shapes, ACM Trans. Graphics, 13:43-72, [2] H. Edelsbrunner, The union of balls and its dual shape, Discrete Comput. Geometry, 13: , [3] H. Edelsbrunner, M.A. Facello and J. Liang, On the definition and the construction of pockets in macromolecules, Discrete Appl. Math, 88:83-102, [4] J. Liang, H. Edelsbrunner, P. Fu, P.V. Sudharkar and S. Subramaniam, Analytic shape computation of macromolecules I: molecular area and volume through alpha shape, Proteins: Structure, Function and Genetics 33:1-17, 1998.

9 CS273: Handout # 12 9 [5] J. Liang, H. Edelsbrunner, and C. Woodward, Anatomy of protein pockets and cavities: measurement of binding site geometry and implications for ligand design, Protein Science 7: , 1998.