Doctoral Dissertation: Topics in Chemical Reaction Modeling Matthew - - PowerPoint PPT Presentation

Background Original Results Conclusions and Future Work

Doctoral Dissertation: Topics in Chemical Reaction Modeling

Matthew D. Johnston Department of Applied Mathematics University of Waterloo December 9, 2011

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work

1 Background

Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work

1 Background

Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

2 Original Results

Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work

1 Background

Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

2 Original Results

Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

3 Conclusions and Future Work

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

1 Background

Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

2 Original Results

Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

3 Conclusions and Future Work

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

A chemical reaction network consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O /

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

A chemical reaction network consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Species/Reactants

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

A chemical reaction network consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Reactant Complex/

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

A chemical reaction network consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Product Complex/

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

A chemical reaction network consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O Reaction Constant/

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

A chemical reaction network consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

A chemical reaction network consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions. Has connections with, and applications to, industrial chemistry

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

A chemical reaction network consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions. Has connections with, and applications to, industrial chemistry, pharmaceutics

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

A chemical reaction network consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / Chemical kinetics is the study of the rates/dynamics resulting from systems of such reactions. Has connections with, and applications to, industrial chemistry, pharmaceutics, systems biology

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

A chemical reaction network consists of a set of reactants which turn into a set of products, e.g. 2H2 + O2

k

− → 2H2O / Chemical kinetics is the study of the rates/dynamics resulting from 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 other areas....

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the general network N given by Ci

ki

− → C′

i, i = 1, . . . , r.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the general network N given by Ci

ki

− → C′

i, i = 1, . . . , r.

Under mass-action kinetics, this network is governed by the system of autonomous, polynomial, ordinary differential equations ˙ x =

r

• i=1

ki(z′

i − zi)xzi.

(1)

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the general network N given by Ci

ki

− → C′

i, i = 1, . . . , r.

Under mass-action kinetics, this network is governed by the system of autonomous, polynomial, ordinary differential equations ˙ x =

r

• i=1

ki(z′

i − zi)xzi.

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the general network N given by Ci

ki

− → C′

i, i = 1, . . . , r.

Under mass-action kinetics, this network is governed by the system 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 ith reaction.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the general network N given by Ci

ki

− → C′

i, i = 1, . . . , r.

Under mass-action kinetics, this network is governed by the system 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 ith reaction. (z′

i − zi) is the reaction vector for the ith reaction.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the general network N given by Ci

ki

− → C′

i, i = 1, . . . , r.

Under mass-action kinetics, this network is governed by the system 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 ith reaction. (z′

i − zi) is the reaction vector for the ith reaction.

xzi = m

j=1 xzij j

is the mass-action term for the ith reaction.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the reversible system A1

k1

⇄

k2

2A2.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the reversible system A1

k1

⇄

k2

2A2. This has the governing dynamics ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2,

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the reversible system A1

k1

⇄

k2

2A2. This has the governing dynamics ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2,

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the reversible system A1

k1

⇄

k2

2A2. This has the governing dynamics ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2,

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the reversible system A1

k1

⇄

k2

2A2. This has the governing dynamics ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2,

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Consider the reversible system A1

k1

⇄

k2

2A2. This has the governing dynamics ˙ x1 ˙ x2

• = k1

−1 2

• x1 + k2
• 1

−2

• x2

2,

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Figure: Previous system with k1 = k2 = 1.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Figure: Previous system with k1 = k2 = 1.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

Figure: Previous system with k1 = k2 = 1.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph. The particular class of networks I have been interested in are weakly reversible networks.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph. The particular class of networks I 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 Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph. The particular class of networks I 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 Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph. The particular class of networks I 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 Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph. The particular class of networks I 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 Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

The dynamical behaviour of chemical reaction networks is heavily dependent on the structure of its reaction graph. The particular class of networks I 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 Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

An important subset of weakly reversible networks are complex balanced networks.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

An important subset of weakly reversible networks are complex balanced 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.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

An important subset of weakly reversible networks are complex balanced 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.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

An important subset of weakly reversible networks are complex balanced 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.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

An important subset of weakly reversible networks are complex balanced 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 there exists within each invariant space of the network a unique positive equilibrium concentration, and this equilibrium concentration is locally asymptotically stable relative to that invariant space.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

1 Background

Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

2 Original Results

Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

3 Conclusions and Future Work

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy 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.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy 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 that d dt L(x) = ∇L(x) · f(x) < 0 for all x ∈ Rm

>0 such that f(x) = 0.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy 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 that d dt L(x) = ∇L(x) · f(x) < 0 for all x ∈ Rm

>0 such that f(x) = 0.

This means that all trajectories converge locally to their respective equilibrium concentrations!

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

This problem has not been attempted by the method of linearization about equilibrium concentrations before. For the method of linearization, we consider the decomposition f(x) = Df(x∗)(x − x∗) + O((x − x∗)2) and the corresponding linear problem dy dt = Ay where y = x − x∗ and A = Df(x∗).

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

This problem has not been attempted by the method of linearization about equilibrium concentrations before. For the method of linearization, we consider the decomposition f(x) = Df(x∗)(x − x∗) + O((x − x∗)2) and the corresponding linear problem dy dt = Ay where y = x − x∗ and A = Df(x∗). This method has the advantage of guaranteeing local exponential convergence to x∗ if Df(x∗) has non-degenerate eigenvalues with negative real part.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

>0:

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

>0: 1 the local stable manifold W s loc coincides locally with the

compatible linear invariant space

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

>0: 1 the local stable manifold W s loc coincides locally with the

compatible linear invariant space; and

2 the local centre manifold W c loc coincides locally with the

tangent plane to the equilibrium set at x∗

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

>0: 1 the local stable manifold W s loc coincides locally with the

compatible linear invariant space; and

2 the local centre manifold W c loc 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−Mtx0 − x∗, ∀ t ≥ 0 for all x0 sufficiently close to x∗ in the compatible linear invariant space.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Consider the network 2A1 + A2

1/2

− → 3A1

1/8 ↑

↓ 1 3A2 ← −

1/4 A1 + 2A2.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Consider the network 2A1 + A2

1/2

− → 3A1

1/8 ↑

↓ 1 3A2 ← −

1/4 A1 + 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

• .

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Consider the equilibrium concentration x∗

1 = 1/2, x∗ 2 = 1.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Consider the equilibrium concentration x∗

1 = 1/2, x∗ 2 = 1.

This network has the Jacobian Df(x∗) = − 5

4 5 8 5 4

− 5

8

• .

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Consider the equilibrium concentration x∗

1 = 1/2, x∗ 2 = 1.

This network has the Jacobian Df(x∗) = − 5

4 5 8 5 4

− 5

8

• .

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Consider the equilibrium concentration x∗

1 = 1/2, x∗ 2 = 1.

This network has the Jacobian Df(x∗) = − 5

4 5 8 5 4

− 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 invariant space x1 + x2 = constant, and the local centre manifold is tangent to the equilibrium set x2 = 2x1!

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Global Attractor Conjecture A significant amount of research has been conducted on the question of persistence of chemical reaction networks.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Global Attractor Conjecture A significant amount of research has been conducted on the question of persistence of chemical reaction networks. Persistence is the property that if all species are initially present then none will tend toward zero.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Global Attractor Conjecture A significant amount of research has been conducted on the question of persistence of chemical reaction networks. Persistence is the property that if all species are initially present then none will tend toward zero. This is a very important property of networks! (e.g. Chemical reactors using up reactants, biological circuits losing intermediates, etc.)

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Reconsider complex balanced networks: it is known that d dt L(x) = ∇L(x) · f(x) < 0 for all x ∈ Rm

>0 such that f(x) = 0.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Reconsider complex balanced networks: it is known that d dt L(x) = ∇L(x) · f(x) < 0 for all x ∈ Rm

>0 such that f(x) = 0.

QUESTION: Is this sufficient to show x∗ is globally asymptotically stable relative to its corresponding linear invariant space?

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Reconsider complex balanced networks: it is known that d dt L(x) = ∇L(x) · f(x) < 0 for all x ∈ Rm

>0 such that f(x) = 0.

QUESTION: Is this sufficient to show x∗ is globally asymptotically stable relative to its corresponding linear invariant space? ANSWER: Only if the network is also persistent (since L(x) is not radially unbounded with respect to Rm

>0).

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

One approach to persistence is identifying certain special regions

• f the boundary and showing they must repel trajectories.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

One approach to persistence is identifying certain special regions

• f 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. Suppose that every semilocking set I is weakly dynamically

• nonemptiable. Then the system is persistent.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

One approach to persistence is identifying certain special regions

• f 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. Suppose that every semilocking set I is weakly dynamically

• nonemptiable. Then the system is persistent.

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

>0, which was introduced in [5] for detailed

balanced networks (a subset of complex balanced networks).

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

>0, which was introduced in [5] for detailed

balanced networks (a subset of complex balanced networks). We generalized this to complex balanced networks in [6] and proved the following: Theorem (Theorem 3.12, [6]) Consider a complex balanced system and an arbitrary permutation

• perator µ. If Sµ ∩ LI = ∅ then there exists an α ∈ Rm

≤0 satisfying

αi < 0 for i ∈ I and αi = 0 for i ∈ I such that α, f(x) ≤ 0 for every x ∈ Sµ.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

>0, which was introduced in [5] for detailed

balanced networks (a subset of complex balanced networks). We generalized this to complex balanced networks in [6] and proved the following: Theorem (Theorem 3.12, [6]) Consider a complex balanced system and an arbitrary permutation

• perator µ. If Sµ ∩ LI = ∅ then there exists an α ∈ Rm

≤0 satisfying

αi < 0 for i ∈ I and αi = 0 for i ∈ I such that α, f(x) ≤ 0 for every x ∈ Sµ.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

>0, which was introduced in [5] for detailed

balanced networks (a subset of complex balanced networks). We generalized this to complex balanced networks in [6] and proved the following: Theorem (Theorem 3.12, [6]) Consider a complex balanced system and an arbitrary permutation

• perator µ. If Sµ ∩ LI = ∅ then there exists an α ∈ Rm

≤0 satisfying

αi < 0 for i ∈ I and αi = 0 for i ∈ I such that α, f(x) ≤ 0 for every x ∈ Sµ.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

>0, which was introduced in [5] for detailed

balanced networks (a subset of complex balanced networks). We generalized this to complex balanced networks in [6] and proved the following: Theorem (Theorem 3.12, [6]) Consider a complex balanced system and an arbitrary permutation

• perator µ. If Sµ ∩ LI = ∅ then there exists an α ∈ Rm

≤0 satisfying

αi < 0 for i ∈ I and αi = 0 for i ∈ I such that α, f(x) ≤ 0 for every x ∈ Sµ.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

The strata Sµ represent a partition of the state space Rm

>0.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

The strata Sµ represent a partition of the state space Rm

>0.

x* S1 S2 S3 S4 MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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 from portions of the boundary intersecting the strata!

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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 from portions of the boundary intersecting the strata!

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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 from portions of the boundary intersecting the strata! Supplemental conditions guarantee persistence.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Conjugacy of Chemical Reaction Networks A lot of recent research has been conducted on determining conditions under which two networks with disparate network structure share the same qualitative dynamics.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Conjugacy of Chemical Reaction Networks A lot of recent research has been conducted on determining conditions under which two networks with disparate network structure share the same qualitative dynamics. We can often relate the dynamics of networks based on properties

• f their reaction graphs.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Conjugacy of Chemical Reaction Networks A lot of recent research has been conducted on determining conditions under which two networks with disparate network structure share the same qualitative dynamics. We can often relate the dynamics of networks based on properties

• f their reaction graphs.

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

It is known that two chemical reaction networks can give rise to the same set of differential equations. Consider the chemical reaction networks N : 2A1

1

− → 2A2

1

− → A1 + A2 N ′ : 2A1

1

⇄

0.5

2A2.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

It is known that two chemical reaction networks can give rise to the same set of differential equations. Consider the chemical reaction networks N : 2A1

1

− → 2A2

1

− → A1 + A2 N ′ : 2A1

1

⇄

0.5

2A2. Both of these networks are governed by ˙ x1 = −2x2

1 + x2 2

˙ x2 = 2x2

1 − x2 2.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

It is known that two chemical reaction networks can give rise to the same set of differential equations. Consider the chemical reaction networks N : 2A1

1

− → 2A2

1

− → A1 + A2 N ′ : 2A1

1

⇄

0.5

2A2. (Well structured!) Both of these networks are governed by ˙ x1 = −2x2

1 + x2 2

˙ x2 = 2x2

1 − x2 2.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

It is known that two chemical reaction networks can give rise to the same set of differential equations. Consider the chemical reaction networks N : 2A1

1

− → 2A2

1

− → A1 + A2 N ′ : 2A1

1

⇄

0.5

2A2. (Well structured!) Both of these networks are governed by ˙ x1 = −2x2

1 + x2 2

˙ x2 = 2x2

1 − x2 2.

(Known dynamics!)

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

It is known that two chemical reaction networks can give rise to the same set of differential equations. Consider the chemical reaction networks N : 2A1

1

− → 2A2

1

− → A1 + A2 N ′ : 2A1

1

⇄

0.5

2A2. (Well structured!) Both of these networks are governed by ˙ x1 = −2x2

1 + x2 2

˙ x2 = 2x2

1 − x2 2.

(Known dynamics!)

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

A complete theoretical analysis of dynamical equivalence was conducted in [6]. The problem of finding dynamically equivalent networks using computer software has been investigated in [7] and subsequent papers.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

A complete theoretical analysis of dynamical equivalence was conducted in [6]. The problem of finding dynamically equivalent networks using computer software has been investigated in [7] and subsequent papers. The procedure uses mixed-integer linear programming (MILP) methods to find dynamically equivalent networks with a maximal and minimal number of reactions.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

QUESTION #1: Can we find more general conditions under which two networks have the same qualitative dynamics?

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

QUESTION #1: Can we find more general conditions under which two networks have the same qualitative dynamics? QUESTION #2: Given a network with “bad” structure, can we find a network with “good” structure with related dynamics?

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

We extended the notion of dynamical equivalence to linear conjugacy with the following result: Theorem (Theorem 3.2 of [2] 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 where T =diag{c}. Then N is linearly conjugate to N ′ with kinetics matrix A′

k = Ab · diag {Ψ(c)} .

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Two networks N and N ′ are said to be linearly conjugate to one another if there is a linear mapping which takes the flow from one in the flow of the other. The qualitative dynamics (e.g. number and stability of equilibria, persistence, boundedness, etc.) of linearly conjugate networks are identical! (i.e. The answer to Question #1 is YES.)

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Two networks N and N ′ are said to be linearly conjugate to one another if there is a linear mapping which takes the flow from one in the flow of the other. The qualitative dynamics (e.g. number and stability of equilibria, persistence, boundedness, etc.) of linearly conjugate networks are identical! (i.e. The answer to Question #1 is YES.) Linear conjugacy includes dynamical equivalence as a special case (identity transformation).

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

PROBLEM: If we are just given a single network N, can we find a linearly conjugate network N ′ with known dynamics?

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

PROBLEM: If we are just given a single network N, can we find a linearly conjugate network N ′ with known dynamics? The space of possible networks is typically too large to consider by hand.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

PROBLEM: If we are just given a single network N, can we find a linearly conjugate network N ′ with known dynamics? The space of possible networks is typically too large to consider by hand. The development of computer algorithms is vital to making this theory applicable.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

The linear conjugacy condition for N ′, Y · Ak = T · Y · Ab where Y and Ak are given, can be rewritten as a linear constraint in a linear programming problem! Linear Conjugacy Y · Ab = T −1 · Y · Ak

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 Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

We can also impose weak reversibility on N ′ with a linear constraint! (But we have to be sneaky...) We introduce a matrix ˜ Ak with the same structure as Ab and use properties of the kernel of Ab for weakly reversible networks.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

We can also impose weak reversibility on N ′ with a linear constraint! (But we have to be sneaky...) We introduce a matrix ˜ Ak with the same structure as Ab and use properties of the kernel of Ab for weakly reversible networks. Weak Reversibility

m

• i=1,i=j

[˜ Ak]ij =

m

• i=1,i=j

[˜ Ak]ji, j = 1, . . . , m [˜ Ak]ij ≥ 0, i, j = 1, . . . , m, i = j.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

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

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

We can keep track of whether a network is in N ′ or not by introducing binary variables δij ∈ {0, 1}. Sparse/Dense Realizations minimize

m

• i,j=1,i=j

δij

• r

minimize

m

• i,j=1,i=j

−δij 0 ≤ [Ak]ij − ǫδij 0 ≤ −[Ak]ij + uijδij δij ∈ {0, 1} i, j = 1, . . . , m, i = j

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Example: Consider the reaction network N given by A1 + 2A2

1

− → 2A1 + 2A2

1

− → 2A1 + A2 A1

2

← − 2A1

1

− → 2A1 + A3 2A1 + 2A3

1

← − A1 + 2A3

1

− → A1 + A2 + 2A3 ↓3 A1 + A3.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

Example: Consider the reaction network N given by A1 + 2A2

1

− → 2A1 + 2A2

1

− → 2A1 + A2 A1

2

← − 2A1

1

− → 2A1 + A3 2A1 + 2A3

1

← − A1 + 2A3

1

− → A1 + A2 + 2A3 ↓3 A1 + A3. Question: Can we find a weakly reversible network N ′ which is linearly conjugate to N?

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

YES! We can find several of them very quickly with GLPK.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

YES! We can find several of them very quickly with GLPK.

X1+2X2 2X1+2X2 2X1 X1+2X3

4 400 25 40 125

X1+2X2 2X1+2X2 2X1 X1+2X3 2X1+X2

0.367 13.9 0.926 13.1 1.35 0.816 13.3 1.35 0.926 0.926

(a) (b)

Figure: Weakly reversible networks which are linearly conjugate to N. The network in (a) is sparse while the network in (b) is dense.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work

1 Background

Chemical Reaction Networks Weakly Reversible Networks Complex Balanced Networks

2 Original Results

Linearization of Complex Balanced Networks Global Attractor Conjecture Conjugacy of Chemical Reaction Networks

3 Conclusions and Future Work

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work

I have presented results on linearization of complex balanced networks, and persistence and linear conjugacy of chemical reaction networks.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work

I have presented results on linearization of complex balanced networks, and persistence and linear conjugacy of chemical reaction networks. A number of avenues for future work are apparent:

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work

I have presented results on linearization of complex balanced networks, and persistence and linear conjugacy of chemical reaction networks. A number of avenues for future work are apparent: Persistence: While significant work has been conducted recently [1, 2, 3, 7, 5], the Global Attractor Conjecture remains unproved in general.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work

I have presented results on linearization of complex balanced networks, and persistence and linear conjugacy of chemical reaction networks. A number of avenues for future work are apparent: Persistence: While significant work has been conducted recently [1, 2, 3, 7, 5], the Global Attractor Conjecture remains unproved in general. Linear Conjugacy: Numerous topic areas, including considering nonlinear conjugacies, alternative kinetic schemes, parameter-free networks, etc.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work

Thanks for coming out! Special thanks for my advisor David Siegel, my advisory committee Brian Ingalls and Xinzhi Liu, and the rest of my examining committee Henry Wolkowicz and Gheorghe Craciun.

MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work 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. MD Johnston Topics in Chemical Reaction Modeling

Background Original Results Conclusions and Future Work 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 G´ abor Szederk´ enyi. 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 Szederk´ enyi. Computing sparse and dense realizations of reaction kinetic systems.

• J. Math. Chem., 47:551–568, 2010.

MD Johnston Topics in Chemical Reaction Modeling