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Stoichiometric Network Analysis and Associated Dimensionless Kinetic
Equations. Application to a Model of the Bray#Liebhafsky Reaction
Guy Schmitz, Ljiljana Z. Kolar-Anic#, Slobodan R. Anic#, and Z#eljko D. C#upic#
J. Phys. Chem. A, 2008, 112 (51), 13452-13457 • DOI: 10.1021/jp8056674 • Publication Date (Web): 05 December 2008
Downloaded from http://pubs.acs.org on December 22, 2008
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Stoichiometric Network Analysis and Associated Dimensionless Kinetic Equations.
Application to a Model of the Bray-Liebhafsky Reaction
Guy Schmitz,†Ljiljana Z. Kolar-Anic ´,‡Slobodan R. Anic ´,‡and Zˇeljko D. Cˇupic ´*,§
Faculté des Sciences Appliquées, UniVersté Libre de Bruxelles, CP165/63, AV. F. RooseVelt 50,
1050 Bruxelles, Belgium, Faculty of Physical Chemistry, UniVersity of Belgrade, P.O.Box 47,
11000 Belgrade, Serbia, and Department of Catalysis and Chemical Engineering, IChTM,
UniVersity of Belgrade, Njegos ˇeVa 12, 11000 Belgrade, Serbia
ReceiVed: June 27, 2008; ReVised Manuscript ReceiVed: NoVember 3, 2008
The stoichiometric network analysis (SNA) introduced by B. L. Clarke is applied to a simplified model of the
complex oscillating Bray-Liebhafsky reaction under batch conditions, which was not examined by this method
earlier. This powerful method for the analysis of steady-states stability is also used to transform the classical
differential equations into dimensionless equations. This transformation is easy and leads to a form of the
equations combining the advantages of classical dimensionless equations with the advantages of the SNA.
The used dimensionless parameters have orders of magnitude given by the experimental information about
concentrations and currents. This simplifies greatly the study of the slow manifold and shows which parameters
are essential for controlling its shape and consequently have an important influence on the trajectories. The
effectiveness of these equations is illustrated on two examples: the study of the bifurcations points and a
simple sensitivity analysis, different from the classical one, more based on the chemistry of the studied system.
The stoichiometric network analysis (SNA) introduced by
Clarke1is a powerful method for the examination of complex
systems and for the stability analysis of steady-states. It is based
on the definition of new variables and parameters leading to
general equations of motion with several advantages over the
classical ones. This paper discusses its application to complex
reactions under batch conditions and show how important results
can be easily obtained. Moreover, the SNA equations can be
written in a dimensionless form that combines the advantages
of this kind of equations with the advantages of the SNA. They
simplify greatly not only the stability analysis but also the study
of the state space properties. Our topic is illustrated using one
variant of the model of the oscillating Bray-Liebhafsky
reaction.2,3It was selected because it reproduces the main
features of this reaction, allowing a connection between the
theory and a real oscillating system, but avoids complications
related with kinetics details clearly outside the scope of the
present work. This paper shows how to apply the SNA and
associated dimensionless equations in a batch reactor, referring
the readers to the original papers for the underlying theory.1
The Bray-Liebhafsky reaction is the decomposition (D) of
hydrogen peroxide in the presence of iodate and hydrogen ions.
This decomposition is the result of two complex reactions in
which hydrogen peroxide acts as either a reducing (R) or an
oxidizing (O) agent.
The sum of reactions R and O gives reaction D. When the rates
of these two reactions are equal, the decomposition of hydrogen
peroxide is monotonous. However, under some conditions
discussed in this paper, the reactions R and O dominate
alternately, resulting in a cascading consumption of hydrogen
peroxide and an oscillatory evolution of the intermediates.2-5
These reactions are themselves complex and numerous inves-
tigations of the role of possible intermediates appeared.3-9As
our aim is to show the SNA usefulness when studying such
systems, we will not discuss details requiring more complex
models and will use the simple model presented in Table 1.
The reactions numbers are taken from our earlier publications.10-16
The Extreme Currents
The fundamental idea of the SNA is to express the rates of
reactions using new sets of variables and parameters.1The
variables are the ratios between the actual concentrations and
their values at a steady-state and the parameters are some rates
at this steady-state. Thus we need to define the steady-state
chosen as reference. Here it is the smooth decomposition of
hydrogen peroxide (D) called “the disproportionation steady-
state”. The decomposition of hydrogen peroxide under batch
conditions is slow at the considered time scale, and therefore
we can assume that the concentration of hydrogen peroxide, as
well as iodate and acid concentrations, is constant during the
time of interest. These compounds are called external. The
evolutions of the concentrations of the other compounds
appearing in Table 1 (I2, I-, IOH, IO2H, and I2O) occur on a
faster time scale and these compounds are called internal.1
At the disproportionation steady state, we have specific
relations between the rates of the steps of the model, where
steps means complete stoichiometric form of reactions with
* Corresponding author. Telephone: (+) 381 11 2630 213. Fax: (+) 381
11 2637 977. E-mail: email@example.com.
†Faculte des Sciences Aplique, Universte Libre de Bruxelles.
‡Faculty of Physical Chemistry, University of Belgrade.
§Department of Catalysis and Chemical Engineering, IChTM, University
J. Phys. Chem. A 2008, 112, 13452–13457
10.1021/jp8056674 CCC: $40.75
2008 American Chemical Society
Published on Web 12/05/2008
power law kinetics, even if they are not elementary. The SNA
express these relations in the rates space instead of the
concentrations space. Clarke1has proved that any set of rates
values satisfying the steady state equations can be represented
in the rates space as a linear combination of vectors with non-
negative coefficients, named the extreme currents Ei.17The
extreme currents can be interpreted geometrically as the edges
of the corresponding cone in the reaction rates space.1,18-20
Denoting by S the matrix of the stoichiometric coefficients in
the model and by s its rank, all the extreme currents are obtained
looking for all nontrivial solutions with no negative component
of the s independent equations S Ei) 0. The Eivectors are
determined only up to a positive factor that is usually chosen
to get round numbers. The set of extreme currents for a given
factor is unique and is represented by the E matrix where each
row comes from one step of the model. A MATLAB program
Ematrix.m calculating this matrix is given in the Supporting
Information. For the model in Table 1, we have the following
In the SNA, the rates at the steady state rssare expressed as
linear combinations of the columns of E, rss) E j, giving in
where the subscripts ss denote the values of the concentrations
and rates at the disproportionation steady state. The concentra-
tions of the external compounds are included in the rate
constants. An essential characteristic of the theory is that the
components of j, the current rates ji,21,22are non-negative.
Clarke23has underlined that, since stoichiometry is the most
essential element of a reaction network, and since the matrix E
is determined solely by stoichiometry, E plays a fundamental
role in the theory of reaction networks. In our example, the
sum (R2) + (R5) + (R6) and the sum (R-1) + (R2) + (R6) +
(R8) give the global reaction (D). Thus, the extreme currents
E4and E5are two pathways leading to the observed stoichi-
ometry. Therefore, we call them stoichiometric currents. On
the other hand, reversible reactions are represented in the SNA
by two reactions giving columns of E with no net contribution
to the stoichiometry such as (R1) + (R-1), (R3) + (R-3) and
(R4) + (R-4) giving E1, E2and E3in our example. These
currents are characteristic of the reversibility of the correspond-
ing reactions and we call them exchange currents. By analogy
with a terminology used in electrochemistry, a high value of
exchange current is characteristic of a highly reversible reaction.
The name equilibrium currents used formerly1could be confus-
ing because the corresponding reactions are not necessarily at
equilibrium. For example, (r+1)ss) j1is not equal to (r-1)ss)
j1+ j5. Our terminology underlines an important aspect of the
SNA discussed hereafter, the relation between the currents and
the experimental information.
Rate Equations and Stability Analysis
The stability of a steady state can be analyzed by linearization
of the stoichiometric network general equation of motion about
this steady state.1,20,24-27The SNA theory simplifies greatly this
analysis using the ji’s as parameters and the ratios between the
actual concentrations and their values at the steady-state as
variables.1Thus, we define x1) [I2]/[I2]ss, x2) [I-]/[I-]ss, x3)
[IOH]/[IOH]ss, x4) [IO2H]/[IO2H]ss, x5) [I2O]/[I2O]ss, write
the rate equations as functions of these variables, for example
r+1) k+1[I-]ssx2) j1x2, and obtain the equations of motion in
the following form.
The matrix of currents V(j) is defined as
V(j))-S(diag E j)KT
where KTis the transpose of the matrix of the order of reactions
K. Since rss) E j, diag E j is a diagonal matrix whose elements
are the reaction rates at the steady states. The stability depends
on the sign of the real part of the eigenvalues of the matrix M
) -(diag h) V(j) where diag h is a diagonal matrix whose
elements are the reciprocals steady state concentrations (hi)
1/[Xi]ss, Xi) I2, I-, HIO, HIO2, I2O). Although the stability
analysis by this method is much simpler than by direct
linearization of the kinetic equations, it becomes limited for real
models by the number and size of the required polynomials and
TABLE 1: Model of the Bray-Liebhafsky Reaction
IO3-+ I-+ 2 H+h IOH + IO2H
IO2H + I-+ H+f I2O + H2O
I2O + H2O h 2 IOH
IOH + I-+ H+h I2+ H2O
IOH + H2O2f I-+ H++ O2+ H2O
I2O + H2O2f IOH + IO2H
IO3-+ H++ H2O2f IO2H + O2+ H2O
J. Phys. Chem. A, Vol. 112, No. 51, 2008 13453
the following sufficient instability condition is used: If at least
one negative term exists in a principal minor of V(j), the steady
state is unstable for some values of the parameters. The
MATLAB program Clarkestab.m given in the electronic supple-
ment computes the coefficients of the jiin V(j) and uses symbolic
variables to locate all the destabilizing terms. For the model in
Table 1, it reveals three negative terms proving that the steady
state can be unstable.
When we have found that a steady state can be unstable, we
would like to know when it is actually unstable. This remains
a difficult task unless we use an a priori knowledge of the orders
of magnitude of the parameters to locate the dominating negative
terms. For the BL reaction, we know that the iodine concentra-
tion is much larger than the concentration of the other internal
compounds. This means that h1) 1/[I2]ssis much smaller than
the other hiand that we can discard all the terms multiplied by
h1in the development of the characteristic equation. We also
know that reaction (R4) remains nearly at equilibrium, meaning
that its exchange current rate j3is much larger than the values
of stoichiometric current rates. This reduces drastically the
number of terms in the instability condition given by the
program Clarkestab.m, leaving
The identification of the dominating negative terms based on
experimentally known orders of magnitude is a striking advan-
tage of the SNA.
The expressions (2) of the kinetic equations throw some light
on classical concepts in chemical kinetics, quasi-steady state
approximation, nullclines and slow manifold. They are closely
related and their use rests on the relative magnitude of some
concentrations. We illustrate this after a transformation of eqs
2. The variables are already dimensionless and, since the number
of dimensionless parameters is always less than classical ones,28
it is useful to define dimensionless parameters and dimensionless
time based on a proper choice of the reference values:
We take [I2]ssas the reference concentration because it is larger
than the concentrations of the other internal compounds [I-]ss,
[IOH]ss, [IO2H]ss, and [I2O]ss, such that c2, c3, c4, and c5are
small parameters. We take (j4+ j5) as reference current rate
because it is the rate of reaction (D) at the catalytic dispropor-
tionation steady state. We will see that this choice simplifies
the relations between the dimensionless parameters and the
properties of the chemical system. The corresponding dimen-
sionless time is τ ) t × (j4 + j5)/[I2]ss. Introducing these
parameters, the equations of motion (2) take the dimensionless
These equations are similar to the classical SNA equations (eqs
2) and have all their advantages together with the ones coming
from their dimensionless form resulting from Pi theorem.28
Relations between the dimensionless parameters and the kinetic
constants are easily derived from eqs 1. The following study of
the slow manifold and the sensitivity analysis show the
advantages of these equations over the classical ones.
Slow Manifold and Time Evolutions
The iodine concentration is much larger than the concentra-
tions of the other internal compounds and taking [I2]ss as
reference concentration in eqs 5 simplifies greatly the study of
the motion. The last four equations have the form ? dxi/dt )
fi(x, c) where ? is small, so that the trajectories in the state space
are strongly attracted by the nullclines fi(x, c) ) 0. The equations
of the four nullclines define the slow manifold. It is one-
dimensional in the five-dimensional state space. This is a first
advantage of eqs 5 over the classical ones: they give directly
simple equations of the slow manifold and show that the main
effect of the parameters c2, c3, c4, and c5is to determine its
attracting power. The equations (eqs 5) simplify also greatly
the study of the shape of the slow manifold. They reveal that
this shape depends only on the four parameters R, ?, γ, and δ
and not on the individual values of the ten rate constants.
At this point, we would like to underline the relation between
the concepts of nullclines or slow manifold and the classical
quasi-steady state approximation. The equations of the nullclines
are identical to the equations we would write using the quasi-
steady state approximation but their meaning is clearer and this
approach reveal a frequent misunderstanding in chemical
kinetics. Taking for example the concentration [I-], the steady-
state approximation does not mean that d[I-]/dt is equal to zero,
what is clearly untrue. The slow manifold approach gives the
exact condition: c2is small.
Figure 1 shows examples of time evolutions calculated by
numerical integration of eqs 5 and of slow manifolds calculated
analytically when the disproportionation steady state is stable
or unstable. The instability condition (eq 6) is obtained replacing
the jiin eq 4 with the dimensionless parameters. The simplifica-
tion is striking.
The parameter R, equal to k+3/k6, is the main parameter
controlling the stability. The orders of magnitude of the other
parameters are dictated by experimentally known orders of
magnitude. The equilibrium of reaction (R4) being only weakly
disturbed during the Bray-Liebhafsky reaction, its exchange
current rate j3is much larger than the sum of current rates j4+
j5. The value of γ ) j3/(j4+ j5) must be much larger than one
and has a minor influence on the shape of the calculated curves.
The current rate j3does not appear in eq 4 and the parameter γ
does not appear in eq 6 for the same reason. On the contrary,
the equilibrium of reaction (R1) is strongly disturbed, its
exchange current rate j1must be relatively small and ? ) j1/ (j4
J. Phys. Chem. A, Vol. 112, No. 51, 2008
Schmitz et al.
+ j5) must be small. The parameter δ, equal to j5/(j4+ j5), is
the relative contribution of reaction (R8) to reaction D. The
values of c2and c3used in Figure 1 are based on experimentally
known orders of magnitude of the concentrations while the
values of c4and c5are unknown. However, as explained before,
their values have nearly no influence on the calculated curves
as long as they remain small.
Parts a and b of Figure 1 illustrate the motion when the
disproportionation steady state is stable. As expected, the
trajectories calculated numerically are close to the slow manifold
calculated analytically. The evolution begins with reaction R
because the initial value of x1(iodine) is zero. When the initial
value of x1is larger than one, the evolution begins with reaction
O. Parts c and d of Figure 1 illustrate the motion when the
disproportionation steady state is unstable. The x1oscillations
obtained by numerical integration of the eqs 5 are very similar
to the experimental ones.4,5,29The oscillations can be divided
into two periods separated by transition points. During the period
R the rate of reaction R is larger than the rate of reaction O and
x1 increases; during the period O, it is the opposite and x1
decreases. Figure 1d shows the projections of the slow manifold
and of the trajectory from the five dimensional state space onto
the x3- x1plane explaining the transition points T1and T2.
The slow manifold has an S shape with upper and lower stable
branches and an intermediate unstable branch between points
T1and T2. The calculated trajectory follows the lower branch
until it reaches point T1. At this point, dx1/dt is still positive
and the trajectory must leave the slow manifold. It jumps quickly
(more or less quickly depending on the smallness of the ci) to
the stable upper branch where dx1/dt is negative and follows
this branch to point T2. Then it must again leave the slow
manifold, jumps to its lower branch and closes the limit cycle.
The Bifurcation Points
We have studied the transitions between stability and instabil-
ity using R ) j2/(j4+ j5) ) k+3/k6as bifurcation parameter.
When the steady state is stable and is far from a bifurcation,
the slow manifold has a shape like in Figure 1b leading to the
smooth disproportionation. When the disproportionation steady
state is unstable, the slow manifold has an S shape as in Figure
1d leading to oscillations. Between these two situations surpris-
ing behaviours are observed. At the transitions between stability
and instability, the slow manifold has still an S shape and the
steady state is close to one of the points T1or T2. The first case
is favored by low δ values and will be illustrated by the example
in Figure 2. The second case is favored by high δ values and
will be illustrated by the example in Figure 3. The parameter δ
has always a stabilizing effect and the meaning of “high δ
values” is defined by the instability condition 6. As δ ) j5/(j4
+j5) < 1, its left-hand side is positive and the instability
condition 6 cannot be satisfied if its right-hand side is negative,
that is, if δ2- 8δ + 1 < 0 or δ > 0.127. The parameter ? has
also a stabilizing effect and the amplitude of the studied
phenomena increases when ? decreases because the left-hand
side term decreases. The other parameters were chosen as
Figure 2 gives an example of bifurcation when the steady
state is close to T1. A supercritical Hopf bifurcation is found
by numerical simulations for R ) 4.0473 × 10-3, very close to
R ) 4.0465 × 10-3given by condition 6. The small difference
comes from the numerous small terms neglected in the
characteristic equation. When R increases over R ) 4.0473 ×
10-3the limit cycle born at the Hopf bifurcation grows and a
situation similar to that in Figure 1d is finally obtained.
However, this growth can be more or less fast depending on
the values of the other parameters. Figure 2a gives an example
of abrupt increase of the oscillations amplitude near RC )
4.04897475 × 10-3, known as a canard explosion.30-33This
phenomenon is characteristic of systems with very different time
scales, which is our case, and can be continuous or not.30The
insert in Figure 2a suggests that it is discontinuous for the used
values of the parameters. Such canard explosions are often
associated with excitability,31and Figure 2b shows the behavior
for R ) 4.0489 × 10-3, just before the explosion. The unstable
Figure 1. (a, b) Time evolution for R ) 0.2, ? ) 0.01, γ ) 100, δ ) 0.05, c2) 2 × 10-4, c3) 5 × 10-4,c4) 2 × 10-4, and c5) 2 × 10-6and
projections of the trajectory (s) and the slow manifold (+ + +) onto the x3-x1plane. The disproportionation steady state (b) is stable. (c, d) Same
as parts a and b, except R ) 2. The disproportionation steady state (b) is unstable.
J. Phys. Chem. A, Vol. 112, No. 51, 2008 13455
steady state is surrounded by a very small limit cycle and the
system is highly excitable because the trajectory is strongly
attracted by the slow manifold. For the chosen initial values,
the system makes a large excursion near the slow manifold
before cycling toward the small limit cycle. This excursion
announces the large limit cycle that will appear at the canard
Figure 3 shows an example of bifurcation when the steady
state is close to T2. When R increases from 8.2710 to 8.2711,
a large limit cycle seems to appear from nowhere and to
understand what happens it is easier to consider decreasing R
values. For high R values the disproportionation steady state is
unstable and surrounded by a limit cycle. Near R ) 8.284 a
subcritical Hopf bifurcation is observed. The steady state
becomes stable but is still surrounded by a limit cycle. For R <
8.284 bistability is observed as in Figure 3b. They are two basins
of attraction, one for the steady state, the other for the limit
cycle. They are separated in the five-dimensional state space
by a four-dimensional manifold called the separatrix. Figure
3a shows a section in it. When R continue to decrease a new
bifurcation is encountered: the separatrix collides with the limit
cycle near R ) 8.2711 and breaks it. The large limit cycle
disappears suddenly. For lower R values the only attractor is
the stable steady state but the system can perform large
excursions near the former limit cycle before reaching it. Let
us note that the difference between the R values at the two
bifurcations is so small that they could probably not be resolved
experimentally. Only a sudden transition between a stable steady
state and a limit cycle with a finite size would be observed.
Our example shows that this sudden transition would be an
illusion and that normal transitions lay under it.
The dimensionless equations derived from the SNA offer a
simple approach to the sensitivity analysis different from the
classical one and more based on the chemistry of the studied
system. Instead of ten rate constants and three external
concentrations (iodate, acidity and hydrogen peroxide), we have
two scaling factors, [I2]ssand the reference current rate (j4+
j5), four main parameters, R, ?, γ and δ, and four ci. The
reference concentration [I2]ssand the reference current rate (j4
+ j5) have no effect on the structure of the state space or on the
shape of the trajectories. The four main parameters determine
the stability of the steady state and the properties of the slow
manifold. The four cidetermine the attracting power of the slow
manifold. Their values affect weakly the trajectories, as long
as they remain small. Thus, the sensitivity analysis reduces to
the study of the effect of the four main dimensionless param-
eters. Moreover, if reaction (R4) is at quasi-equilibrium, the
value of γ is large, the trajectories are rather insensitive to it,
and we have only three important parameters. The SNA
parameters are related by the instability condition 4 and the
introduction of the dimensionless parameters simplifies it further
to condition 6. This offers some qualitative sensitivity analysis:
the oscillations are highly sensitive to only three parameters,
R, ? and δ, and exist only when they satisfy condition 6. Some
Figure 2. Bifurcations for ? ) 0.001, γ ) 100, δ ) 0, c2) 2 × 10-4,
c3) 5 × 10-4, c4) 2 × 10-4, c5) 2 × 10-6. (a) Amplitudes of the
x1oscillations vs R. The insert, where ∆R ) (R - RC) × 109, shows
the discontinuity. (b) Projections of the trajectory (s) and the slow
manifold (+ + +) on the x1- x3plane for R ) 4.0489 × 10-3and
initial values vector xi) [1.001 1 1 1 1].
Figure 3. Bifurcations for ? ) 0.001, δ ) 0.120 and other parameters
as in Figure 2. (a) Maximum (upper part) and minimum values of x1
during the oscillations (s) and section in the separatrix (- - -). (b)
Projections of the trajectory (s) and the slow manifold (+ + +)
showing the bistability for R ) 8.272.
J. Phys. Chem. A, Vol. 112, No. 51, 2008
Schmitz et al.
direct conclusions about the rate constants can also be obtained.
For example, the ratio R ) k+3/k6is important, not the individual
values of these two rate constants. A direct conclusion is also
obtained about reaction (R8). If k8is such that δ is very small,
this reaction has no effect and can be neglected. If k8is such
that δ is larger than 0.127, it suppress any possibility of
oscillations. After the analysis of the sensitivity to the dimen-
sionless parameters it is easy to go back to the kinetic constants
solving the eqs 1. Hence, if we want to adjust rate constants it
is easier to begin with the dimensionless parameters, to adjust
independently the shape and the scales of the calculated curves
and calculate afterward the rate constants using the eqs 1.
Since it was proposed by Clarke,1the SNA appears occasion-
ally in the literature34-40but the importance of the concept of
currents in chemical kinetics remains underestimated. We have
illustrated the main features of the method taking as an example
a simplified model of the BL reaction and proposed MATLAB
programs in the electronic supplement computing the extreme
currents E matrix and locating the destabilizing terms in matrix
of currents V(j). Then, we have shown that the usefulness of
the SNA is not limited to the stability analysis of steady states.
This powerful method for the examination of complex systems
gives equations of motion that can be written easily in a
dimensionless form simplifying greatly the study of the slow
manifold, revealing the parameters controlling its shape and
showing which parameters have or have not a noticeable
influence on the trajectories. Moreover, the orders of magnitude
of most SNA parameters, and consequently of the derived
dimensionless parameters, are related to experimental informa-
tion. This relation is important for locating the dominating terms
in the instability condition and selecting the parameters values
used in the numerical simulations.
We have given two examples of the effectiveness of these
equations, the study of the bifurcations points and the sensitivity
analysis. The observed bifurcations are well-known theoretically
but we are not aware of another example of such a complex
behavior in a realistic chemical reaction mechanism consisting
of stoichiometric steps with integer coefficients and mass action
kinetics. It occurs in a small range of parameters values and
would have probably not been discovered by integration of the
usual differential equations using the rate constants as param-
eters. We also establish here that the dimensionless parameters
derived from the SNA offer a nonclassical sensitivity analysis
showing directly which functions of the rate constants have an
important influence on the dynamics of the studied model.
Acknowledgment. This work was partially supported by the
Ministry for Science of the Republic of Serbia (Grant Nos.
142025 and 142019).
Supporting Information Available: Matlab file Ematrix.m
used for determination of the extreme currents in the SNA theory
and files SNAStab.m and OutSNAStab.m used for the search
of the destabilizing terms. This material is available free of
charge via the Internet at http://pubs.acs.org.
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