Algebraic and Combinatorial Computational Biology introduces students and researchers to a panorama of powerful and current methods for mathematical problem-solving in modern computational biology. Presented in a modular format, each topic introduces the biological foundations of the field, covers specialized mathematical theory, and concludes by highlighting connections with ongoing research, particularly open questions. The work addresses problems from gene regulation, neuroscience, phylogenetics, molecular networks, assembly and folding of biomolecular structures, and the use of clustering methods in biology. A number of these chapters are surveys of new topics that have not been previously compiled into one unified source. These topics were selected because they highlight the use of technique from algebra and combinatorics that are becoming mainstream in the life sciences.
1. 4.1 Basic Biology Introduction - p. 90 2. 4.2 Continuous and Discrete Models of Biological Networks - p. 98 3. 4.3 Local Models - p. 106 4. 4.4 Local Models of Operons - p. 118 5. 4.5 Analyzing Local Models With Computational Algebra - p. 124 6. 4.6 Software for Local Models - p. 130 7. 4.7 Concluding Remarks - p. 140 8. Chapter 5: Modeling the Stochastic Nature of Gene Regulation With Boolean Networks - p. 146 9. Chapter 6: Inferring Interactions in Molecular Networks via Primary Decompositions of Monomial Ideals - p. 173 10. Abstract - p. 175 11. 6.1 Introduction - p. 175 12. 6.2 Stanley-Reisner Theory - p. 180 13. 6.3 Finding Min-Sets of Local Models - p. 191 14. 6.4 Finding Signed Min-Sets of Local Models - p. 199 15. 6.5 Applications to a Real Gene Network - p. 203 16. 6.6 Concluding Remarks - p. 207 17. Chapter 7: Analysis of Combinatorial Neural Codes: An Algebraic Approach - p. 211 18. Chapter 8: Predicting Neural Network Dynamics via Graphical Analysis - p. 240 19. Chapter 9: Multistationarity in Biochemical Networks: Results, Analysis, and Examples - p. 277 20. Abstract - p. 279 21. 9.1 Introduction - p. 280 22. 9.2 Reaction Network Terminology and Background - p. 281 23. Exercises - p. 294 24. 9.3 Necessary Conditions for Multistationarity I: Injective CRNs - p. 295 25. Exercises - p. 299 26. 9.4 Necessary Conditions for Multistationarity II: The DSR Graph - p. 300 27. Exercises - p. 303 28. 9.5 Sufficient Conditions for Multistationarity: Inheritance of Multiple Equilibria - p. 304 29. Exercises - p. 306 30. 9.6 Sufficient Conditions for Multistationarity II: The Determinant Optimization Method - p. 308 31. Exercises - p. 309 32. 9.7 Results Based on Deficiency Theory - p. 309 33. Exercises - p. 313 34. Acknowledgments - p. 313 35. Chapter 10: The Minimum Evolution Problem in Phylogenetics: Polytopes, Linear Programming, and Interpretation - p. 317 36. Chapter 11: Data Clustering and Self-Organizing Maps in Biology - p. 349 37. Chapter 12: Toward Revealing Protein Function: Identifying Biologically Relevant Clusters With Graph Spectral Methods - p. 374 38. Index - p. 411
Algebraic and Combinatorial Computational Biology introduces students and researchers to a panorama of powerful and current methods for mathematical problem-solving in modern computational biology. Presented in a modular format, each topic introduces the biological foundations of the field, covers specialized mathematical theory, and concludes by highlighting connections with ongoing research, particularly open questions. The work addresses problems from gene regulation, neuroscience, phylogenetics, molecular networks, assembly and folding of biomolecular structures, and the use of clustering methods in biology. A number of these chapters are surveys of new topics that have not been previously compiled into one unified source. These topics were selected because they highlight the use of technique from algebra and combinatorics that are becoming mainstream in the life sciences.
1. 4.1 Basic Biology Introduction - p. 90 2. 4.2 Continuous and Discrete Models of Biological Networks - p. 98 3. 4.3 Local Models - p. 106 4. 4.4 Local Models of Operons - p. 118 5. 4.5 Analyzing Local Models With Computational Algebra - p. 124 6. 4.6 Software for Local Models - p. 130 7. 4.7 Concluding Remarks - p. 140 8. Chapter 5: Modeling the Stochastic Nature of Gene Regulation With Boolean Networks - p. 146 9. Chapter 6: Inferring Interactions in Molecular Networks via Primary Decompositions of Monomial Ideals - p. 173 10. Abstract - p. 175 11. 6.1 Introduction - p. 175 12. 6.2 Stanley-Reisner Theory - p. 180 13. 6.3 Finding Min-Sets of Local Models - p. 191 14. 6.4 Finding Signed Min-Sets of Local Models - p. 199 15. 6.5 Applications to a Real Gene Network - p. 203 16. 6.6 Concluding Remarks - p. 207 17. Chapter 7: Analysis of Combinatorial Neural Codes: An Algebraic Approach - p. 211 18. Chapter 8: Predicting Neural Network Dynamics via Graphical Analysis - p. 240 19. Chapter 9: Multistationarity in Biochemical Networks: Results, Analysis, and Examples - p. 277 20. Abstract - p. 279 21. 9.1 Introduction - p. 280 22. 9.2 Reaction Network Terminology and Background - p. 281 23. Exercises - p. 294 24. 9.3 Necessary Conditions for Multistationarity I: Injective CRNs - p. 295 25. Exercises - p. 299 26. 9.4 Necessary Conditions for Multistationarity II: The DSR Graph - p. 300 27. Exercises - p. 303 28. 9.5 Sufficient Conditions for Multistationarity: Inheritance of Multiple Equilibria - p. 304 29. Exercises - p. 306 30. 9.6 Sufficient Conditions for Multistationarity II: The Determinant Optimization Method - p. 308 31. Exercises - p. 309 32. 9.7 Results Based on Deficiency Theory - p. 309 33. Exercises - p. 313 34. Acknowledgments - p. 313 35. Chapter 10: The Minimum Evolution Problem in Phylogenetics: Polytopes, Linear Programming, and Interpretation - p. 317 36. Chapter 11: Data Clustering and Self-Organizing Maps in Biology - p. 349 37. Chapter 12: Toward Revealing Protein Function: Identifying Biologically Relevant Clusters With Graph Spectral Methods - p. 374 38. Index - p. 411
