Doctoral Dissertation: Topics in Chemical Reaction Modeling · Doctoral Dissertation: Topics in Chemical Reaction Modeling Matthew D. Johnston Department of Applied Mathematics University

BackgroundOriginal Results

Conclusions and Future Work

Doctoral Dissertation:Topics in Chemical Reaction Modeling

Matthew D. JohnstonDepartment of Applied Mathematics

University of Waterloo

December 9, 2011

MD Johnston Topics in Chemical Reaction Modeling



1 BackgroundChemical Reaction NetworksWeakly Reversible NetworksComplex Balanced Networks

2 Original ResultsLinearization of Complex Balanced NetworksGlobal Attractor ConjectureConjugacy of Chemical Reaction Networks

3 Conclusions and Future Work
















Chemical Reaction NetworksWeakly Reversible NetworksComplex Balanced Networks








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.

Has connections with, and applications to, industrial chemistry,pharmaceutics, systems biology, gene regulation, enzyme kinetics,dynamical systems, graph theory, algebraic geometry, stochastics,the study of polynomial differential equations, and many otherareas....






2H2 + O2k−→ 2H2O

Species/Reactants








2H2 + O2k−→ 2H2O

Reactant Complex/








2H2 + O2k−→ 2H2O

Product Complex/








2H2 + O2k−→ 2H2O

Reaction Constant/








2H2 + O2k−→ 2H2O








2H2 + O2k−→ 2H2O


Has connections with, and applications to, industrial chemistry

,pharmaceutics, systems biology, gene regulation, enzyme kinetics,dynamical systems, graph theory, algebraic geometry, stochastics,the study of polynomial differential equations, and many otherareas....






2H2 + O2k−→ 2H2O


Has connections with, and applications to, industrial chemistry,pharmaceutics

, systems biology, gene regulation, enzyme kinetics,dynamical systems, graph theory, algebraic geometry, stochastics,the study of polynomial differential equations, and many otherareas....






2H2 + O2k−→ 2H2O


Has connections with, and applications to, industrial chemistry,pharmaceutics, systems biology

, gene regulation, enzyme kinetics,dynamical systems, graph theory, algebraic geometry, stochastics,the study of polynomial differential equations, and many otherareas....






2H2 + O2k−→ 2H2O







Consider the general network N given by

Ciki−→ C′i , i = 1, . . . , r .

Under mass-action kinetics, this network is governed by thesystem of autonomous, polynomial, ordinary differential equations

x =r∑

i=1

ki (z′i − zi )xzi . (1)

Sum over the i = 1, . . . , r reactions.

ki is the rate constant for the i th reaction.

(z′i − zi ) is the reaction vector for the i th reaction.

xzi =∏m

j=1 xzijj is the mass-action term for the i th reaction.






Ciki−→ C′i , i = 1, . . . , r .


x =r∑

i=1

ki (z′i − zi )xzi . (1)




xzi =∏m







Ciki−→ C′i , i = 1, . . . , r .


x =r∑

i=1

ki (z′i − zi )xzi . (1)




xzi =∏m







Ciki−→ C′i , i = 1, . . . , r .


x =r∑

i=1

ki (z′i − zi )xzi . (1)




xzi =∏m







Ciki−→ C′i , i = 1, . . . , r .


x =r∑

i=1

ki (z′i − zi )xzi . (1)




xzi =∏m







Ciki−→ C′i , i = 1, . . . , r .


x =r∑

i=1

ki (z′i − zi )xzi . (1)




xzi =∏m






Consider the reversible system

A1

k1

�k2

2A2.

This has the governing dynamics[x1

x2

]= k1

[−12

]x1 + k2

[1−2

]x2

2 ,

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






A1

k1

�k2

2A2.


x2

]= k1

[−12

]x1 + k2

[1−2

]x2

2 ,







A1

k1

�k2

2A2.


x2

]= k1

[−12

]x1 + k2

[1−2

]x2

2 ,







A1

k1

�k2

2A2.


x2

]= k1

[−12

]x1 + k2

[1−2

]x2

2 ,







A1

k1

�k2

2A2.


x2

]= k1

[−12

]x1 + k2

[1−2

]x2

2 ,







A1

k1

�k2

2A2.


x2

]= k1

[−12

]x1 + k2

[1−2

]x2

2 ,






Figure: Previous system with k1 = k2 = 1.















The dynamical behaviour of chemical reaction networks is heavilydependent on the structure of its reaction graph.

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.







C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

C3

C4

k(4,5)

�k(5,4)

C5








C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

C3

C4

k(4,5)

�k(5,4)

C5








C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

C3

C4

k(4,5)

�k(5,4)

C5








C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

C3

C4

k(4,5)

�k(5,4)

C5








C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

C3

C4

k(4,5)

�k(5,4)

C5








C1k(1,2)−→ C2

k(3,1) ↖ ↙k(2,3)

C3

C4

k(4,5)

�k(5,4)

C5






An important subset of weakly reversible networks are complexbalanced networks.

A network is complex balanced at x∗ if

n∑j=1

k(j , i)(x∗)zj = (x∗)zi

n∑j=1

k(i , j), for i = 1, . . . , n.

Theorem (Lemma 4C and Theorem 6A, [1])

If a chemical reaction network is complex balanced, then thereexists within each invariant space of the network a unique positiveequilibrium concentration, and this equilibrium concentration islocally asymptotically stable relative to that invariant space.







n∑j=1

k(j , i)(x∗)zj = (x∗)zi

n∑j=1

k(i , j), for i = 1, . . . , n.









n∑j=1

k(j , i)(x∗)zj = (x∗)zi

n∑j=1

k(i , j), for i = 1, . . . , n.









n∑j=1

k(j , i)(x∗)zj = (x∗)zi

n∑j=1

k(i , j), for i = 1, . . . , n.









n∑j=1

k(j , i)(x∗)zj = (x∗)zi

n∑j=1

k(i , j), for i = 1, . . . , n.






Linearization of Complex Balanced NetworksGlobal Attractor ConjectureConjugacy of Chemical Reaction Networks








Linearization of Complex Balanced Network

The proof of Theorem 1 was dependent on the Lyapunov function

L(x) =m∑i=1

xi [ln(xi )− ln(x∗i )− 1] + x∗i

where x∗ ∈ Rm>0 is a positive equilibrium concentration.

For complex balanced networks, it can be shown thatd

dtL(x) = ∇L(x) · f(x) < 0 for all x ∈ Rm

>0 such that f(x) 6= 0.

This means that all trajectories converge locally to theirrespective equilibrium concentrations!







L(x) =m∑i=1

xi [ln(xi )− ln(x∗i )− 1] + x∗i












L(x) =m∑i=1

xi [ln(xi )− ln(x∗i )− 1] + x∗i










This problem has not been attempted by the method oflinearization about equilibrium concentrations before.

For the method of linearization, we consider the decompositionf(x) = Df(x∗)(x− x∗) + O((x− x∗)2) and the correspondinglinear problem

dy

dt= Ay

where y = x− x∗ and A = Df(x∗).

This method has the advantage of guaranteeing local exponentialconvergence to x∗ if Df(x∗) has non-degenerate eigenvalues withnegative real part.







dy

dt= Ay









dy

dt= Ay







Theorem

A complex balanced mass-action system satisfies the followingproperties about any positive equilibrium concentration x∗ ∈ Rm

>0:

1 the local stable manifold W sloc coincides locally with the

compatible linear invariant space; and

2 the local centre manifold W cloc coincides locally with the

tangent plane to the equilibrium set at x∗; and

3 for any M > 0 satisfying

max {Re(λi ) | Re(λi ) < 0} < −M < 0

there exists a k ≥ 1 such that

‖x(t)− x∗‖ ≤ ke−Mt‖x0 − x∗‖, ∀ t ≥ 0

for all x0 sufficiently close to x∗ in the compatible linearinvariant space.





Theorem


>0:


compatible linear invariant space

; and




max {Re(λi ) | Re(λi ) < 0} < −M < 0


‖x(t)− x∗‖ ≤ ke−Mt‖x0 − x∗‖, ∀ t ≥ 0






Theorem


>0:




tangent plane to the equilibrium set at x∗

; and


max {Re(λi ) | Re(λi ) < 0} < −M < 0


‖x(t)− x∗‖ ≤ ke−Mt‖x0 − x∗‖, ∀ t ≥ 0






Theorem


>0:






max {Re(λi ) | Re(λi ) < 0} < −M < 0


‖x(t)− x∗‖ ≤ ke−Mt‖x0 − x∗‖, ∀ t ≥ 0






Consider the network

2A1 +A21/2−→ 3A1

1/8 ↑ ↓ 1

3A2 ←−1/4A1 + 2A2.

This system is governed by the dynamics

dx1

dt= (x2 − 2x1)

(1

4x2

2 +1

4x1x2 + x2

1

)dx2

dt= (2x1 − x2)

(1

4x2

2 +1

4x1x2 + x2

1

).





Consider the network

2A1 +A21/2−→ 3A1

1/8 ↑ ↓ 1

3A2 ←−1/4A1 + 2A2.

This system is governed by the dynamics

dx1

dt= (x2 − 2x1)

(1

4x2

2 +1

4x1x2 + x2

1

)dx2

dt= (2x1 − x2)

(1

4x2

2 +1

4x1x2 + x2

1

).





Figure: Vector field plot of previous system. Invariant spaces satisfyx1 + x2 = constant, and equilibria are along the line x2 = 2x1.





Consider the equilibrium concentration x∗1 = 1/2, x∗2 = 1.

This network has the Jacobian

Df(x∗) =

[−5

458

54 −5

8

].

This matrix has the eigenvalue/eigenvector pairs λ1 = −15/8,v1 = [−1 1]T , and λ2 = 0, v2 = [1 2]T .

As expected, the local stable manifold is tangent to the invariantspace x1 + x2 = constant, and the local centre manifold istangent to the equilibrium set x2 = 2x1!







Df(x∗) =

[−5

458

54 −5

8

].









Df(x∗) =

[−5

458

54 −5

8

].









Df(x∗) =

[−5

458

54 −5

8

].







t

||x(t) - x*||ke-Mt||x0 - x*||

Figure: The exponential bound is shown with values M = 1.85 andk = 1.25. For an initial condition chosen sufficiently close to x∗, we cansee that the correspondence between the convergence of x(t) to x∗ andthe upper bound is nearly exact.





Global Attractor Conjecture

A significant amount of research has been conducted on thequestion of persistence of chemical reaction networks.

Persistence is the property that if all species are initially presentthen none will tend toward zero.

This is a very important property of networks! (e.g. Chemicalreactors using up reactants, biological circuits losing intermediates,etc.)





















Reconsider complex balanced networks: it is known that

d



QUESTION:

Is this sufficient to show x∗ is globally asymptotically stablerelative to its corresponding linear invariant space?

ANSWER:

Only if the network is also persistent (since L(x) is not radiallyunbounded with respect to Rm

>0).






d



QUESTION:


ANSWER:


>0).






d



QUESTION:


ANSWER:


>0).





One approach to persistence is identifying certain special regionsof the boundary and showing they must repel trajectories.

The main result of [3] is the following:

Theorem (Theorem 3.7, [3])

Consider a mass-action system with bounded solutions. Supposethat every semilocking set I is weakly dynamicallynonemptiable. Then the system is persistent.

Weak dynamical non-emptiability is a technical condition and is ageneralization of dynamical non-emptiability introduced in [4].























Another approach to persistence is the idea of stratifying thestate space Rm

>0, which was introduced in [5] for detailedbalanced networks (a subset of complex balanced networks).

We generalized this to complex balanced networks in [6] andproved the following:


Consider a complex balanced system and an arbitrary permutationoperator µ. If Sµ ∩ LI 6= ∅ then there exists an α ∈ Rm

≤0 satisfying

αi < 0 for i ∈ I and αi = 0 for i 6∈ I

such that 〈α, f(x)〉 ≤ 0 for every x ∈ Sµ.










≤0 satisfying












≤0 satisfying












≤0 satisfying












≤0 satisfying







The strata Sµ represent a partition of the state space Rm>0.

x*

S1

S2

S3

S4

This result guarantees trajectories within strata are kept away fromportions of the boundary intersecting the strata!

Supplemental conditions guarantee persistence.






x*

S1

S2

S3

S4








x*

S1

S2

S3

S4








x*

S1

S2

S3

S4








x*

S1

S2

S3

S4







Conjugacy of Chemical Reaction Networks

A lot of recent research has been conducted on determiningconditions under which two networks with disparate networkstructure share the same qualitative dynamics.

We can often relate the dynamics of networks based on propertiesof their reaction graphs.

In the case where one network has “good” structure and the otherdoes not, we can extend the dynamics to the poorly structurednetwork!





















It is known that two chemical reaction networks can give rise tothe same set of differential equations.

Consider the chemical reaction networks

N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5

2A2.

(Well structured!)

Both of these networks are governed by

x1 = −2x21 + x2

2

x2 = 2x21 − x2

2 .

(Known dynamics!)







N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5

2A2.

(Well structured!)


x1 = −2x21 + x2

2

x2 = 2x21 − x2

2 .

(Known dynamics!)







N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5

2A2.

(Well structured!)


x1 = −2x21 + x2

2

x2 = 2x21 − x2

2 .

(Known dynamics!)







N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5

2A2. (Well structured!)


x1 = −2x21 + x2

2

x2 = 2x21 − x2

2 .

(Known dynamics!)







N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5



x1 = −2x21 + x2

2

x2 = 2x21 − x2

2 .(Known dynamics!)







N : 2A11−→ 2A2

1−→ A1 +A2

N ′ : 2A1

1�0.5



x1 = −2x21 + x2

2

x2 = 2x21 − x2

2 .(Known dynamics!)





A complete theoretical analysis of dynamical equivalence wasconducted in [6].

The problem of finding dynamically equivalent networks usingcomputer software has been investigated in [7] and subsequentpapers.

The procedure uses mixed-integer linear programming (MILP)methods to find dynamically equivalent networks with a maximaland minimal number of reactions.



















QUESTION #1:

Can we find more general conditions under which two networkshave the same qualitative dynamics?

QUESTION #2:

Given a network with “bad” structure, can we find a networkwith “good” structure with related dynamics?





QUESTION #1:

Can we find more general conditions under which two networkshave the same qualitative dynamics?

QUESTION #2:

Given a network with “bad” structure, can we find a networkwith “good” structure with related dynamics?





We extended the notion of dynamical equivalence to linearconjugacy with the following result:

Theorem (Theorem 3.2 of [2] 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

where T =diag{c}. Then N is linearly conjugate to N ′ withkinetics matrix

A′k = Ab · diag {Ψ(c)} .





Two networks N and N ′ are said to be linearly conjugate to oneanother if there is a linear mapping which takes the flow from onein the flow of the other.

The qualitative dynamics (e.g. number and stability of equilibria,persistence, boundedness, etc.) of linearly conjugate networks areidentical! (i.e. The answer to Question #1 is YES.)

Linear conjugacy includes dynamical equivalence as a specialcase (identity transformation).



















PROBLEM:

If we are just given a single network N , can we find a linearlyconjugate network N ′ with known dynamics?

The space of possible networks is typically too large to consider byhand.

The development of computer algorithms is vital to making thistheory applicable.





PROBLEM:








PROBLEM:








The linear conjugacy condition for N ′, Y · Ak = T · Y · Ab whereY and Ak are given, can be rewritten as a linear constraint in alinear programming problem!

Linear Conjugacy

Y · Ab = T−1 · Y · Akm∑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}





The linear conjugacy condition for N ′, Y · Ak = T · Y · Ab whereY and Ak are given, can be rewritten as a linear constraint in alinear programming problem!

Linear Conjugacy

Y · Ab = T−1 · Y · Akm∑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}





We can also impose weak reversibility on N ′ with a linearconstraint! (But we have to be sneaky...)

We introduce a matrix Ak with the same structure as Ab and useproperties of the kernel of Ab for weakly reversible networks.

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 .





We can keep track of whether a network is in N ′ or not byintroducing binary variables δij ∈ {0, 1}.

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





We can keep track of whether a network is in N ′ or not byintroducing binary variables δij ∈ {0, 1}.

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





Example: 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.

Question:

Can we find a weakly reversible network N ′ which is linearlyconjugate to N ?





Example: 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.

Question:

Can we find a weakly reversible network N ′ which is linearlyconjugate to N ?





YES! We can find several of them 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.





YES! We can find several of them 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.










I have presented results on linearization of complex balancednetworks, and persistence and linear conjugacy of chemicalreaction networks.

A number of avenues for future work are apparent:

Persistence: While significant work has been conductedrecently [1, 2, 3, 7, 5], the Global Attractor Conjectureremains unproved in general.

Linear Conjugacy: Numerous topic areas, includingconsidering nonlinear conjugacies, alternative kinetic schemes,parameter-free networks, etc.

























Thanks for coming out!

Special thanks for my advisor David Siegel, my advisory committeeBrian Ingalls and Xinzhi Liu, and the rest of my examining

committee Henry Wolkowicz and Gheorghe Craciun.




David Anderson.

Global asymptotic stability for a class of nonlinear chemical equations.SIAM J. Appl. Math., 68(5):1464–1476, 2008.

David Anderson.

A proof of the global attractor conjecture in the single linkage class case.SIAM J. Appl. Math., to appear, 2011.

David Anderson and Anne Shiu.

The dynamics of weakly reversible population processes near facets.SIAM J. Appl. Math., 70(6):1840–1858, 2010.

David Angeli, Patrick Leenheer, and Eduardo Sontag.

A petri net approach to the study of persistence in chemical reaction networks.Math. Biosci., 210(2):598–618, 2007.

Gheorghe Craciun, Alicia Dickenstein, Anne Shiu, and Bernd Sturmfels.

Toric dynamical systems.J. Symbolic Comput., 44(11):1551–1565, 2009.

Gheorghe Craciun and Casian Pantea.

Identifiability of chemical reaction networks.J. Math Chem., 44(1):244–259, 2008.

Gheorghe Craciun, Casian Pantea, and Fedor Nazarov.

Persistence and permanence of mass-action and power-law dynamical systems.Available on the ArXiv at arxiv:1010.3050.




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 and David Siegel.

Weak dynamic non-emptiability and persistence of chemical kinetics systems.SIAM J. Appl. Math., 71(4):1263–1279, 2011.Available on the arXiv at arxiv:1009.0720.

Matthew D. Johnston, David Siegel, and Gabor Szederkenyi.

A linear programming approach to weak reversibility and linear conjugacy of chemical reaction networks.J. Math. Chem., to appear.Available on the arXiv at arxiv:1107.1659.

Casian Pantea.

On the persistence and global stability of mass-action systems.Available on the ArXiv at arxiv:1103.0603.

David Siegel and Matthew D. Johnston.

A stratum approach to global stability of complex balanced systems.Dyn. Syst., 26(2):125–146, 2011.Available on the arXiv at arxiv:1008.1622.

Gabor Szederkenyi.

Computing sparse and dense realizations of reaction kinetic systems.J. Math. Chem., 47:551–568, 2010.


Doctoral Dissertation: Topics in Chemical Reaction Modeling · Doctoral Dissertation: Topics in Chemical Reaction Modeling Matthew D. Johnston Department of Applied Mathematics University

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