Untangling Complex Systems, page 43
oscillations. This first group of reactions provides two essential intermediates: HO•
+
−
2 and Cu
SC
{
N } n.
The key steps of the second group involve the oxidation reactions of KSCN by H O :
2
2
2H
+
− →
( ) − +
2O2
SCN
OS O CN
2H O
2
[8.22]
2OS(O)CN− → OOS(O)CN− + OSCN−
In [8.22] some new intermediates are produced such as cyanosulfite, OS(O)CN−, and peroxocya-
nosulfite, OOS(O)CN− (the oxygen in parentheses is doubled bonded to sulfur). The third group is
the pivotal one for oscillations and becomes relevant when the concentrations of OS(O)CN− and
Cu+ SCN−
{
} n become high enough:
H
( ) −
( ) −
( ) •
−
2O + OS O CN + OOS O CN
→ O
2 S O CN + OH
2
OS(O)CN• + Cu+ S
{ CN−} →OS(O)CN− +Cu2+ + n SCN− [8.23]
n
OS(O)CN− + HO•
2
2 → SO −
3 + HOCN
In the mechanism of the Orbán reaction, OS(O)CN− plays like species X of the Oregonator model
[8.20], being generated auto-catalytically through formation and reduction of the OS(O)CN• radi-
cal. Y is HO•2 that depletes X and is produced by the coproduct Z = HO2Cu(I). The intermediates SO2−
2−
−
+
3 and HOCN end up as final products SO4 , HCO3 and NH4 by further oxidations. The overall
transformation is
2+
4H
−
Cu
−
+
−
2O2 + SCN
→
HSO4 + NH4 + HCO3 + H2O [8.24]
The oscillations in the Orbán reaction are not perturbed by addition of luminol (Sattar and Epstein
1990). The release of a photon by a molecule of luminol is a multi-stage process (Figure 8.5).
208
Untangling Complex Systems
NH2 O
NH
O–
NH
2 O
NH2
2
O
N–
+HO2Cu(I)
N–
–
N
O–
+O2
O O
+ N2
NH
N–
N
O–
O
O
O–
O
FIGURE 8.5 Generation of the chemiluminescent state of luminol.
First, a conjugated base of luminol is oxidized by copper(I)-peroxide complex. Then superoxide
binds to it, and finally it decomposes to give 3-aminophthalate in its excited state and nitrogen.
The excited state of 3-aminophthalate relaxes to the ground state by emitting a blue photon.4 You can experience the Orbán reaction by doing exercise 8.5.
8.4.2 The modified loTka-volTerra or PredaTor-Prey “Primary oscillaTor”
The Lotka-Volterra model proposed originally in an ecological context (see Chapter 5), suffers from the deficiency that the oscillations are sensitive to initial conditions and perturbations. In fact, the
direct coupling between the two variables, prey and predator, H and C, X and Y, and the lack of sufficient time delay give rise to an infinite set of oscillatory solutions. Fluctuations push the system
from one solution to another, resulting in irregular behavior in any real system.
By adding a third variable that reacts with one of the two autocatalytic species, we obtain a modi-
fied Lotka-Volterra model that generates a considerably stable limit cycle.
A + X
k
1→
2 X
k 2
Y + X →
Y
2
[8.25]
Z + X
k
3→
P
The third step of [8.25] replaces the first-order consumption of Y in the original Lotka-Volterra
model. If we introduce flow terms for X, Y, and Z and assume that the other elements are constant, we can solve the system of three differential equations by numerical integration and find that, for
certain contour conditions, the system oscillates as shown in Figure 8.6.
Since the reagent A is maintained constant, the first step of [8.25] is an “explosive” autocatalytic
step. The “explosion” is damped by the consumption of X by Y in the second step. This second step
is a non-explosive autocatalytic process and terminates when X is consumed. The third variable Z
reacts with X and provides the essential time delay between X and Y. Z accumulates sharply as Y
consumes X, then decreases until X regains its high value and the cycle repeats again.
TRY EXERCISE 8.6
The oscillations can also emerge when the third species, Z, depletes Y instead of X:
A + X
k
1→
2 X
Y + X
k
2→
Y
2 [8.26]
Z + Y
k
3→
P
4 The energy of a photon having 430 nm as wavelength is E = hc
= 6 63 × −
10 34
3× 8
−
10
1 4 3× −
10 7
= 4 6× −
λ ( .
)(
.
)
.
10 19
Js
ms
m
J.
The energy of one mole of such photons is about 277 KJ. This is a huge amount of energy! Oxidation reactions can release such amount of energy. However, the oxidation reaction must occur very fast and the energy produced must not be dispersed in the surrounding molecules, otherwise the product will not be electronically excited. The concerted process of peroxide rings decomposition has the double property of being highly exothermic and ultrafast. In fact, many chemiluminescent events are based on peroxide rings decomposition.
The Emergence of Temporal Order in a Chemical Laboratory
209
X
Time
Y
Z
Time
Z
X
Y
(a)
Time
(b)
FIGURE 8.6 Simulated oscillations5 for the modified Lotka-Volterra model described by the mechanism
[8.25]. In graph (a), there are the trends of X, Y, and Z versus time. In graph (b), the behavior of the system in its three-dimensional phase space.
X
Time
Z
Y
Time
Z
X
Y
(a)
Time
(b)
FIGURE 8.7 Simulated oscillations6 for the modified Lotka-Volterra model described by the scheme of reactions [8.26]. Graph (a) shows the trends of X, Y, and Z versus time. Graph (b) represents the behavior of the system in its three-dimensional phase space.
The essential time delay between the two autocatalytic steps is provided by the third reaction that
is a sort of negative feedback of a negative feedback (i.e., the second step). A graphical example of
oscillations for the system represented by [8.26] is plotted in Figure 8.7.
Whenever Z is high, Y is low, and Y becomes high only after X has reached its maximum.
A concrete example of the “modified Lotka-Volterra oscillator” is the Briggs-Rauscher reaction.
It is an oxidation of malonic acid by a mixture of hydrogen peroxide and iodate catalyzed by manga-
nese ion (Mn2+) in acidic solution. When appropriate amounts of the reactants are mixed in the presence
of starch as an indicator, the system repeats more than ten times the sequence colorless, yellow, and
blue, in a stirred batch reactor before expiring as a purplish solution with a strong odor of iodine.
5 The results shown in Figure 8.6 can be achieved by using the following values for the parameters of mechanism [8.25]: k
1 A = 0.153; k 2 = 0.055; k 3 = 0.18; k 0 = 0.1 (representing the flow rate). The concentrations of X, Y and Z constantly injected inside the reactor have been fixed to X
in = 1.8, Y in = 0.001, and Z in = 3.5. These values of the parameters and
concentrations have also been used by Franck (1985). The initial conditions were [ X ]
0 Y 0 Z 0 = [1 1 1].
6 The results depicted in Figure 8.7 have been obtained by using the following values of the parameters characterizing model [8.26]: k
1 A = 0.0055, k 2 = 0.00028, k 3 = 0.00084, k 0 = 0.005 (representing the flow rate). The concentrations of X, Y and Z constantly injected inside the reactor have been fixed to X
in = 40, Y in = 15, and Z in = 95. The initial conditions
were [ X ]
0 Y 0 Z 0 = [10 10 10].
210
Untangling Complex Systems
The mechanism of the Briggs-Rauscher reaction consists of many elementary steps (Luo and Epstein
1990) where we distinguish two autocatalytic reactions. The first involves the iodous acid HIO :7
2
HIO
−
+
2+
2
+
+
+
→
+
+
2
IO3 H
2Mn
2HIO2 2MnOH [8.27]
When the concentration of HIO is high enough, the second autocatalytic process is activated; it
2
exerts a negative feedback on HIO because it consumes iodous acid by producing iodide:
2
I− HIO2 2H2O2
2I− 2O2 H+
+
+
→
+
+
+ 2H2O [8.28]
The crucial time delay or phase shift, between the two autocatalytic reactions is provided by a
reaction that consumes iodide in analogy to the third step of [8.26]. It is a negative feedback on the
species that exerts the negative feedback on HIO :
2
HOI + I− + H+ → I2 + H2O [8.29]
The calculated profiles of [HIO ], [I−], and [HOI] reproduce those shown in Figure 8.7, wherein the 2
two autocatalytic processes are well separated in time.
Another concrete example of the modified Lotka-Volterra model is offered by DNA biochemis-
try (Fujii and Rondelez 2013).
BOX 8.1 A FEW NOTES ABOUT DNA
The building block of a DNA strand is a nucleotide whose structure is shown below.
O
–O P O
O–
Base
5’CH2
O
O
–O P O
O–
Base
2
5’ CH2
O
O
pol
O
+OH–
P O
3’
O–
CH
Base
2
OH
O
3’
OH
Note that the carbon atom labeled as 5′ is bound to the phosphate, whereas the 3′ carbon atom
has a hydroxyl group. DNA is synthesized by the enzyme polymerase (pol) that binds a new
nucleotide through the 5′-phosphate group to the 3′-hydroxyl group of the terminal nucleotide
of the strand. Two complementary strands of nucleic acids hybridize when they link to each
other through hydrogen bonds establishing between them complementary bases: Cytosine
(C, a Pyrimidine) with Guanine (G, a Purine), and Thymine (T, a Pyrimidine) with Adenine
(A, a Purine). The nicking enzyme, or nicking endonuclease (nick), recognizes specific
nucleotide sequences and cleaves only to one of the strands. The exonuclease (exo) is an
enzyme that cleaves nucleotides one at a time from the end of a DNA strand by hydrolyzing
phosphodiester bonds at either the 3′ or the 5′ end.
7 The iodous acid is produced in the first step: I− IO−3 2H+
+
+
→ HOI + HIO2.
The Emergence of Temporal Order in a Chemical Laboratory
211
Grass ( A)
Prey ( X )
Predator ( Y )
b
5′
A
5′
a
CGGCCG
A a∗
b
5′
a
CATT
3′
CGG
GTAA
T
b
GCCGGC
CATT CGGCCG
GTAA GCC
3′
a∗
b
G
3′
a∗
A(5′- b- a∗- b- a∗-3′)
X(5′- a- b-3′)
Y(5′- a- b- a∗-3′)
FIGURE 8.8 Schematic structures of the main single-stranded DNA protagonists of the prey-predator
process.
The main characters of the biochemical version of the predator-prey interaction are presented in
Figure 8.8. The prey X is a 10-base single-stranded DNA whose sequence is labeled as (5′-a-b-3′).
The food of the prey ( A) is a 20-base single-stranded DNA that is composed of four domains: two
four-base domains a* that are the complement of the four-base domain a, and two six-base domains
b that are self-complementary. Species A has the sequence 5′-b-a*-b-a*-3′. Finally, the predator Y is a palindromic 14-base single-stranded DNA with sequence 5′-a-b-a*-3′.
The mechanism of this biochemical prey and predator system is depicted in Figure 8.9. The spe-
cies A serves as a template for the growth of the prey X. Prey proliferation is a multistep process: X
hybridizes to the 3′ end of A to form the complex A: X, which is extended by a polymerase (pol) to yield the double strand A:2 X. The complex A:2 X bears a recognition site for a nicking enzyme (nick), which cuts its top strand into two equal parts, yielding two copies of X upon dehybridization. During
predation, X hybridizes over Y, and polymerase extends this adduct to form the double-stranded Y: Y.
Upon dehybridization, Y: Y yields two copies of Y. The two main characters, X and Y, are degraded by a specific exonuclease.
To ascertain the oscillatory dynamics of such biochemical system, try to solve exercise 8.7.
Prey proliferation
X
A: X
+
X
A
pol
A:2 X
+
nick
A
+
X
X
Predator proliferation
+
pol
Y
+
Y
Y
Prey consumption
X
exo
waste
Predator consumption
Y
exo
waste
FIGURE 8.9 Mechanism of the biochemical version of the predator-prey process. Harpoon-ended arrows
denote DNA strands. The elementary steps are both reversible hybridization/dehybridization reactions and
irreversible enzymatic transformations.
212
Untangling Complex Systems
8.4.3 The “flow conTrol Primary oscillaTor”
There is a third group of oscillatory reactions wherein the inflow of reagents plays a crucial role,
beyond the fundamental function of simple replenishment (Luo and Epstein 1990). Its model, named
as “primary oscillator with flow control” consists of three main processes:
X + Z
k
1→
2 X
X + Y
k
2→
P 1
[8.30]
A + X
k
3→
P 2
In the first reaction, there is a situation of antagonistic feedback. In fact, it combines a positive feed-
back action regarding X and a negative feedback action regarding Z. The variable X is, also, cross-coupled with Y independently from Z. Y is consumed first when X grows, as shown in Figure 8.10.
If the flow rate and the inflow concentrations are such that Y recovers before Z, then, a sufficient phase-shift between X and Z can be established to obtain sustained oscillations. If the chosen parameters do not guarantee the proper delay, the first reaction of [8.30] resumes too early, and the cycle is
never completed. The third process is a necessary consumption reaction for X to avoid the autocata-
lytic explosion. Concrete examples of oscillatory reactions complying with the “flow control” model
are (Luo and Epstein 1990): (1) the chlorite-iodide reaction where ClO−2 is reduced to Cl−, whereas
I− is oxidized to I ; (2) the minimal bromate reaction, where bromate, bromide, and the catalyst
2
flow in the reactor without the organic substrate; (3) an autocatalytic process that produces thiols
from thioesters and diallyl disulfides, inhibited by the inflow of acrylamide (Semenov et al. 2016);
(4) the BrO−
2−
4−
3 − SO3 − Fe(CN)6 reaction that is an example of pH oscillator. In a pH oscillator, the
concentration of protons plays a crucial role in the kinetic behavior of redox chemical reactions.
The reactants are one oxidant species (like BrO−
−
−
3 , IO3 , H2O2 , IO4) and either one or two reduc-
tants (like SO2− 2−
4−
