Area and Volume of Molecular Skin Surfaces Xinwei Shi Patrice Koehl 1 Introduction In this paper, we propose a new method for computing the geometric measures of bio-molecules using skin surfaces to represent their shapes. Specifically, we give the formu- las for measuring the total area A and volume V of the skin surface defined by n weighted points, as well as the contributions of the individual points to A and V . Motivation. Bio-molecules interact and function accord- ing to their shapes: understanding of the latter is therefore crucial in life sciences. A common way to represent the shape of a bio-molecule is to use a union of balls in which each ball corresponds to an atom. Geometric measures of this union of balls (mainly surface area and volume) give access to energetics properties of the molecule, such as its stability in water (see [3] for a complete review). Among all analytical and numerical methods that have been pro- posed to compute these measures, the approach based on the Alpha Shape theory is the most promising [3]. Its applications for modeling however have been limited be- cause of the discontinuous nature of the derivatives of the surface area and volume of a union of balls. As an alterna- tive, the skin surface introduced by Edelsbrunner [2] has a number of desirable properties for molecular shape repre- sentation such as smoothness, free of self-intersection and deformation with smooth transitions. Cheng and Edels- brunner [1] developed formulas for the area, perimeter and derivatives of skin curves. Here we generalize these formulas to skin surfaces in three dimensions. Main Results. We give the formulas for measuring the total area A and volume V of the skin surface defined by n weighted points, as well as for measuring the contribu- tions of the individual points to A and V . All the formulas except part of the area calculation can be evaluated ana- lytically. We first introduce geometric structures defining skin surfaces. Then, we give the formulas for comput- ing the area and volume as well as the formulas of the individual contributions of each point. We briefly discuss their applications and the calculation of the derivatives. 2 Geometric Structures Given a set of weighted points B = {b i =(z i , w i ) ∈ R 3 × R | i = 1..n}, the skin surface F B is defined as the en- velope of the convex hull of B after shrinking, namely, F B = env(  conv(B)), in which the addition and the scalar multiplication operations of two weighted points follows the addition and multiplication of weighted distance func- tions π (x)= ‖x − z i ‖ 2 − w i in vector space, and the shrink- ing operation for a set of weighted points X is defined as √ X = { √ b i =(z i , w i /2)|b i ∈ X }. We consider each weighted point as a sphere centered at z i with a radius √ w i ; the convex hull of B is then an in- finite family of spheres. After shrinking these spheres by a factor of 1/ √ 2, the boundary of the union of spheres, namely, the skin surface, is a tangent continuous surface that blends adjacent spheres smoothly. A skin surface can be decomposed into a collection of simple quadratic patches based on the framework of the weighted Delaunay triangulation and Voronoi diagram. Let D B be the weighted Delaunay triangulation of B, V B be its dual Voronoi diagram, and K B the dual complex of B. For a simplex σ ∗ ∈ D B , we denote its vertices σ i = z i , its edges σ ij = z i z j , its triangles σ ijk = z i z j z k and its tetra- hedra σ i jkl = z i z j z k z l . Similarly, the dual Voronoi cell v ∗ of σ ∗ are polytopes v i , polygons v ij , edges v ijk and ver- tices v i jkl respectively. The Minkowski sum of σ ∗ and its dual v ∗ scaled down by half is called the mixed cell μ ∗ , namely, μ ∗ = 1 2 (σ ∗ + v ∗ ). The center f ∗ of μ ∗ is defined as aff(σ ∗ )  aff(v ∗ ), in which aff(σ ) is the affine hull of σ . The size R ∗ of μ ∗ is defined as the absolute value of the radius of the orthosphere of σ ∗ divided by √ 2. As proved in [2], the skin surface within a mixed cell, S ∗ = F B  μ ∗ , is a quadratic patch defined by f ∗ and R ∗ . Specifically, S i and S i jkl are the parts of spheres ( f i , R i ) and ( f i jkl , R i jkl ) within μ i and μ i jkl , respectively, and S ij and S ijk are the parts of hyperboloids with focus f ij and f ijk . 3 Surface Area and Volume of a skin surface Let A ∗ and V ∗ be the surface area and volume of S ∗ re- spectively. We denote A i ∗ and V i ∗ the portions of A ∗ and V ∗ that belong to the sphere b i if z i ⊂ σ ∗ . The surface area A i and volume V i of a sphere b i is defined as follows, A i = A i i + ∑ σ ij ∈D B A i ij + ∑ σ ijk ∈D B A i ijk + ∑ σ i jkl ∈D B A i i jkl , V i = V i i + ∑ σ ij ∈D B V i ij + ∑ σ ijk ∈D B V i ijk + ∑ σ i jkl ∈D B V i i jkl . 1