Untangling Complex Systems, page 45
regulatory gene of transposon Tn 10, whose protein product, in turn, inhibits the expression of
a third gene, cI from λ phage. Finally, CI inhibits lac I expression, completing the loop. Don’t worry if you are not familiar with the main characters of this synthetic biochemical circuit.
What is important is to note that with three repressors connected in a negative feedback loop
we can observe temporal oscillations.
TetR
λ c I
CI
lac I
LacI
tet R
FIGURE 8.12 Scheme of the “repressilator.”
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Untangling Complex Systems
The experimental temporal oscillations of the repressilator may be confirmed from a simple
model of transcriptional regulation (remember the relevant steps of a process of transcriptional
regulation we learned in Chapter 7). Three repressor-proteins, LacI ( Z ), TetR ( ) and CI (
0
Z 2
Y), and
their corresponding RNA messengers, lac I ( X), tet R ( Z ) and
), participate in transcription,
1
c I ( Z 3
translation and degradation reactions. The kinetics of the system are determined by six coupled
first-order differential equations:
dX =α
α
0 +
− kdmX
dt
Y n
1+ Kn
dZ 0 = kdmX − kdpZ
dt
0
dZ 1 =α
α
0 +
− kdmZ
dt
Z n
1
1+ 0 n
K
[8.39]
dZ 2 = kdm 1 Z − kdpZ 2
dt
dZ 3 =α
α
0 +
− kdmZ 3
dt
Z n 2
1+ Kn
dY = kdmZ 3 − k Y
dt
dp
In [8.39], we consider a simplified symmetrical case in which all three repressors are identi-
cal except for their DNA-binding specificities. The constants kdm, kdp are relative to the decays
of the mRNAs and proteins, respectively. K is the concentration of the repressor needed to
repress a promoter half-maximally and n is a Hill coefficient. α0 is the constant rate of “leaky”
initiation, which is the rate of protein production from a given promoter type in the presence
of saturating amount of repressor, whereas α is the same rate but in the absence of repressor.
For certain conditions, the model shows stable limit cycle oscillations like those shown in
Figure 8.13.
The repressilator is the biochemical counterpart of the ring oscillator in electronics. A ring oscil-
lator is a device composed of an odd number of NOT gates. A scheme is shown in Figure 8.14.
A NOT gate is a single input device that inverts the logic level of its input signal. In fact, it gives the
output logic level 1 when the input is at the logic level 0, and the output logic level 0 when the input
is at the logic level 1. In the ring oscillator, the output of the last inverter is fed back into the first.
From Figure 8.14 it is evident that the last output of the chain is the logical NOT of the first input.
This situation occurs whenever the chain contains an odd number of inverters. The final output of
the ring oscillator is asserted a finite amount of time after the first input is asserted; the feedback of
this last output to the input produces oscillations.
A ring oscillator requires electric power to operate; a repressilator requires a positive chemical
affinity to work. The general rule of thumb is that for electronic/biochemical circuits with a ring
architecture, there must be an odd number of inverters/repressors for the entire loop to be a delayed
negative feedback circuit and give oscillations. To test this rule try exercise 8.9.
The Emergence of Temporal Order in a Chemical Laboratory
219
LacI
CI
TetR
lac I
c I
tet R
70
60
10
50
Proteins
r cell
s pe 40
per cell
30
mRNA
5
20
10
0
0
190
195
200
205
Time (min)
FIGURE 8.13 Temporal oscillations of the number of mRNA and proteins participating in the repressilator.10
NOT
1
0
1
0
0
1
0
1
FIGURE 8.14 Scheme of the three-inverter ring oscillator.
10 The oscillations of the repressilator depicted in Figure 8.13 have been obtained by numerical integration of the ODEs
[8.39]. In MATLAB, the function file used was
function dy = repressilator(t,y)
dy = zeros(6,1);
a0 = 0.1;
a = 100;
n = 2.5;
kdp = 5;
kdm = 1;
K = 1;
dy(1) = a0+[a/(1+((y(6))^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);
dy(5) = a0+[a/(1+((y(4))^n/K^n))]-kdm*y(5);
dy(6) = kdm*y(5)-kdp*y(6);
The script file was
[t,y]=ode45(‘repressilator’,[0 1000], [5 5 0 0 0 0]);
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Untangling Complex Systems
8.5 OVERVIEW AND HINTS FOR FURTHER READING
In this chapter, we have learned a classification of the homogeneous chemical oscillatory reactions,
which is based on the types of involved feedback processes. A schematic view of the proposed five
“primary oscillators” is shown in Figure 8.15. The schemes report only the main processes involv-
ing the essential species X, Y, and Z.
Scheme (a) represents the coproduct autocontrol model where we have autocatalysis for
X and a negative feedback action exerted by Y on X, which is delayed because Y is produced by Z.
Scheme (b) describes the “modified predator-prey oscillator.” In (b), we have two auto-
catalytic processes, for X and Y, and a negative feedback action played by Z on either
Y or X.
Scheme (c) regards the flow-control model. X is the autocatalytic species that requires the
inflow of Z; Y is the inhibitor of X. In a pH oscillator, X is H+, Z is both Ox and Red A
(see equation [8.32]), and Y is either Red (see equation [8.33]) or a second oxidized state
B
of the reductant Red .
A
Scheme (d) depicts a “flow-control oscillator” coupled to an equilibrium reaction through
the oscillating concentration of species X. The equilibrium A B swings back and forth
synchronized with the oscillations of X.
Finally, scheme (e) represents the delayed negative feedback model wherein X is continuously
fed from outside.
The list of the five “primary oscillators” presented in this chapter might not be exhaustive. Other
ways might exist in which homogeneous chemical oscillations arise. For example, an oscillating
chemical reaction may ground on more than one “primary oscillator.”
Z
Z
X
Y
X
Y
(a)
(b)
Z
Z
X
Y
X
Y
(c)
(d)
A
B
Z
(e)
X
Y
FIGURE 8.15 Schematic representation of the five “primary oscillators.” Scheme (a) is for coproduct
autocontrol model; (b) for the modified prey-predator model; scheme (c) for the flow-control model; scheme
(d) for the “equilibrium-coupled-to-a-flow-control oscillator” model; and (e) for the delayed negative feed-
back model.
The Emergence of Temporal Order in a Chemical Laboratory
221
Finally, the classification of oscillating chemical reactions proposed in this book is just one
of many others. For instance, Eiswirth et al. (1991) offered another classification based on the
stoichiometric network analysis of the chemical mechanisms. Readers who want to know more
about such classification of oscillatory reactions can read a review written by Ross and Vlad
(1999).
To check if you have a clear picture of the different “primary oscillators” presented in this
chapter, try exercise 8.10.
Hopefully, the study of the main features of the known chemical oscillators will favor the design
of new families of periodic reactions. For this purpose, it is worthwhile reading the review about the
mechanisms of autocatalysis by Bissette and Fletcher (2013).
BOX 8.2 THE IMPORTANCE OF MIXING A CHEMICAL REACTION
It is really hard to have perfect mixing of a chemical reaction. However, we always need to
strive to achieve it. In fact, it guarantees that the concentration of a species is the same every-
where into the reactor. The effect of imperfect mixing in the presence of nonlinear kinet-
ics can be awesome (for a quantitative proof of this statement, try to solve exercise 9.2 in
Chapter 9). The effects of imperfect mixing are exacerbated by those reactions that involve
more than one phase. If we have liquid and solid phases, the mixing efficiency determines
the rate at which fresh material from the liquid phase reaches the surface. On the other hand,
if a reaction, occurring in liquid phase, produces a gas, or a gas in the air can participate to
the chemical transformation, the rate of mixing exerts a concrete effect on the rate of the
chemical reaction. For example, in the case of the BZ reaction, a high rate of stirring tends
to increase (I) the rate at which the gaseous bromine generated by the reaction leaves the
solution, and (II) the rate at which oxygen can oxidize some of the organic intermediates is
taken up by the solution. Since both bromine and oxygen play key roles in the BZ reaction, it
is clear that changes in stirring rate affect its dynamical behavior. In fact, there are two sorts
of mixing: macro-mixing and micro-mixing. They occur at two different spatial scales and
at two different time scales. Macro-mixing guarantees a homogeneous system at the mac-
roscopic level. A solution is homogeneous at the macroscopic level if the average composi-
tions of its macroscopic portions (each having a volume of roughly one cubic centimeter) are
identical. The characteristic time needed for a successful macro-mixing can be determined
by experiments with colored species. Micro-mixing is the process that guarantees homo-
geneity at the molecular level. For instance, if we prepare a diluted solution of species X in
the solvent L, we may consider the micro-mixing as complete when all the X molecules are
surrounded by a shell consisting of at least a layer of L molecules. This situation does not
necessarily hold for a solution that is homogeneous at the macroscopic level. Micro-mixing
requires much more time than macro-mixing because it occurs through molecular diffu-
sion. Model calculations explain the observation that in a number of systems, very different
outputs are obtained if reactants are premixed in a small mixing chamber before entering
a CSTR (Puhl and Nicolis 1987). There are even experimental proofs demonstrating that
imperfect micro-mixing leads to shifts in bifurcation points and the appearance of aperiodic,
transient oscillations in bistable systems (read, e.g., Hu et al. 2010). For more information
about stirring and mixing effects, read Chapter 15 of the fascinating book by Epstein and
Pojman (1998).
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Untangling Complex Systems
8.6 KEY QUESTIONS
• Why did it take time to accept oscillations in chemistry?
• What does feedback action mean in chemistry?
• Which are the conditions favorable for generating chemical oscillations?
• Which are the features of an excitable system?
• Describe the features of the “primary oscillators.”
• Why is so important to mix properly reactive solutions?
8.7 KEY WORDS
Positive and negative feedback; Time delay; Excitability; Oscillations; “Primary Oscillators.”
8.8 EXERCISES
8.1. Considering the elementary reactions [8.3], [8.4] and the flow rate k [
0 X], find the possible
steady-state solutions for X. Build the plot of the rates of the X formation and depletion as
functions of ([ X ]/ C )
A = x, where C is the analytical concentration of A flowing inside the
A
CSTR with a flow rate constant k . Assuming that k
−1
2
1
r = 1 t
, ′ r =
− −
k
100 M t and CA = 1 M
0
,
find at least three values of k giving (I) a steady-state solution characterized by low [
,
0
X]ss
(II) a steady-state solution characterized by large [ X] , (III) three steady-state solutions.
ss
8.2. The goals of this exercise are to (I) observe the phenomenon of spontaneous temporal self-
organization in a chemical laboratory, (II) determine the dependence of the period (T) of
the oscillations of the Belousov-Zhabotinsky reaction on the concentration of KBrO , and
3
(III) become aware of errors that can be made in an experiment carried out in a laboratory.
Go to the wet laboratory, wear a white coat, gloves, safety glasses and prepare the follow-
ing solutions using deionized water as the solvent.
• Solution A: H SO (sulfuric acid) 0.76 M.
2
4
• Solution B: KBrO (potassium bromate) 0.35 M.
3
• Solution C: CH (COOH) (malonic acid) 1.2 M.
2
2
• Solution D: Ce(SO ) (cerium(IV) sulphate) 0.005 M in an aqueous solution containing
4 2
H SO 0.76 M.
2
4
• Solution E: Ferroin (tris(1,10-phenanthroline)iron(III) sulphate) 0.025 M.
Insert a magnetic stir bar into a dry 25 mL glass beaker and place the beaker on a stirrer.
Using pipettes and/or automatic pipettor and tips, add the following amounts of stock solu-
tions into the beaker: 2 mL of A, 2 mL of B, 2 mL of C, 2 mL of D, and 200 μL of E. After
adding all the reagents and waiting for a few minutes under stirring, transfer 2.5 mL of the
reactive solution from the beaker inside a cuvette (with 1 cm as optical path length) having
a tiny stir bar. Place the cuvette in a UV-visible spectrophotometer, and maintain it under
stirring. If you have a spectrophotometer whose detector is a diode array (that permits
simultaneous measurements at multiple wavelengths), fix the spectral range in the inter-
val 200–800 nm, and record how the absorption spectrum of the solution changes over
time (by recording one spectrum every 3 seconds). Alternatively, if you have a traditional
double beam spectrophotometer, fix the monochromator at one wavelength (I suggest you
choose 511 nm), and record how the absorbance of the solution at that specific wavelength
changes over time. Monitor the spectral evolution of the system over 40′. By using the trend
of absorbance (at a specific wavelength) as a function of time, determine the periods of the
oscillations and calculate its average value ( T 1) and the standard deviation of the average
(σ T ). Now, repeat the experiment by changing only the amount of solution B: instead of
1
2 mL of solution B, add 1.5 mL of it plus 0,5 mL of deionized water (the addition of water is
needed to maintain constant the total volume of the solution). Record the spectral evolution
of the new BZ reaction and determine the new average period ( T 2) and the new standard
The Emergence of Temporal Order in a Chemical Laboratory
223
deviation of the average period (σ T ). Finally, repeat the entire procedure after changing
2
only the amount of solution B. Use 1 mL of B + 1 mL of deionized water. Determine the
new average period ( T 3) and its uncertainty.
The final goal of this experiment is the determination of the exponent n of the following
relation: T
n
∝ KBrO3 . Build a log-log plot for its determination.
Warning: for the success of this experiment it is essential to mix the reagents properly.
For more information about the importance of stirring solutions, read Box 2 of this chapter.
8.3. The five steps of the Oregonator model [8.20] represent the following five elementary steps
of the FKN mechanism (Pellitero et al. 2013):
BrO−
−
+
1
k
3 + Br + 2H →
HBrO2 + HOBr
HBrO
−
+
k 2
2 + Br + H →
2HOBr
BrO−
+
3
k
4+
3 + HBrrO2 + H →
2HBrO2 + Ce
2HBrO
