Untangling Complex Systems, page 48
1000
1500
2000
2500
3000
3500
Emitt
Exp. 4
3.0 × 108
0.0
0
500
1000
1500
2000
2500
3000
3500
6.0 × 108
Exp. 5
3.0 × 108
0.0
0
500
1000
1500
2000
2500
3000
3500
Time (s)
FIGURE 8.24 Chemiluminescent signals recorded in the five experiments whose conditions are listed in
Table 8.9.
The time evolutions of the chemiluminescent signals are shown in Figure 8.24.
The most intense peaks have been achieved in “Exp. 3” when the concentration of
KSCN is at its maximum value. However, the oscillations die within less than one hour.
The shortest periods of the oscillations have been observed in “Exp. 1” when [CuSO ] is at
4
its maximum value. In fact, in “Exp. 1” the average period is 291 seconds. Unfortunately,
the intensity of peaks progressively decreases. In “Exp. 2” and “Exp. 4” (when the con-
centration of NaOH is at its maximum value), the “artificial firefly” lasts longer: it does
not stop to give blue chemiluminescent flashes even after one hour. When we use a very
high [H O ] in “Exp. 5,” we record only two peaks and, then, “the artificial firefly”
2
2
dies. According to these results, we may optimize the intensity and the persistence of
the chemiluminescent signal by selecting the following two combinations of reagents
concentrations.
Combination 1: [H O ]
2
2
= 0.32 M, [KSCN] = 0.03 M, [luminol] = 1.1·10 − 3 M,
[NaOH] = 0.03 M, [CuSO ]
4 = 2.4·10−4 M;
Combination 2: [H O ]
2
2 = 0.21 M, [KSCN] = 0.03 M, [luminol] = 1.2·10−3 M, [NaOH] =
0.033 M, [CuSO ]
4 = 2.4·10−4 M.
For these two conditions, the chemiluminescence becomes visible by sight in a dark room.
The recorded chemiluminescent traces are shown in Figure 8.25.
We confirm that a higher [NaOH] guarantees a longer persistence of the oscillations,
whereas a larger [H O ] assures shorter periods of oscillations at the beginning of the
2
2
reaction.
The Emergence of Temporal Order in a Chemical Laboratory
235
6.0 × 108
Combination 1
3.0 × 108
0.0 0
500
1000
1500
2000
2500
3000
3500
d intensity
6.0 × 108
Combination 2
Emitte
3.0 × 108
0.0 0
500
1000
1500
2000
2500
3000
3500
Time (s)
FIGURE 8.25 Chemiluminescent signals recorded for two new conditions of the Orbán reaction (read the
text for the details).
8.6. The system of differential equations will be:
dX = k 1 AX − k 2 XY − k 3 XZ + k 0( X
)
in − X
dt
dY = k
2 XY + k 0 ( Y
)
in − Y
dt
dZ = − k
k 0 ( Zin Z )
3 XZ +
−
dt
If we solve the system by numerical integration with the solver “ode45” of MATLAB, the
results are those shown in Figure 8.26.
Since the value of k is larger in this example than in the case of Figure 8.6, X reacts
1
more quickly, and the profile of its spikes appears sharper. On the other hand, k and are
2
k 3
much smaller in this example. Therefore, Y changes more smoothly than before.
X
Time
Z
Y
Time
Z
X
Y
(a)
Time
(b)
FIGURE 8.26 Simulated oscillations for the modified Lotka-Volterra model. Graph (a) shows the temporal
trends of X, Y, and Z. Graph (b) is the representation of the three-dimensional phase space for the system.
236
Untangling Complex Systems
300
200
Y (nM) 100
0 0
200
400
600
800
Time (min)
150
100
X (nM) 50
0 0
200
400
600
800
Time (min)
FIGURE 8.27 Temporal trends of the concentrations of predator (upper graph) and prey (lower graph).
8.7. In MATLAB, the function file for the abstract model should look like this:
function dy = DNA(t, y)
dy = zeros(2,1);
k1 = 0.003;
pol = 1.7;
A = 140;
b = 0.00006;
k2 = 0.004;
kN = 0.01;
exo = 25;
K = 34;
kp = 0.004;
dy(1) = (k1*pol*A*y(1)/(1+b*A*y(1)))-(k2*pol*y(1)*y(2))-(kN*exo*y(1)/
(1+y(2)/K))
dy(2) = (k2*pol*y(1)*y(2))-(kp*exo*y(2)/(1+y(2)/K))
wherein y(1) is the prey X, and y(2) is the predator Y.
The script file is:
[t, y] = ode15s( DNA
‘
’,[0
1000], [ .
3 0 .
0 ]
1 );
The results are plotted in Figure 8.27. There are evident oscillations of the biochemical
predator and prey species.
8.8. The function file to integrate the system of differential equations in MATLAB should look
like this
function dy = pHoscillator(t, y)
dy = zeros(4,1);
k1 = 50000000000;
k11 = 3000;
k2 = 3077000;
k3 = 1000000;
k4 = 11;
k5 = 2.5;
k0 = 0.001;
The Emergence of Temporal Order in a Chemical Laboratory
237
X0 = 0.06;
Y0 = 0.02;
A = 0.065;
B = 0.02;
dy(1) = -k1*y(1)*y(2)+k11*y(3)+k0*(X0-y(1));
dy(2) = -k1*y(1)*y(2)+k11*y(3)-k2*A*y(3)*y(2)+3*k3*y(3)*y(4)-
k5*B*y(2)+k0*(Y0-y(2));
dy(3) = k1*y(1)*y(2)-k11*y(3)-k2*A*y(3)*y(2)-k3*y(3)*y(4)-k0*y(3);
dy(4) = k2*A*y(3)*y(2)-k3*y(3)*y(4)-k4*y(4)-k0*y(4);
wherein y(1) = RedA, y(2) = H, y(3) = HRedA, and y(4) = Y.
The script file should look like the following:
[t, y] = ode15s(‘pHoscillator’,[0 1000], [0.065 0.025 0.001 0.001]);
The results of the simulation are plotted in Figure 8.28. It is evident that the mechanistic
model of the pH oscillator reproduces quite well the experimental oscillations.
8.9. If we use MATLAB to solve this exercise, the function file to integrate the system of ordi-
nary differential equations should look like the following one:
function dy = tworepressors(t, y)
dy = zeros(4,1);
a0 = 0.1;
a = 100;
n = 2.5;
kdp = 5;
kdm = 1;
K = 1;
dy(1) = a0+[a/(1+((y(4))^n/K^n))]-kdm*y(1);
dy(2) = kdm*y(1)-kdp*y(2);
dy(3) = a0+[a/(1+((y(2))^n/K^n))]-kdm*y(3);
dy(4) = kdm*y(3)-kdp*y(4);
Such file is valid for a circuit having two repressors linked in a ring. The results are shown
in Figure 8.29. The system reaches a fixed point where the concentrations of the second
mRNA and protein vanish.
When the number of repressors is three, we have the repressilator we studied in para-
graph 8.4.5. The results are depicted in Figure 8.30. We do have oscillations and the period
is 7 minutes and 24 seconds.
0.0002
[ Y] 0.0000
200
400
600
Time (s)
] A 0.002
8
7
[ HRed 0.000
200
400
600
6
Time (s)
0.002
]
pH 5
[ H 0.000
4
200
400
600
Time (s)
3
] 0.003
d A
2
[ Re 0.000
1
200
400
600
0
200
400
600
800
1000
(a)
Time (s)
(b)
Time (s)
FIGURE 8.28 Time evolution of Y, HRedA, H, and Red
A in graph (a); pH oscillations in graph (b).
238
Untangling Complex Systems
mRNA(1)
100
20
mRNA(2)
Protein(1)
Pr(1)
80
Protein(2)
16 Proteins
s 60
12
mRNA(1)
mRNA(2)
mRNA 40
8
20
4
Pr(2)
0
0
0
2
4
6
8
Time
FIGURE 8.29 Scheme of a biochemical circuit with two repressors linked in a ring (on the left) and its
dynamical evolution (on the right).
mRNA(1)
Protein(1)
mRNA(2)
Protein(2)
mRNA(3)
Protein(3)
70
Pr(1)
Pr(2)
60
10
s 50
Protein
mRNA(1)
mRNA(2)
mRNA(3)
40
mRNA
s
30
5
Pr(3)
20
10
0
0
0
10
20
30
Time
FIGURE 8.30 Scheme of a biochemical circuit with three repressors linked in a ring (on the left) and its
dynamical evolution (on the right).
When the network consists of four repressors, the circuit reaches a fixed point (see
Figure 8.31). In the fixed point, the concentrations of the components of repressors (2) and
(4) vanish, whereas those of repressors (1) and (3) reach not-null steady state values.
When the biochemical circuit consists of five repressors, oscillations emerge, again (see
Figure 8.32). The period of the oscillations is 14 minutes, i.e., longer than that found for the
repressilator. The larger the number of repressors, the longer the period of the oscillations.
8.10. The solution of the first part requires MATLAB or an analogous software. In case you use
MATLAB, the function file to integrate the system of differential equations should look
like this:
function dy = bacteria(t, y)
dy = zeros(4,1);
k1 = 0.7;
kf = 0.1;
K1 = 3 * 10^9;
kd = 0.004;
kE = 1.0;
dE= 0.7;
kR = 1.0;
dR = 0.7;
kA = 4 * 10^-7;
dA = 0.1;
The Emergence of Temporal Order in a Chemical Laboratory
239
dy(1) = k1*y(1)*(1-y(1)/K1)-kd*y(1)*y(4)-kf*y(1);
dy(2) = kA*y(1)-(dA+kf)*y(2);
dy(3) = kR*y(2)-dR*y(3);
dy(4) = kE*y(3)-dE*y(4);
The script file is
[t, y]=ode45(‘bacteria’,[0 500], [1 * 10^7 0 0 0]);
mRNA(1)
100
20
mRNA(2)
mRNA(3)
Protein(1)
80
mRNA(4)
Protein(2)
Protein(3)
15
Pr(1)
Pr(2)
Pr(3)
Protein(4)
Proteins
s 60
mRNA(1)
mRNA(2)
mRNA(3)
mRNA(4)
10
mRNA 40
Pr(4)
5
20
0
0
0
2
4
6
8
10
12
Time
FIGURE 8.31 Scheme of a biochemical circuit with four repressors linked in a ring (on the left) and its
dynamical evolution (on the right).
Pr(1)
Pr(2)
Pr(3)
Pr(4)
mRNA(1)
mRNA(2)
mRNA(3)
mRNA(4)
mRNA(5)
Pr(1)
Pr(5)
mRNA(1)
Pr(2)
mRNA(2)
Pr(3)
mRNA(3)
Pr(4)
mRNA(4)
Pr(5)
mRNA(5)
80
16
60
12 Proteins
s
40
8
mRNA
20
4
0
0
0
5
10
15
20
25
30
35
Time
FIGURE 8.32 Scheme of a biochemical circuit with five repressors linked in a ring (on top) and its dynami-
cal evolution (at the bottom).
240
Untangling Complex Systems
n° cells/mL
[ A]
300
2.00E+008
[ R]
[ E]
C (nM)
1.00E+008
150
n° cells/mL
0.00E+000
0
100
150
Time (hours)
FIGURE 8.33 Oscillations in the number of bacteria and in the concentrations of the molecular signal ( A), the killer gene ( R) and protein ( E).
k1
I)
N + F
2N
k−1
II)
N
kA
A + N
II)
A
dA
P
III)
A
kR
A + R
III)
R
dR
P′
IV)
R
kE
R + E
IV)
E
dE
P″
I)
N + E
kd
D + E
FIGURE 8.34 The mechanism of the proliferation of bacteria sending chemical messages that induce
inhibition.
The results are plotted in Figure 8.33.
The second part of the exercise requires to assign the synthetic biochemical circuit
to one of the “primary oscillators” we studied in this chapter. If we transform the dif-
ferential equations of the text in elementary steps, we may write the mechanism shown in
Figure 8.34.
The numbers I, II, III and IV, appearing in Figure 8.34, refer to the number of the differ-
ential equations. The first step is autocatalytic in N. Then, a bunch of steps follows, where
the bacteria produce the signal molecule A; A activates the killer gene R, and, finally, R
produces the killer enzyme E. E exerts a negative feedback on N. Such negative feedback
is delayed by the processes needed to produce E. Therefore, such synthetic biochemical
circuit may be embedded in the “coproduct autocontrolled oscillator” that is at the basis of
the Oregonator model.
The Emergence of Order
9 in Space
The scientist does not study nature because it is useful; he studies it because he delights in it,
and he delights in it because it is beautiful.
Henri Poincaré (1854–1912 AD)
9.1 INTRODUCTION
What happens if an autocatalytic process or an oscillating reaction are carried out in a medium
that is not under stirring? Well, surprising phenomena and structures appear, like those shown
in Figure 9.1. In this chapter, we will discover that systems in out-of-equilibrium conditions may self-organize even in space. Beautiful phenomena and patterns, like chemical waves, and Turing
structures and periodic precipitations (Figure 9.1) may emerge even if we start from initial spatially homogeneous conditions. The spontaneous pattern formation occurs not only in a chemistry laboratory, but also everywhere in nature (Ball 2001), in both living and azoic systems.
A spontaneous question arises: how is it possible that chemical structures emerge from a
spatially homogeneous system? Such phenomena of self-organization (or dynamic self-assembly)
are in sharp contrast with Curie’s symmetry principle, which we learned in Chapter 3. In fact, they are examples of spatial “Symmetry Breaking” transitions. The Symmetry Breaking is characteristic
of the nonlinear regime.
In this chapter, we will discover that diffusion combined with proper reaction kinetics may
contribute to the development of instabilities beyond which the system gives rise to spatial
or spatiotemporal ordered patterns. The resulting patterns have a reduced number of symme-
try elements if compared with the initial uniform states. They are puzzling because it seems
that entropy is dissipated rather than produced. But, they do not violate the Second Law of
Thermodynamics. They produce entropy that is discharged in the surrounding environment, like
any other “Dissipative Structure” that we have already known.
