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Background Linearly Conjugate Networks Computational Approach Determining Weakly Reversible Linearly Conjugate Chemical Reaction Networks with Minimal Deficiency Matthew D. Johnston a , David Siegel a and G. Szederk enyi b a University of

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Background Linearly Conjugate Networks Computational Approach

Determining Weakly Reversible Linearly Conjugate Chemical Reaction Networks with Minimal Deficiency

Matthew D. Johnstona, David Siegela and G. Szederk´ enyib

a University of Waterloo b Hungarian Academy of Sciences

August 7, 2012

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach

1 Background

Objectives Mass-Action Kinetics Chemical Reaction Networks

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach

1 Background

Objectives Mass-Action Kinetics Chemical Reaction Networks

2 Linearly Conjugate Networks

Realizations Linearly Conjugate Networks

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 4

Background 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

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SLIDE 5

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

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

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SLIDE 6

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

An elementary reaction is given by a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O /

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

An elementary reaction is given by a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Species/Reactants

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

An elementary reaction is given by a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Reactant Complex/

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 9

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

An elementary reaction is given by a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Product Complex/

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

An elementary reaction is given by a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Reaction Constant/

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

An elementary reaction is given by a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / A set of simultaneously occurring elementary reactions is called a chemical reaction network and is denoted by N = (S, C, R, k).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

An elementary reaction is given by a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / A set of simultaneously occurring elementary reactions is called a chemical reaction network and is denoted by N = (S, C, R, k). Chemical kinetics is the study of the time evolution of concentrations of such networks as a result of some kinetic assumption (e.g. mass-action, Michaelis-Menten, etc.)

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Significant recent work has been done on the connection between network structure and dynamical behaviour (of mass-action systems, in particular).

CRN with "good" graph CRN with "bad" graph Understood dynamical behaviour! same

dynamics MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Significant recent work has been done on the connection between network structure and dynamical behaviour (of mass-action systems, in particular).

CRN with "good" graph CRN with "bad" graph Understood dynamical behaviour! same

dynamics MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 15

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Significant recent work has been done on the connection between network structure and dynamical behaviour (of mass-action systems, in particular).

CRN with "good" graph CRN with "bad" graph Understood dynamical behaviour! same

dynamics MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 16

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Significant recent work has been done on the connection between network structure and dynamical behaviour (of mass-action systems, in particular).

CRN with "good" graph CRN with "bad" graph Understood dynamical behaviour! same

dynamics MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 17

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Significant recent work has been done on the connection between network structure and dynamical behaviour (of mass-action systems, in particular).

CRN with "good" graph CRN with "bad" graph Understood dynamical behaviour! same

dynamics YES!

We can say something about the dynamical behaviour without having to analyse the model’s governing dynamics!

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

QUESTION #1 Can we find, from within some class of systems with identical qualitative dynamics, a system with “good” network structure?

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

QUESTION #1 Can we find, from within some class of systems with identical qualitative dynamics, a system with “good” network structure? This was the main topic of a series of papers published by G. Szederk´ enyi (and collaborators).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

QUESTION #1 Can we find, from within some class of systems with identical qualitative dynamics, a system with “good” network structure? This was the main topic of a series of papers published by G. Szederk´ enyi (and collaborators). The problem of finding networks can be formulate as a mixed-integer linear programming (MILP) problem [5].

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

QUESTION #1 Can we find, from within some class of systems with identical qualitative dynamics, a system with “good” network structure? This was the main topic of a series of papers published by G. Szederk´ enyi (and collaborators). The problem of finding networks can be formulate as a mixed-integer linear programming (MILP) problem [5]. Algorithms exist for a wide class of structural properties (maximal/minimal number of reactions, weak/full reversibility, specific equilibrium properties, etc.).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Another structural parameter of a chemical reaction network which is particularly strongly related to the dynamics is the deficiency.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

  • scillations/multistability, etc.)

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

  • scillations/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

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the general network N given by N : Ci

k(i,j)

− → Cj, (i, j) ∈ R.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the general network N given by N : Ci

k(i,j)

− → Cj, (i, j) ∈ R. Under mass-action kinetics, this network is governed by dx dt = Y Ak Ψ(x). (1)

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the general network N given by N : Ci

k(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 complex matrix,

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the general network N given by N : Ci

k(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 complex matrix, Ak is the kinetics matrix (keeps track of connections),

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the general network N given by N : Ci

k(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 complex matrix, Ak is the kinetics matrix (keeps track of connections), Ψ(x) is the mass-action vector (with entries Ψi(x) = m

j=1(xj)zij).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the reversible network A1

k(1,2)

⇄

k(2,1)

2A2.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the reversible network A1

k(1,2)

⇄

k(2,1)

2A2. This has the governing dynamics     dx1 dt dx2 dt     = 1 2 −k(1, 2) k(2, 1) k(1, 2) −k(2, 1) x1 x2

2

  • = k(1, 2)

−1 2

  • 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 Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the reversible network A1

k(1,2)

⇄

k(2,1)

2A2. This has the governing dynamics     dx1 dt dx2 dt     = 1 2 −k(1, 2) k(2, 1) k(1, 2) −k(2, 1) x1 x2

2

  • = k(1, 2)

−1 2

  • 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 Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 33

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the reversible network A1

k(1,2)

⇄

k(2,1)

2A2. This has the governing dynamics     dx1 dt dx2 dt     = 1 2 −k(1, 2) k(2, 1) k(1, 2) −k(2, 1) x1 x2

2

  • = k(1, 2)

−1 2

  • 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 Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 34

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the reversible network A1

k(1,2)

⇄

k(2,1)

2A2. This has the governing dynamics     dx1 dt dx2 dt     = 1 2 −k(1, 2) k(2, 1) k(1, 2) −k(2, 1) x1 x2

2

  • = k(1, 2)

−1 2

  • 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 Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 35

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Consider the reversible network A1

k(1,2)

⇄

k(2,1)

2A2. This has the governing dynamics     dx1 dt dx2 dt     = 1 2 −k(1, 2) k(2, 1) k(1, 2) −k(2, 1) x1 x2

2

  • = k(1, 2)

−1 2

  • 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 Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Figure: Previous system with k1 = k2 = 1.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Figure: Previous system with k1 = k2 = 1.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 38

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Figure: Previous system with k1 = k2 = 1.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Chemical reaction networks can be treated as weighted directed graphs G(V , E): C1

k(1,2)

− → C2

k(3,1) տ

ւk(2,3) C3 C4

k(4,5)

⇄

k(5,4)

C5

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 41

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Chemical reaction networks can be treated as weighted directed graphs G(V , E): C1

k(1,2)

− → C2

k(3,1) տ

ւk(2,3) C3 C4

k(4,5)

⇄

k(5,4)

C5 The vertexes are the distinct complexes

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 42

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Chemical reaction networks can be treated as weighted directed graphs G(V , E): C1

k(1,2)

− → C2

k(3,1) տ

ւk(2,3) C3 C4

k(4,5)

⇄

k(5,4)

C5 The vertexes are the distinct complexes, V = C.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 43

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Chemical reaction networks can be treated as weighted directed graphs G(V , E): C1

k(1,2)

− → C2

k(3,1) տ

ւk(2,3) C3 C4

k(4,5)

⇄

k(5,4)

C5 The vertexes are the distinct complexes, V = C. The edges are the reactions

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 44

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Chemical reaction networks can be treated as weighted directed graphs G(V , E): C1

k(1,2)

− → C2

k(3,1) տ

ւk(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.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 45

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

Chemical reaction networks can be treated as weighted directed graphs G(V , E): C1

k(1,2)

− → C2

k(3,1) տ

ւk(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 Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 46

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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SLIDE 47

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

The particular class of networks we have been interested in are weakly reversible networks. C1

k(1,2)

− → C2

k(3,1) տ

ւk(2,3) C3 C4

k(4,5)

⇄

k(5,4)

C5

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 48

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

The particular class of networks we have been interested in are weakly reversible networks. C1

k(1,2)

− → C2

k(3,1) տ

ւk(2,3) C3 C4

k(4,5)

⇄

k(5,4)

C5 / 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

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SLIDE 49

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

The particular class of networks we have been interested in are weakly reversible networks. C1

k(1,2)

− → C2

k(3,1) տ

ւk(2,3) C3 C4

k(4,5)

⇄

k(5,4)

C5 / A network is weakly reversible if a path from one another complex to another implies a path back, e.g. / C1 − → C2

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 50

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

The particular class of networks we have been interested in are weakly reversible networks. C1

k(1,2)

− → C2

k(3,1) տ

ւk(2,3) C3 C4

k(4,5)

⇄

k(5,4)

C5 / A network is weakly reversible if a path from one another complex to another implies a path back, e.g. / C1 − → C2 − → C3 − → C1

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 51

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics Chemical Reaction Networks

The particular class of networks we have been interested in are weakly reversible networks. C1

k(1,2)

− → C2

k(3,1) տ

ւk(2,3) C3 C4

k(4,5)

⇄

k(5,4)

C5 / A network is weakly reversible if a path from one another complex to another implies a path back, e.g. / C1 − → C2 − → C3 − → C1, C4 − → C5 − → C4.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 52

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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SLIDE 53

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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SLIDE 54

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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SLIDE 55

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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SLIDE 56

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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SLIDE 57

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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SLIDE 58

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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SLIDE 59

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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SLIDE 60

Background Linearly Conjugate Networks Computational Approach Objectives Mass-Action Kinetics 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

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Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

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

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SLIDE 62

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

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

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SLIDE 63

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

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

  • r vice-versa.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 64

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

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

  • r 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

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SLIDE 65

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

Consider the set of polynomial differential equations ˙ x1 = −2x2

1 + x2 2

˙ x2 = 2x2

1 − x2 2.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-66
SLIDE 66

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

Consider the set of polynomial differential equations ˙ x1 = −2x2

1 + x2 2

˙ x2 = 2x2

1 − x2 2.

This governs the dynamics of both of the mass-action systems N : 2A1

1

− → 2A2

1

− → A1 + A2 N ′ : 2A1

1

⇄

0.5

2A2.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-67
SLIDE 67

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

Consider the set of polynomial differential equations ˙ x1 = −2x2

1 + x2 2

˙ x2 = 2x2

1 − x2 2.

This governs the dynamics of both of the mass-action systems N : 2A1

1

− → 2A2

1

− → A1 + A2 N ′ : 2A1

1

⇄

0.5

2A2. N ′ has “good” underlying structure...

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-68
SLIDE 68

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

Consider the set of polynomial differential equations ˙ x1 = −2x2

1 + x2 2

˙ x2 = 2x2

1 − x2 2.

This governs the dynamics of both of the mass-action systems N : 2A1

1

− → 2A2

1

− → A1 + A2 N ′ : 2A1

1

⇄

0.5

2A2. N ′ has “good” underlying structure... / ...we know its dynamics...

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-69
SLIDE 69

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

Consider the set of polynomial differential equations ˙ x1 = −2x2

1 + x2 2

˙ x2 = 2x2

1 − x2 2.

This governs the dynamics of both of the mass-action systems N : 2A1

1

− → 2A2

1

− → A1 + A2 N ′ : 2A1

1

⇄

0.5

2A2. 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

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SLIDE 70

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

/ Consider the networks N : A1

k1

− → A2 2A2

k2

− → 2A1 N ′ : A1

˜ k1

⇄

˜ k2

2A2

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-71
SLIDE 71

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

/ Consider the networks N : A1

k1

− → A2 2A2

k2

− → 2A1 = ⇒ dx1 dt = −k1x1 + 2k2x2

2

dx2 dt = k1x1 − 2k2x2

2.

N ′ : A1

˜ k1

⇄

˜ k2

2A2

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-72
SLIDE 72

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

/ Consider the networks N : A1

k1

− → A2 2A2

k2

− → 2A1 = ⇒ dx1 dt = −k1x1 + 2k2x2

2

dx2 dt = k1x1 − 2k2x2

2.

N ′ : A1

˜ k1

⇄

˜ k2

2A2 = ⇒ dy1 dt = −˜ k1y1 + ˜ k2y2

2

dy2 dt = 2˜ k1y1 − 2˜ k2y2

2

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-73
SLIDE 73

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

/ Consider the networks N : A1

k1

− → A2 2A2

k2

− → 2A1 = ⇒ dx1 dt = −k1x1 + 2k2x2

2

dx2 dt = k1x1 − 2k2x2

2.

N ′ : A1

˜ k1

⇄

˜ k2

2A2 = ⇒ dy1 dt = −˜ k1y1 + ˜ k2y2

2

dy2 dt = 2˜ k1y1 − 2˜ k2y2

2

The governing dynamics look very similar (but not identical).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-74
SLIDE 74

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

/ Consider the networks N : A1

k1

− → A2 2A2

k2

− → 2A1 = ⇒ dx1 dt = −k1x1 + 2k2x2

2

dx2 dt = k1x1 − 2k2x2

2.

N ′ : A1

˜ k1

⇄

˜ k2

2A2 = ⇒ dy1 dt = −˜ k1y1 + ˜ k2y2

2

dy2 dt = 2˜ k1y1 − 2˜ k2y2

2

The governing dynamics look very similar (but not identical).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-75
SLIDE 75

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

/ Consider the networks N : A1

k1

− → A2 2A2

k2

− → 2A1 = ⇒ dx1 dt = −k1x1 + 2k2x2

2

dx2 dt = k1x1 − 2k2x2

2.

N ′ : A1

˜ k1

⇄

˜ k2

2A2 = ⇒ dy1 dt = −˜ k1y1 + ˜ k2y2

2

dy2 dt = 2˜ k1y1 − 2˜ k2y2

2

The governing dynamics look very similar (but not identical).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 76

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

y1 y2 x1 x2

N: N':

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 77

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

y1 y2 x1 x2

N: N':

We can relate these systems with the transformation x1 = 2y1, x2 = y2 (a linear transformation)!

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 78

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

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

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SLIDE 79

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

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

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SLIDE 80

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

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

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SLIDE 81

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

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

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SLIDE 82

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

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 Ak corresponding to N and suppose that 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 ′ with kinetics matrix A′

k = Ab · diag {Ψ(c)} .

(3)

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 83

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

N : A1

k1

− → A2 2A2

k2

− → 2A1 = ⇒ dx1 dt = −k1x1 + 2k2x2

2

dx2 dt = k1x1 − 2k2x2

2.

N ′ : A1

˜ k1

⇄

˜ k2

2A2 = ⇒ dy1 dt = −˜ k1y1 + ˜ k2y2

2

dy2 dt = 2˜ k1y1 − 2˜ k2y2

2

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-84
SLIDE 84

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

N : A1

k1

− → A2 2A2

k2

− → 2A1 = ⇒ dx1 dt = −k1x1 + 2k2x2

2

dx2 dt = k1x1 − 2k2x2

2.

N ′ : A1

˜ k1

⇄

˜ k2

2A2 = ⇒ dy1 dt = −˜ k1y1 + ˜ k2y2

2

dy2 dt = 2˜ k1y1 − 2˜ k2y2

2

N ′ has “good” structure...

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-85
SLIDE 85

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

N : A1

k1

− → A2 2A2

k2

− → 2A1 = ⇒ dx1 dt = −k1x1 + 2k2x2

2

dx2 dt = k1x1 − 2k2x2

2.

N ′ : A1

˜ k1

⇄

˜ k2

2A2 = ⇒ dy1 dt = −˜ k1y1 + ˜ k2y2

2

dy2 dt = 2˜ k1y1 − 2˜ k2y2

2

N ′ has “good” structure, so it has known dynamical behaviour...

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-86
SLIDE 86

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

N : A1

k1

− → A2 2A2

k2

− → 2A1 = ⇒ dx1 dt = −k1x1 + 2k2x2

2

dx2 dt = k1x1 − 2k2x2

2.

N ′ : A1

˜ k1

⇄

˜ k2

2A2 = ⇒ dy1 dt = −˜ k1y1 + ˜ k2y2

2

dy2 dt = 2˜ k1y1 − 2˜ k2y2

2

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

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SLIDE 87

Background Linearly Conjugate Networks Computational Approach Realizations Linearly Conjugate Networks

N : A1

k1

− → A2 2A2

k2

− → 2A1 = ⇒ dx1 dt = −k1x1 + 2k2x2

2

dx2 dt = k1x1 − 2k2x2

2.

N ′ : A1

˜ k1

⇄

˜ k2

2A2 = ⇒ dy1 dt = −˜ k1y1 + ˜ k2y2

2

dy2 dt = 2˜ k1y1 − 2˜ k2y2

2

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

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SLIDE 88

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency 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

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SLIDE 89

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

PROBLEM: Suppose we are given a single mass-action system N, e.g. N : 2A1

1

− → 2A2

1

− → A1 + A2. 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

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SLIDE 90

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

PROBLEM: Suppose we are given a single mass-action system N, e.g. N : 2A1

1

− → 2A2

1

− → A1 + A2. 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

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SLIDE 91

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

PROBLEM: Suppose we are given a single mass-action system N, e.g. N : 2A1

1

− → 2A2

1

− → A1 + A2. 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

slide-92
SLIDE 92

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency 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

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SLIDE 93

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

For dynamical equivalence, G. Szederk´ enyi placed this problem within 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 = j [Ak]ii ≤ 0, i = 1, . . . , m.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 94

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

Reconsider the network N : 2A1

1

− → 2A2

1

− → A1 + A2.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-95
SLIDE 95

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

Reconsider the network N : 2A1

1

− → 2A2

1

− → A1 + A2. Running a MILP procedure in GLPK gives the following alternative realizations: 2A1 2A2 A1+A2

0.1 0.1 0.1 1.8 0.1 0.45

2A1 2A2

1 0.5

(a) (b)

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

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SLIDE 96

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency 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

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SLIDE 97

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency 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

slide-98
SLIDE 98

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency 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

slide-99
SLIDE 99

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency 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

slide-100
SLIDE 100

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency 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

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SLIDE 101

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency 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? (Solved!) 2 Can we handle weak reversibility? (Solved!) 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

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SLIDE 102

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We know that networks with lower deficiency (in particular zero and one) are generally more desireable.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-103
SLIDE 103

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We know that networks with lower deficiency (in particular zero and one) are generally more desireable. We can extend this framework to find a mass-action system for which the underlying network is weakly reversible and has minimal deficiency!

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-104
SLIDE 104

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We know that networks with lower deficiency (in particular zero and one) are generally more desireable. We can extend this framework to find a mass-action system for which the underlying network is weakly reversible and has minimal deficiency! General flavour of the extension:

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 105

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We know that networks with lower deficiency (in particular zero and one) are generally more desireable. We can extend this framework to find a mass-action system for which the underlying network is weakly reversible and has minimal deficiency! General flavour of the extension:

1 Force linear conjugacy

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 106

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We know that networks with lower deficiency (in particular zero and one) are generally more desireable. We can extend this framework to find a mass-action system for which the underlying network is weakly reversible and has minimal deficiency! General flavour of the extension:

1 Force linear conjugacy 2 Partition the complexes

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-107
SLIDE 107

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We know that networks with lower deficiency (in particular zero and one) are generally more desireable. We can extend this framework to find a mass-action system for which the underlying network is weakly reversible and has minimal deficiency! General flavour of the extension:

1 Force linear conjugacy 2 Partition the complexes 3 Correspond the partitions to linkage classes

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-108
SLIDE 108

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We know that networks with lower deficiency (in particular zero and one) are generally more desireable. We can extend this framework to find a mass-action system for which the underlying network is weakly reversible and has minimal deficiency! General flavour of the extension:

1 Force linear conjugacy 2 Partition the complexes 3 Correspond the partitions to linkage classes 4 Remove redundant partitions

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

slide-109
SLIDE 109

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We know that networks with lower deficiency (in particular zero and one) are generally more desireable. We can extend this framework to find a mass-action system for which the underlying network is weakly reversible and has minimal deficiency! General flavour of the extension:

1 Force linear conjugacy 2 Partition the complexes 3 Correspond the partitions to linkage classes 4 Remove redundant partitions 5 Maximize the number of linkage classes

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 110

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

The mixed-integer linear programming framework was extended to include linear conjugacy in [4].

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 111

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

The mixed-integer linear programming framework was extended to include linear conjugacy in [4]. Linear Conjugacy Y · Ab = T −1 · M

m

  • i=1

[Ab]ij = 0, j = 1, . . . , m [Ab]ij ≥ 0, i, j = 1, . . . , m, i = j [Ab]ii ≤ 0, i = 1, . . . , m ǫ ≤ cj ≤ 1/ǫ, j = 1, . . . , n T = diag {c}

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can assign the ith complex to the kth partition (the γik’s) and count the number of partitions (the θk’s) with the following:

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 113

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can assign the ith complex to the kth partition (the γik’s) and count the number of partitions (the θk’s) with the following: Complete Partition

m−s

  • k=1

γik = 1, i = 1, . . . , m

m

  • i=1

γik − ǫθk ≥ 0, k = 1, . . . , m − s −

m

  • k=1

γik + 1 ǫ θk ≥ 0, k = 1, . . . , m − s γik ∈ {0, 1} , i = 1, . . . , m, k = 1, . . . , m − s θk ∈ [0, 1] , k = 1, . . . , m − s.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 114

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can assign the ith complex to the kth partition (the γik’s) and count the number of partitions (the θk’s) with the following: Complete Partition

m−s

  • k=1

γik = 1, i = 1, . . . , m

m

  • i=1

γik − ǫθk ≥ 0, k = 1, . . . , m − s −

m

  • k=1

γik + 1 ǫ θk ≥ 0, k = 1, . . . , m − s γik ∈ {0, 1} , i = 1, . . . , m, k = 1, . . . , m − s θk ∈ [0, 1] , k = 1, . . . , m − s.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 115

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can assign the ith complex to the kth partition (the γik’s) and count the number of partitions (the θk’s) with the following: Complete Partition

m−s

  • k=1

γik = 1, i = 1, . . . , m

m

  • i=1

γik − ǫθk ≥ 0, k = 1, . . . , m − s −

m

  • k=1

γik + 1 ǫ θk ≥ 0, k = 1, . . . , m − s γik ∈ {0, 1} , i = 1, . . . , m, k = 1, . . . , m − s θk ∈ [0, 1] , k = 1, . . . , m − s.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 116

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can correspond the partitions to linkage classes with:

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 117

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can correspond the partitions to linkage classes with: Correpondence with Linkage Classes

m

  • l=1

l=i

Φil =

m

  • l=1

l=i

Φli Φij ≤ 1 ǫ (γik − γjk + 1) Φij ≥ ǫ[Ab]ij Φij ≤ 1 ǫ [Ab]ij i, j = 1, . . . , m, i = j, k = 1, . . . , m − s.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can correspond the partitions to linkage classes with: Correpondence with Linkage Classes

m

  • l=1

l=i

Φil =

m

  • l=1

l=i

Φli (Weak Reversibility) Φij ≤ 1 ǫ (γik − γjk + 1) Φij ≥ ǫ[Ab]ij Φij ≤ 1 ǫ [Ab]ij i, j = 1, . . . , m, i = j, k = 1, . . . , m − s.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can correspond the partitions to linkage classes with: Correpondence with Linkage Classes

m

  • l=1

l=i

Φil =

m

  • l=1

l=i

Φli Φij ≤ 1 ǫ (γik − γjk + 1) Φij ≥ ǫ[Ab]ij (No Cross Φij ≤ 1 ǫ [Ab]ij Reactions) i, j = 1, . . . , m, i = j, k = 1, . . . , m − s.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can remove redundant partition structures with:

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can remove redundant partition structures with: Unique Partition

i−1

  • j=1

γjk ≥

m−s

  • l=k+1

γil, i = 1, . . . , m, k = 1, . . . , m − s, k ≤ i.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

We can remove redundant partition structures with: Unique Partition

i−1

  • j=1

γjk ≥

m−s

  • l=k+1

γil, i = 1, . . . , m, k = 1, . . . , m − s, k ≤ i. Lastly, we can minimize the deficiency with: minimize

m−1

  • k=1

−θk.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

Consider the following chemical reaction network:

2T100 2T010 2T001 T100+T010 T100+T001 T010+T001

Figure: Network of singularly bound enzymes with three binding sites. Interactions between the enzymes allows transfer of substrates from one binding site to another.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

This network is weakly reversible and deficiency three (not amenable to most deficiency results).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

This network is weakly reversible and deficiency three (not amenable to most deficiency results). The Chemical Reaction Toolbox gives some useful results (e.g. exhibits multistationarity - i.e. multiple positive steady states - for some rate constant values)

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

This network is weakly reversible and deficiency three (not amenable to most deficiency results). The Chemical Reaction Toolbox gives some useful results (e.g. exhibits multistationarity - i.e. multiple positive steady states - for some rate constant values) QUESTION: Do there exist linearly conjugate systems for which the underlying network has a lower deficiency and, if so, what are the rate constant values?

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example 2T100 2T010 2T001 T100+T010 T100+T001 T010+T001

1 1 2 2 3 3 4 4 4 4 5 5 5 5 6 6 6 6

2T100

2.5 2 2 2.5

2T010

3

2T001 T100+T010 T100+T001 T010+T001

3 4 6 8 7 Linearly conjugate 3 2

(a)

2T100 2T010 2T001 T100+T010 T100+T001 T010+T001

5 4 4 6 5 6 5 6 2 1 4 3 1 6 5 3 4 2

2T100 2T010

5

2T001 T100+T010 T100+T001 T010+T001

4 Linearly conjugate 2 3 6 1

(b)

Figure: Weakly reversible networks which are dynamically equivalent to the one contained in Figure 3. The network in (a) has a deficiency of two while the network in (b) has a deficiency of one.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

Running the algorithm in GLPK gives an answer very quickly (tenths of seconds).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

Running the algorithm in GLPK gives an answer very quickly (tenths of seconds). The deficiency one system is amenable to deficiency one algorithms (the system has a unique positive equilibrium state in each linear invariant space of the system).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

Running the algorithm in GLPK gives an answer very quickly (tenths of seconds). The deficiency one system is amenable to deficiency one algorithms (the system has a unique positive equilibrium state in each linear invariant space of the system). It should be noted that the rate constants of the original system need to be specified.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

SUMMARY: We can find the weakly reversible system within a particular class

  • f systems with identical qualitative dynamics which has the

minimal deficiency. /

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

SUMMARY: We can find the weakly reversible system within a particular class

  • f systems with identical qualitative dynamics which has the

minimal deficiency. / Areas of future work include:

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

SUMMARY: We can find the weakly reversible system within a particular class

  • f systems with identical qualitative dynamics which has the

minimal deficiency. / Areas of future work include:

1 Expand scope to search through rate constant choices for

both the initial and target network.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 134

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

SUMMARY: We can find the weakly reversible system within a particular class

  • f systems with identical qualitative dynamics which has the

minimal deficiency. / Areas of future work include:

1 Expand scope to search through rate constant choices for

both the initial and target network.

2 Extend linear conjugacy to non-linear transformations and

alternate kinetic assumptions.

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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SLIDE 135

Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

SUMMARY: We can find the weakly reversible system within a particular class

  • f systems with identical qualitative dynamics which has the

minimal deficiency. / Areas of future work include:

1 Expand scope to search through rate constant choices for

both the initial and target network.

2 Extend linear conjugacy to non-linear transformations and

alternate kinetic assumptions.

3 Apply to reaction networks arising in specific applications

(especially biological examples).

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example

Special thanks go out to D. Siegel, G. Szederk´ enyi, and the

  • rganizers of the SIAM Conference on the Life Sciences!

Thanks for coming out!

MD Johnston, D Siegel, G Szederk´ enyi SIAM Life Sciences 2012

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Background Linearly Conjugate Networks Computational Approach Generating Realizations Minimizing the Deficiency Example Gheorghe Craciun and Casian Pantea. Identifiability of chemical reaction networks.

  • J. Math Chem., 44(1):244–259, 2008.

Fritz Horn and Roy Jackson. General mass action kinetics.

  • Arch. Ration. Mech. Anal., 47:187–194, 1972.

Matthew D. Johnston and David Siegel. Linear conjugacy of chemical reaction networks.

  • J. Math. Chem., 49(7):1263–1282, 2011.

Matthew D. Johnston, David Siegel, and G´ abor Szederk´ enyi. A linear programming approach to weak reversibility and linear conjugacy of chemical reaction networks. Available on the arXiv at arXiv:1107.1659. Gabor Szederk´ enyi. Computing sparse and dense realizations of reaction kinetic systems.

  • J. Math. Chem., 47:551–568, 2010.

Gabor Szederk´ enyi, Katalin Hangos, and Tamas P´ eni. 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 Szederk´ enyi, 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 Szederk´ enyi SIAM Life Sciences 2012