Mathematical Approaches to Molecular Structural Biology offers a comprehensive overview of the mathematical foundations behind the study of biomolecular structure. Initial chapters provide an introduction to the mathematics associated with the study of molecular structure, such as vector spaces and matrices, linear systems, matrix decomposition, vector calculus, probability and statistics. The book then moves on to more advanced areas of molecular structural biology based on the mathematical concepts discussed in earlier chapters. Here, key methods such as X-ray crystallography and cryo-electron microscopy are explored, in addition to biomolecular structure dynamics within the context of mathematics and physics.
This book equips readers with an understanding of the fundamental principles behind structural biology, providing researchers with a strong groundwork for further investigation in both this and related fields.
1. Mathematical Approaches to Molecular Structural Biology - p. iv 2. Cover - p. iv 3. Title Page - p. iv 4. Copyright - p. iv 5. Dedication - p. v 6. Table of Contents - p. v 7. About the Author - p. v 8. Preface - p. xi 9. Acknowledgments - p. xv 10. Table of symbols - p. xvii 11. Chapter 1 Mathematical preliminaries - p. 1 12. 1.1: Functions - p. 1 13. 1.2: Vectors - p. 8 14. 1.3: Matrices and determinants - p. 14 15. 1.3.1: Systems of linear equations - p. 14 16. 1.3.2: Matrices - p. 20 17. 1.3.3: Determinants - p. 26 18. 1.4: Calculus - p. 29 19. 1.5: Series and limits - p. 40 20. Exercise 1 - p. 44 21. Chapter 2 Vector spaces and matrices - p. 47 22. 2.1: Linear systems - p. 47 23. Exercises 2.1 - p. 48 24. 2.2: Sets and subsets - p. 48 25. Exercise 2.2 - p. 53 26. 2.3: Vector spaces and subspaces - p. 53 27. Exercise 2.3 - p. 59 28. 2.4: Liner combination/linear independence - p. 60 29. Exercise 2.4 - p. 63 30. 2.5: Basis vectors - p. 64 31. Exercise 2.5 - p. 67 32. 2.6: Dimension and rank - p. 68 33. Exercise 2.6 - p. 72 34. 2.7: Inner product space - p. 73 35. Exercise 2.7 - p. 78 36. 2.8: Orthogonality - p. 78 37. Exercise 2.8 - p. 89 38. 2.9: Mapping and transformation - p. 90 39. Exercise 2.9 - p. 97 40. 2.10: Change of basis - p. 98 41. Exercise 2.10 - p. 100 42. Chapter 3 Matrix decomposition - p. 103 43. 3.1: Eigensystems from different perspectives - p. 103 44. Exercise 3.1 - p. 107 45. 3.2: Eigensystem basics - p. 107 46. Nonuniqueness of eigenvectors - p. 107 47. Computing eigenvectors - p. 108 48. Linear independence of eigenvectors - p. 112 49. Eigendecomposition - p. 113 50. Geometric intuition for eigendecomposition - p. 114 51. Diagonalization - p. 116 52. Invertibility of matrix P - p. 117 53. Diagonalizability of a matrix - p. 118 54. Orthogonal diagonalization - p. 121 55. Projection matrix and spectral decomposition - p. 124 56. Exercise 3.2 - p. 128 57. 3.3: Singular value decomposition - p. 129 58. Exercises 3.3 - p. 136 59. Chapter 4 Vector calculus - p. 137 60. Chapter 5 Integral transform - p. 159 61. Chapter 6 Probability and statistics - p. 173 62. 6.1: Probability—definitions and properties - p. 173 63. 6.2: Random variables and distribution - p. 177 64. 6.3: Multivariate distribution - p. 189 65. 6.3.1: Bivariate distribution - p. 189 66. 6.3.2: Generalized multivariate distribution - p. 194 67. 6.4: Covariance and correlation - p. 195 68. 6.5: Principal component analysis - p. 202 69. Exercise 6 - p. 208 70. Chapter 7 X-ray crystallography - p. 211 71. Chapter 8 Cryo-electron microscopy - p. 235 72. Chapter 9 Biomolecular structure and dynamics - p. 257 73. Index - p. 283
Mathematical Approaches to Molecular Structural Biology offers a comprehensive overview of the mathematical foundations behind the study of biomolecular structure. Initial chapters provide an introduction to the mathematics associated with the study of molecular structure, such as vector spaces and matrices, linear systems, matrix decomposition, vector calculus, probability and statistics. The book then moves on to more advanced areas of molecular structural biology based on the mathematical concepts discussed in earlier chapters. Here, key methods such as X-ray crystallography and cryo-electron microscopy are explored, in addition to biomolecular structure dynamics within the context of mathematics and physics.
This book equips readers with an understanding of the fundamental principles behind structural biology, providing researchers with a strong groundwork for further investigation in both this and related fields.
1. Mathematical Approaches to Molecular Structural Biology - p. iv 2. Cover - p. iv 3. Title Page - p. iv 4. Copyright - p. iv 5. Dedication - p. v 6. Table of Contents - p. v 7. About the Author - p. v 8. Preface - p. xi 9. Acknowledgments - p. xv 10. Table of symbols - p. xvii 11. Chapter 1 Mathematical preliminaries - p. 1 12. 1.1: Functions - p. 1 13. 1.2: Vectors - p. 8 14. 1.3: Matrices and determinants - p. 14 15. 1.3.1: Systems of linear equations - p. 14 16. 1.3.2: Matrices - p. 20 17. 1.3.3: Determinants - p. 26 18. 1.4: Calculus - p. 29 19. 1.5: Series and limits - p. 40 20. Exercise 1 - p. 44 21. Chapter 2 Vector spaces and matrices - p. 47 22. 2.1: Linear systems - p. 47 23. Exercises 2.1 - p. 48 24. 2.2: Sets and subsets - p. 48 25. Exercise 2.2 - p. 53 26. 2.3: Vector spaces and subspaces - p. 53 27. Exercise 2.3 - p. 59 28. 2.4: Liner combination/linear independence - p. 60 29. Exercise 2.4 - p. 63 30. 2.5: Basis vectors - p. 64 31. Exercise 2.5 - p. 67 32. 2.6: Dimension and rank - p. 68 33. Exercise 2.6 - p. 72 34. 2.7: Inner product space - p. 73 35. Exercise 2.7 - p. 78 36. 2.8: Orthogonality - p. 78 37. Exercise 2.8 - p. 89 38. 2.9: Mapping and transformation - p. 90 39. Exercise 2.9 - p. 97 40. 2.10: Change of basis - p. 98 41. Exercise 2.10 - p. 100 42. Chapter 3 Matrix decomposition - p. 103 43. 3.1: Eigensystems from different perspectives - p. 103 44. Exercise 3.1 - p. 107 45. 3.2: Eigensystem basics - p. 107 46. Nonuniqueness of eigenvectors - p. 107 47. Computing eigenvectors - p. 108 48. Linear independence of eigenvectors - p. 112 49. Eigendecomposition - p. 113 50. Geometric intuition for eigendecomposition - p. 114 51. Diagonalization - p. 116 52. Invertibility of matrix P - p. 117 53. Diagonalizability of a matrix - p. 118 54. Orthogonal diagonalization - p. 121 55. Projection matrix and spectral decomposition - p. 124 56. Exercise 3.2 - p. 128 57. 3.3: Singular value decomposition - p. 129 58. Exercises 3.3 - p. 136 59. Chapter 4 Vector calculus - p. 137 60. Chapter 5 Integral transform - p. 159 61. Chapter 6 Probability and statistics - p. 173 62. 6.1: Probability—definitions and properties - p. 173 63. 6.2: Random variables and distribution - p. 177 64. 6.3: Multivariate distribution - p. 189 65. 6.3.1: Bivariate distribution - p. 189 66. 6.3.2: Generalized multivariate distribution - p. 194 67. 6.4: Covariance and correlation - p. 195 68. 6.5: Principal component analysis - p. 202 69. Exercise 6 - p. 208 70. Chapter 7 X-ray crystallography - p. 211 71. Chapter 8 Cryo-electron microscopy - p. 235 72. Chapter 9 Biomolecular structure and dynamics - p. 257 73. Index - p. 283
