Mathematical Concepts and Methods in Modern Biology: Using Modern Discrete Models, Second Edition presents an updated, quantitative framework for analyzing, predicting, and modulating complex biological systems. Structured around project-based learning and problem-solving, it integrates essential mathematical concepts and methods in the context of key biological questions. Topics include gene regulatory and neuronal networks, CpG islands, plant population growth, interacting populations, phylogenetics, and phylogenomics. Modeling tools span discrete dynamical systems (including deterministic and stochastic Boolean networks), ordinary differential equations, projection matrices, agent-based modeling, polynomial ideals, Gröbner bases, tropical algebra, and other algebraic approaches. Computation is supported by a range of software tools, including some developed for this book, as well as open source, freely available, and proprietary platforms. Updated code examples and step-by-step instructions illustrate software use, making this a valuable resource for health professionals, scientists, researchers, and students in related fields.
1. 1.1 Introduction - p. 1 2. 1.2 The Lactose operon in E. coli - p. 3 3. 1.3 Boolean network models of the lac operon - p. 7 4. 1.4 Determining the fixed points of Boolean networks - p. 27 5. 1.5 A Boolean model of the arabinose operon - p. 35 6. 1.6 Dynamics of Boolean models under different update regimes - p. 41 7. 1.7 Discussion - p. 42 8. Supplementary materials - p. 44 9. Chapter 2. Bistability in bacterial genetic networks: A comparison of differential equations and Boolean network models∗ - p. 47 10. 2.1 Introduction - p. 47 11. 2.2 Modeling biochemical reactions with differential equations - p. 49 12. 2.3 The lactose operon of Escherichia coli - p. 56 13. 2.4 The Yildirim-Mackey lactose operon models - p. 58 14. 2.5 Boolean models with different degradation times, delays, and bistability - p. 68 15. 2.6 Boolean analogues of the Yildirim-Mackey models - p. 72 16. 2.7 Modeling the arabinose operon of E. coli - p. 82 17. 2.8 Conclusions and discussion - p. 91 18. Supplementary materials - p. 93 19. Chapter 3. Inferring the topology of gene regulatory networks: An algebraic approach to reverse engineering∗ - p. 97 20. 3.1 Introduction - p. 97 21. 3.2 Polynomial dynamical systems (PDSs) - p. 100 22. 3.3 Computational algebra preliminaries - p. 105 23. 3.4 Construction of the model space: A reverse engineering algorithm - p. 108 24. 3.5 Model selection - p. 113 25. 3.6 Discretization - p. 119 26. 3.7 Further readings - p. 123 27. Appendix - p. 124 28. Supplementary materials - p. 124 29. Chapter 4. Stochastic Boolean Networks∗ - p. 129 30. 4.1 Introduction - p. 129 31. 4.2 Stochastic Discrete Dynamical Systems (SDDS) - p. 130 32. 4.3 Optimal control - p. 138 33. 4.4 Simulations - p. 144 34. 4.5 Parameter estimation - p. 147 35. 4.6 Discussion and conclusions - p. 151 36. Supplementary materials - p. 151 37. Chapter 5. Global dynamics emerging from local interactions: Agent-based modeling for the life sciences∗ - p. 153 38. Chapter 6. Agent-based models and optimal control in biology: A discrete approach∗ - p. 191 39. Chapter 7. Neuronal networks: A discrete model∗ - p. 227 40. 7.1 Introduction and overview - p. 227 41. 7.2 Neuroscience in a nutshell - p. 228 42. 7.3 The discrete model - p. 232 43. 7.4 Exploring the model for some simple connectivities - p. 238 44. 7.5 Exploring the model for some random connectivities - p. 247 45. 7.6 Another interpretation of the model: Disease dynamics - p. 252 46. 7.7 MORE neuroscience: Connection with ODE models - p. 255 47. 7.8 Directions of further research - p. 259 48. Supplementary materials - p. 260 49. Chapter 8. Predicting population growth: Modeling with projection matrices∗ - p. 263 50. 8.1 Introduction - p. 263 51. 8.2 Life cycles and population growth - p. 264 52. 8.3 Determining stages and transition rates in a life cycle - p. 265 53. 8.4 Predicting the future number of individuals in each stage - p. 268 54. 8.5 Constructing a projection matrix - p. 270 55. 8.6 Predicting how a population changes after 1year - p. 275 56. 8.7 The stable distribution of individuals across stages - p. 277 57. 8.8 Theory supporting the calculation of stable distributions - p. 279 58. 8.9 Determining population growth rate and the stable distribution - p. 288 59. 8.10 Modeling stochastic environmental change - p. 290 60. 8.11 Further applications of the projection matrix - p. 295 61. Supplementary materials - p. 295 62. Chapter 9. Identifying CpG islands: Sliding window and hidden Markov model approaches - p. 299 63. 9.1 Introduction - p. 299 64. 9.2 Quantitative characteristics of the CpG island regions and sliding window algorithms - p. 304 65. 9.3 Definition and basic properties of Markov chains and hidden Markov models - p. 308 66. 9.4 Three canonical problems for HMMs with applications to CGI identification - p. 317 67. 9.5 Conclusions and discussion - p. 336 68. Supplementary materials - p. 338 69. Chapter 10. Phylogenetic tree reconstruction: Geometric approaches∗ - p. 343 70. 10.1 Introduction - p. 343 71. 10.2 An overview of distance-based methods - p. 344 72. 10.3 Basics on trees and phylogenetic trees - p. 347 73. 10.4 Tree space - p. 353 74. 10.5 Neighbor-Joining and BME - p. 366 75. 10.6 From phylogenetics to phylogenomics - p. 378 76. Supplementary materials - p. 380 77. Chapter 11. Tropical linear algebra and its applications to phylogenomics∗ - p. 383 78. 11.1 Introduction - p. 383 79. 11.2 Max-plus algebra - p. 384 80. 11.3 Tropical polynomial - p. 386 81. 11.4 Tropical matrix - p. 390 82. 11.5 Tropical linear spaces and tree spaces - p. 396 83. 11.6 Tropical polytopes - p. 402 84. 11.7 Tropical unsupervised learning - p. 407 85. Supplementary materials - p. 411 86. Index - p. 413
Mathematical Concepts and Methods in Modern Biology: Using Modern Discrete Models, Second Edition presents an updated, quantitative framework for analyzing, predicting, and modulating complex biological systems. Structured around project-based learning and problem-solving, it integrates essential mathematical concepts and methods in the context of key biological questions. Topics include gene regulatory and neuronal networks, CpG islands, plant population growth, interacting populations, phylogenetics, and phylogenomics. Modeling tools span discrete dynamical systems (including deterministic and stochastic Boolean networks), ordinary differential equations, projection matrices, agent-based modeling, polynomial ideals, Gröbner bases, tropical algebra, and other algebraic approaches. Computation is supported by a range of software tools, including some developed for this book, as well as open source, freely available, and proprietary platforms. Updated code examples and step-by-step instructions illustrate software use, making this a valuable resource for health professionals, scientists, researchers, and students in related fields.
1. 1.1 Introduction - p. 1 2. 1.2 The Lactose operon in E. coli - p. 3 3. 1.3 Boolean network models of the lac operon - p. 7 4. 1.4 Determining the fixed points of Boolean networks - p. 27 5. 1.5 A Boolean model of the arabinose operon - p. 35 6. 1.6 Dynamics of Boolean models under different update regimes - p. 41 7. 1.7 Discussion - p. 42 8. Supplementary materials - p. 44 9. Chapter 2. Bistability in bacterial genetic networks: A comparison of differential equations and Boolean network models∗ - p. 47 10. 2.1 Introduction - p. 47 11. 2.2 Modeling biochemical reactions with differential equations - p. 49 12. 2.3 The lactose operon of Escherichia coli - p. 56 13. 2.4 The Yildirim-Mackey lactose operon models - p. 58 14. 2.5 Boolean models with different degradation times, delays, and bistability - p. 68 15. 2.6 Boolean analogues of the Yildirim-Mackey models - p. 72 16. 2.7 Modeling the arabinose operon of E. coli - p. 82 17. 2.8 Conclusions and discussion - p. 91 18. Supplementary materials - p. 93 19. Chapter 3. Inferring the topology of gene regulatory networks: An algebraic approach to reverse engineering∗ - p. 97 20. 3.1 Introduction - p. 97 21. 3.2 Polynomial dynamical systems (PDSs) - p. 100 22. 3.3 Computational algebra preliminaries - p. 105 23. 3.4 Construction of the model space: A reverse engineering algorithm - p. 108 24. 3.5 Model selection - p. 113 25. 3.6 Discretization - p. 119 26. 3.7 Further readings - p. 123 27. Appendix - p. 124 28. Supplementary materials - p. 124 29. Chapter 4. Stochastic Boolean Networks∗ - p. 129 30. 4.1 Introduction - p. 129 31. 4.2 Stochastic Discrete Dynamical Systems (SDDS) - p. 130 32. 4.3 Optimal control - p. 138 33. 4.4 Simulations - p. 144 34. 4.5 Parameter estimation - p. 147 35. 4.6 Discussion and conclusions - p. 151 36. Supplementary materials - p. 151 37. Chapter 5. Global dynamics emerging from local interactions: Agent-based modeling for the life sciences∗ - p. 153 38. Chapter 6. Agent-based models and optimal control in biology: A discrete approach∗ - p. 191 39. Chapter 7. Neuronal networks: A discrete model∗ - p. 227 40. 7.1 Introduction and overview - p. 227 41. 7.2 Neuroscience in a nutshell - p. 228 42. 7.3 The discrete model - p. 232 43. 7.4 Exploring the model for some simple connectivities - p. 238 44. 7.5 Exploring the model for some random connectivities - p. 247 45. 7.6 Another interpretation of the model: Disease dynamics - p. 252 46. 7.7 MORE neuroscience: Connection with ODE models - p. 255 47. 7.8 Directions of further research - p. 259 48. Supplementary materials - p. 260 49. Chapter 8. Predicting population growth: Modeling with projection matrices∗ - p. 263 50. 8.1 Introduction - p. 263 51. 8.2 Life cycles and population growth - p. 264 52. 8.3 Determining stages and transition rates in a life cycle - p. 265 53. 8.4 Predicting the future number of individuals in each stage - p. 268 54. 8.5 Constructing a projection matrix - p. 270 55. 8.6 Predicting how a population changes after 1year - p. 275 56. 8.7 The stable distribution of individuals across stages - p. 277 57. 8.8 Theory supporting the calculation of stable distributions - p. 279 58. 8.9 Determining population growth rate and the stable distribution - p. 288 59. 8.10 Modeling stochastic environmental change - p. 290 60. 8.11 Further applications of the projection matrix - p. 295 61. Supplementary materials - p. 295 62. Chapter 9. Identifying CpG islands: Sliding window and hidden Markov model approaches - p. 299 63. 9.1 Introduction - p. 299 64. 9.2 Quantitative characteristics of the CpG island regions and sliding window algorithms - p. 304 65. 9.3 Definition and basic properties of Markov chains and hidden Markov models - p. 308 66. 9.4 Three canonical problems for HMMs with applications to CGI identification - p. 317 67. 9.5 Conclusions and discussion - p. 336 68. Supplementary materials - p. 338 69. Chapter 10. Phylogenetic tree reconstruction: Geometric approaches∗ - p. 343 70. 10.1 Introduction - p. 343 71. 10.2 An overview of distance-based methods - p. 344 72. 10.3 Basics on trees and phylogenetic trees - p. 347 73. 10.4 Tree space - p. 353 74. 10.5 Neighbor-Joining and BME - p. 366 75. 10.6 From phylogenetics to phylogenomics - p. 378 76. Supplementary materials - p. 380 77. Chapter 11. Tropical linear algebra and its applications to phylogenomics∗ - p. 383 78. 11.1 Introduction - p. 383 79. 11.2 Max-plus algebra - p. 384 80. 11.3 Tropical polynomial - p. 386 81. 11.4 Tropical matrix - p. 390 82. 11.5 Tropical linear spaces and tree spaces - p. 396 83. 11.6 Tropical polytopes - p. 402 84. 11.7 Tropical unsupervised learning - p. 407 85. Supplementary materials - p. 411 86. Index - p. 413
