Determining Weakly Reversible Linearly Conjugate Chemical Reaction - PowerPoint PPT Presentation

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Another structural parameter of a chemical reaction network which is particularly strongly related to the dynamics is the deficiency . In general, a network with a lower deficiency exhibits more regular behaviour (i.e. fewer equilibrium states, lower capacity for oscillations/multistability, etc.) QUESTION #2 Can we extend this MILP framework to search for the system, within some class of systems with identical qualitative dynamics, which has the minimal deficiency ? MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the general network N given by k ( i , j ) N : C i − → C j , ( i , j ) ∈ R . MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the general network N given by k ( i , j ) N : C i − → C j , ( i , j ) ∈ R . Under mass-action kinetics , this network is governed by d x dt = Y A k Ψ( x ) . (1) MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the general network N given by k ( i , j ) N : C i − → C j , ( i , j ) ∈ R . Under mass-action kinetics , this network is governed by d x dt = Y A k Ψ( x ) . (1) We have the following important components: Y is the complex matrix, MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the general network N given by k ( i , j ) N : C i − → C j , ( i , j ) ∈ R . Under mass-action kinetics , this network is governed by d x dt = Y A k Ψ( x ) . (1) We have the following important components: Y is the complex matrix, A k is the kinetics matrix (keeps track of connections), MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the general network N given by k ( i , j ) N : C i − → C j , ( i , j ) ∈ R . Under mass-action kinetics , this network is governed by d x dt = Y A k Ψ( x ) . (1) We have the following important components: Y is the complex matrix, A k is the kinetics matrix (keeps track of connections), Ψ( x ) is the mass-action vector (with entries Ψ i ( x ) = � m j =1 ( x j ) z ij ). MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the reversible network k (1 , 2) A 1 2 A 2 . ⇄ k (2 , 1) MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the reversible network k (1 , 2) A 1 2 A 2 . ⇄ k (2 , 1) This has the governing dynamics  dx 1  � � − k (1 , 2) � 1 � � x 1 � 0 k (2 , 1) dt    =   x 2 0 2 k (1 , 2) − k (2 , 1) dx 2  2 dt � − 1 � � � 1 x 2 = k (1 , 2) x 1 + k (2 , 1) 2 − 2 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the reversible network k (1 , 2) A 1 2 A 2 . ⇄ k (2 , 1) This has the governing dynamics  dx 1  � � − k (1 , 2) � 1 � � x 1 � 0 k (2 , 1) dt    =   x 2 0 2 k (1 , 2) − k (2 , 1) dx 2  2 dt � − 1 � � � 1 x 2 = k (1 , 2) x 1 + k (2 , 1) 2 − 2 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the reversible network k (1 , 2) A 1 2 A 2 . ⇄ k (2 , 1) This has the governing dynamics  dx 1  � � − k (1 , 2) � 1 � � x 1 � 0 k (2 , 1) dt    =   x 2 0 2 k (1 , 2) − k (2 , 1) dx 2  2 dt � − 1 � � � 1 x 2 = k (1 , 2) x 1 + k (2 , 1) 2 − 2 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the reversible network k (1 , 2) A 1 2 A 2 . ⇄ k (2 , 1) This has the governing dynamics  dx 1  � � − k (1 , 2) � 1 � � x 1 � 0 k (2 , 1) dt    =   x 2 0 2 k (1 , 2) − k (2 , 1) dx 2  2 dt � − 1 � � � 1 x 2 = k (1 , 2) x 1 + k (2 , 1) 2 − 2 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Consider the reversible network k (1 , 2) A 1 2 A 2 . ⇄ k (2 , 1) This has the governing dynamics  dx 1  � � − k (1 , 2) � 1 � � x 1 � 0 k (2 , 1) dt    =   x 2 0 2 k (1 , 2) − k (2 , 1) dx 2  2 dt � − 1 � � � 1 x 2 = k (1 , 2) x 1 + k (2 , 1) 2 − 2 2 where x 1 and x 2 are the concentrations of A 1 and A 2 respectively. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Figure: Previous system with k 1 = k 2 = 1. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Figure: Previous system with k 1 = k 2 = 1. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Figure: Previous system with k 1 = k 2 = 1. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Chemical reaction networks can be treated as weighted directed graphs G ( V , E ): MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Chemical reaction networks can be treated as weighted directed graphs G ( V , E ): k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Chemical reaction networks can be treated as weighted directed graphs G ( V , E ): k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 The vertexes are the distinct complexes MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Chemical reaction networks can be treated as weighted directed graphs G ( V , E ): k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 The vertexes are the distinct complexes, V = C . MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Chemical reaction networks can be treated as weighted directed graphs G ( V , E ): k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 The vertexes are the distinct complexes, V = C . The edges are the reactions MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Chemical reaction networks can be treated as weighted directed graphs G ( V , E ): k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 The vertexes are the distinct complexes, V = C . The edges are the reactions, E = R . MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Chemical reaction networks can be treated as weighted directed graphs G ( V , E ): k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 The vertexes are the distinct complexes, V = C . The edges are the reactions, E = R . We are interested in all the standard components of graphs, e.g. paths, cycles, connected components, trees, etc. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The particular class of networks we have been interested in are weakly reversible networks . MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The particular class of networks we have been interested in are weakly reversible networks . k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The particular class of networks we have been interested in are weakly reversible networks . k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 / A network is weakly reversible if a path from one another complex to another implies a path back MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The particular class of networks we have been interested in are weakly reversible networks . k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 / A network is weakly reversible if a path from one another complex to another implies a path back, e.g. C 1 − → C 2 / MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The particular class of networks we have been interested in are weakly reversible networks . k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 / A network is weakly reversible if a path from one another complex to another implies a path back, e.g. C 1 − → C 2 − → C 3 − → C 1 / MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The particular class of networks we have been interested in are weakly reversible networks . k (1 , 2) C 1 − → C 2 k (4 , 5) C 4 C 5 ⇄ k (3 , 1) տ ւ k (2 , 3) k (5 , 4) C 3 / A network is weakly reversible if a path from one another complex to another implies a path back, e.g. C 1 − → C 2 − → C 3 − → C 1 , C 4 − → C 5 − → C 4 . / MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The deficiency is a nonnegative integer parameter value defined by δ = m − ℓ − s MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The deficiency is a nonnegative integer parameter value defined by δ = m − ℓ − s where m is the number of distinct complexes (i.e. nodes), MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The deficiency is a nonnegative integer parameter value defined by δ = m − ℓ − s where m is the number of distinct complexes (i.e. nodes), ℓ is the number of linkage classes (i.e. connected components), MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The deficiency is a nonnegative integer parameter value defined by δ = m − ℓ − s where m is the number of distinct complexes (i.e. nodes), ℓ is the number of linkage classes (i.e. connected components), and s is the dimension of the stoichiometric space (i.e. the dimension of the linear invariant spaces). MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks The deficiency is a nonnegative integer parameter value defined by δ = m − ℓ − s where m is the number of distinct complexes (i.e. nodes), ℓ is the number of linkage classes (i.e. connected components), and s is the dimension of the stoichiometric space (i.e. the dimension of the linear invariant spaces). All of these values can be easily determined based on the structure of the reaction graph alone ! MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Many strong dynamical properties are known to hold for weakly reversible networks with low deficiency! MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Many strong dynamical properties are known to hold for weakly reversible networks with low deficiency! Deficiency Zero Theorem - Weakly reversible deficiency zero networks are complex balanced. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Many strong dynamical properties are known to hold for weakly reversible networks with low deficiency! Deficiency Zero Theorem - Weakly reversible deficiency zero networks are complex balanced. Complex Balanced Networks - Each positive invariant space has precisedly one equilibrium state (asymptotically stable). MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Objectives Linearly Conjugate Networks Mass-Action Kinetics Computational Approach Chemical Reaction Networks Many strong dynamical properties are known to hold for weakly reversible networks with low deficiency! Deficiency Zero Theorem - Weakly reversible deficiency zero networks are complex balanced. Complex Balanced Networks - Each positive invariant space has precisedly one equilibrium state (asymptotically stable). Deficiency One Theorem - Weakly reversible networks under certain technical assumptions have precisely one equilibrium state in each positive invariant space (variable stability). MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach 1 Background Objectives Mass-Action Kinetics Chemical Reaction Networks 2 Linearly Conjugate Networks Realizations Linearly Conjugate Networks 3 Computational Approach Generating Realizations Minimizing the Deficiency Example MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach It has long been known that two mass-action system with distinct underlying network structure can be governed by identical mass-action kinetics ! MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach It has long been known that two mass-action system with distinct underlying network structure can be governed by identical mass-action kinetics ! Definition ([1, 2]) Two mass-action systems N and N ′ which give rise to the same mass-action kinetics (1) are said to be dynamically equivalent . We will alternatively say that N ′ is an alternative realization of N or vice-versa. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach It has long been known that two mass-action system with distinct underlying network structure can be governed by identical mass-action kinetics ! Definition ([1, 2]) Two mass-action systems N and N ′ which give rise to the same mass-action kinetics (1) are said to be dynamically equivalent . We will alternatively say that N ′ is an alternative realization of N or vice-versa. IMPORTANT: If one realization has known dynamics while another does not, the dynamics are transferred to the unknown network! MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach Consider the set of polynomial differential equations x 1 = − 2 x 2 1 + x 2 ˙ 2 x 2 = 2 x 2 1 − x 2 ˙ 2 . MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach Consider the set of polynomial differential equations x 1 = − 2 x 2 1 + x 2 ˙ 2 x 2 = 2 x 2 1 − x 2 ˙ 2 . This governs the dynamics of both of the mass-action systems 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 1 N ′ : 2 A 1 2 A 2 . ⇄ 0 . 5 MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach Consider the set of polynomial differential equations x 1 = − 2 x 2 1 + x 2 ˙ 2 x 2 = 2 x 2 1 − x 2 ˙ 2 . This governs the dynamics of both of the mass-action systems 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 1 N ′ : 2 A 1 2 A 2 . ⇄ 0 . 5 N ′ has “good” underlying structure... MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach Consider the set of polynomial differential equations x 1 = − 2 x 2 1 + x 2 ˙ 2 x 2 = 2 x 2 1 − x 2 ˙ 2 . This governs the dynamics of both of the mass-action systems 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 1 N ′ : 2 A 1 2 A 2 . ⇄ 0 . 5 N ′ has “good” underlying structure... / ...we know its dynamics... MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach Consider the set of polynomial differential equations x 1 = − 2 x 2 1 + x 2 ˙ 2 x 2 = 2 x 2 1 − x 2 ˙ 2 . This governs the dynamics of both of the mass-action systems 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 1 N ′ : 2 A 1 2 A 2 . ⇄ 0 . 5 N ′ has “good” underlying structure... / ...we know its dynamics... / ... so we know the dynamics of N as well. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach / Consider the networks k 1 A 1 − → A 2 N : k 2 2 A 2 − → 2 A 1 ˜ k 1 N ′ : A 1 2 A 2 ⇄ ˜ k 2 MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach / Consider the networks dx 1 = − k 1 x 1 + 2 k 2 x 2 k 1 A 1 − → A 2 2 dt N : ⇒ = k 2 dx 2 2 A 2 − → 2 A 1 = k 1 x 1 − 2 k 2 x 2 2 . dt ˜ k 1 N ′ : A 1 2 A 2 ⇄ ˜ k 2 MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach / Consider the networks dx 1 = − k 1 x 1 + 2 k 2 x 2 k 1 A 1 − → A 2 2 dt N : ⇒ = k 2 dx 2 2 A 2 − → 2 A 1 = k 1 x 1 − 2 k 2 x 2 2 . dt dy 1 = − ˜ k 1 y 1 + ˜ k 2 y 2 ˜ k 1 2 N ′ : dt A 1 2 A 2 ⇒ ⇄ = dy 2 ˜ = 2˜ k 1 y 1 − 2˜ k 2 y 2 k 2 2 dt MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach / Consider the networks dx 1 = − k 1 x 1 + 2 k 2 x 2 k 1 A 1 − → A 2 2 dt N : ⇒ = k 2 dx 2 2 A 2 − → 2 A 1 = k 1 x 1 − 2 k 2 x 2 2 . dt dy 1 = − ˜ k 1 y 1 + ˜ k 2 y 2 ˜ k 1 2 N ′ : dt A 1 2 A 2 ⇒ ⇄ = dy 2 ˜ = 2˜ k 1 y 1 − 2˜ k 2 y 2 k 2 2 dt The governing dynamics look very similar (but not identical). MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach / Consider the networks dx 1 = − k 1 x 1 + 2 k 2 x 2 k 1 A 1 − → A 2 2 dt N : ⇒ = k 2 dx 2 2 A 2 − → 2 A 1 = k 1 x 1 − 2 k 2 x 2 2 . dt dy 1 = − ˜ k 1 y 1 + ˜ k 2 y 2 ˜ k 1 2 N ′ : dt A 1 2 A 2 ⇒ ⇄ = dy 2 ˜ = 2˜ k 1 y 1 − 2˜ k 2 y 2 k 2 2 dt The governing dynamics look very similar (but not identical). MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach / Consider the networks dx 1 = − k 1 x 1 + 2 k 2 x 2 k 1 A 1 − → A 2 2 dt N : ⇒ = k 2 dx 2 2 A 2 − → 2 A 1 = k 1 x 1 − 2 k 2 x 2 2 . dt dy 1 = − ˜ k 1 y 1 + ˜ k 2 y 2 ˜ k 1 2 N ′ : dt A 1 2 A 2 ⇒ ⇄ = dy 2 ˜ = 2˜ k 1 y 1 − 2˜ k 2 y 2 k 2 2 dt The governing dynamics look very similar (but not identical). MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach N: N': x 2 y 2 x 1 y 1 MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach N: N': x 2 y 2 x 1 y 1 We can relate these systems with the transformation x 1 = 2 y 1 , x 2 = y 2 (a linear transformation)! MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach Definition We will say N and N ′ are linearly conjugate if their flows under mass-action kinetics are related by a linear transformation. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach Definition We will say N and N ′ are linearly conjugate if their flows under mass-action kinetics are related by a linear transformation. In terms of many key dynamical properties, linearly conjugate systems exhibit identical qualitative dynamics (e.g. number and stability of equilibria, persistence, boundedness, etc.)! MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach Definition We will say N and N ′ are linearly conjugate if their flows under mass-action kinetics are related by a linear transformation. In terms of many key dynamical properties, linearly conjugate systems exhibit identical qualitative dynamics (e.g. number and stability of equilibria, persistence, boundedness, etc.)! Linear conjugacy includes dynamical equivalence as a special case (identity transformation). MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach Linearly conjugate networks were the central focus of study in the paper of M. D. Johnston and D. Siegel [3]. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach Linearly conjugate networks were the central focus of study in the paper of M. D. Johnston and D. Siegel [3]. Theorem (Theorem 3.2 of [3] and Theorem 2 of [4]) Consider the kinetics matrix A k corresponding to N and suppose that there is a kinetics matrix A b with the same structure as N ′ and a vector c ∈ R n > 0 such that Y · A k = T · Y · A b (2) where T = diag { c } . Then N is linearly conjugate to N ′ with kinetics matrix A ′ k = A b · diag { Ψ( c ) } . (3) MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach dx 1 = − k 1 x 1 + 2 k 2 x 2 k 1 A 1 − → A 2 2 dt N : = ⇒ k 2 dx 2 2 A 2 − → 2 A 1 = k 1 x 1 − 2 k 2 x 2 2 . dt dy 1 = − ˜ k 1 y 1 + ˜ k 2 y 2 ˜ k 1 2 N ′ : dt A 1 2 A 2 = ⇒ ⇄ dy 2 ˜ = 2˜ k 1 y 1 − 2˜ k 2 y 2 k 2 2 dt MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach dx 1 = − k 1 x 1 + 2 k 2 x 2 k 1 A 1 − → A 2 2 dt N : = ⇒ k 2 dx 2 2 A 2 − → 2 A 1 = k 1 x 1 − 2 k 2 x 2 2 . dt dy 1 = − ˜ k 1 y 1 + ˜ k 2 y 2 ˜ k 1 2 N ′ : dt A 1 2 A 2 = ⇒ ⇄ dy 2 ˜ = 2˜ k 1 y 1 − 2˜ k 2 y 2 k 2 2 dt N ′ has “good” structure... MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach dx 1 = − k 1 x 1 + 2 k 2 x 2 k 1 A 1 − → A 2 2 dt N : = ⇒ k 2 dx 2 2 A 2 − → 2 A 1 = k 1 x 1 − 2 k 2 x 2 2 . dt dy 1 = − ˜ k 1 y 1 + ˜ k 2 y 2 ˜ k 1 2 N ′ : dt A 1 2 A 2 = ⇒ ⇄ dy 2 ˜ = 2˜ k 1 y 1 − 2˜ k 2 y 2 k 2 2 dt N ′ has “good” structure, so it has known dynamical behaviour... MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach dx 1 = − k 1 x 1 + 2 k 2 x 2 k 1 A 1 − → A 2 2 dt N : = ⇒ k 2 dx 2 2 A 2 − → 2 A 1 = k 1 x 1 − 2 k 2 x 2 2 . dt dy 1 = − ˜ k 1 y 1 + ˜ k 2 y 2 ˜ k 1 2 N ′ : dt A 1 2 A 2 = ⇒ ⇄ dy 2 ˜ = 2˜ k 1 y 1 − 2˜ k 2 y 2 k 2 2 dt N ′ has “good” structure, so it has known dynamical behaviour, which is linearly conjugate to the dynamical behaviour of the other system... MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Realizations Linearly Conjugate Networks Linearly Conjugate Networks Computational Approach dx 1 = − k 1 x 1 + 2 k 2 x 2 k 1 A 1 − → A 2 2 dt N : = ⇒ k 2 dx 2 2 A 2 − → 2 A 1 = k 1 x 1 − 2 k 2 x 2 2 . dt dy 1 = − ˜ k 1 y 1 + ˜ k 2 y 2 ˜ k 1 2 N ′ : dt A 1 2 A 2 = ⇒ ⇄ dy 2 ˜ = 2˜ k 1 y 1 − 2˜ k 2 y 2 k 2 2 dt N ′ has “good” structure, so it has known dynamical behaviour, which is linearly conjugate to the dynamical behaviour of the other system, so N has known dynamics as well! MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example 1 Background Objectives Mass-Action Kinetics Chemical Reaction Networks 2 Linearly Conjugate Networks Realizations Linearly Conjugate Networks 3 Computational Approach Generating Realizations Minimizing the Deficiency Example MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example PROBLEM: Suppose we are given a single mass-action system N , e.g. 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 . How do we find a linearly conjugate mass-action system N ′ with known dynamics? MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example PROBLEM: Suppose we are given a single mass-action system N , e.g. 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 . How do we find a linearly conjugate mass-action system N ′ with known dynamics? The space of possible underlying network structures is typically too large to consider by hand. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example PROBLEM: Suppose we are given a single mass-action system N , e.g. 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 . How do we find a linearly conjugate mass-action system N ′ with known dynamics? The space of possible underlying network structures is typically too large to consider by hand. The development of computer algorithms is vital to tackling this problem. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example For dynamical equivalence, G. Szederk´ enyi placed this problem within a linear programming optimization framework [5]. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example For dynamical equivalence, G. Szederk´ enyi placed this problem within a linear programming optimization framework [5]. Dynamical Equivalence (= Y A ′ Y A k = M k ) m � [ A k ] ij = 0 , j = 1 , . . . , m i =1 [ A k ] ij ≥ 0 , i , j = 1 , . . . , m , i � = j [ A k ] ii ≤ 0 , i = 1 , . . . , m . MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example Reconsider the network 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 . MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example Reconsider the network 1 1 N : 2 A 1 − → 2 A 2 − → A 1 + A 2 . Running a MILP procedure in GLPK gives the following alternative realizations : 0.1 2A 1 2A 2 (a) (b) 1 0.45 2A 1 2A 2 1.8 0.1 0.5 0.1 0.1 A 1 +A 2 Figure: Alternative mass-action systems which generate the same mass-action kinetics as N . The system in (a) has the fewest number of reactions, while the system in (b) has the greatest number of reactions. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example In subsequent papers, G. Szederkenyi et al. imposed further conditions on the networks (detailed and complex balancing, weak and full reversibility) [6, 7]. MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example In subsequent papers, G. Szederkenyi et al. imposed further conditions on the networks (detailed and complex balancing, weak and full reversibility) [6, 7]. Further questions spring to mind: MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example In subsequent papers, G. Szederkenyi et al. imposed further conditions on the networks (detailed and complex balancing, weak and full reversibility) [6, 7]. Further questions spring to mind: 1 Can we adapt this to linearly conjugate systems ? MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example In subsequent papers, G. Szederkenyi et al. imposed further conditions on the networks (detailed and complex balancing, weak and full reversibility) [6, 7]. Further questions spring to mind: 1 Can we adapt this to linearly conjugate systems ? 2 Can we handle weak reversibility ? MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

Background Generating Realizations Linearly Conjugate Networks Minimizing the Deficiency Computational Approach Example In subsequent papers, G. Szederkenyi et al. imposed further conditions on the networks (detailed and complex balancing, weak and full reversibility) [6, 7]. Further questions spring to mind: 1 Can we adapt this to linearly conjugate systems ? 2 Can we handle weak reversibility ? 3 Can we adapt these techniques to find the system for which the underlying network has the minimal deficiency ? MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012