Linear Conjugacy of Chemical Reaction Networks Linearly Conjugate Networks Computation Approach...

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

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

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.

2H2 + O2k−→ 2H2O

Species/Reactants

2H2 + O2k−→ 2H2O

Reactant Complex/

2H2 + O2k−→ 2H2O

Product Complex/

2H2 + O2k−→ 2H2O

Reaction Constant/

2H2 + O2k−→ 2H2O

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!

dynamics

We can say something about the dynamics without having toanalyse the governing set of differential equations!

dynamics

dynamicsYES!

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).

QUESTION #1:

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.

QUESTION #2:

Consider the general network N given by

N : Cik(i ,j)−→ Cj , (i , j) ∈ R.

Under mass-action kinetics, this network is governed by

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 ).

N : Cik(i ,j)−→ Cj , (i , j) ∈ R.

dt= Y Ak Ψ(x). (1)

∏mj=1(xj)

zij ).

N : Cik(i ,j)−→ Cj , (i , j) ∈ R.

dt= Y Ak Ψ(x). (1)

∏mj=1(xj)

zij ).

N : Cik(i ,j)−→ Cj , (i , j) ∈ R.

dt= Y Ak Ψ(x). (1)

∏mj=1(xj)

zij ).

N : Cik(i ,j)−→ Cj , (i , j) ∈ R.

dt= Y Ak Ψ(x). (1)

∏mj=1(xj)

zij ).

Consider the reversible network

k(1,2)

�k(2,1)

This has the governing dynamicsdx1

[1 00 2

] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)

= k(1, 2)

[−12

]x1 + k(2, 1)

[1−2

where x1 and x2 are the concentrations of A1 and A2 respectively.

k(1,2)

�k(2,1)

[1 00 2

] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)

= k(1, 2)

[−12

]x1 + k(2, 1)

[1−2

k(1,2)

�k(2,1)

[1 00 2

] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)

= k(1, 2)

[−12

]x1 + k(2, 1)

[1−2

k(1,2)

�k(2,1)

[1 00 2

] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)

= k(1, 2)

[−12

]x1 + k(2, 1)

[1−2

k(1,2)

�k(2,1)

[1 00 2

] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)

= k(1, 2)

[−12

]x1 + k(2, 1)

[1−2

k(1,2)

�k(2,1)

[1 00 2

] [−k(1, 2) k(2, 1)k(1, 2) −k(2, 1)

= k(1, 2)

[−12

]x1 + k(2, 1)

[1−2

Figure: Previous system with k1 = k2 = 1.

Chemical reaction networks can be treated as weighted directedgraphs G(V ,E ):

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

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.

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

The vertexes are the distinct complexes

, V = C.

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

The edges are the reactions

, E = R.

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

The particular class of networks I have been interested in areweakly reversible networks.

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

/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!

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

/ C1 −→ C2 −→ C3 −→ C1, C4 −→ C5 −→ C4.

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

/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.

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

/ C1 −→ C2

−→ C3 −→ C1, C4 −→ C5 −→ C4.

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

/ C1 −→ C2 −→ C3 −→ C1

, C4 −→ C5 −→ C4.

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

/ C1 −→ C2 −→ C3 −→ C1, C4 −→ C5 −→ C4.

C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

k(4,5)

�k(5,4)

/ C1 −→ C2 −→ C3 −→ C1, C4 −→ C5 −→ C4.

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!

Definition

Consider the set of polynomial differential equations

x1 = −2x21 + x2

x2 = 2x21 − x2

This governs the dynamics of both of the networks

N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5

N ′ has “good” structure.../ ...we know its dynamics.../ ... so we know the dynamics of N as well.

x1 = −2x21 + x2

x2 = 2x21 − x2

N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5

x1 = −2x21 + x2

x2 = 2x21 − x2

N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5

N ′ has “good” structure...

/ ...we know its dynamics.../ ... so we know the dynamics of N as well.

x1 = −2x21 + x2

x2 = 2x21 − x2

N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5

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

/ ... so we know the dynamics of N as well.

x1 = −2x21 + x2

x2 = 2x21 − x2

N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5

/ Consider the networks

k1−→ A2

2A2k2−→ 2A1

dt= −k1x1 + 2k2x

dt= k1x1 − 2k2x

N ′ : A1

dt= −k1y1 + k2y

dt= 2k1y1 − 2k2y

These networks look very similar (but not identical).

k1−→ A2

2A2k2−→ 2A1

dt= −k1x1 + 2k2x

dt= k1x1 − 2k2x

N ′ : A1

dt= −k1y1 + k2y

dt= 2k1y1 − 2k2y

k1−→ A2

2A2k2−→ 2A1

dt= −k1x1 + 2k2x

dt= k1x1 − 2k2x

N ′ : A1

2A2 =⇒

dt= −k1y1 + k2y

dt= 2k1y1 − 2k2y

k1−→ A2

2A2k2−→ 2A1

dt= −k1x1 + 2k2x

dt= k1x1 − 2k2x

N ′ : A1

2A2 =⇒

dt= −k1y1 + k2y

dt= 2k1y1 − 2k2y

k1−→ A2

2A2k2−→ 2A1

dt= −k1x1 + 2k2x

dt= k1x1 − 2k2x

N ′ : A1

2A2 =⇒

dt= −k1y1 + k2y

dt= 2k1y1 − 2k2y

k1−→ A2

2A2k2−→ 2A1

dt= −k1x1 + 2k2x

dt= k1x1 − 2k2x

N ′ : A1

2A2 =⇒

dt= −k1y1 + k2y

dt= 2k1y1 − 2k2y

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)!

N: N':

QUESTION:

N: N':

QUESTION:

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).

Definition

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)

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)

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.

PROBLEM:

N : 2A11−→ 2A2

1−→ A1 +A2,

PROBLEM:

N : 2A11−→ 2A2

1−→ A1 +A2,

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.

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.

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.

1 For δij = 0 we have [Ak ]ij = 0

=⇒ reaction is ‘off’.

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.

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

minimizem∑

i ,j=1,i 6=j

−δij

δij ∈ {0, 1}i , j = 1, . . . ,m, i 6= j

minimizem∑

i ,j=1,i 6=j

−δij

δij ∈ {0, 1}i , j = 1, . . . ,m, i 6= j

minimizem∑

i ,j=1,i 6=j

−δij

δij ∈ {0, 1}i , j = 1, . . . ,m, i 6= j

Reconsider the network

N : 2A11−→ 2A2

1−→ A1 +A2.

Running the optimization procedure in GLPK gives the followingrealizations:

2A1 2A2

0.10.1

1.8 0.1

2A1 2A2

(a) (b)

Figure: Sparse (a) and dense (b) networks which generate the samekinetics as N .

N : 2A11−→ 2A2

1−→ A1 +A2.

Running the optimization procedure in GLPK gives the followingrealizations:

2A1 2A2

0.10.1

1.8 0.1

2A1 2A2

(a) (b)

Figure: Sparse (a) and dense (b) networks which generate the samekinetics as N .

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?

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}

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}

1−→ A2

2A21−→ 2A1.

Running this in GLPK for both a sparse and dense linearlyconjugate network gives

N ′ : A1

1�1/2

with conjugacy constants c1 = 1, c2 = 1/2.

1−→ A2

2A21−→ 2A1.

/Running this in GLPK for both a sparse and dense linearlyconjugate network gives

N ′ : A1

1�1/2

with conjugacy constants c1 = 1, c2 = 1/2.

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.

QUESTION:

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...

ANSWER:

Ak b = 0, b ∈ Rm>0.

ANSWER:

Ak b = 0, b ∈ Rm>0.

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 .

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 .

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 .

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

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.

A1 + 2A21−→ 2A1 + 2A2

1−→ 2A1 +A2

A12←− 2A1

1−→ 2A1 +A3

2A1 + 2A31←− A1 + 2A3

1−→ A1 +A2 + 2A3

A1 +A3.

A1 + 2A21−→ 2A1 + 2A2

1−→ 2A1 +A2

A12←− 2A1

1−→ 2A1 +A3

2A1 + 2A31←− A1 + 2A3

1−→ A1 +A2 + 2A3

A1 +A3.

There are weakly reversible networks which are linearly conjugateto N (found very quickly with GLPK)!

X1+2X2 2X1+2X2

2X1X1+2X3

X1+2X2 2X1+2X2

2X1X1+2X3 2X1+X2

13.9 0.926 13.11.35

13.3 1.35

(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.

There are weakly reversible networks which are linearly conjugateto N (found very quickly with GLPK)!

X1+2X2 2X1+2X2

2X1X1+2X3

X1+2X2 2X1+2X2

2X1X1+2X3 2X1+X2

13.9 0.926 13.11.35

13.3 1.35

(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.

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.

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.

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

2/32/3

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.

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

2/32/3

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.

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.

SUMMARY:

/ Areas of future work include:

SUMMARY:

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!

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 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.

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