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Graduate StudentSeminar:
Behaviour of Chemical ReactionNetworks
Matthew Douglas Johnston
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
Graduate Student Seminar: – p. 2/26
1. IntroductionIn a chemical reaction, a set of reactants reacts at agiven rate to form a product, e.g.
2H2 + O2k
−→ 2H2O
Graduate Student Seminar: – p. 3/26
1. IntroductionIn a chemical reaction, a set of reactants reacts at agiven rate to form a product, e.g.
2H2 + O2k
−→ 2H2O
Species/Reactants
Graduate Student Seminar: – p. 3/26
1. IntroductionIn a chemical reaction, a set of reactants reacts at agiven rate to form a product, e.g.
2H2 + O2k
−→ 2H2O
Reactant Complex
Graduate Student Seminar: – p. 3/26
1. IntroductionIn a chemical reaction, a set of reactants reacts at agiven rate to form a product, e.g.
2H2 + O2k
−→ 2H2O
Product Complex
Graduate Student Seminar: – p. 3/26
1. IntroductionIn a chemical reaction, a set of reactants reacts at agiven rate to form a product, e.g.
2H2 + O2k
−→ 2H2O
Reaction Constant
Graduate Student Seminar: – p. 3/26
1. IntroductionIn a chemical reaction, a set of reactants reacts at agiven rate to form a product, e.g.
2H2 + O2k
−→ 2H2O
We will denote the species set by
S = {A1,A2, . . . ,Am} .
Graduate Student Seminar: – p. 3/26
1. IntroductionIn a chemical reaction, a set of reactants reacts at agiven rate to form a product, e.g.
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} .
Graduate Student Seminar: – p. 3/26
1. IntroductionChemical kinetics systems can be representedgraphically in several ways.
Graduate Student Seminar: – p. 4/26
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)
Graduate Student Seminar: – p. 4/26
1. IntroductionChemical kinetics systems can be representedgraphically in several ways.
Can index system according to reactions:
2A1k1−→ A2 (1)
A2k2−→ A3 + A4 (2)
A3 + A4k3−→ 2A1 (3)
Graduate Student Seminar: – p. 4/26
1. IntroductionChemical kinetics systems can be representedgraphically in several ways.
Can index system according to reactions:
2A1k1−→ A2 (1)
A2k2−→ A3 + A4 (2)
A3 + A4k3−→ 2A1 (3)
We let Cp−(1) = 2A1, Cp+(1) = A2, Cp−(2) = A2, etc.
Graduate Student Seminar: – p. 4/26
1. IntroductionChemical kinetics systems can be representedgraphically in several ways.
Can index system according to reactions:
2A1k1−→ A2 (1)
A2k2−→ A3 + A4 (2)
A3 + A4k3−→ 2A1 (3)
We let Cp−(1) = 2A1, Cp+(1) = A2, Cp−(2) = A2, etc.
Graduate Student Seminar: – p. 4/26
1. IntroductionChemical kinetics systems can be representedgraphically in several ways.
Can index system according to reactions:
2A1k1−→ A2 (1)
A2k2−→ A3 + A4 (2)
A3 + A4k3−→ 2A1 (3)
We let Cp−(1) = 2A1, Cp+(1) = A2, Cp−(2) = A2, etc.
Graduate Student Seminar: – p. 4/26
1. IntroductionChemical kinetics systems can be representedgraphically in several ways.
Can index system according to reactions:
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)...
Graduate Student Seminar: – p. 4/26
1. IntroductionChemical kinetics systems can be representedgraphically in several ways.
Can index system according to complexes:
2A1k(1,2)−→ A2
k(3,1) ↖ ↙k(2,3)
A3 + A4.
Graduate Student Seminar: – p. 4/26
1. IntroductionChemical kinetics systems can be representedgraphically in several ways.
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.
Graduate Student Seminar: – p. 4/26
1. IntroductionChemical kinetics is the study of the rates/dynamicsresulting from chemical reactions.
Graduate Student Seminar: – p. 5/26
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;
Graduate Student Seminar: – p. 5/26
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;
Graduate Student Seminar: – p. 5/26
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.
Graduate Student Seminar: – p. 5/26
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.
Applications can be found in chemical engineering,enzyme study, atmospherics, cell biology,developmental biology, etc.
Graduate Student Seminar: – p. 5/26
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).
Graduate Student Seminar: – p. 6/26
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).
ki is the reaction rate
Graduate Student Seminar: – p. 6/26
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).
(zp+(i) − zp−(i)) is the reaction vector
Graduate Student Seminar: – p. 6/26
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).
xz
p−(i) =∏m
j=1(xj)z
j
p−(i) is the mass-action term.
Graduate Student Seminar: – p. 6/26
1. IntroductionConsider the (reversible) system:
A1
k1
�k2
2A2.
Graduate Student Seminar: – p. 7/26
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.
Graduate Student Seminar: – p. 7/26
1. IntroductionWhat kind of properties does this system have?
Graduate Student Seminar: – p. 8/26
1. IntroductionWhat kind of properties does this system have?
The (positive) equilibrium set is given by
E =
{
x ∈ R2+ | x2 =
√
k1
k2x1
}
.
Graduate Student Seminar: – p. 8/26
1. IntroductionWhat kind of properties does this system have?
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
.
Graduate Student Seminar: – p. 8/26
1. Introduction
x1
x2E
(x0+S)
Figure 1: Previous system with k1 = k2 = 1.
Graduate Student Seminar: – p. 9/26
1. Introduction
x1
x2E
(x0+S)
Figure 3: Previous system with k1 = k2 = 1.
Graduate Student Seminar: – p. 9/26
1. Introduction
x1
x2E
(x0+S)
Figure 5: Previous system with k1 = k2 = 1.
Graduate Student Seminar: – p. 9/26
1. Introduction
x1
x2E
(x0+S)
Figure 7: Previous system with k1 = k2 = 1.
Graduate Student Seminar: – p. 9/26
1. IntroductionThis example illustrates some general properties ofchemical kinetics systems.
Graduate Student Seminar: – p. 10/26
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+ ).
Graduate Student Seminar: – p. 10/26
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+ ).
This example exhibits locally stable dynamics: unique,asymptotically stable equilibria relative to compatibilityclasses.
Graduate Student Seminar: – p. 10/26
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+ ).
This example exhibits locally stable dynamics: unique,asymptotically stable equilibria relative to compatibilityclasses.
This is the type of behaviour I study!
Graduate Student Seminar: – p. 10/26
2. BehaviourWe know chemical kinetics systems can exhibit stablebehaviour – what else?
Graduate Student Seminar: – p. 11/26
2. BehaviourWe know chemical kinetics systems can exhibit stablebehaviour – what else?
Examples are known exhibiting:
periodic behaviour,
Graduate Student Seminar: – p. 11/26
2. BehaviourWe know chemical kinetics systems can exhibit stablebehaviour – what else?
Examples are known exhibiting:
periodic behaviour,
chaotic behaviour,
Graduate Student Seminar: – p. 11/26
2. BehaviourWe know chemical kinetics systems can exhibit stablebehaviour – what else?
Examples are known exhibiting:
periodic behaviour,
chaotic behaviour,
switch-like behaviour, and
Graduate Student Seminar: – p. 11/26
2. BehaviourWe know chemical kinetics systems can exhibit stablebehaviour – what else?
Examples are known exhibiting:
periodic behaviour,
chaotic behaviour,
switch-like behaviour, and
pattern formation.
Graduate Student Seminar: – p. 11/26
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 .
Graduate Student Seminar: – p. 12/26
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 .
Periodic behaviour is common: seasonal systems,predator-prey interactions, evolutionary games, etc.
Graduate Student Seminar: – p. 12/26
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 .
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.
Graduate Student Seminar: – p. 12/26
2. Behaviour
(Pause for clip)
Graduate Student Seminar: – p. 13/26
2. BehaviourChemical kinetics models also permitoscillatory/periodic behaviour.
Graduate Student Seminar: – p. 14/26
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.
Graduate Student Seminar: – p. 14/26
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
Graduate Student Seminar: – p. 15/26
2. BehaviourA system is said to exhibit chaotic behaviour iftrajectories/solutions are:
Graduate Student Seminar: – p. 16/26
2. BehaviourA system is said to exhibit chaotic behaviour iftrajectories/solutions are:
nonconvergent and nonperiodic,
Graduate Student Seminar: – p. 16/26
2. BehaviourA system is said to exhibit chaotic behaviour iftrajectories/solutions are:
nonconvergent and nonperiodic,
bounded, and
Graduate Student Seminar: – p. 16/26
2. BehaviourA system is said to exhibit chaotic behaviour iftrajectories/solutions are:
nonconvergent and nonperiodic,
bounded, and
extremely sensitive to initial conditions.
Graduate Student Seminar: – p. 16/26
2. BehaviourA system is said to exhibit chaotic behaviour iftrajectories/solutions are:
nonconvergent and nonperiodic,
bounded, and
extremely sensitive to initial conditions.
Chaos is not very well understood: system needs to atleast three-dimensional, nonlinear, etc.
Graduate Student Seminar: – p. 16/26
2. BehaviourA system is said to exhibit chaotic behaviour iftrajectories/solutions are:
nonconvergent and nonperiodic,
bounded, and
extremely sensitive to initial conditions.
Chaos is not very well understood: system needs to atleast three-dimensional, nonlinear, etc.
Forced mechanical systems are often chaotic.
Graduate Student Seminar: – p. 16/26
2. BehaviourChaos (or what appears to be chaos) can occur as aresult of chemical reactions.
Graduate Student Seminar: – p. 17/26
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.
Graduate Student Seminar: – p. 17/26
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
Graduate Student Seminar: – p. 18/26
2. BehaviourIf we remove some of our model assumptions, morevaried behaviour can occur.
Graduate Student Seminar: – p. 19/26
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.
Graduate Student Seminar: – p. 19/26
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)).
Graduate Student Seminar: – p. 19/26
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)).
Diffusion term
Graduate Student Seminar: – p. 19/26
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)).
Reaction term
Graduate Student Seminar: – p. 19/26
2. BehaviourHow does the behaviour of reaction-diffusion systemsdiffer from standard ones?
Graduate Student Seminar: – p. 20/26
2. BehaviourHow does the behaviour of reaction-diffusion systemsdiffer from standard ones?
Consider a locally stable system, i.e. reaction partsettles down.
Graduate Student Seminar: – p. 20/26
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.
Graduate Student Seminar: – p. 20/26
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?
Graduate Student Seminar: – p. 20/26
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? ... Well, not always.
Graduate Student Seminar: – p. 20/26
2. BehaviourConsider the system
A1
k1
�k2
Ok3−→ A2
2A1 + A2k4−→ 3A1
Graduate Student Seminar: – p. 21/26
2. BehaviourConsider the 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.
Graduate Student Seminar: – p. 21/26
2. BehaviourConsider the 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!
Graduate Student Seminar: – p. 21/26
2. Behaviour
(Pause for clip)
Graduate Student Seminar: – p. 22/26
3. ResearchMy research (so far) has dealt with systems exhibitinglocally stable dynamics.
Graduate Student Seminar: – p. 23/26
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).
Graduate Student Seminar: – p. 23/26
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.
Graduate Student Seminar: – p. 23/26
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.
Graduate Student Seminar: – p. 23/26
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.
New approaches - linearization.
Graduate Student Seminar: – p. 23/26
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
Graduate Student Seminar: – p. 24/26
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.
Graduate Student Seminar: – p. 25/26
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.
Graduate Student Seminar: – p. 26/26