Graduate Student Seminar - WordPress.com...Graduate Student Seminar: – p. 2/26 1. Introduction In a chemical reaction, a set of reactants reacts at a given rate to form a product,

Graduate StudentSeminar:

Behaviour of Chemical ReactionNetworks

Matthew Douglas Johnston

[email protected]

Department of Applied Mathematics

University of Waterloo

Waterloo, ON, Canada

Graduate Student Seminar: – p. 1/26

Graduate StudentSeminar

1. Introduction to Chemical Reaction Systems

Compatibility Classes

2. Behaviour of Chemical Reaction Systems

Periodic Behaviour

Chaotic Behaviour

Pattern Formation

3. Research

Locally Stable Dynamics

References


1. IntroductionIn a chemical reaction, a set of reactants reacts at agiven rate to form a product, e.g.

2H2 + O2k

−→ 2H2O



2H2 + O2k

−→ 2H2O

Species/Reactants



2H2 + O2k

−→ 2H2O

Reactant Complex



2H2 + O2k

−→ 2H2O

Product Complex



2H2 + O2k

−→ 2H2O

Reaction Constant



2H2 + O2k

−→ 2H2O

We will denote the species set by

S = {A1,A2, . . . ,Am} .



2H2 + O2k

−→ 2H2O

We will denote the species set by

S = {A1,A2, . . . ,Am} .

We will denote the complex set by

C = {C1, C2, . . . , Cn} .


1. IntroductionChemical kinetics systems can be representedgraphically in several ways.



Consider the system of elementary reactions:

2A1k

−→ A2 (1)

A2k

−→ A3 + A4 (2)

A3 + A4k

−→ 2A1 (3)



Can index system according to reactions:

2A1k1−→ A2 (1)

A2k2−→ A3 + A4 (2)

A3 + A4k3−→ 2A1 (3)




2A1k1−→ A2 (1)

A2k2−→ A3 + A4 (2)

A3 + A4k3−→ 2A1 (3)

We let Cp−(1) = 2A1, Cp+(1) = A2, Cp−(2) = A2, etc.




2A1k1−→ A2 (1)

A2k2−→ A3 + A4 (2)

A3 + A4k3−→ 2A1 (3)





2A1k1−→ A2 (1)

A2k2−→ A3 + A4 (2)

A3 + A4k3−→ 2A1 (3)





2A1k1−→ A2 (1)

A2k2−→ A3 + A4 (2)

A3 + A4k3−→ 2A1 (3)

But some of the complexes are repeated (e.g. 2A1

appears in reaction 1 and 3)...



Can index system according to complexes:

2A1k(1,2)−→ A2

k(3,1) ↖ ↙k(2,3)

A3 + A4.



Can index system according to complexes:

2A1k(1,2)−→ A2

k(3,1) ↖ ↙k(2,3)

A3 + A4.

We let C1 = 2A1, C2 = A2, and C3 = A3 + A4.


1. IntroductionChemical kinetics is the study of the rates/dynamicsresulting from chemical reactions.



To build the model, we assume that:

the chemical species are well-mixed;





temperature and volume are constant;






the law of mass action applies.






the law of mass action applies.

Applications can be found in chemical engineering,enzyme study, atmospherics, cell biology,developmental biology, etc.


1. IntroductionFor the elementary reaction set

Cp−(i)ki−→ Cp+(i), i = 1, . . . , r

we have the system of differential equations

x =r

∑

i=1

ki(zp+(i) − zp−(i))xz

p−(i).



Cp−(i)ki−→ Cp+(i), i = 1, . . . , r


x =r

∑

i=1


p−(i).

ki is the reaction rate



Cp−(i)ki−→ Cp+(i), i = 1, . . . , r


x =r

∑

i=1


p−(i).

(zp+(i) − zp−(i)) is the reaction vector



Cp−(i)ki−→ Cp+(i), i = 1, . . . , r


x =r

∑

i=1


p−(i).

xz

p−(i) =∏m

j=1(xj)z

j

p−(i) is the mass-action term.


1. IntroductionConsider the (reversible) system:

A1

k1

�k2

2A2.


1. IntroductionConsider the (reversible) system:

A1

k1

�k2

2A2.

This has dynamics

x1

x2

= k1

−1

2

x1 + k2

1

−2

x22,

where x1 and x2 are the concentrations of A1 and A2

respectively.


1. IntroductionWhat kind of properties does this system have?



The (positive) equilibrium set is given by

E =

{

x ∈ R2+ | x2 =

√

k1

k2x1

}

.



The (positive) equilibrium set is given by

E =

{

x ∈ R2+ | x2 =

√

k1

k2x1

}

.

For any k1, k2, x1, x2 we have f(x) ∈ S where

S = span

1

−2

.


1. Introduction

x1

x2E

(x0+S)

Figure 1: Previous system with k1 = k2 = 1.


1. Introduction

x1

x2E

(x0+S)



1. Introduction

x1

x2E

(x0+S)



1. Introduction

x1

x2E

(x0+S)



1. IntroductionThis example illustrates some general properties ofchemical kinetics systems.



Solutions are restricted to positive translates of S,called stoichiometric compatibility classes((x0 + S) ∩ R

m+ ).




m+ ).

This example exhibits locally stable dynamics: unique,asymptotically stable equilibria relative to compatibilityclasses.




m+ ).

This example exhibits locally stable dynamics: unique,asymptotically stable equilibria relative to compatibilityclasses.

This is the type of behaviour I study!


2. BehaviourWe know chemical kinetics systems can exhibit stablebehaviour – what else?



Examples are known exhibiting:

periodic behaviour,




periodic behaviour,

chaotic behaviour,




periodic behaviour,

chaotic behaviour,

switch-like behaviour, and




periodic behaviour,

chaotic behaviour,

switch-like behaviour, and

pattern formation.


2. BehaviourA trajectory (solution) x(t) of an ODE is said to beperiodic with period T > 0 if

x(t + T ) = x(t) ∀ t ≥ 0

but not for any 0 < τ < T .



x(t + T ) = x(t) ∀ t ≥ 0


Periodic behaviour is common: seasonal systems,predator-prey interactions, evolutionary games, etc.



x(t + T ) = x(t) ∀ t ≥ 0


Periodic behaviour is common: seasonal systems,predator-prey interactions, evolutionary games, etc.

A few chemical reactions closely approximate periodicbehaviour (many oscillations): e.g.Belousov-Zhabotinskii and Briggs-Rauscher reactions.


2. Behaviour

(Pause for clip)


2. BehaviourChemical kinetics models also permitoscillatory/periodic behaviour.


2. BehaviourChemical kinetics models also permitoscillatory/periodic behaviour.

Consider the system

A1 + A2k1−→ 2A2

A2 + A3k2−→ 2A3

A1 + A3k3−→ 2A1

with k1 = 1, k2 = 0.1, k3 = 1.8, x1(0) = 0.1, x2(0) = 0.3,and x3(0) = 0.25.


2. Behaviour

0 50 100 150 2000

0.05

0.1

0.15

0.2Reactant 1

t

conc

entr

atio

n

0 50 100 150 2000.2

0.4

0.6

0.8Reactant 2

t

conc

entr

atio

n

0 50 100 150 2000.1

0.2

0.3

0.4

0.5Reactant 3

t

conc

entr

atio

n


2. BehaviourA system is said to exhibit chaotic behaviour iftrajectories/solutions are:



nonconvergent and nonperiodic,




bounded, and




bounded, and

extremely sensitive to initial conditions.




bounded, and


Chaos is not very well understood: system needs to atleast three-dimensional, nonlinear, etc.




bounded, and


Chaos is not very well understood: system needs to atleast three-dimensional, nonlinear, etc.

Forced mechanical systems are often chaotic.


2. BehaviourChaos (or what appears to be chaos) can occur as aresult of chemical reactions.


2. BehaviourChaos (or what appears to be chaos) can occur as aresult of chemical reactions.

Consider the system

A1

k+1

�

k−

1

2A1 A1 + A2

k+2

�

k−

2

2A2

Ok+3

�

k−

3

A2 Ok+4

�

k−

4

A1 + A3

A3

k+5

�

k−

5

2A3.


2. Behaviour

0 5 100

20

40

60Reactant 1

t

conc

entr

atio

n

0 5 100

20

40

60Reactant 2

t

conc

entr

atio

n

0 5 100

10

20

30Reactant 3

t

conc

entr

atio

n

0

50

0

500

20

40

Reactant 1

Attractor

Reactant 2

Rea

ctan

t 3


2. BehaviourIf we remove some of our model assumptions, morevaried behaviour can occur.



We assumed the reactants were well mixed so we didnot have to consider diffusive effects.




Reaction-Diffusion systems are governed by thedynamics

d

dtc(t,x) = d∆

xc(t,x) + f(c(t,x)).





d

dtc(t,x) = d∆


Diffusion term





d

dtc(t,x) = d∆


Reaction term


2. BehaviourHow does the behaviour of reaction-diffusion systemsdiffer from standard ones?



Consider a locally stable system, i.e. reaction partsettles down.




Diffusion has an averaging effect over time, i.e.diffusion part settles down.





Adding a stabilizing effect to a stable system shouldguarantee stability, right?





Adding a stabilizing effect to a stable system shouldguarantee stability, right? ... Well, not always.


2. BehaviourConsider the system

A1

k1

�k2

Ok3−→ A2

2A1 + A2k4−→ 3A1



A1

k1

�k2

Ok3−→ A2

2A1 + A2k4−→ 3A1

Reaction system is locally stable; however, conditionscan be derived for the spatial instability of theReaction-Diffusion system.



A1

k1

�k2

Ok3−→ A2

2A1 + A2k4−→ 3A1

Reaction system is locally stable; however, conditionscan be derived for the spatial instability of theReaction-Diffusion system.

In other words, a non-constant pattern emerges overtime!


2. Behaviour

(Pause for clip)


3. ResearchMy research (so far) has dealt with systems exhibitinglocally stable dynamics.



These systems are experimentally “nice”(concentrations tend to equilibrium regardless of ICsand rate constants).




Topics of research include:

Finding general systems which exhibit l.s.d.






Extending stability globally.






Extending stability globally.

New approaches - linearization.


Thanks!Thanks for your attention, and I hope you learnedsomething.

My work is made possible by:

My advisor, David Siegel

My committee members, Professors Liu and Ingalls

Support of family and friends

NSERC


ReferencesReferences

[1] A. Bamberger and E. Billette. Quelques extensions d’un théorème de Horn etJackson. C. R. Acad. Sci. Paris Sér. I Math., 319(12):1257-1262, 1994.

[2] P. Erdi and J. Toth. Mathematical Models of Chemical Reactions. PrincetonUniversity Press, 1989.

[3] M. Feinberg. Complex balancing in general kinetic systems. Archive ForRational Mechanics and Analysis, 49:187-194, 1972.

[4] F. Horn. Necessary and sufficient conditions for complex balancing in chemicalkinetics. Archive for Rational Mechanics and Analysis, 49:172-186, 1972.

[5] F. Horn and R. Jackson. General mass action kinetics. Archive for RationalMechanics and Analysis, 47:187-194, 1972.


ReferencesReferences

[6] D. MacLean. Positivity and Stability of Chemical Kinetics Systems. Master’sthesis, University of Waterloo, 1998.

[7] Murray, J.D. Mathematical Biology: II. Spatial Models and BiomedicalApplications, Springer-Verlag, 2003.

[8] D. Siegel and D. MacLean. Global stability of complex balanced mechanisms.Journal of Mathematical Chemistry, 27(1-2): 89-110, 2000.

[9] V.M. Vasil’ev, A.I. Vol’pert, and S.I. Khudyaev. A method of quasistationaryconcentrations for the equations of chemical kinetics. USSR Computatl. Math.and Math. Phys., 13(3):187-206, 1973.


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Graduate Student Seminar - WordPress.com...Graduate Student Seminar: – p. 2/26 1. Introduction In a chemical reaction, a set of reactants reacts at a given rate to form a product,