Fractals in Molecular Biophysics

Contents......Page 10
1 What Are Fractals?......Page 16
1.1 Basic Concepts......Page 18
1.2 Fractals and Set Theory......Page 28
1.3 Self-Affine Fractals......Page 32
1.4 Multifractals and the Partition Function......Page 34
1.5 Fractals and the Renormalization Group......Page 38
1.6 Summary......Page 41
2 Fractal Aspects of Protein Structure......Page 43
2.1.1 The Radius of Gyration and the Fractal Dimension......Page 45
2.1.2 Scaling of Excluded Volume, Ideal, and Collapsed Polymers......Page 46
2.2.1 Measuring the Fractal Dimension......Page 49
2.2.2 Significance of the Fractal Dimension......Page 53
2.3 Scaling Properties of Protein Surfaces......Page 57
2.4.1 Basic Concepts......Page 62
2.4.2 Low-Angle Scattering from Proteins......Page 64
2.5 Scaling Laws for Membrane Proteins......Page 66
2.6 Summary......Page 69
3 Loops, Polymer Statistics, and Helix-Coil Transitions......Page 72
3.1 Random Walks and Loops......Page 73
3.2 Real Space Renormalization Groups......Page 75
3.2.1 The Self-Avoiding Walk......Page 76
3.2.2 Ring Closure......Page 78
3.3.1 Loop Formation in Proteins......Page 80
3.3.2 Loop Formation in Nucleic Acids......Page 81
3.3.3 Comparison with Theory......Page 83
3.4 Constraints on the Size of Enzymes and Receptors......Page 84
3.5 The Statistical Mechanics of Biopolymers......Page 88
3.5.1 Sequence-Generating Functions......Page 90
3.5.2 Calculations on the Alpha Helix......Page 92
3.5.3 The Finite Perfect Matching Double Strand......Page 94
3.6 Summary......Page 97
4 The Multifractality of Biomacromolecules......Page 100
4.1.1 Random Multiplicative Processes......Page 101
4.1.2 Random Walks on Random Media......Page 103
4.2 Multifractal Behavior of the Generalized Cantor Set......Page 104
4.3 Multifractality of Order–Disorder Transitions in Biopolymers......Page 106
4.3.1 Sequence-Generating Functions and Multifractals......Page 107
4.3.2 Alpha Helical Model......Page 110
4.3.3 Perfect Matched Double Helix......Page 112
4.4 Protein Structure and Multifractals......Page 116
4.4.1 Algorithm for Calculating Multifractal Spectra......Page 117
4.4.2 Multifractal Spectra of Solvent Accessibilities from Proteins......Page 120
4.4.3 Origins of Multifractality in Proteins......Page 124
Appendix: Extraction Procedures and Comparison with Transfer Matrix Methods......Page 125
5 Fractal Diffusion and Chemical Kinetics......Page 133
5.1 Diffusion and Dimensionality......Page 134
5.2.1 Fluctuations and Anomalous Scaling......Page 137
5.2.2 Scaling of Depletion Zones......Page 138
5.2.4 Scaling in Chemically Controlled Reactions......Page 140
5.3 Isotope Exchange Kinetics in Proteins......Page 142
5.3.1 Models of Exchange in Lysozyme......Page 144
5.3.2 Reactions on Spheres and Surface Fractals......Page 149
5.3.4 Fractal Effects on Chemically Controlled Reactions......Page 150
5.4 Diffusion and Reaction in an Archipelago......Page 152
5.6 Summary......Page 158
6 Are Protein Dynamics Fractal?......Page 162
6.1 Introduction to Protein Dynamics......Page 163
6.2.1 Random Multiplicative Processes and the Log–Normal Distribution......Page 166
6.2.2 Static and Dynamic Disorder Models......Page 167
6.3.1 Long Tail Distributions and Fractal Time......Page 171
6.3.3 Hierarchical and Defect-Diffusion Models......Page 173
6.4 Ion Channel Kinetics......Page 176
6.5 Summary......Page 179
7 Fractons and Vibrational Relaxation in Proteins......Page 182
7.1.1 Lattice Dynamics and Scaling Arguments......Page 183
7.1.2 Fractons......Page 185
7.2.1 Heat Capacity Measurements on Biological Systems......Page 186
7.2.2 Electron Spin Relaxation and Proteins with Iron Centers......Page 188
7.2.3 Neutron Scattering from Proteins......Page 190
7.3 Density of States of Simple Polymer Models......Page 191
7.4 Random Walks on Multiply Connected Linear Arrays......Page 194
7.5 Summary......Page 198
8 Encoded Walks and Correlations in Sequence Data......Page 200
8.1 Encoded Walks and Scaling Laws......Page 201
8.2 Correlations in DNA Sequences......Page 205
8.3 The Brownian Bridge and Protein Sequences......Page 208
8.4 The Connection with Multifractals......Page 211
8.5 Information Theory and the Ergodicity of Protein Sequences......Page 215
8.6 Summary......Page 218
9 Percolation......Page 220
9.1.1 Scaling Laws for Percolation......Page 223
9.1.3 Percolation Clusters as Fractals......Page 228
9.2.1 Clustering and the Immune Response System......Page 232
9.2.2 Percolation on the Cayley Tree......Page 234
9.3.1 Fluorescence Recovery after Photobleaching......Page 236
9.3.2 Finite Size Scaling......Page 240
9.3.3 Continuum and Correlated Percolation Thresholds......Page 242
9.4 Percolation of Proton Transport in Hydrated Proteins......Page 245
Appendix: A Correlated Percolation Model for Biomembranes......Page 248
10 Chaos in Biochemical Systems......Page 254
10.1.1 Chemical Kinetics in Simple Systems......Page 255
10.1.2 Open Systems, Oscillations, and Chaos......Page 257
10.1.3 Origins of Chaos......Page 260
10.2.1 Mechanisms involving Michaelis–Menten Enzymes......Page 264
10.2.2 Mechanisms involving Allosteric Enzymes......Page 265
10.3.1 Glycolytic Oscillations......Page 271
10.3.2 Horseradish Peroxidase......Page 275
10.4 Physiological Advantages of Chaos......Page 278
10.5 Summary......Page 279
B......Page 282
C......Page 283
F......Page 284
L......Page 285
P......Page 286
S......Page 288
W......Page 289