Background Linearly Conjugacy Linear Conjugacy of Chemical Reaction Networks Matthew Douglas Johnston University of Waterloo April 25, 2011 Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Background Linearly Conjugacy 1 Background Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Background Linearly Conjugacy 1 Background Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks 2 Linearly Conjugacy Main Theorem Examples Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks 1 Background Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks 2 Linearly Conjugacy Main Theorem Examples Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks An elementary reaction consists of a set of reactants which turn into a set of products, e.g. k 2H 2 + O 2 − → 2H 2 O / Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks An elementary reaction consists of a set of reactants which turn into a set of products, e.g. k 2H 2 + O 2 − → 2H 2 O Species/Reactants Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks An elementary reaction consists of a set of reactants which turn into a set of products, e.g. k 2H 2 + O 2 − → 2H 2 O Reactant Complex/ Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks An elementary reaction consists of a set of reactants which turn into a set of products, e.g. k 2H 2 + O 2 − → 2H 2 O Product Complex/ Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks An elementary reaction consists of a set of reactants which turn into a set of products, e.g. k 2H 2 + O 2 − → 2H 2 O Reaction Constant/ Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks An elementary reaction consists of a set of reactants which turn into a set of products, e.g. k 2H 2 + O 2 − → 2H 2 O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions. Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks An elementary reaction consists of a set of reactants which turn into a set of products, e.g. k 2H 2 + O 2 − → 2H 2 O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions. In order to build a model, we assume the mixture is spatially homogeneous, temperature and volume are held constant, and the law of mass action applies. Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks Consider the general network N given by k i → C ′ C i − i , i = 1 , . . . , r . Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks Consider the general network N given by k i → C ′ C i − i , i = 1 , . . . , r . Under mass-action kinetics, this network is governed by the system of autonomous, polynomial, ordinary differential equations r � k i ( z ′ i − z i ) x z i x = ˙ (1) i =1 where x i , i = 1 , . . . , m , are the reactant concentrations. Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks Consider the general network N given by k i → C ′ C i − i , i = 1 , . . . , r . Under mass-action kinetics, this network is governed by the system of autonomous, polynomial, ordinary differential equations r � k i ( z ′ i − z i ) x z i x = ˙ (1) i =1 where x i , i = 1 , . . . , m , are the reactant concentrations. Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks Consider the general network N given by k i → C ′ C i − i , i = 1 , . . . , r . Under mass-action kinetics, this network is governed by the system of autonomous, polynomial, ordinary differential equations r � k i ( z ′ i − z i ) x z i x = ˙ (1) i =1 where x i , i = 1 , . . . , m , are the reactant concentrations. Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks Consider the general network N given by k i → C ′ C i − i , i = 1 , . . . , r . Under mass-action kinetics, this network is governed by the system of autonomous, polynomial, ordinary differential equations r � k i ( z ′ i − z i ) x z i x = ˙ (1) i =1 where x i , i = 1 , . . . , m , are the reactant concentrations. Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks Consider the general network N given by k i → C ′ C i − i , i = 1 , . . . , r . Under mass-action kinetics, this network is governed by the system of autonomous, polynomial, ordinary differential equations r � k i ( z ′ i − z i ) x z i x = ˙ (1) i =1 where x i , i = 1 , . . . , m , are the reactant concentrations. Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks Consider the general network N given by k i → C ′ C i − i , i = 1 , . . . , r . Under mass-action kinetics, this network is governed by the system of autonomous, polynomial, ordinary differential equations r � k i ( z ′ i − z i ) x z i x = ˙ (1) i =1 where x i , i = 1 , . . . , m , are the reactant concentrations. Model is applied to systems biology, enzyme kinetics, industrial reactors, neural networks, atmospherics, etc., and is related to predator-prey and epidemic growth models in biology. Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks The particular class of networks which I have been interested in are weakly reversible networks . Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks The particular class of networks which I have been interested in are weakly reversible networks . A network is weakly reversible if a path from one complex to another in the reaction graph implies a path back Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks The particular class of networks which I have been interested in are weakly reversible networks . A network is weakly reversible if a path from one complex to another in the reaction graph implies a path back , e.g. k 1 C 1 − → C 2 k 3 տ ւ k 2 C 3 . Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks The particular class of networks which I have been interested in are weakly reversible networks . A network is weakly reversible if a path from one complex to another in the reaction graph implies a path back , e.g. k 1 C 1 − → C 2 k 3 տ ւ k 2 C 3 . Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks The particular class of networks which I have been interested in are weakly reversible networks . A network is weakly reversible if a path from one complex to another in the reaction graph implies a path back , e.g. k 1 C 1 − → C 2 k 3 տ ւ k 2 C 3 . Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
Chemical Reactions Background Mass-Action Kinetics Linearly Conjugacy Weakly Reversible Networks The particular class of networks which I have been interested in are weakly reversible networks . A network is weakly reversible if a path from one complex to another in the reaction graph implies a path back , e.g. k 1 C 1 − → C 2 k 3 տ ւ k 2 C 3 . Under the assumption of mass-action kinetics, strong properties are known about the dynamics of weakly reversible networks. Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks
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