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Foundations of Chemical Reaction Network Theory 1st ed. 2019 [Kõva köide]

  • Formaat: Hardback, 454 pages, kõrgus x laius: 235x155 mm, kaal: 893 g, 42 Illustrations, color; 8 Illustrations, black and white; XXIX, 454 p. 50 illus., 42 illus. in color., 1 Hardback
  • Sari: Applied Mathematical Sciences 202
  • Ilmumisaeg: 28-Feb-2019
  • Kirjastus: Springer Nature Switzerland AG
  • ISBN-10: 3030038572
  • ISBN-13: 9783030038571
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Foundations of Chemical Reaction Network Theory 1st ed. 2019Suurem pilt
  • Formaat: Hardback, 454 pages, kõrgus x laius: 235x155 mm, kaal: 893 g, 42 Illustrations, color; 8 Illustrations, black and white; XXIX, 454 p. 50 illus., 42 illus. in color., 1 Hardback
  • Sari: Applied Mathematical Sciences 202
  • Ilmumisaeg: 28-Feb-2019
  • Kirjastus: Springer Nature Switzerland AG
  • ISBN-10: 3030038572
  • ISBN-13: 9783030038571
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783030038571.html
Märksõnad:

This book provides an authoritative introduction to the rapidly growing field of chemical reaction network theory. In particular, the book presents deep and surprising theorems that relate the graphical and algebraic structure of a reaction network to qualitative properties of the intricate system of nonlinear differential equations that the network induces. Over the course of three main parts, Feinberg provides a gradual transition from a tutorial on the basics of reaction network theory, to a survey of some of its principal theorems, and, finally, to a discussion of the theory’s more technical aspects. Written with great clarity, this book will be of value to mathematicians and to mathematically-inclined biologists, chemists, physicists, and engineers who want to contribute to chemical reaction network theory or make use of its powerful results.

Arvustused

This book comprehensively describes network theory applied to chemical reactions, delivering powerful conclusions surprisingly following from hypotheses that are essentially formulated in terms of linear algebra and graph theory. ... The presentation is rich in material including motivating examples, applications to practical problems, guide to the literature, and mathematical proofs. It deserves a salient place in the section on mathematical chemistry of any library. (Dieter Erle, zbMATH 1420.92001, 2019)

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