Contents -- 1 What Are Fractals? -- 1.1 Basic Concepts -- 1.2 Fractals and Set Theory -- 1.3 Self-Affine Fractals -- 1.4 Multifractals and the Partition Function -- 1.5 Fractals and the Renormalization Group -- 1.6 Summary -- 2 Fractal Aspects of Protein Structure -- 2.1 Polymers as Mass Fractals -- 2.1.1 The Radius of Gyration and the Fractal Dimension -- 2.1.2 Scaling of Excluded Volume, Ideal, and Collapsed Polymers -- 2.2 Fractal Dimension of the Protein Backbone -- 2.2.1 Measuring the Fractal Dimension -- 2.2.2 Significance of the Fractal Dimension -- 2.3 Scaling Properties of Protein Surfaces -- 2.4 Determining Fractal Dimensions by Scattering Techniques -- 2.4.1 Basic Concepts -- 2.4.2 Low-Angle Scattering from Proteins -- 2.5 Scaling Laws for Membrane Proteins -- 2.6 Summary -- 3 Loops, Polymer Statistics, and Helix-Coil Transitions -- 3.1 Random Walks and Loops -- 3.2 Real Space Renormalization Groups -- 3.2.1 The Self-Avoiding Walk -- 3.2.2 Ring Closure -- 3.3 The Entropy of Loop Formation -- 3.3.1 Loop Formation in Proteins -- 3.3.2 Loop Formation in Nucleic Acids -- 3.3.3 Comparison with Theory -- 3.4 Constraints on the Size of Enzymes and Receptors -- 3.5 The Statistical Mechanics of Biopolymers -- 3.5.1 Sequence-Generating Functions -- 3.5.2 Calculations on the Alpha Helix -- 3.5.3 The Finite Perfect Matching Double Strand -- 3.6 Summary -- 4 The Multifractality of Biomacromolecules -- 4.1 Origins of Multifractality -- 4.1.1 Random Multiplicative Processes -- 4.1.2 Random Walks on Random Media -- 4.2 Multifractal Behavior of the Generalized Cantor Set -- 4.3 Multifractality of Order-Disorder Transitions in Biopolymers -- 4.3.1 Sequence-Generating Functions and Multifractals -- 4.3.2 Alpha Helical Model -- 4.3.3 Perfect Matched Double Helix -- 4.4 Protein Structure and Multifractals.

4.4.1 Algorithm for Calculating Multifractal Spectra -- 4.4.2 Multifractal Spectra of Solvent Accessibilities from Proteins -- 4.4.3 Origins of Multifractality in Proteins -- 4.5 Summary -- Appendix: Extraction Procedures and Comparison with Transfer Matrix Methods -- 5 Fractal Diffusion and Chemical Kinetics -- 5.1 Diffusion and Dimensionality -- 5.2 Fractal Chemical Kinetics -- 5.2.1 Fluctuations and Anomalous Scaling -- 5.2.2 Scaling of Depletion Zones -- 5.2.3 Anomalous Reaction Order -- 5.2.4 Scaling in Chemically Controlled Reactions -- 5.3 Isotope Exchange Kinetics in Proteins -- 5.3.1 Models of Exchange in Lysozyme -- 5.3.2 Reactions on Spheres and Surface Fractals -- 5.3.3 Reaction Widths in Proteins -- 5.3.4 Fractal Effects on Chemically Controlled Reactions -- 5.3.5 Summary of Hydrogen Exchange in Proteins -- 5.4 Diffusion and Reaction in an Archipelago -- 5.5 Other Diffusional Processes -- 5.6 Summary -- 6 Are Protein Dynamics Fractal? -- 6.1 Introduction to Protein Dynamics -- 6.2 The Origins of Nonexponential Behavior -- 6.2.1 Random Multiplicative Processes and the Log-Normal Distribution -- 6.2.2 Static and Dynamic Disorder Models -- 6.3 Fractal Time -- 6.3.1 Long Tail Distributions and Fractal Time -- 6.3.2 A Chemical Kinetic Model with Fractal Time -- 6.3.3 Hierarchical and Defect-Diffusion Models -- 6.4 Ion Channel Kinetics -- 6.5 Summary -- 7 Fractons and Vibrational Relaxation in Proteins -- 7.1 Lattice Diffusion and Fracton Dynamics -- 7.1.1 Lattice Dynamics and Scaling Arguments -- 7.1.2 Fractons -- 7.2 Experimental Techniques for Determining the Density of States -- 7.2.1 Heat Capacity Measurements on Biological Systems -- 7.2.2 Electron Spin Relaxation and Proteins with Iron Centers -- 7.2.3 Neutron Scattering from Proteins -- 7.3 Density of States of Simple Polymer Models.

7.4 Random Walks on Multiply Connected Linear Arrays -- 7.5 Summary -- 8 Encoded Walks and Correlations in Sequence Data -- 8.1 Encoded Walks and Scaling Laws -- 8.2 Correlations in DNA Sequences -- 8.3 The Brownian Bridge and Protein Sequences -- 8.4 The Connection with Multifractals -- 8.5 Information Theory and the Ergodicity of Protein Sequences -- 8.6 Summary -- 9 Percolation -- 9.1 Introduction to Percolation Phenomena -- 9.1.1 Scaling Laws for Percolation -- 9.1.2 Real-Space Renormalization -- 9.1.3 Percolation Clusters as Fractals -- 9.2 Antibody-Receptor Clustering and the Cayley Tree -- 9.2.1 Clustering and the Immune Response System -- 9.2.2 Percolation on the Cayley Tree -- 9.3 Percolation in Biomembranes -- 9.3.1 Fluorescence Recovery after Photobleaching -- 9.3.2 Finite Size Scaling -- 9.3.3 Continuum and Correlated Percolation Thresholds -- 9.4 Percolation of Proton Transport in Hydrated Proteins -- 9.5 Summary -- Appendix: A Correlated Percolation Model for Biomembranes -- 10 Chaos in Biochemical Systems -- 10.1 Chaos in Chemical Systems -- 10.1.1 Chemical Kinetics in Simple Systems -- 10.1.2 Open Systems, Oscillations, and Chaos -- 10.1.3 Origins of Chaos -- 10.2 Theoretical Models of Enzymatic Chaos -- 10.2.1 Mechanisms involving Michaelis-Menten Enzymes -- 10.2.2 Mechanisms involving Allosteric Enzymes -- 10.3 Experimental Systems -- 10.3.1 Glycolytic Oscillations -- 10.3.2 Horseradish Peroxidase -- 10.4 Physiological Advantages of Chaos -- 10.5 Summary -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- P -- R -- S -- U -- V -- W.