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BackgroundLinearly Conjugate Networks
Computation Approach
Linear Conjugacy of Chemical Reaction Networks
Matthew D. Johnstona, David Siegela and G. Szederkenyiba University of Waterloo
b Hungarian Academy of Sciences
October 8, 2011
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
1 BackgroundChemical Reaction NetworksMass-Action KineticsReaction Graphs
2 Linearly Conjugate NetworksRealizationsLinearly Conjugate Networks
3 Computation ApproachSparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
1 BackgroundChemical Reaction NetworksMass-Action KineticsReaction Graphs
2 Linearly Conjugate NetworksRealizationsLinearly Conjugate Networks
3 Computation ApproachSparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
1 BackgroundChemical Reaction NetworksMass-Action KineticsReaction Graphs
2 Linearly Conjugate NetworksRealizationsLinearly Conjugate Networks
3 Computation ApproachSparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
1 BackgroundChemical Reaction NetworksMass-Action KineticsReaction Graphs
2 Linearly Conjugate NetworksRealizationsLinearly Conjugate Networks
3 Computation ApproachSparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
A chemical reaction network consists of a set of reactants whichturn into a set of products, e.g.
2H2 + O2k−→ 2H2O
/
/Chemical kinetics is the study of the rates/dynamics resultingfrom systems of such reactions.
Applications in industrial chemistry, pharmaceutics, systemsbiology, gene regulation, and many other areas.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
A chemical reaction network consists of a set of reactants whichturn into a set of products, e.g.
2H2 + O2k−→ 2H2O
Species/Reactants
/Chemical kinetics is the study of the rates/dynamics resultingfrom systems of such reactions.
Applications in industrial chemistry, pharmaceutics, systemsbiology, gene regulation, and many other areas.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
A chemical reaction network consists of a set of reactants whichturn into a set of products, e.g.
2H2 + O2k−→ 2H2O
Reactant Complex/
/Chemical kinetics is the study of the rates/dynamics resultingfrom systems of such reactions.
Applications in industrial chemistry, pharmaceutics, systemsbiology, gene regulation, and many other areas.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
A chemical reaction network consists of a set of reactants whichturn into a set of products, e.g.
2H2 + O2k−→ 2H2O
Product Complex/
/Chemical kinetics is the study of the rates/dynamics resultingfrom systems of such reactions.
Applications in industrial chemistry, pharmaceutics, systemsbiology, gene regulation, and many other areas.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
A chemical reaction network consists of a set of reactants whichturn into a set of products, e.g.
2H2 + O2k−→ 2H2O
Reaction Constant/
/Chemical kinetics is the study of the rates/dynamics resultingfrom systems of such reactions.
Applications in industrial chemistry, pharmaceutics, systemsbiology, gene regulation, and many other areas.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
A chemical reaction network consists of a set of reactants whichturn into a set of products, e.g.
2H2 + O2k−→ 2H2O
/Chemical kinetics is the study of the rates/dynamics resultingfrom systems of such reactions.
Applications in industrial chemistry, pharmaceutics, systemsbiology, gene regulation, and many other areas.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
A chemical reaction network consists of a set of reactants whichturn into a set of products, e.g.
2H2 + O2k−→ 2H2O
/Chemical kinetics is the study of the rates/dynamics resultingfrom systems of such reactions.
Applications in industrial chemistry, pharmaceutics, systemsbiology, gene regulation, and many other areas.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Significant recent work has been done on the connection betweennetwork structure and dynamical behaviour.
CRN with "good" graph
CRN with "bad" graph
Understood dynamical behaviour!
same
dynamics
We can say something about the dynamics without having toanalyse the governing set of differential equations!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Significant recent work has been done on the connection betweennetwork structure and dynamical behaviour.
CRN with "good" graph
CRN with "bad" graph
Understood dynamical behaviour!
same
dynamics
We can say something about the dynamics without having toanalyse the governing set of differential equations!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Significant recent work has been done on the connection betweennetwork structure and dynamical behaviour.
CRN with "good" graph
CRN with "bad" graph
Understood dynamical behaviour!
same
dynamics
We can say something about the dynamics without having toanalyse the governing set of differential equations!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Significant recent work has been done on the connection betweennetwork structure and dynamical behaviour.
CRN with "good" graph
CRN with "bad" graph
Understood dynamical behaviour!
same
dynamics
We can say something about the dynamics without having toanalyse the governing set of differential equations!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Significant recent work has been done on the connection betweennetwork structure and dynamical behaviour.
CRN with "good" graph
CRN with "bad" graph
Understood dynamical behaviour!
same
dynamicsYES!
We can say something about the dynamics without having toanalyse the governing set of differential equations!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
QUESTION #1:
Can we find more general conditions under which two networkshave the same qualitative dynamics?
Current results require the governing differential equations to beidentical.
This could be used to widen the umbrella of classes of networkswith understood dynamics (e.g. weakly reversible networks).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
QUESTION #1:
Can we find more general conditions under which two networkshave the same qualitative dynamics?
Current results require the governing differential equations to beidentical.
This could be used to widen the umbrella of classes of networkswith understood dynamics (e.g. weakly reversible networks).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
QUESTION #1:
Can we find more general conditions under which two networkshave the same qualitative dynamics?
Current results require the governing differential equations to beidentical.
This could be used to widen the umbrella of classes of networkswith understood dynamics (e.g. weakly reversible networks).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
QUESTION #2:
Given a network with “bad” structure, can we find a network with“good” structure with related dynamics?
We are typically given a single network and the related networkmay not be apparent.
The development of computer algorithms is essential.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
QUESTION #2:
Given a network with “bad” structure, can we find a network with“good” structure with related dynamics?
We are typically given a single network and the related networkmay not be apparent.
The development of computer algorithms is essential.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
QUESTION #2:
Given a network with “bad” structure, can we find a network with“good” structure with related dynamics?
We are typically given a single network and the related networkmay not be apparent.
The development of computer algorithms is essential.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the general network N given by
N : Cik(i ,j)−→ Cj , (i , j) ∈ R.
Under mass-action kinetics, this network is governed by
dx
dt= Y Ak Ψ(x). (1)
We have the following important components:
Y is the stoichiometric matrix,
Ak is the kinetics matrix (keeps track of connections),
Ψ(x) is the mass-action vector (with entriesΨi (x) =
∏mj=1(xj)
zij ).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the general network N given by
N : Cik(i ,j)−→ Cj , (i , j) ∈ R.
Under mass-action kinetics, this network is governed by
dx
dt= Y Ak Ψ(x). (1)
We have the following important components:
Y is the stoichiometric matrix,
Ak is the kinetics matrix (keeps track of connections),
Ψ(x) is the mass-action vector (with entriesΨi (x) =
∏mj=1(xj)
zij ).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the general network N given by
N : Cik(i ,j)−→ Cj , (i , j) ∈ R.
Under mass-action kinetics, this network is governed by
dx
dt= Y Ak Ψ(x). (1)
We have the following important components:
Y is the stoichiometric matrix,
Ak is the kinetics matrix (keeps track of connections),
Ψ(x) is the mass-action vector (with entriesΨi (x) =
∏mj=1(xj)
zij ).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the general network N given by
N : Cik(i ,j)−→ Cj , (i , j) ∈ R.
Under mass-action kinetics, this network is governed by
dx
dt= Y Ak Ψ(x). (1)
We have the following important components:
Y is the stoichiometric matrix,
Ak is the kinetics matrix (keeps track of connections),
Ψ(x) is the mass-action vector (with entriesΨi (x) =
∏mj=1(xj)
zij ).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the general network N given by
N : Cik(i ,j)−→ Cj , (i , j) ∈ R.
Under mass-action kinetics, this network is governed by
dx
dt= Y Ak Ψ(x). (1)
We have the following important components:
Y is the stoichiometric matrix,
Ak is the kinetics matrix (keeps track of connections),
Ψ(x) is the mass-action vector (with entriesΨi (x) =
∏mj=1(xj)
zij ).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the reversible network
A1
k(1,2)
�k(2,1)
2A2.
This has the governing dynamicsdx1
dt
dx2
dt
=
[1 00 2
] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)
] [x1
x22
]
= k(1, 2)
[−12
]x1 + k(2, 1)
[1−2
]x2
2
where x1 and x2 are the concentrations of A1 and A2 respectively.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the reversible network
A1
k(1,2)
�k(2,1)
2A2.
This has the governing dynamicsdx1
dt
dx2
dt
=
[1 00 2
] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)
] [x1
x22
]
= k(1, 2)
[−12
]x1 + k(2, 1)
[1−2
]x2
2
where x1 and x2 are the concentrations of A1 and A2 respectively.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the reversible network
A1
k(1,2)
�k(2,1)
2A2.
This has the governing dynamicsdx1
dt
dx2
dt
=
[1 00 2
] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)
] [x1
x22
]
= k(1, 2)
[−12
]x1 + k(2, 1)
[1−2
]x2
2
where x1 and x2 are the concentrations of A1 and A2 respectively.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the reversible network
A1
k(1,2)
�k(2,1)
2A2.
This has the governing dynamicsdx1
dt
dx2
dt
=
[1 00 2
] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)
] [x1
x22
]
= k(1, 2)
[−12
]x1 + k(2, 1)
[1−2
]x2
2
where x1 and x2 are the concentrations of A1 and A2 respectively.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the reversible network
A1
k(1,2)
�k(2,1)
2A2.
This has the governing dynamicsdx1
dt
dx2
dt
=
[1 00 2
] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)
] [x1
x22
]
= k(1, 2)
[−12
]x1 + k(2, 1)
[1−2
]x2
2
where x1 and x2 are the concentrations of A1 and A2 respectively.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Consider the reversible network
A1
k(1,2)
�k(2,1)
2A2.
This has the governing dynamicsdx1
dt
dx2
dt
=
[1 00 2
] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)
] [x1
x22
]
= k(1, 2)
[−12
]x1 + k(2, 1)
[1−2
]x2
2
where x1 and x2 are the concentrations of A1 and A2 respectively.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Figure: Previous system with k1 = k2 = 1.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Figure: Previous system with k1 = k2 = 1.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Figure: Previous system with k1 = k2 = 1.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Chemical reaction networks can be treated as weighted directedgraphs G(V ,E ):
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
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 Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Chemical reaction networks can be treated as weighted directedgraphs G(V ,E ):
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
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 Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Chemical reaction networks can be treated as weighted directedgraphs G(V ,E ):
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
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 Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Chemical reaction networks can be treated as weighted directedgraphs G(V ,E ):
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
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 Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Chemical reaction networks can be treated as weighted directedgraphs G(V ,E ):
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
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 Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Chemical reaction networks can be treated as weighted directedgraphs G(V ,E ):
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
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 Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
Chemical reaction networks can be treated as weighted directedgraphs G(V ,E ):
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
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 Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
The particular class of networks I have been interested in areweakly reversible networks.
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
/A network is weakly reversible if a path from one another complexto another implies a path back, e.g.
/ C1 −→ C2 −→ C3 −→ C1, C4 −→ C5 −→ C4.
Strong dynamical properties follow from weak reversibility!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
The particular class of networks I have been interested in areweakly reversible networks.
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
/A network is weakly reversible if a path from one another complexto another implies a path back, e.g.
/ C1 −→ C2 −→ C3 −→ C1, C4 −→ C5 −→ C4.
Strong dynamical properties follow from weak reversibility!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
The particular class of networks I have been interested in areweakly reversible networks.
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
/A network is weakly reversible if a path from one another complexto another implies a path back
, e.g.
/ C1 −→ C2 −→ C3 −→ C1, C4 −→ C5 −→ C4.
Strong dynamical properties follow from weak reversibility!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
The particular class of networks I have been interested in areweakly reversible networks.
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
/A network is weakly reversible if a path from one another complexto another implies a path back, e.g.
/ C1 −→ C2
−→ C3 −→ C1, C4 −→ C5 −→ C4.
Strong dynamical properties follow from weak reversibility!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
The particular class of networks I have been interested in areweakly reversible networks.
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
/A network is weakly reversible if a path from one another complexto another implies a path back, e.g.
/ C1 −→ C2 −→ C3 −→ C1
, C4 −→ C5 −→ C4.
Strong dynamical properties follow from weak reversibility!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
The particular class of networks I have been interested in areweakly reversible networks.
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
/A network is weakly reversible if a path from one another complexto another implies a path back, e.g.
/ C1 −→ C2 −→ C3 −→ C1, C4 −→ C5 −→ C4.
Strong dynamical properties follow from weak reversibility!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Chemical Reaction NetworksMass-Action KineticsReaction Graphs
The particular class of networks I have been interested in areweakly reversible networks.
C1k(1,2)−→ C2
k(3,1) ↖ ↙k(2,3)
C3
C4
k(4,5)
�k(5,4)
C5
/A network is weakly reversible if a path from one another complexto another implies a path back, e.g.
/ C1 −→ C2 −→ C3 −→ C1, C4 −→ C5 −→ C4.
Strong dynamical properties follow from weak reversibility!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
1 BackgroundChemical Reaction NetworksMass-Action KineticsReaction Graphs
2 Linearly Conjugate NetworksRealizationsLinearly Conjugate Networks
3 Computation ApproachSparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Under the assumption of mass-action kinetics, it is possible for twochemical reaction networks to be dynamically equivalent [1, 2].
Definition
In the case that two networks N and N ′ give rise to the samemass-action kinetics (1) we will say that N ′ is an alternativerealization of N and vice-versa.
If one realization has known dynamics while another does not, thedynamics are transferred to the unknown network!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Under the assumption of mass-action kinetics, it is possible for twochemical reaction networks to be dynamically equivalent [1, 2].
Definition
In the case that two networks N and N ′ give rise to the samemass-action kinetics (1) we will say that N ′ is an alternativerealization of N and vice-versa.
If one realization has known dynamics while another does not, thedynamics are transferred to the unknown network!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Under the assumption of mass-action kinetics, it is possible for twochemical reaction networks to be dynamically equivalent [1, 2].
Definition
In the case that two networks N and N ′ give rise to the samemass-action kinetics (1) we will say that N ′ is an alternativerealization of N and vice-versa.
If one realization has known dynamics while another does not, thedynamics are transferred to the unknown network!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Consider the set of polynomial differential equations
x1 = −2x21 + x2
2
x2 = 2x21 − x2
2 .
This governs the dynamics of both of the networks
N : 2A11−→ 2A2
1−→ A1 +A2
N ′ : 2A1
1�0.5
2A2.
N ′ has “good” structure.../ ...we know its dynamics.../ ... so we know the dynamics of N as well.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Consider the set of polynomial differential equations
x1 = −2x21 + x2
2
x2 = 2x21 − x2
2 .
This governs the dynamics of both of the networks
N : 2A11−→ 2A2
1−→ A1 +A2
N ′ : 2A1
1�0.5
2A2.
N ′ has “good” structure.../ ...we know its dynamics.../ ... so we know the dynamics of N as well.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Consider the set of polynomial differential equations
x1 = −2x21 + x2
2
x2 = 2x21 − x2
2 .
This governs the dynamics of both of the networks
N : 2A11−→ 2A2
1−→ A1 +A2
N ′ : 2A1
1�0.5
2A2.
N ′ has “good” structure...
/ ...we know its dynamics.../ ... so we know the dynamics of N as well.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Consider the set of polynomial differential equations
x1 = −2x21 + x2
2
x2 = 2x21 − x2
2 .
This governs the dynamics of both of the networks
N : 2A11−→ 2A2
1−→ A1 +A2
N ′ : 2A1
1�0.5
2A2.
N ′ has “good” structure.../ ...we know its dynamics...
/ ... so we know the dynamics of N as well.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Consider the set of polynomial differential equations
x1 = −2x21 + x2
2
x2 = 2x21 − x2
2 .
This governs the dynamics of both of the networks
N : 2A11−→ 2A2
1−→ A1 +A2
N ′ : 2A1
1�0.5
2A2.
N ′ has “good” structure.../ ...we know its dynamics.../ ... so we know the dynamics of N as well.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
/ Consider the networks
N :A1
k1−→ A2
2A2k2−→ 2A1
=⇒
dx1
dt= −k1x1 + 2k2x
22
dx2
dt= k1x1 − 2k2x
22 .
N ′ : A1
k1
�k2
2A2
=⇒
dy1
dt= −k1y1 + k2y
22
dy2
dt= 2k1y1 − 2k2y
22
These networks look very similar (but not identical).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
/ Consider the networks
N :A1
k1−→ A2
2A2k2−→ 2A1
=⇒
dx1
dt= −k1x1 + 2k2x
22
dx2
dt= k1x1 − 2k2x
22 .
N ′ : A1
k1
�k2
2A2
=⇒
dy1
dt= −k1y1 + k2y
22
dy2
dt= 2k1y1 − 2k2y
22
These networks look very similar (but not identical).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
/ Consider the networks
N :A1
k1−→ A2
2A2k2−→ 2A1
=⇒
dx1
dt= −k1x1 + 2k2x
22
dx2
dt= k1x1 − 2k2x
22 .
N ′ : A1
k1
�k2
2A2 =⇒
dy1
dt= −k1y1 + k2y
22
dy2
dt= 2k1y1 − 2k2y
22
These networks look very similar (but not identical).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
/ Consider the networks
N :A1
k1−→ A2
2A2k2−→ 2A1
=⇒
dx1
dt= −k1x1 + 2k2x
22
dx2
dt= k1x1 − 2k2x
22 .
N ′ : A1
k1
�k2
2A2 =⇒
dy1
dt= −k1y1 + k2y
22
dy2
dt= 2k1y1 − 2k2y
22
These networks look very similar (but not identical).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
/ Consider the networks
N :A1
k1−→ A2
2A2k2−→ 2A1
=⇒
dx1
dt= −k1x1 + 2k2x
22
dx2
dt= k1x1 − 2k2x
22 .
N ′ : A1
k1
�k2
2A2 =⇒
dy1
dt= −k1y1 + k2y
22
dy2
dt= 2k1y1 − 2k2y
22
These networks look very similar (but not identical).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
/ Consider the networks
N :A1
k1−→ A2
2A2k2−→ 2A1
=⇒
dx1
dt= −k1x1 + 2k2x
22
dx2
dt= k1x1 − 2k2x
22 .
N ′ : A1
k1
�k2
2A2 =⇒
dy1
dt= −k1y1 + k2y
22
dy2
dt= 2k1y1 − 2k2y
22
These networks look very similar (but not identical).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
y1
y2
x1
x2
N: N':
QUESTION:
Is there a way we can relate the dynamics of N to N ′?
YES! These networks are related by the transformation x1 = 2y1,x2 = y2 (a linear transformation)!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
y1
y2
x1
x2
N: N':
QUESTION:
Is there a way we can relate the dynamics of N to N ′?
YES! These networks are related by the transformation x1 = 2y1,x2 = y2 (a linear transformation)!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
y1
y2
x1
x2
N: N':
QUESTION:
Is there a way we can relate the dynamics of N to N ′?
YES! These networks are related by the transformation x1 = 2y1,x2 = y2 (a linear transformation)!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Definition
We will say N and N ′ are linearly conjugate if their flows undermass-action kinetics are related by a linear transformation.
The qualitative dynamics (e.g. number and stability of equilibria,persistence, boundedness, etc.) of linearly conjugate networks areidentical!
Linear conjugacy includes dynamical equivalence as a specialcase (identity transformation).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Definition
We will say N and N ′ are linearly conjugate if their flows undermass-action kinetics are related by a linear transformation.
The qualitative dynamics (e.g. number and stability of equilibria,persistence, boundedness, etc.) of linearly conjugate networks areidentical!
Linear conjugacy includes dynamical equivalence as a specialcase (identity transformation).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Definition
We will say N and N ′ are linearly conjugate if their flows undermass-action kinetics are related by a linear transformation.
The qualitative dynamics (e.g. number and stability of equilibria,persistence, boundedness, etc.) of linearly conjugate networks areidentical!
Linear conjugacy includes dynamical equivalence as a specialcase (identity transformation).
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Linearly conjugate networks were the central focus of study in thepaper of M. D. Johnston and D. Siegel [3].
Theorem (Theorem 3.2 of [3] and Theorem 2 of [4])
Consider the kinetics matrix Ak corresponding to N and supposethat there is a kinetics matrix Ab with the same structure as N ′and a vector c ∈ Rn
>0 such that
Y · Ak = T · Y · Ab (2)
where T =diag{c}. Then N is linearly conjugate to N ′ withkinetics matrix
A′k = Ab · diag {Ψ(c)} . (3)
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
RealizationsLinearly Conjugate Networks
Linearly conjugate networks were the central focus of study in thepaper of M. D. Johnston and D. Siegel [3].
Theorem (Theorem 3.2 of [3] and Theorem 2 of [4])
Consider the kinetics matrix Ak corresponding to N and supposethat there is a kinetics matrix Ab with the same structure as N ′and a vector c ∈ Rn
>0 such that
Y · Ak = T · Y · Ab (2)
where T =diag{c}. Then N is linearly conjugate to N ′ withkinetics matrix
A′k = Ab · diag {Ψ(c)} . (3)
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
1 BackgroundChemical Reaction NetworksMass-Action KineticsReaction Graphs
2 Linearly Conjugate NetworksRealizationsLinearly Conjugate Networks
3 Computation ApproachSparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
PROBLEM:
If we are just given a single network N , e.g.
N : 2A11−→ 2A2
1−→ A1 +A2,
how do we find a linearly conjugate network N ′ with knowndynamics?
The space of possible networks is typically too large to consider byhand.
The development of computer algorithms is vital to making thistheory applicable.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
PROBLEM:
If we are just given a single network N , e.g.
N : 2A11−→ 2A2
1−→ A1 +A2,
how do we find a linearly conjugate network N ′ with knowndynamics?
The space of possible networks is typically too large to consider byhand.
The development of computer algorithms is vital to making thistheory applicable.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
PROBLEM:
If we are just given a single network N , e.g.
N : 2A11−→ 2A2
1−→ A1 +A2,
how do we find a linearly conjugate network N ′ with knowndynamics?
The space of possible networks is typically too large to consider byhand.
The development of computer algorithms is vital to making thistheory applicable.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
For dynamical equivalence, G. Szederkenyi placed this problemwithin a linear programming optimization framework [5].
Dynamical Equivalence
Y Ak = M (= Y A′k)m∑i=1
[Ak ]ij = 0, j = 1, . . . ,m
[Ak ]ij ≥ 0, i , j = 1, . . . ,m, i 6= j[Ak ]ii ≤ 0, i = 1, . . . ,m.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
For dynamical equivalence, G. Szederkenyi placed this problemwithin a linear programming optimization framework [5].
Dynamical Equivalence
Y Ak = M (= Y A′k)m∑i=1
[Ak ]ij = 0, j = 1, . . . ,m
[Ak ]ij ≥ 0, i , j = 1, . . . ,m, i 6= j[Ak ]ii ≤ 0, i = 1, . . . ,m.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Consider the binary variables δij ∈ {0, 1}, let uij , ε > 0 beconstants, and consider the constraints
0 ≤ εδij ≤ [Ak ]ij ≤ uijδij , i , j = 1, . . . ,m, i 6= j .
1 For δij = 0 we have [Ak ]ij = 0
=⇒ reaction is ‘off’.
2 For δij = 1 we have 0 < ε ≤ [Ak ]ij ≤ uij
=⇒ reaction is ‘on’.
/
The δij ’s tell us whether the reaction Cj −→ Ci is in the network ornot.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Consider the binary variables δij ∈ {0, 1}, let uij , ε > 0 beconstants, and consider the constraints
0 ≤ εδij ≤ [Ak ]ij ≤ uijδij , i , j = 1, . . . ,m, i 6= j .
1 For δij = 0 we have [Ak ]ij = 0
=⇒ reaction is ‘off’.
2 For δij = 1 we have 0 < ε ≤ [Ak ]ij ≤ uij
=⇒ reaction is ‘on’.
/
The δij ’s tell us whether the reaction Cj −→ Ci is in the network ornot.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Consider the binary variables δij ∈ {0, 1}, let uij , ε > 0 beconstants, and consider the constraints
0 ≤ εδij ≤ [Ak ]ij ≤ uijδij , i , j = 1, . . . ,m, i 6= j .
1 For δij = 0 we have [Ak ]ij = 0 =⇒ reaction is ‘off’.
2 For δij = 1 we have 0 < ε ≤ [Ak ]ij ≤ uij
=⇒ reaction is ‘on’.
/
The δij ’s tell us whether the reaction Cj −→ Ci is in the network ornot.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Consider the binary variables δij ∈ {0, 1}, let uij , ε > 0 beconstants, and consider the constraints
0 ≤ εδij ≤ [Ak ]ij ≤ uijδij , i , j = 1, . . . ,m, i 6= j .
1 For δij = 0 we have [Ak ]ij = 0 =⇒ reaction is ‘off’.
2 For δij = 1 we have 0 < ε ≤ [Ak ]ij ≤ uij
=⇒ reaction is ‘on’.
/
The δij ’s tell us whether the reaction Cj −→ Ci is in the network ornot.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Consider the binary variables δij ∈ {0, 1}, let uij , ε > 0 beconstants, and consider the constraints
0 ≤ εδij ≤ [Ak ]ij ≤ uijδij , i , j = 1, . . . ,m, i 6= j .
1 For δij = 0 we have [Ak ]ij = 0 =⇒ reaction is ‘off’.
2 For δij = 1 we have 0 < ε ≤ [Ak ]ij ≤ uij =⇒ reaction is ‘on’.
/
The δij ’s tell us whether the reaction Cj −→ Ci is in the network ornot.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Consider the binary variables δij ∈ {0, 1}, let uij , ε > 0 beconstants, and consider the constraints
0 ≤ εδij ≤ [Ak ]ij ≤ uijδij , i , j = 1, . . . ,m, i 6= j .
1 For δij = 0 we have [Ak ]ij = 0 =⇒ reaction is ‘off’.
2 For δij = 1 we have 0 < ε ≤ [Ak ]ij ≤ uij =⇒ reaction is ‘on’.
/The δij ’s tell us whether the reaction Cj −→ Ci is in the network ornot.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Summing over δij , i , j = 1, . . . ,m, i 6= j , tells us how manyreactions there are in the network!
Sparse/Dense Realizations
minimizem∑
i ,j=1,i 6=j
δij or minimizem∑
i ,j=1,i 6=j
−δij
0 ≤ [Ak ]ij − εδij0 ≤ −[Ak ]ij + uijδij
δij ∈ {0, 1}i , j = 1, . . . ,m, i 6= j
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Summing over δij , i , j = 1, . . . ,m, i 6= j , tells us how manyreactions there are in the network!
Sparse/Dense Realizations
minimizem∑
i ,j=1,i 6=j
δij or minimizem∑
i ,j=1,i 6=j
−δij
0 ≤ [Ak ]ij − εδij0 ≤ −[Ak ]ij + uijδij
δij ∈ {0, 1}i , j = 1, . . . ,m, i 6= j
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Summing over δij , i , j = 1, . . . ,m, i 6= j , tells us how manyreactions there are in the network!
Sparse/Dense Realizations
minimizem∑
i ,j=1,i 6=j
δij or minimizem∑
i ,j=1,i 6=j
−δij
0 ≤ [Ak ]ij − εδij0 ≤ −[Ak ]ij + uijδij
δij ∈ {0, 1}i , j = 1, . . . ,m, i 6= j
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Summing over δij , i , j = 1, . . . ,m, i 6= j , tells us how manyreactions there are in the network!
Sparse/Dense Realizations
minimizem∑
i ,j=1,i 6=j
δij or minimizem∑
i ,j=1,i 6=j
−δij
0 ≤ [Ak ]ij − εδij0 ≤ −[Ak ]ij + uijδij
δij ∈ {0, 1}i , j = 1, . . . ,m, i 6= j
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Reconsider the network
N : 2A11−→ 2A2
1−→ A1 +A2.
Running the optimization procedure in GLPK gives the followingrealizations:
2A1 2A2
A1+A2
0.10.1
0.1
1.8 0.1
0.45
2A1 2A2
1
0.5
(a) (b)
Figure: Sparse (a) and dense (b) networks which generate the samekinetics as N .
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Reconsider the network
N : 2A11−→ 2A2
1−→ A1 +A2.
Running the optimization procedure in GLPK gives the followingrealizations:
2A1 2A2
A1+A2
0.10.1
0.1
1.8 0.1
0.45
2A1 2A2
1
0.5
(a) (b)
Figure: Sparse (a) and dense (b) networks which generate the samekinetics as N .
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
This is a mixed integer linear programming (MILP) problem whichis known to be NP-hard.
In subsequent papers, further conditions have been imposed on thenetworks (detailed and complex balancing, weak and fullreversibility) [6, 7].
There are still several remaining questions:
1 Can we adapt this to linearly conjugate networks?
2 Can we simplify the requirement for weak reversibility?
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
This is a mixed integer linear programming (MILP) problem whichis known to be NP-hard.
In subsequent papers, further conditions have been imposed on thenetworks (detailed and complex balancing, weak and fullreversibility) [6, 7].
There are still several remaining questions:
1 Can we adapt this to linearly conjugate networks?
2 Can we simplify the requirement for weak reversibility?
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
This is a mixed integer linear programming (MILP) problem whichis known to be NP-hard.
In subsequent papers, further conditions have been imposed on thenetworks (detailed and complex balancing, weak and fullreversibility) [6, 7].
There are still several remaining questions:
1 Can we adapt this to linearly conjugate networks?
2 Can we simplify the requirement for weak reversibility?
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
This is a mixed integer linear programming (MILP) problem whichis known to be NP-hard.
In subsequent papers, further conditions have been imposed on thenetworks (detailed and complex balancing, weak and fullreversibility) [6, 7].
There are still several remaining questions:
1 Can we adapt this to linearly conjugate networks?
2 Can we simplify the requirement for weak reversibility?
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
This is a mixed integer linear programming (MILP) problem whichis known to be NP-hard.
In subsequent papers, further conditions have been imposed on thenetworks (detailed and complex balancing, weak and fullreversibility) [6, 7].
There are still several remaining questions:
1 Can we adapt this to linearly conjugate networks?
2 Can we simplify the requirement for weak reversibility?
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
In order to incorporate linear conjugacy, we need to satisfy
M = Y · Ak = T · Y · Ab.
Linear Conjugacy
Y · Ab = T−1 ·Mm∑i=1
[Ab]ij = 0, j = 1, . . . ,m
[Ab]ij ≥ 0, i , j = 1, . . . ,m, i 6= j[Ab]ii ≤ 0, i = 1, . . . ,mε ≤ cj ≤ 1/ε, j = 1, . . . , nT = diag {c}
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
In order to incorporate linear conjugacy, we need to satisfy
M = Y · Ak = T · Y · Ab.
Linear Conjugacy
Y · Ab = T−1 ·Mm∑i=1
[Ab]ij = 0, j = 1, . . . ,m
[Ab]ij ≥ 0, i , j = 1, . . . ,m, i 6= j[Ab]ii ≤ 0, i = 1, . . . ,mε ≤ cj ≤ 1/ε, j = 1, . . . , nT = diag {c}
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Reconsider the network
N :A1
1−→ A2
2A21−→ 2A1.
/
Running this in GLPK for both a sparse and dense linearlyconjugate network gives
N ′ : A1
1�1/2
2A2
with conjugacy constants c1 = 1, c2 = 1/2.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Reconsider the network
N :A1
1−→ A2
2A21−→ 2A1.
/Running this in GLPK for both a sparse and dense linearlyconjugate network gives
N ′ : A1
1�1/2
2A2
with conjugacy constants c1 = 1, c2 = 1/2.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
QUESTION:
Can we incorporate weak reversibility as a linear constraint in theMILP procedure?
Existing method requires checking for weak reversibility as aseparate step and requires (potentially) multiple MILPoptimizations [7].
It can also only handle dense networks not sparse.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
QUESTION:
Can we incorporate weak reversibility as a linear constraint in theMILP procedure?
Existing method requires checking for weak reversibility as aseparate step and requires (potentially) multiple MILPoptimizations [7].
It can also only handle dense networks not sparse.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
QUESTION:
Can we incorporate weak reversibility as a linear constraint in theMILP procedure?
Existing method requires checking for weak reversibility as aseparate step and requires (potentially) multiple MILPoptimizations [7].
It can also only handle dense networks not sparse.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
ANSWER:
Yes we can!... But we have to be sneaky...
The kinetics matrix Ak of a weakly reversible network has apositive vector in its kernel, i.e.
Ak b = 0, b ∈ Rm>0.
We can introduce bj , j = 1, . . . ,m as decision variables and imposethat bj > 0, but the constraint is non-linear...
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
ANSWER:
Yes we can!... But we have to be sneaky...
The kinetics matrix Ak of a weakly reversible network has apositive vector in its kernel, i.e.
Ak b = 0, b ∈ Rm>0.
We can introduce bj , j = 1, . . . ,m as decision variables and imposethat bj > 0, but the constraint is non-linear...
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
ANSWER:
Yes we can!... But we have to be sneaky...
The kinetics matrix Ak of a weakly reversible network has apositive vector in its kernel, i.e.
Ak b = 0, b ∈ Rm>0.
We can introduce bj , j = 1, . . . ,m as decision variables and imposethat bj > 0, but the constraint is non-linear...
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Define the matrix Ak with entries [Ak ]ij = [Ak ]ij · bj (i.e. multiplyb through Ak).
The constraints are linear in these new variables!
Weak Reversibility
m∑i=1,i 6=j
[Ak ]ij =m∑
i=1,i 6=j
[Ak ]ji , j = 1, . . . ,m
[Ak ]ij ≥ 0, i , j = 1, . . . ,m, i 6= j .
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Define the matrix Ak with entries [Ak ]ij = [Ak ]ij · bj (i.e. multiplyb through Ak).
The constraints are linear in these new variables!
Weak Reversibility
m∑i=1,i 6=j
[Ak ]ij =m∑
i=1,i 6=j
[Ak ]ji , j = 1, . . . ,m
[Ak ]ij ≥ 0, i , j = 1, . . . ,m, i 6= j .
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Define the matrix Ak with entries [Ak ]ij = [Ak ]ij · bj (i.e. multiplyb through Ak).
The constraints are linear in these new variables!
Weak Reversibility
m∑i=1,i 6=j
[Ak ]ij =m∑
i=1,i 6=j
[Ak ]ji , j = 1, . . . ,m
[Ak ]ij ≥ 0, i , j = 1, . . . ,m, i 6= j .
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Example 1: Consider the reaction network N given by
A1 + 2A21−→ 2A1 + 2A2
1−→ 2A1 +A2
A12←− 2A1
1−→ 2A1 +A3
2A1 + 2A31←− A1 + 2A3
1−→ A1 +A2 + 2A3
↓3
A1 +A3.
In terms of network structure, this is a mess!
We can say very little about the dynamics of the mass-actionsystem without directly analysing it.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Example 1: Consider the reaction network N given by
A1 + 2A21−→ 2A1 + 2A2
1−→ 2A1 +A2
A12←− 2A1
1−→ 2A1 +A3
2A1 + 2A31←− A1 + 2A3
1−→ A1 +A2 + 2A3
↓3
A1 +A3.
In terms of network structure, this is a mess!
We can say very little about the dynamics of the mass-actionsystem without directly analysing it.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Example 1: Consider the reaction network N given by
A1 + 2A21−→ 2A1 + 2A2
1−→ 2A1 +A2
A12←− 2A1
1−→ 2A1 +A3
2A1 + 2A31←− A1 + 2A3
1−→ A1 +A2 + 2A3
↓3
A1 +A3.
In terms of network structure, this is a mess!
We can say very little about the dynamics of the mass-actionsystem without directly analysing it.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
There are weakly reversible networks which are linearly conjugateto N (found very quickly with GLPK)!
X1+2X2 2X1+2X2
2X1X1+2X3
4
40025
40
125
X1+2X2 2X1+2X2
2X1X1+2X3 2X1+X2
0.367
13.9 0.926 13.11.35
0.816
13.3 1.35
0.926
0.926
(a) (b)
Figure: Weakly reversible networks which are linearly conjugate to N .The network in (a) is sparse while the network in (b) is dense.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
There are weakly reversible networks which are linearly conjugateto N (found very quickly with GLPK)!
X1+2X2 2X1+2X2
2X1X1+2X3
4
40025
40
125
X1+2X2 2X1+2X2
2X1X1+2X3 2X1+X2
0.367
13.9 0.926 13.11.35
0.816
13.3 1.35
0.926
0.926
(a) (b)
Figure: Weakly reversible networks which are linearly conjugate to N .The network in (a) is sparse while the network in (b) is dense.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Example 2: Consider the chemical reaction network N given by
X1 + 2X21.5−→ X1
N : 2X1 + X21−→ 3X2
X1 + 3X21−→ X1 + X2
1−→ 3X1 + X2.
Again, the network structure does not tell us anything about thedynamics.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Example 2: Consider the chemical reaction network N given by
X1 + 2X21.5−→ X1
N : 2X1 + X21−→ 3X2
X1 + 3X21−→ X1 + X2
1−→ 3X1 + X2.
Again, the network structure does not tell us anything about thedynamics.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Running the optimization problem in GLPK very quickly producesweakly reversible realizations.
X1+2X2 X1+X2
X1+3X2 2X1+X2
(a) X1+2X2 X1+X2
X1+3X2 2X1+X2
(b)
2/3
11/3
2/32/3
2/3
2/32/3
2
150
10
100
500
Figure: Weakly reversible networks which are linearly conjugate to N .The network in (a) is dense while the network in (b) is sparse. The denserealizations has a trivial linear conjugacy while the sparse realization doesnot.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Running the optimization problem in GLPK very quickly producesweakly reversible realizations.
X1+2X2 X1+X2
X1+3X2 2X1+X2
(a) X1+2X2 X1+X2
X1+3X2 2X1+X2
(b)
2/3
11/3
2/32/3
2/3
2/32/3
2
150
10
100
500
Figure: Weakly reversible networks which are linearly conjugate to N .The network in (a) is dense while the network in (b) is sparse. The denserealizations has a trivial linear conjugacy while the sparse realization doesnot.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
SUMMARY:
We can find sparse and dense weakly reversible networks which arelinearly conjugate to a given network in a single MILP step.
/
Areas of future work include:
1 Extend linear conjugacy to non-linear cases and alternatedynamics.
2 Search for applied systems where these results are applicable.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
SUMMARY:
We can find sparse and dense weakly reversible networks which arelinearly conjugate to a given network in a single MILP step.
/ Areas of future work include:
1 Extend linear conjugacy to non-linear cases and alternatedynamics.
2 Search for applied systems where these results are applicable.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
SUMMARY:
We can find sparse and dense weakly reversible networks which arelinearly conjugate to a given network in a single MILP step.
/ Areas of future work include:
1 Extend linear conjugacy to non-linear cases and alternatedynamics.
2 Search for applied systems where these results are applicable.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
SUMMARY:
We can find sparse and dense weakly reversible networks which arelinearly conjugate to a given network in a single MILP step.
/ Areas of future work include:
1 Extend linear conjugacy to non-linear cases and alternatedynamics.
2 Search for applied systems where these results are applicable.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
Special thanks go out to D. Siegel, G. Szederkenyi, M.Gopalkrishnan, C. Pantea, A. Shiu, and the organizers of the SIAM
Conference!
Thanks for coming out!
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks
BackgroundLinearly Conjugate Networks
Computation Approach
Sparse and Dense RealizationsLinear ConjugacyWeak Reversibility as a Linear Constraint
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Linear conjugacy of chemical reaction networks.J. Math. Chem., 49(7):1263–1282, 2011.
Matthew D. Johnston, David Siegel, and Gabor Szederkenyi.
A linear programming approach to weak reversibility and linear conjugacy of chemical reaction networks.Available on the arXiv at arXiv:1107.1659.
Gabor Szederkenyi.
Computing sparse and dense realizations of reaction kinetic systems.J. Math. Chem., 47:551–568, 2010.
Gabor Szederkenyi, Katalin Hangos, and Tamas Peni.
Maximal and minimal realizations of chemical kinetics systems: computation and properties.MATCH Commun. Math. Comput. Chem., 65:309–332, 2011.Available on the arXiv at arxiv:1005.2913.
Gabor Szederkenyi, Katalin Hangos, and Zsolt Tuza.
Finding weakly reversible realizations of chemical reaction networks using optimization.MATCH Commun. Math. Comput. Chem., 67:193–212, 2012.Available on the arXiv at arxiv:1103.4741.
MD Johnston, D Siegel, G Szederkenyi Linear Conjugacy of Chemical Reaction Networks