| Preface |
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vii | |
| Acknowledgments |
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xiii | |
| Part I: Preliminaries |
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1 Anticipating the Big Picture: Some Clues |
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5 | (8) |
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1.1 The Strange Relationship of Mathematics and Chemistry |
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6 | (2) |
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1.1.1 The Differential Equations of Chemistry |
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6 | (1) |
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1.1.2 Is There a Mathematical Tradition in Chemistry? |
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7 | (1) |
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1.1.3 A Historical Puzzle |
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8 | (1) |
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1.2 Fruit Flies and Platypuses |
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8 | (1) |
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1.3 Chemical Education: The Tacit Doctrine of Stable Reactor Behavior |
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9 | (1) |
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1.4 Biochemistry Is Different from "Regular" Chemistry |
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10 | (1) |
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1.5 The Big Picture, Veiled |
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11 | (2) |
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2 Chemical and Notational Preliminaries |
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13 | (10) |
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2.1 How the Differential Equations of Chemistry Come About |
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13 | (6) |
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15 | (1) |
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2.1.2 Mass Action Kinetics |
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16 | (1) |
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17 | (1) |
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2.1.4 Our Long-Term Objectives |
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18 | (1) |
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2.2 About Setting and Notation |
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19 | (4) |
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2.2.1 What's the Problem? |
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19 | (1) |
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20 | (2) |
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2.2.3 A Special Case: The Vector Space Generated by the Species |
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22 | (1) |
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3 Reaction Networks, Kinetics, and the Induced Differential Equations |
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23 | (20) |
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23 | (2) |
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25 | (2) |
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3.2.1 General Kinetics Revisited |
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25 | (1) |
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3.2.2 Mass Action Kinetics Revisited |
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26 | (1) |
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3.3 The Differential Equations for a Kinetic System |
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27 | (2) |
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3.4 Stoichiometric Compatibility |
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29 | (5) |
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3.5 An Elementary Necessary Condition for the Existence of a Positive Equilibrium or a Cyclic Composition Trajectory Passing Through a Positive Composition |
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34 | (1) |
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3.6 About the Derivative of the Species-Formation-Rate Function |
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35 | (1) |
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3.7 Elementary Boundary Behavior of the Differential Equations: Where Can Equilibria and Cyclic Composition Trajectories Reside? |
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36 | (2) |
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Appendix 3.A The Kinetic Subspace |
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38 | (5) |
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3.A.1 When the Kinetic Subspace Is Smaller than the Stoichiometric Subspace |
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39 | (1) |
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3.A.2 Should We Focus on the Kinetic Rather than the Stoichiometric Subspace9 |
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40 | (2) |
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3.A.3 Thinking (and Not Thinking) About the Kinetic Subspace |
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42 | (1) |
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4 Open Systems: Why Study Nonconservative and Otherwise Peculiar Reaction Networks? |
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43 | (14) |
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4.1 Conservative Networks |
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43 | (2) |
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4.2 Reaction Network Descriptions of Open Systems |
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45 | (8) |
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4.2.1 The Continuous-Flow Stirred-Tank Reactor (CFSTR) |
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47 | (1) |
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4.2.2 Semi-Open Reactors: An Enzyme Example |
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48 | (2) |
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4.2.3 Reactors with Certain Species Concentrations Regarded Constant |
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50 | (2) |
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4.2.4 Interconnected Cells |
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52 | (1) |
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53 | (4) |
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5 A Toy-Reaction-Network Zoo: Varieties of Behavior and Some Questions |
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57 | (16) |
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5.1 Multiple Stoichiometrically Compatible Equilibria and Unstable Equilibria |
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59 | (2) |
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5.1.1 The Horn and Jackson Example |
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59 | (1) |
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5.1.2 The Edelstein Example |
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60 | (1) |
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5.1.3 An Example Having Stoichiometric Compatibility Classes with Zero, One, and Two Positive Equilibria |
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60 | (1) |
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61 | (4) |
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5.2.1 Left-Handed and Right-Handed Molecules: Origins of Enantiomeric Excess |
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62 | (2) |
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64 | (1) |
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5.3 Cyclic Composition Trajectories |
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65 | (2) |
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5.3.1 The Lotka Example: Rabbits and Wolves |
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65 | (1) |
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5.3.2 The Brusselator Example |
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66 | (1) |
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5.4 Questions: Fruit Flies and Platypuses Again |
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67 | (6) |
| Part II: Some Principal Theorems: A First Look |
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6 Aspects of Reaction Network Structure |
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73 | (16) |
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6.1 The Linkage Classes of a Reaction Network |
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74 | (1) |
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6.2 The Rank of a Reaction Network Revisited |
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75 | (1) |
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6.3 The Deficiency of a Reaction Network |
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76 | (1) |
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6.4 The Rank and Deficiency of a Linkage Class |
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77 | (2) |
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6.5 Reversibility and Weak Reversibility |
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79 | (2) |
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6.6 The Strong-Linkage Classes of a Reaction Network and Terminal Strong-Linkage Classes |
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81 | (1) |
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6.7 Cuts, Trees, and Forests in the Standard Reaction Diagram |
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82 | (2) |
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Appendix 6.A Independent Subnetworks |
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84 | (5) |
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7 The Deficiency Zero Theorem |
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89 | (16) |
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7.1 A Statement of the Deficiency Zero Theorem |
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89 | (1) |
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90 | (3) |
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7.3 A Derivative Version of the Deficiency Zero Theorem |
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93 | (1) |
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7.4 Boundary Equilibria for Deficiency Zero Networks: Only Certain Combinations of Species Can Coexist |
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94 | (3) |
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7.5 Multicell Systems with Deficiency Zero Intracellular Chemistry |
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97 | (1) |
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7.6 A Striking Counterpoint to the Deficiency Zero Theorem: Star-Like Networks |
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98 | (2) |
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7.7 Complex Balancing and Origins of the Deficiency Zero Theorem |
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100 | (3) |
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7.8 The Global Attractor Conjecture and the Persistence Conjecture |
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103 | (2) |
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105 | (22) |
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8.1 Motivation for the Deficiency One Theorem |
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105 | (1) |
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8.2 The Deficiency One Theorem |
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106 | (2) |
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8.3 What the Deficiency One Theorem Does Not Preclude |
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108 | (1) |
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8.4 All of the Network-Structural Conditions of the Deficiency One Theorem Are Essential |
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109 | (2) |
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8.5 The Deficiency One Algorithm: A First Glimpse |
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111 | (4) |
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8.5.1 Describing Reaction Network Behavior in Broad, Bold Terms: When Do the Fine Details of Network Structure Matter? |
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111 | (2) |
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8.5.2 What the Deficiency One Algorithm Does: Translating Difficult Questions About Nonlinear Equations into Easy Questions About Linear Inequalities |
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113 | (2) |
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8.6 About Higher Deficiency Networks and Irregular Deficiency One Networks |
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115 | (1) |
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8.7 Two Remarks About the Existence of Positive Equilibria for Mass Action Systems |
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116 | (1) |
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Appendix 8.A Why Mass Action Models with an Excess of Terminal Strong-Linkage Classes Are Problematic |
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117 | (10) |
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8.A.1 The Kinetic Subspace Revisited |
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118 | (1) |
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8.A.2 The Kinetic Subspace for a Mass Action System |
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119 | (1) |
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8.A.3 An Instructive Mass Action Example Revisited |
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120 | (2) |
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8.A.4 The Importance of the Number of Terminal Strong-Linkage Classes |
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122 | (2) |
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8.A.5 The Fragility of Phenomena Emerging from Mass Action Models Having an Excess of Terminal Strong-Linkage Classes |
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124 | (1) |
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8.A.6 Summary: The Good News That Theorem 8.A.2 Contains |
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125 | (2) |
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9 Concentration Robustness and Its Importance in Biology: Some More Deficiency-Oriented Theorems |
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127 | (26) |
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9.1 A Motivating Thought Experiment |
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127 | (2) |
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9.2 A Problem for the Cell |
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129 | (1) |
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9.3 A Toy Network Exhibiting Absolute Concentration Robustness |
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130 | (2) |
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9.4 More Substantial Toy Networks: Biochemical Concentration Robustness Models Inspired by Experiments |
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132 | (2) |
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9.4.1 The EnvZ-OmpR Signaling System in E. coli |
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132 | (2) |
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9.4.2 The IDHKP-IDH Glyoxylate Bypass Regulation System |
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134 | (1) |
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9.5 A Theorem: Structural Sources of Absolute Concentration Robustness |
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134 | (3) |
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9.6 How the Robust Concentration Depends on Rate Constants: Implications for Biology |
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137 | (7) |
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9.6.1 First Theorem on Rate Constant Independence |
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139 | (2) |
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9.6.2 Second Theorem on Rate Constant Independence |
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141 | (2) |
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9.6.3 A Useful but Less Interesting Observation About Rate Constant Independence |
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143 | (1) |
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9.7 Reaction Network Architectures That Thwart Concentration Robustness |
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144 | (9) |
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9.7.1 First Robustness-Thwarting Theorem |
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145 | (1) |
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9.7.2 Second Robustness-Thwarting Theorem: Architectures That Deny Even Approximate Concentration Robustness |
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146 | (5) |
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9.7.3 Networks Involving Large Complex Molecules Made from Many Distinct Gregarious Subunits |
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151 | (2) |
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10 Concordant Reaction Networks: Architectures That Promote Dull, Reliable Behavior Across Broad Kinetic Classes |
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153 | (52) |
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10.1 Some Advice for the Reader |
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153 | (1) |
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10.2 A Puzzle: The Seemingly Unreasonable Stability of Classical Isothermal Continuous-Flow Stirred-Tank Reactors |
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154 | (2) |
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10.3 Fully Open Networks and a Network's Fully Open Extension |
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156 | (2) |
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10.4 Reaction Network Concordance: Definition and Examples |
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158 | (2) |
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10.5 Behavioral Consequences of Concordance and Discordance, Part I |
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160 | (10) |
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10.5.1 Weakly Monotonic Kinetics |
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161 | (1) |
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10.5.2 Concordance, Injectivity, and Multiple Equilibria |
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162 | (3) |
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10.5.3 Concordance and the Nonsingularity of Derivatives |
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165 | (1) |
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10.5.4 A Substantial Biological Example: The Wnt Pathway |
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166 | (2) |
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10.5.5 Weakly Reversible Concordant Networks: Absence of Boundary Equilibria or Cycles; Existence of a Unique Nondegenerate Equilibrium in Each Positive Stoichiometric Compatibility Class |
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168 | (2) |
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10.6 Digression: Degenerate and Nondegenerate Networks |
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170 | (12) |
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10.6.1 Nondegenerate (and Degenerate) Networks Defined |
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171 | (1) |
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10.6.2 Reversibility and Nondegeneracy: Every Weakly Reversible Network Is Nondegenerate |
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172 | (1) |
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10.6.3 Mild Network-Structural Conditions Equivalent to Nondegeneracy |
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173 | (1) |
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10.6.4 Every Fully Open Network Is Nondegenerate |
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174 | (1) |
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10.6.5 NNondegeneracy of Semi-open Reaction Networks: An Enzyme Example and the Wnt Pathway |
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175 | (2) |
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10.6.6 A Nondegenerate Network with a Concordant Fully Open Extension Is Itself Concordant |
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177 | (1) |
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10.6.7 Surprising "All-or-Nothing" Properties of Networks with Concordant Fully Open Extensions |
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178 | (4) |
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10.7 Behavioral Consequences of Concordance and Discordance, Part II |
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182 | (7) |
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10.7.1 Concordance and Eigenvalues |
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183 | (2) |
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10.7.2 A Summary of Some Results for Nondegenerate Networks with Concordant Fully Open Extensions |
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185 | (1) |
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10.7.3 Discordance and Eigenvalues: The Certainty of a Kinetics for Which There Is an Unstable Positive Equilibrium |
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185 | (2) |
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10.7.4 Concordant Weakly Reversible Networks with Discordant Fully Open Extensions: Sustained Periodic Composition Oscillations |
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187 | (2) |
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10.8 Discordance, Degeneracy, and an Excess of Terminal Strong-Linkage Classes |
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189 | (1) |
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10.9 Strong Concordance and Product Inhibition |
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190 | (4) |
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10.10 More General Species Influences on Reaction Rates |
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194 | (2) |
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10.11 A Concluding Remark About Concordance and Mathematical Aesthetics |
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196 | (1) |
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Appendix 10.A Deducing Behavior from Fully Open Behavior |
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197 | (1) |
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Appendix 10.B Mass Action Injectivity |
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198 | (7) |
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10.B.1 Two Similar Theorems, One Broad and One Mass-Action-Specific |
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198 | (2) |
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10.B.2 Two Instructive Examples |
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200 | (1) |
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201 | (3) |
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10.B.4 A Route to the Determination of Mass Action Injectivity |
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204 | (1) |
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11 The Species-Reaction Graph |
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205 | (36) |
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11.1 About the Connection of This
Chapter to the Preceding One |
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205 | (1) |
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206 | (1) |
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11.3 How the Species-Reaction Graph Is Drawn |
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207 | (2) |
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11.3.1 Species Vertices and Reaction Vertices |
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207 | (1) |
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11.3.2 How Edges Are Drawn |
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208 | (1) |
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11.4 Some Species-Reaction Graph Vocabulary |
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209 | (3) |
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11.4.1 Complex Pairs, Odd Cycles, and Even Cycles |
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209 | (1) |
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209 | (1) |
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11.4.3 Stoichiometrically Expansive Orientations |
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210 | (1) |
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11.4.4 The Intersection of Two Cycles |
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211 | (1) |
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11.5 First Preliminary Theorem |
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212 | (3) |
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11.6 First Principal Theorem |
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215 | (1) |
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216 | (7) |
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11.7.1 Revisiting Some of Our First Species-Reaction Graphs |
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216 | (1) |
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11.7.2 Subtleties Present Even When There Is Only One Cycle |
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217 | (2) |
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11.7.3 The Wnt Pathway Again |
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219 | (2) |
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11.7.4 Even Cycles Intersecting in Two Directed Species-to-Reaction Paths |
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221 | (2) |
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11.8 Second Preliminary Theorem |
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223 | (1) |
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11.9 Second Principal Theorem: What the Species-Reaction Graph Tells Us About Behavior (Two-Way Monotonic Kinetics) |
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224 | (1) |
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225 | (2) |
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11.10.1 Some of Our First Species-Reaction Graph Examples Revisited, Yet Again |
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225 | (1) |
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11.10.2 The Wnt Pathway Again, This Time with Kinetics Admitting Product Inhibition |
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225 | (1) |
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11.10.3 A More Incisive Example Illustrating Condition (ii) |
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226 | (1) |
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11.11 Special Considerations for Mass Action Kinetics: The Knot Graph and the Species-Reaction Graph |
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227 | (8) |
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11.11.1 The Knot Graph for a Reaction Network |
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229 | (3) |
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11.11.2 What the Knot Graph Can Tell Us |
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232 | (1) |
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11.11.3 Connections Between the Knot Graph and the Species-Reaction Graph |
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233 | (1) |
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11.11.4 Mass Action Kinetics and Acyclic Species-Reaction Graphs |
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234 | (1) |
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Appendix 11.A Proof of Theorem 11.11.7 |
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235 | (6) |
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12 The Big Picture Revisited |
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241 | (32) |
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12.1 Back to the Beginning: Mysteries |
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241 | (1) |
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12.2 Reprise: Mathematical Roots of Enforced Dull, Stable Behavior in Complex Reaction Networks |
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242 | (2) |
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12.3 Reprise: Answers to Some Earlier Questions |
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244 | (1) |
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12.4 Useful Prejudices That Can Mislead in Fundamental Ways |
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245 | (1) |
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12.5 Biochemistry: What Makes Life Interesting? |
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245 | (3) |
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12.6 Are There Deeper and More Pervasive Sources of Rich Behavior in Biochemistry? |
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248 | (14) |
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12.6.1 Pathway Diagrams Often Mask Crucial Mechanistic Detail |
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248 | (1) |
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12.6.2 Sources of Rich Behavior Can Lurk in the Most Fundamental Mechanisms of Enzyme Catalysis |
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249 | (5) |
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12.6.3 Another Example: Single-Substrate Enzyme Catalysis with an Inhibitor |
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254 | (2) |
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12.6.4 The Example of Dihydrofolate Reductase, a Chemotherapy Target, with Rate Constants Taken from the Literature |
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256 | (6) |
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12.7 Rich Behavior Rooted in Classical Mechanisms of Catalysis on Metal Surfaces |
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262 | (5) |
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12.8 Remarks About the Big Picture |
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267 | (6) |
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12.8.1 Some Things We Don't Know |
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267 | (1) |
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12.8.2 Why the Big Picture Is Already Satisfying, Even Beautiful |
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268 | (5) |
| Part III: Going Deeper |
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13 Quasi-Thermodynamic Kinetic Systems |
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273 | (20) |
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13.1 A Little Classical Thermodynamics |
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273 | (8) |
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274 | (1) |
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274 | (1) |
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13.1.3 The Non-increasing Helmholtz Free Energy |
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274 | (1) |
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13.1.4 Laws vs. Constitutive Equations |
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275 | (1) |
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13.1.5 A Constitutive Description of the Helmholtz Free Energy |
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275 | (1) |
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13.1.6 Some Consequences of a Non-increasing Helmholtz Free Energy |
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275 | (3) |
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13.1.7 Monday-Wednesday-Friday Equilibria vs. Tuesday-Thursday Equilibria |
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278 | (1) |
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13.1.8 The Second Law Imposes an Orchestration of Constitutive Equations |
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279 | (2) |
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13.2 Quasi-Thermodynamics |
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281 | (1) |
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13.3 Quasi-Thermostatic Kinetic Systems |
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282 | (1) |
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13.4 Quasi-Thermodynamic Kinetic Systems |
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283 | (2) |
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13.5 Application: Proof of the Star-Like Network Theorem |
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285 | (2) |
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287 | (1) |
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Appendix 13.A Proof of Theorem 13.3.3 |
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288 | (2) |
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Appendix 13.B Existence of a Positive Equilibrium for a Reversible Star-Like Mass Action System |
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290 | (3) |
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293 | (16) |
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14.1 Detailed Balancing in Kinetic Systems |
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294 | (1) |
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14.2 Consequences of Detailed Balancing in Mass Action Systems |
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295 | (3) |
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14.3 Reversible Deficiency Zero Forest-Like Reaction Networks |
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298 | (2) |
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14.4 Necessary and Sufficient Conditions for Detailed Balancing in Mass Action Systems: Rate Constant Orchestration |
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300 | (9) |
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14.4.1 Spanning Forests in the Standard Reaction Diagram |
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301 | (1) |
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14.4.2 Independent Cycle Conditions |
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302 | (2) |
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14.4.3 Spanning Forest Conditions |
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304 | (1) |
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14.4.4 The Detailed Balance Rate Constant Theorem |
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305 | (4) |
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309 | (12) |
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15.1 Complex Balancing in Kinetic Systems |
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309 | (2) |
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15.2 Consequences of Complex Balancing in Mass Action Systems |
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311 | (4) |
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15.3 Multicell Mass Action Systems in Which the Intracellular Chemistry Is Complex Balanced |
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315 | (6) |
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15.3.1 A Two-Cell Example |
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316 | (2) |
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15.3.2 A Three-Cell Example |
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318 | (1) |
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15.3.3 A Multicell-Dynamics Theorem Based on Complex Balancing |
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319 | (1) |
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15.3.4 The Path to the Deficiency Zero Multicell Theorem |
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320 | (1) |
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16 Deficiency Zero Theory Foundations and Some Key Propositions |
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321 | (38) |
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16.1 A Decomposition of the Species-Formation-Rate Function |
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322 | (3) |
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16.1.1 The Stoichiometric Map |
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322 | (1) |
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16.1.2 The Complex-Formation-Rate Function |
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322 | (1) |
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16.1.3 The Linear Subspace Span (a) and Its Properties |
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323 | (2) |
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16.2 The Deficiency as a Measure of How Tightly an Equilibrium Complex-Formation-Rate Vector Is Constrained |
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325 | (3) |
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16.2.1 Another Interpretation of the Deficiency |
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325 | (2) |
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16.2.2 Deficiency Zero Implies Complex Balancing at Every Equilibrium |
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327 | (1) |
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16.3 Decomposing the Mass Action Species-Formation-Rate Function |
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328 | (2) |
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16.4 The First Salt Theorem |
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330 | (5) |
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16.4.1 Scalar Equations Describing ker Ak |
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330 | (1) |
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16.4.2 The Salt-Barrel Picture |
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331 | (1) |
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16.4.3 Salt Concentrations at Steady State |
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332 | (1) |
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16.4.4 The First Salt Theorem and Some Corollaries |
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333 | (2) |
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16.5 Salt Theorem Consequences for General Kinetic Systems |
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335 | (10) |
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16.5.1 Some Key Propositions, Useful for General Kinetic Systems [ 71] |
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335 | (3) |
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16.5.2 Proof of the Deficiency Zero Theorem, Part (i) |
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338 | (1) |
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16.5.3 Complex Balancing and Weak Reversibility |
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339 | (1) |
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16.5.4 Complex Balancing in Networks of Nonzero Deficiency |
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339 | (2) |
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16.5.5 Weakly Reversible Networks Are Invariably Nondegenerate: Proof |
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341 | (1) |
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16.5.6 Which Species Can Coexist at an Equilibrium of a Deficiency Zero Kinetic System? |
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341 | (4) |
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16.6 Positive Equilibria for Weakly Reversible Deficiency Zero Mass Action Systems |
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345 | (2) |
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16.7 Every Weakly Reversible Deficiency Zero Mass Action System Is Complex Balanced and Quasi-Thermodynamic |
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347 | (1) |
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16.8 Proofs of the Deficiency Zero Theorem and Its Offshoots: Summing Up |
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347 | (2) |
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16.8.1 Proof of the Deficiency Zero Theorem, Part (ii) |
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347 | (1) |
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16.8.2 Proof of the Derivative Version of the Deficiency Zero Theorem |
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347 | (1) |
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16.8.3 Proof of the Deficiency Zero Multicell Theorem |
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348 | (1) |
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Appendix 16.A Proof of the First Salt Theorem |
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349 | (4) |
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Appendix 16.B The Kinetic and Stoichiometric Subspaces: Salt-Theorem Insights |
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353 | (6) |
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17 Deficiency One Theory Foundations |
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359 | (40) |
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17.1 Proof of the Deficiency One Theorem |
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359 | (14) |
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17.1.1 A Little About Proof Strategy |
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360 | (1) |
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17.1.2 Some Preliminaries |
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361 | (3) |
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17.1.3 Proof of the "If" Part of Proposition 17.1.7 |
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364 | (1) |
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17.1.4 Beginning the "Only If" Part of Proposition 17.1.7 |
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364 | (2) |
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17.1.5 The Case of One Linkage Class and Motivation for the Second Salt Theorem |
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366 | (1) |
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17.1.6 The Second Salt Theorem |
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367 | (3) |
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17.1.7 Proof of Proposition 17.1.10 (and the Deficiency One Theorem) in the Single-Linkage-Class Case |
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370 | (1) |
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17.1.8 Proof of Proposition 17.1.7 (and the Deficiency One Theorem) with Multiple Linkage Classes |
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371 | (2) |
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17.2 The Deficiency One Algorithm |
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373 | (23) |
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17.2.1 Getting Started: Two Useful Propositions |
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373 | (2) |
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17.2.2 A Key Overarching Question |
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375 | (1) |
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17.2.3 The Key Question, Posed for a Regular Deficiency One Network |
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376 | (2) |
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17.2.4 Three Observations Resulting in Some Linear Equations and Inequalities |
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378 | (1) |
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17.2.5 Digression: Another (Minor) Salt Theorem |
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379 | (2) |
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17.2.6 Observation 3 Rewritten |
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381 | (1) |
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17.2.7 Confluence Vectors and Confluence Vector Orientations |
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381 | (3) |
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17.2.8 The Inequality System Induced by a {U,M,L} Partition and a Confluence Vector Orientation |
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384 | (3) |
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17.2.9 A Theorem Underlying the Deficiency One Algorithm |
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387 | (1) |
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17.2.10 Algorithm Overview: Converting Questions About Nonlinear Equations into Questions About Linear Inequalities |
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388 | (2) |
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17.2.11 Some Earlier Examples Reconsidered |
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390 | (5) |
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17.2.12 A Brief Remark About Higher-Deficiency Mass Action Theory |
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395 | (1) |
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Appendix 17.A Proof of the Second Salt Theorem |
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396 | (3) |
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18 Concentration Robustness Foundations |
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399 | (20) |
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18.1 The Deficiency One Concentration Robustness Theorem |
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399 | (4) |
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18.2 Proof of the First Rate-Constant-Independence Theorem |
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403 | (9) |
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18.2.1 Motivation for the Proof |
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404 | (3) |
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18.2.2 A Consequence of Cut-Link Removal |
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407 | (1) |
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18.2.3 Proof of Theorem 18.2.2 When y-bar is a resonance isomer of y-bar' Consists of a Single Irreversible Reaction y-bar yields y-bar' |
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408 | (1) |
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18.2.4 Proof of Theorem 18.2.2 When y-bar is a resonance isomer of y-bar' Consists of Reversible Reactions y-bar is in equilibrium with y-bar' |
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409 | (3) |
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18.3 Proof of the Second Rate-Constant-Independence Theorem |
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412 | (7) |
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18.3.1 Proof of Theorem 18.3.2 When y-bar is a resonance isomer of y-bar' Consists of a Single Reaction y-bar yields y-bar' |
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413 | (2) |
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18.3.2 Proof of Theorem 18.3.2 When y-bar is a resonance isomer of y-bar' Consists of a Reversible Reaction Pair y-bar is in equilibrium with y-bar' |
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415 | (1) |
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18.4 About Reaction Network Architectures that Thwart Concentration Robustness |
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416 | (3) |
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19 Species-Reaction Graph Foundations |
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419 | (17) |
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420 | (1) |
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19.2 A Useful Supposition |
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420 | (1) |
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19.3 Signed Species and Signed Reactions |
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421 | (1) |
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19.4 The Sign-Causality Graph Associated with a Discordance |
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422 | (3) |
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422 | (1) |
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19.4.2 How the Sign-Causality Graph Is drawn |
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423 | (2) |
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19.5 Why Even Cycles Are Necessary for a Discordance |
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425 | (1) |
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19.6 Sources in the Sign-Causality Graph |
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425 | (1) |
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19.7 Inequalities Associated with a Source |
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426 | (1) |
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19.8 Stoichiometric Coefficients in the Sign-Causality Graph |
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427 | (1) |
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19.9 Motivation: Proof of Theorem 19.1.1 for a Simple Special Case |
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427 | (2) |
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19.10 The Reflection of a Source in the Species-Reaction Graph |
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429 | (2) |
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19.11 Proof Strategy Going Forward |
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431 | (1) |
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19.12 When Condition (ii) of Theorem 19.1.1 Is Satisfied, Every Even Cycle Cluster Has a Simple Core |
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432 | (1) |
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19.13 Completion of the Proof of Theorem 19.1.1 |
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433 | (1) |
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19.14 A Species-Reaction Graph Theorem with a Weaker Hypothesis |
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434 | (2) |
| Appendix 19.A Proof of Proposition 19.12.3 |
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436 | (2) |
| Appendix 19.B Proof of Proposition 19.13.1 |
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438 | (3) |
| References |
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441 | (10) |
| Index |
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451 | |
Lisainfo e-raamatute kohta