Linear Conjugacy of Chemical Reaction Networks Matthew Douglas - - PowerPoint PPT Presentation

Background Linearly Conjugacy

Linear Conjugacy of Chemical Reaction Networks

Matthew Douglas Johnston University of Waterloo April 25, 2011

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy

1 Background

Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy

1 Background

Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

2 Linearly Conjugacy

Main Theorem Examples

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

1 Background

Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

2 Linearly Conjugacy

Main Theorem Examples

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

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

k

− → 2H2O /

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

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

k

− → 2H2O Species/Reactants

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

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

k

− → 2H2O Reactant Complex/

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

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

k

− → 2H2O Product Complex/

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

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

k

− → 2H2O Reaction Constant/

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

An elementary reaction 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.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

An elementary reaction 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. In order to build a model, we assume the mixture is spatially homogeneous, temperature and volume are held constant, and the law of mass action applies.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

Consider the general network N given by Ci

ki

− → C′

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

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible 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

• f autonomous, polynomial, ordinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi

(1) where xi, i = 1, . . . , m, are the reactant concentrations.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible 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

• f autonomous, polynomial, ordinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi

(1) where xi, i = 1, . . . , m, are the reactant concentrations.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible 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

• f autonomous, polynomial, ordinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi

(1) where xi, i = 1, . . . , m, are the reactant concentrations.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible 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

• f autonomous, polynomial, ordinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi

(1) where xi, i = 1, . . . , m, are the reactant concentrations.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible 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

• f autonomous, polynomial, ordinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi

(1) where xi, i = 1, . . . , m, are the reactant concentrations.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible 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

• f autonomous, polynomial, ordinary differential equations

˙ x =

r

• i=1

ki(z′

i − zi)xzi

(1) where xi, i = 1, . . . , m, are the reactant concentrations. Model is applied to systems biology, enzyme kinetics, industrial reactors, neural networks, atmospherics, etc., and is related to predator-prey and epidemic growth models in biology.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

The particular class of networks which I have been interested in are weakly reversible networks.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

The particular class of networks which I have been interested in are weakly reversible networks. A network is weakly reversible if a path from one complex to another in the reaction graph implies a path back

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

The particular class of networks which I have been interested in are weakly reversible networks. A network is weakly reversible if a path from one complex to another in the reaction graph implies a path back , e.g. C1

k1

− → C2

k3 տ

ւk2 C3.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

The particular class of networks which I have been interested in are weakly reversible networks. A network is weakly reversible if a path from one complex to another in the reaction graph implies a path back , e.g. C1

k1

− → C2

k3 տ

ւk2 C3.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

The particular class of networks which I have been interested in are weakly reversible networks. A network is weakly reversible if a path from one complex to another in the reaction graph implies a path back , e.g. C1

k1

− → C2

k3 տ

ւk2 C3.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

The particular class of networks which I have been interested in are weakly reversible networks. A network is weakly reversible if a path from one complex to another in the reaction graph implies a path back , e.g. C1

k1

− → C2

k3 տ

ւk2 C3. Under the assumption of mass-action kinetics, strong properties are known about the dynamics of weakly reversible networks.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

Mass-action systems are often very difficult to analyze and many types of behaviour are possible (stable, multi-stable, oscillatory, chaotic behaviours, etc.)

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

Mass-action systems are often very difficult to analyze and many types of behaviour are possible (stable, multi-stable, oscillatory, chaotic behaviours, etc.) However, many classes of systems with strongly predictable behaviour are known (e.g. weakly reversible systems).

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

Mass-action systems are often very difficult to analyze and many types of behaviour are possible (stable, multi-stable, oscillatory, chaotic behaviours, etc.) However, many classes of systems with strongly predictable behaviour are known (e.g. weakly reversible systems). CHALLENGE: Determine conditions under which a system with unknown dynamics can be related to a system with known behaviour.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

1 Background

Chemical Reactions Mass-Action Kinetics Weakly Reversible Networks

2 Linearly Conjugacy

Main Theorem Examples

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

In [1], G. Craciun and C. Pantea give necessary and sufficient conditions under which two different reaction networks N and N ′ generate the same set of differential equations under the assumption of mass-action kinetics.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

In [1], G. Craciun and C. Pantea give necessary and sufficient conditions under which two different reaction networks N and N ′ generate the same set of differential equations under the assumption of mass-action kinetics. Why do we care?

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

In [1], G. Craciun and C. Pantea give necessary and sufficient conditions under which two different reaction networks N and N ′ generate the same set of differential equations under the assumption of mass-action kinetics. Why do we care? The qualitative dynamics of N ′ are transferred to N, even if the graph structure is wildly different!

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

In [1], G. Craciun and C. Pantea give necessary and sufficient conditions under which two different reaction networks N and N ′ generate the same set of differential equations under the assumption of mass-action kinetics. Why do we care? The qualitative dynamics of N ′ are transferred to N, even if the graph structure is wildly different! Further work has been done by G. Szederk´ enyi et al. in developing computer algorithms which determine such equivalent networks [2, 3].

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

We extend this work to networks which do not necessarily generate the same mass-action kinetics.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

We extend this work to networks which do not necessarily generate the same mass-action kinetics. We rely on the well-known theory of conjugacy of dynamical systems.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

We extend this work to networks which do not necessarily generate the same mass-action kinetics. We rely on the well-known theory of conjugacy of dynamical systems. Let Φ(x0, t) denote the flow associated with N and Ψ(y0, t) denote the flow associated with N ′.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

We extend this work to networks which do not necessarily generate the same mass-action kinetics. We rely on the well-known theory of conjugacy of dynamical systems. Let Φ(x0, t) denote the flow associated with N and Ψ(y0, t) denote the flow associated with N ′. We will say N and N ′ are linearly conjugate if there exists a linear mapping h : Rm

>0 → Rm >0 such that

h(Φ(x0, t)) = Ψ(h(x0), t) for all x0 ∈ Rm

>0.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Let Creact denote the set of reactant complexes in either the complex set C or the complex set C′. Theorem Suppose that for the rate constants ki > 0, i = 1, . . . , r, there exist constants bi > 0, i = 1, . . . ,˜ r, and cj > 0, j = 1, . . . , m, such that, for every C0 ∈ Creact, Σr

i=1 Ci=C0 ki(z′ i − zi) = T Σ˜ r i=1 ˜ Ci=C0

bi(˜ z′

i − ˜

zi) where T =diag{cj}m

j=1. Then N is linearly conjugate to N ′ with

rate constants ˜ ki = biΠm

j=1c˜ zij j ,

i = 1, . . . ,˜ r.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Let Creact denote the set of reactant complexes in either the complex set C or the complex set C′. Theorem Suppose that for the rate constants ki > 0, i = 1, . . . , r, there exist constants bi > 0, i = 1, . . . ,˜ r, and cj > 0, j = 1, . . . , m, such that, for every C0 ∈ Creact, Σr

i=1 Ci=C0 ki(z′ i − zi) = T Σ˜ r i=1 ˜ Ci=C0

bi(˜ z′

i − ˜

zi) where T =diag{cj}m

j=1. Then N is linearly conjugate to N ′ with

rate constants ˜ ki = biΠm

j=1c˜ zij j ,

i = 1, . . . ,˜ r.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Let Creact denote the set of reactant complexes in either the complex set C or the complex set C′. Theorem Suppose that for the rate constants ki > 0, i = 1, . . . , r, there exist constants bi > 0, i = 1, . . . ,˜ r, and cj > 0, j = 1, . . . , m, such that, for every C0 ∈ Creact, Σr

i=1 Ci=C0 ki(z′ i − zi) = T Σ˜ r i=1 ˜ Ci=C0

bi(˜ z′

i − ˜

zi) where T =diag{cj}m

j=1. Then N is linearly conjugate to N ′ with

rate constants ˜ ki = biΠm

j=1c˜ zij j ,

i = 1, . . . ,˜ r.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Let Creact denote the set of reactant complexes in either the complex set C or the complex set C′. Theorem Suppose that for the rate constants ki > 0, i = 1, . . . , r, there exist constants bi > 0, i = 1, . . . ,˜ r, and cj > 0, j = 1, . . . , m, such that, for every C0 ∈ Creact, Σr

i=1 Ci=C0 ki(z′ i − zi) = T Σ˜ r i=1 ˜ Ci=C0

bi(˜ z′

i − ˜

zi) where T =diag{cj}m

j=1. Then N is linearly conjugate to N ′ with

rate constants ˜ ki = biΠm

j=1c˜ zij j ,

i = 1, . . . ,˜ r.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 1: Consider the chemical reaction network N : A1 + 2A2

k1

− → A1 + 3A2

k2

− → A1 + A2

k3

− → 3A1 2A1

k4

− → A2.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 1: Consider the chemical reaction network N : A1 + 2A2

k1

− → A1 + 3A2

k2

− → A1 + A2

k3

− → 3A1 2A1

k4

− → A2. This is linearly conjugate to N ′ : A1 + 2A2

˜ k1

⇄

˜ k2

A1 + 3A2 A1 + A2

˜ k3

⇄

˜ k4

2A1.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 1: Consider the chemical reaction network N : A1 + 2A2

k1

− → A1 + 3A2

k2

− → A1 + A2

k3

− → 3A1 2A1

k4

− → A2. This is linearly conjugate to N ′ : A1 + 2A2

˜ k1

⇄

˜ k2

A1 + 3A2 A1 + A2

˜ k3

⇄

˜ k4

2A1. Source complexes are conserved!

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 1: Consider the chemical reaction network N : A1 + 2A2

k1

− → A1 + 3A2

k2

− → A1 + A2

k3

− → 3A1 2A1

k4

− → A2. This is linearly conjugate to N ′ : A1 + 2A2

˜ k1

⇄

˜ k2

A1 + 3A2 A1 + A2

˜ k3

⇄

˜ k4

2A1. Source complexes are conserved!

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 1: Consider the chemical reaction network N : A1 + 2A2

k1

− → A1 + 3A2

k2

− → A1 + A2

k3

− → 3A1 2A1

k4

− → A2. This is linearly conjugate to N ′ : A1 + 2A2

˜ k1

⇄

˜ k2

A1 + 3A2 A1 + A2

˜ k3

⇄

˜ k4

2A1. Source complexes are conserved!

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 1: Consider the chemical reaction network N : A1 + 2A2

k1

− → A1 + 3A2

k2

− → A1 + A2

k3

− → 3A1 2A1

k4

− → A2. This is linearly conjugate to N ′ : A1 + 2A2

˜ k1

⇄

˜ k2

A1 + 3A2 A1 + A2

˜ k3

⇄

˜ k4

2A1. Source complexes are conserved!

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 2: Consider the chemical reaction network A1 + 2A2

ǫ

− → A1 N : 2A1 + A2

1

− → 3A2 A1 + 3A2

1

− → A1 + A2

1

− → 3A1 + A2 for ǫ > 0.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 2: Consider the chemical reaction network A1 + 2A2

ǫ

− → A1 N : 2A1 + A2

1

− → 3A2 A1 + 3A2

1

− → A1 + A2

1

− → 3A1 + A2 for ǫ > 0. This is linearly conjugate to N ′ : A1 + 2A2

˜ k1

− → A1 + A2

˜ k4 ↑ ˜ k5ր

↓˜

k2

A1 + 3A2 ← −

˜ k3

2A1 + A2

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 2: Consider the chemical reaction network A1 + 2A2

ǫ

− → A1 N : 2A1 + A2

1

− → 3A2 A1 + 3A2

1

− → A1 + A2

1

− → 3A1 + A2 for ǫ > 0. This is linearly conjugate to N ′ : A1 + 2A2

˜ k1

− → A1 + A2

˜ k4 ↑ ˜ k5ր

↓˜

k2

A1 + 3A2 ← −

˜ k3

2A1 + A2 The third reaction is split into two!

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 3: Consider the chemical reaction network A1

k1

− → 2A1 + 2A2

k2

− → A2

k3

− → A1 + A2.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 3: Consider the chemical reaction network A1

k1

− → 2A1 + 2A2

k2

− → A2

k3

− → A1 + A2. This is linearly conjugate to N ′ : A1 + A2

˜ k3

⇆

˜ k6

A2

˜ k4 ↓ ˜ k5 ց

↑

˜ k2

A1 − →

˜ k1

2A1 + 2A2.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Example 3: Consider the chemical reaction network A1

k1

− → 2A1 + 2A2

k2

− → A2

k3

− → A1 + A2. This is linearly conjugate to N ′ : A1 + A2

˜ k3

⇆

˜ k6

A2

˜ k4 ↓ ˜ k5 ց

↑

˜ k2

A1 − →

˜ k1

2A1 + 2A2. Under certain conditions, strictly product complexes can become source complexes!

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Outstanding! But what are the next steps?

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Outstanding! But what are the next steps?

1 Generally need to find suitable target networks N ′ - computer

programs are necessary for all but the simplest cases.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Outstanding! But what are the next steps?

1 Generally need to find suitable target networks N ′ - computer

programs are necessary for all but the simplest cases.

2 What about non-linear transformations?

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Outstanding! But what are the next steps?

1 Generally need to find suitable target networks N ′ - computer

programs are necessary for all but the simplest cases.

2 What about non-linear transformations? 3 What about other (i.e. non-mass-action) dynamics?

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

Thanks for coming out!

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks

Background Linearly Conjugacy Main Theorem Examples

• G. Craciun and C. Pantea, Identifiability of chemical reaction

networks, J. Math. Chem. 44 (2008), no. 1, pp. 244–259.

• G. Szederk´

enyi, Computing sparse and dense realizations of reaction kinetic systems, J. Math. Chem. 47 (2010), pp. 551–568.

• G. Szederk´

enyi and K. M. Hangos, Finding complex balanced and detailed balanced realizations of chemical reaction

• networks. Available on the arXiv at arXiv:1010.4477.

Matthew Douglas Johnston Linear Conjugacy of Chemical Reaction Networks