A Mathematical Primer for Computational Structural Biology C. Bajaj October 15, 2014 2 Contents Introduction 9 1 Graphs, Triangulations and Complexes 11 1.1 Graph Theory . 11 1.2 Combinatorial vs. Embedded Graphs . 11 1.2.1 Network Theory . 12 1.2.2 Trees and Spanning Trees . 12 1.3 Topological Complexes . 12 1.3.1 Pointset Topology . 12 1.3.2 CW-complexes . 13 1.4 Primal and Dual Complexes . 13 1.4.1 Primal Meshes . 13 1.4.2 Dual Complexes . 15 1.5 Voronoi and Delaunay Decompositions . 16 1.5.1 Euclidean vs. Power distance. 17 1.5.2 Weighted Alpha Shapes . 18 1.6 Biological Applications . 19 1.6.1 Union of Balls Topology . 19 1.6.2 Meshing of Molecular Interfaces . 20 Summary . 25 References and Further Reading . 25 Exercises . 25 2 Sets, Functions and Mappings 27 2.1 Scalar, Vector and Tensor Functions . 27 2.2 Piecewise-defined Functions . 27 2.3 Height and Morse functions . 27 2.4 The Morse-Smale Complex . 28 2.5 Signed Distance Function and Critical Points of Discrete Distance Functions . 28 2.6 Complementary space topology and geometry . 29 2.6.1 Detection of Tunnels and Pockets . 30 2.7 Algebraic Functions . 33 2.7.1 Polynomial and Rational Parametric . 33 2.7.2 Local and Global Parameterization . 35 2.8 Piecewise Polynomials, Geometric Continuity, Finite Elements . 73 2.8.1 Bernstein-Bezier . 73 2.9 A-Splines, B-Splines, Box-Splines . 76 2.9.1 B-Spline Basis . 76 2.9.2 Implicit Algebraic Surface Patches . 77 2.9.3 Parametric basis . 78 2.10 Biological Applications . 78 3 4 CONTENTS 2.10.1 Tertiary Motif Detection . 78 2.10.2 Ion channel models . 78 2.10.3 Ribosome models . 80 2.10.4 Topological Agreement of Reduced Models . 81 2.10.5 Dynamic Deformation Visualization . 82 Summary . 82 References and Further Reading . 82 Exercises . 82 3 Differential Geometry, Operators 83 3.1 Shape Operators, First and Second Fundamental Forms . 83 3.1.1 Curvature: Gaussian, Mean . 83 3.1.2 The Shape of Space: convex, planar, hyperbolic . 83 3.1.3 Laplacian Eigenfunctions . 83 3.2 Finite Element Basis, Functional Spaces, Inner Products . 83 3.2.1 Hilbert Complexes . 83 3.3 Topology of Function Spaces . 83 3.4 Differential Operators and their Discretization formulas . 83 3.5 Conformal Mappings from Intrinsic Curvature . 84 3.6 Biological Applications . 85 3.6.1 Molecular Surface Analysis . 85 3.6.2 Solving PDEs in Biology . 85 Summary . 85 References and Further Reading . 85 Exercises . 85 4 Differential Forms and Homology of Discrete Functions 87 4.1 Exterior Calculus . 87 4.2 deRham Cohomology . 88 4.3 k-forms and k-cochains . 88 4.3.1 Discrete Differential Forms . 88 4.3.2 Discrete Exterior Derivative . 89 4.4 Types of k-form Finite Elements . 90 4.4.1 Nédélec Elements . 90 4.4.2 Whitney Elements . 91 4.5 Biological Applications . 92 4.5.1 Solving Poisson’s Equation and other PDEs from Biology . 92 Summary . 92 References and Further Reading . 92 Exercises . 92 5 Numerical Integration, Linear Systems 93 5.1 Numerical Quadrature . 93 5.2 Collocation . ..