Untangling Complex Systems, page 41
tion, and he eventually settled for publishing a short abstract in the unrefereed proceedings of a
conference on radiation medicine (Belousov 1958). Belousov’s recipe circulated among Moscow
laboratories, and in 1961 Anatol Zhabotinsky, a graduate student in biophysics at Moscow State
University, began to investigate the reaction. Zhabotinsky was pursuing postgraduate research on
biorhythms. His supervisor, Simon E. Shnoll, recommended that he investigate Belousov’s reaction.
Simon E. Shnoll was a professor of biochemistry at Moscow State University and one of the few
scientists aware of the Belousov’s results at that time (Kiprijanov 2016). Zhabotinsky replaced citric
acid with malonic acid and used the redox indicator ferroin to sharpen the color change during the
oscillations. In fact, ferroin gave a more drastic red-blue color change than the cerous-ceric pair.
He noticed that when, the currently called, Belousov-Zhabotinsky (BZ) reaction is performed in an
unstirred thin layer of solution, it spontaneously gives rise to a “target pattern” or spirals of oxidized
blue ferritin in an initially homogeneous red ferroin-dominated solution. Zhabotinsky presented
some of his results in a conference on “Biological and Biochemical Oscillators” that was held in
Prague in 1968. The meeting motivated many chemists of the Eastern bloc to study the reaction,
and since the proceedings were published in English (Chance et al. 1973), the news of the oscillating
reaction could also spread among the Western chemical community.
Meanwhile, Prigogine and his group in Brussels developed a model, dubbed the Brusselator, which
showed homogeneous oscillations and propagating waves like those seen in the BZ system. The
Brusselator was relevant because it demonstrated that an open chemical system kept far from equilib-
rium can exhibit spontaneous self-organization by dissipating energy to the surroundings. Prigogine
named as “dissipative structures” all the temporal oscillations and the spatial structures that emerge in
very far from equilibrium conditions. When a reaction like the BZ one, is carried out in a closed reactor,
it always reaches the equilibrium; it can exhibit only transitory oscillations as it approaches equilibrium.
1 The citric acid cycle, also known as the Krebs cycle, is a collection of key chemical reactions by which all aerobic organisms generate energy through the oxidation of acetate derived from carbohydrates, fats, and proteins. In eukaryotic cells, the citric acid cycle occurs in the matrix of the mitochondria, whereas in prokaryotic cells, it occurs in the cytoplasm.
The Emergence of Temporal Order in a Chemical Laboratory
199
Stationary oscillations require an open system. In 1977, Nicolis and Prigogine summarized the results
of the Brussels group in a book titled Self-Organization in Nonequilibrium Systems. Ilya Prigogine was
awarded the 1977 Nobel prize in chemistry for his contribution to the nonequilibrium thermodynamics.
Despite the proof offered by the abstract Brusselator model, still missing to legitimize the study
of oscillating reactions was a detailed chemical mechanism for a real oscillating reaction, like the
BZ one. A crucial step in this direction was achieved by Field, Körös, and Noyes (FKN) who suc-
ceeded in explaining the qualitative behavior of the BZ reaction using the grounding principles of
chemical kinetics and thermodynamics that rule “normal” non-oscillating chemical reactions. They
published their mechanism consisting in twenty or so elementary steps, known nowadays as FKN
mechanism, in 1972 (Field et al. 1972). The FKN mechanism was rather large and heavy for com-
putational work. A few years later, Field and Noyes managed to simplify their FKN mechanism;
they abstracted a three-variable model, the “Oregonator,” that maintained the essential features of
the entire BZ reaction (Field and Noyes 1974a).
Finally, in the 1970s, many chemists were convinced that oscillating chemical reactions were
possible, genuine, and even very appealing. From the two parent oscillating reactions, the Bray and
the BZ ones, many variants were developed, changing some of the ingredients. In 1973, two high
school chemistry teachers in San Francisco, Briggs and Rauscher, organized a surprising classroom
demonstration to draw the students’ attention by combining the malonic acid and a metal ion cata-
lyst (Mn+2), typical of the BZ reaction, with the ingredients of the Bray reaction (iodate and hydro-
gen peroxide). They showed to their students how this magic mixture could spontaneously change
color from amber to deep blue to colorless, and again to amber, repeating several cycles until the
solution remains deep blue indefinitely, due to the attainment of the equilibrium.
In the mid-1970s, two groups, one at Brandeis University headed by Epstein and Kustin and the
other at the Paul Pascal Research Center in Bordeaux led by Pacault, strove to find out a systematic
approach to discovering chemical oscillators. Two members of the Bordeaux group, De Kepper and
Boissonade, proposed an abstract model to explain how oscillations may be obtained by perturb-
ing a bistable chemical system in a Continuous-flow Stirred Tank Reactor (CSTR)2 (Boissonade
and De Kepper 1980). Then, De Kepper joined the Brandeis group, and within a few months, the
new team developed the first systematically designed oscillator: the arsenite-iodate-chlorite reac-
tion (De Kepper et al. 1981a, 1981b). The approach was then refined and exploited to discover
dozens of new chemical oscillators spanning much of the periodic table. At the turn of the twen-
tieth and twenty-first century, the implementation of chemical oscillators has been spurred by the
achievements of synthetic biology (Purcell et al. 2010). The fundamental goal of synthetic biology
is to understand the principles of biological circuitry from an engineering perspective to synthesize
biological circuits both in vivo and in vitro. Achieving this goal requires a combination of computa-
tional and experimental efforts. Much attention has been focusing on the synthesis of gene regula-
tory networks. So far, the known chemical oscillators may be grouped into five classes: (1) natural
biological oscillators within living cells (such as the circadian rhythm we studied in Chapter 7); (2) synthetic oscillators engineered into living organisms; (3) biological oscillators reconstructed
in vitro; (4) synthetic oscillators involving bio-molecules in cell-free reactions; (5) synthetic oscillators involving chemical compounds but not bio-molecules.
8.3 THE SYSTEMATIC DESIGN OF CHEMICAL OSCILLATORS
The essential requirements for any chemical system to oscillate are:
1. Far-from-equilibrium conditions;
2. Processes of positive and negative feedback that interplay with a time delay or an ade-
quately delayed negative feedback process.
2 Look back to Chapter 3 for a definition of a CSTR.
200
Untangling Complex Systems
Feedback means “self-influence.” In a chemical reaction, we have feedback when a product affects
the rate at which one or more species involved in the same mechanism are produced or consumed.
The feedback is positive when the self-influence accelerates the rate of the reaction, whereas it is
negative when it slows down the rate.
The positive feedback destabilizes the steady states. The simplest form is the direct autocatalysis.
Let us consider the elementary reaction
A kr
→
X [8.3]
which is accelerated by its own product X according to the following trimolecular autocatalytic
reaction [8.4]:
A
X
kr
+
→
′
2
3 X [8.4]
Of course, k′ r is much greater than kr . Let us suppose that the overall chemical transformation of A to X is carried out in a CSTR. The differential kinetic equation describing how the concentration
of A changes over time will be:
d [ A]
= k
2
0 ( C 0 − [ A]) − k [ ]
′ [ ][ ]
r A − kr
A X [8.5]
dt
where:
k is the flow rate,
0
C 0 is the total analytical concentration of A.
Expressing [ X] as C −
0
A
, equation [8.5] becomes
d [ A]
= k
2
0 ( C 0 − [ A]) − [ A] k
′
{
( 0
]) }
r + kr
C −[ A
[8.6]
dt
If we divide both terms of equation [8.6] by C and we indicate the ratio ([
) with
0
A]/ C 0
a, we obtain
da
= k
2
2
0 (1 − a) − a k
′
{
0 (1
) }
r + kr C
− a [8.7]
dt
At the steady state, k
2
2
′
0 1
( − a) = a k
{ r + kr C 0 1
( − a) }. Note that the term on the left represents a straight
line, whereas the term on the right is a cubic function. Depending on the values of k 0, kr, k′ r and C 0, we can have either one or two or three solutions of steady state. When the flow rate is very high, we
have only one steady state solution ( a ) characterized by a high value of a (see Figure 8.1a). Such SS
solution refers to a stable steady state because when a is larger than a , the rate of A consumption
SS
dominates, whereas when a < aSS, it is the flow rate to dominate (see the inset in Figure 8.1a). For smaller values of k , we have three solutions of steady state. Two of them represent stable steady
0
states, whereas one is unstable as we can easily infer by looking at the plot of Figure 8.1b. With a further reduction of k , only a stable steady state, characterized by a low concentration of
0
A, is
possible as we can notice by looking at Figure 8.1c. When k 0 goes to zero, also [ A] approaches zero because the CSTR becomes similar to a closed system, and the reaction can almost reach the
equilibrium, i.e., the complete depletion of A according to our mechanism. If we plot the a values
SS
as a function of k , we obtain a graph that illustrates the important phenomena of bistability and
0
hysteresis (see Figure 8.1d). The grey squares define an arm that is called thermodynamic, or equilibrium branch, because it extends from the equilibrium condition we have at zero flow rate. The
black circles define a second arm that is called the flow branch because it is traced starting from
The Emergence of Temporal Order in a Chemical Laboratory
201
s
s
te
te
Ra
0.90
0.95
1.00
Ra
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
(a)
a
(b)
a
1.0
0.9
0.8
0.7
s
0.6
te
aSS
Ra
0.5
0.4
0.3
0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30
(c)
a
(d)
k 0
FIGURE 8.1 Steady-state solutions for the chemical system described by equations [8.3], [8.4], working in
a CSTR. The cases of high, medium and low flow rates are shown in (a), (b) and (c), respectively. The steady-
state solutions ( a ) as a function of k
SS
0 are plotted in (d).
high flow rates. The blank triangles are relative to the solutions of unstable steady states. When the
system admits three possible solutions, which state the system resides in depends on from where it
comes. The system exhibits hysteresis—it remembers its history.
TRY EXERCISE 8.1
After finding out positive feedback, oscillations may arise by introducing antagonistic negative
feedback. The positive feedback pushes the system far away from the initial state, whereas the
delayed coupled negative feedback tends to restore the initial state. Therefore, the chain of reac-
tions can start again. Let us assume that species Y exerts a negative feedback effect on the system
described by equations [8.3 and 8.4]. The negative feedback effect consists in depleting X. Let us
assume that the independent differential equations describing the dynamics of the systems are:
dx = f ( x, y) = (1− x)( k ′ 2 2)
r + kr CAx
− k 0 x − k f y
dt
[8.8]
dy = 1 g( x, y)
dt
τ
where:
x = ([ X ] C )
[ ]
)
A , y = ( Y
C , the term kf y represents the negative feedback on x;
τ is a large time constant.
202
Untangling Complex Systems
f = 0
f = 0
g = 0
g = 0
f < 0
f < 0
f > 0
g < 0 g > 0
f > 0
g < 0
g > 0
y
y
P 2
P 1
P 0
(a)
x
(b)
x
f = 0
g = 0
f < 0
P
P
3′
2′
P 3′
P 4′
g < 0
g > 0
f > 0
P 1′
y
x
P 1
P 0
P 2′
P
P
P 0 P
0
1′
P 0
2
P 4′
(c)
x
(d)
Time
FIGURE 8.2 (a) Phase space for a system of two variables, x and y, originating an excitable state. The continuous arrows in (b) and (c) trace the dynamical path of the system after the perturbation highlighted by the
dashed arrows. The temporal evolution of the variable x after the perturbations is plotted in (d).
The presence of a large τ is a way of specifying that x changes much more quickly than y. We get insight on the dynamical behavior of the system if we plot the x and y nullclines (i.e., the curves
on which the rate of change of each variable is zero) on their phase space (see Figure 8.2). The
x-nullcline is S-shaped because f ( x, y) is a cubic equation, and y-nullcline intersects it once. The intersection of the two nullclines represents the steady state of the system. If we know the signs of
the derivatives on either side of each nullcline, we have a qualitative picture of the flow in the phase
space. The time derivatives of the variables change sign as we cross the nullclines. Two situations
are worthwhile to be dealt with and are presented in the next two paragraphs.
8.3.1 exciTabiliTy
The first situation is when the steady-state is on the left or right branches of the S-shaped curve.
This steady state is stable because all trajectories in its neighborhood point in toward the intersec-
tion of the two nullclines. If the system undergoes a small perturbation, for instance from P to
0
P of Figure 8.2b, the system reacts jumping back to the
with a small variation in
1
x-nullcline at P 2
y
because the latter one changes much more slowly. Finally, the system moves along the x-nullcline,
from P to the original steady state . If another perturbation is applied to the steady-state , so
2
P 0
P 0
strong to reach point P′1 (see Figure 8.2c), where f is positive, the effect of the perturbation spontaneously grows up to the right-hand branch of the x-nullcline with a small change in the value of y. As
soon as the system reaches the x-nullcline in P′2, it starts to move up the nullcline as y increases (in fact we are in a region where g > 0), until it attains the maximum in P′3. From P′3, the system jumps quickly to the left-hand branch of the x-nullcline, and finally it slides along the nullcline back to the
The Emergence of Temporal Order in a Chemical Laboratory
203
f = 0
g = 0
f < 0
g < 0
P 3
P 4
y
P 0
P 1
P 2
f > 0
P 5
g > 0
(a)
x
P 2
P 3
P 2
P 3
P
x
1
P 0
P 4
P
P
5
4
(b)
Time
FIGURE 8.3 Representation of the relaxation oscillation in the x − y phase plane in (a); temporal evolution of the x variable in (b).
