Untangling complex syste.., p.40
Untangling Complex Systems, page 40
• How does glycolysis originate oscillations?
• Describe the signal transduction model without feedback.
• Describe the signal transduction model with linear feedback
• Describe the signal transduction model with linear feedback and Michaelis-Menten
de-activation reaction.
• Describe the signal transduction model with ultrasensitive feedback.
• What is an epigenetic event, and which are the protagonists?
• Which are the favorable conditions to observe oscillations in an epigenetic event?
• Make examples of biological rhythms.
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Untangling Complex Systems
• Explain the mechanism of magnitude amplification.
• Which are the mechanisms for having sensitivity amplification?
• How do we adapt to stimuli?
7.8 KEY WORDS
Proteins; Metabolism; Signal Transduction Pathway; Ultrasensitivity; Hysteresis; Epigenesis;
Homeostasis; Ultradian, circadian and infradian rhythms; Amplification in sensing; Adaptation in
sensing.
7.9 HINTS FOR FURTHER READING
• Walsh et al. (2018) pinpoint eight compounds as essential ingredients of metabolism within
a cell. Seven of them use group transfer chemistry to drive otherwise unfavorable biosyn-
thetic equilibria. They are ATP (for phosphoryl transfers), acetyl-CoA and carbamoyl phos-
phate (for acyl transfers), S-adenosylmethionine (for methyl transfers), Δ2-isopentenyl-PP
(for prenyl transfers), UDP-glucose (for glucosyl transfers), and NAD(P)H/NAD(P)+ (for
electron and ADP-ribosyl transfers). The eighth key metabolite is O .
2
• The possible mechanisms for generating ultra-sensitivity can be learned by reading the
series of three papers by Ferrell and Ha (2014a–c).
• If you want to deepen the subject of biochemical oscillators and cellular rhythms, you
can read Goldbeter (1996); Goldbeter and Caplan (1976); Ferrell et al. (2011); Glass and
Mackey (1988).
• A delightful book, about the action of the human brain on physiological processes, such as
hunger, thirst, sex, and sleep, is that by Young (2012).
7.10 EXERCISES
7.1. Compare mechanism A, B, and C of Figure 7.26 and determine which one admits tr(J) = 0
as possible solution.
7.2. Consider the simplest model of a signaling event (Figure 7.10). Determine its response curve, i.e., how (χ * )
[ ] / − [ ]
R
ss changes with the ratio k 1 S
k 1 P .
7.3. Regarding the model of signaling process with linear feedback depicted in Figure 7.12, determine the expression of the maximum for the forward rate. In case the term k [
1 S] is
negligible, which is the expression of the response curve?
7.4. Build the rate balance plot and the stimulus-response curve for the signaling system with
linear feedback and a back reaction that becomes saturated (based on the Michaelis-
Menten mechanism). Such system is described by the mechanism of Figure 7.14. Assume
that the values of the parameters appearing in equation [7.23] are: k [ R]
/
f
tot = 0.7 t−1; ( kh 2 CP
[ R] )
/[ R]
tot = 0.16 t−1; KM 2
tot = 0.029. Plot the curves of the forward reaction for the following
values of the term k [
1 S]: 0; 0.04; 0.073; 0.093.
A
B
C
k 1
k 1
k 1
X
X
X
k 2
k 2
k 2
X + Y
2 Y
X + Y
Z
X
Y
k 3
k 3
k 3
Y
Y
Y
FIGURE 7.26 Three distinct mechanisms.
The Emergence of Temporal Order within a Living Being
193
7.5. Determine the properties of the stationary state for the system of two differential equations
[7.27] describing how the [mRNA] and [ Y] change over time. Assume, for simplicity, that
there is only one TF, its association coefficient is ni = 1 and its concentration [ TF] is fixed.
7.11 SOLUTIONS TO THE EXERCISES
7.1. Regarding mechanism A, the differential equations regarding X and Y are:
d [ X ] = v 1 − k 2[ X] Y[]
dt
d Y
[ ] = k 2[ X] Y[]− k 3 Y[]
dt
The terms of the Jacobian are
X
2 [
]
2 [ ]
2 [
]
2 [ ]
x = − k
Y ; X y = − k X ; Yx = k Y ; Yy = k X
− k
ss
ss
ss
ss
3.
The determinant det( J ) = k 3 k 2[ X ] ss > 0.
The trace is tr( J ) = k 2 ([ X
)
ss
] − Y
[ ss
]
− k 3 can be positive, negative or null.
Regarding mechanism B, the differential equations are
d [ X ] = v 1 − k 2[ X] Y[]
dt
d Y
[ ] = − k 2[ X] Y[]− k 3 Y[]
dt
Therefore, the terms of the Jacobian are
X x = − k 2 Y
[ s] s; X y = − k 2[ X s] s; Yx = − k 2 Y
[ s] s; Yy = − k 2[ X s] s − k 3.
The determinant det( J ) = k 3 k 2[ X ] ss > 0.
The trace is tr( J ) = − k 2 ([ X
)
ss
] + Y
[ ss
]
− k 3 can be only negative.
Regarding mechanism C, the differential equations are
d [ X ] = v 1 − k 2[ X]
dt
d Y
[ ] = k 2[ X]− k 3 Y[]
dt
The terms of the Jacobian are
X x = − k 2; X y = 0; Yx = k 2; Yy = − k 3.
The determinant det( J ) = k 3 k 2 > 0.
The trace is tr( J ) = − k 2 − k 3 can be only negative.
Only mechanism A, having an autocatalytic step, can give a null trace of Jacobian and
hence a limit cycle as a possible stable solution.
7.2. The response curve describes the dependence of (χ *)
R ss on the ratio k 1[ S ] k
/ 1
− [ P]. To obtain
such a relation, we impose equation [7.19] equal to zero. After a few simple mathematical
steps, we obtain
1
(χ ) =
R* ss
k−1[ P]
1+ k 1[ S]
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Untangling Complex Systems
1.0
0.8
0.6
) SS∗
( χ R 0.4
0.2
0.0 0 1 2 3 4 5 6 7 8 9 10
k 1[ S]/ k–1[ P]
FIGURE 7.27 Response curve for the most straightforward signaling system.
This equation is a hyperbolic function. It is shown in Figure 7.27. Note that when the
ratio k
χ
1[ S ] k
/ 1
− [ P] is 1, (
* )
R ss is equal to 0.5, that is 50% of [ R] is in its activated state.
tot
Moreover, (χ *)
R ss goes asymptotically to 1.
7.3. The equation relative to the forward rate ( v ) for a signaling system with linear feedback is
for
v
2
1 [ ]
χ * ( [ ]
1 [ ])
[ ]
for = k
S +
k R
− k S − k R
χ
R
f
tot
f
tot
R*
To find its maximum as a function of χ R*, we calculate the first derivative of vfor with
respect to χ R*. We obtain
dv for = ( k [ ] 1[ ]) 2 [ ]
f R
− k S − k R
χ
tot
f
tot
R*
dχ R*
( k [ ] 1[ ])
f R
− k S
It is equal to zero when χ
tot
=
.
R*
2 k [ ]
f R tot
In
case
k [ R]
[ S], χ ≈ 0 5.
f
tot >> k 1
R*
.
When
k [
1 S] is negligible when compared to the other terms of equation [7.22], the over-
all rate of R* formation becomes:
dχ
R* ≈ χ ( [ ] ) − [ ]
2 − −1[ ]
*
k R
k R
χ * k P χ
dt
R
f
tot
f
tot
R
R*
In such case, the equation of the response curve is
k−1[ P]
χ ≈1−
R*
k [ ]
f R tot
7.4. In the case of a signaling system with a linear feedback and a back reaction with satura-
tion (based on a Michaelis-Menten mechanism), a continuous growth of the stimulus from
zero value, shifts the system from a stable fixed point to a new stable one in an irreversible
The Emergence of Temporal Order within a Living Being
195
0.8
0.7
0.6
0.16
0.5
χR∗ 0.4
te
0.3
0.08
Ra
0.2
0.1
0.0
0.00
0.0
0.2
0.4
0.6
0.8
1.0
0.00
0.02
0.04
0.06
0.08
0.10
(a)
χR∗
(b)
k 1[ S]
FIGURE 7.28 Rate balance plot (a) and stimulus-response curve (b) for a signaling system with linear feed-
back and a Michaelis-Menten back reaction. In both (a) and (b), the stable and unstable fixed points are indi-
cated by squares and circles, respectively.
manner (This behavior is also exhibited in the case of a signaling system with sigmoidal
positive feedback. Read the next part of paragraph 7.3.2). The rate balance plot and the
stimulus-response curve built through the values of the exercise are plotted in Figure 7.28.
7.5. Based on the assumptions of the exercise, the differential equations [7.27] become:
d [mRNA]
TF
[ ]
= ktr
k mRNA
0
dt
TF
− [
] =
[
]+ K
d
d Y
[ ]
[ Y ]
= k [mRNA]− khET
0
dt
tl
KM +
=
[ Y ]
The Jacobian for the system is:
− kd
0
J =
khET K
k
M
tl
−
2
( K [ ] )
M + Y
ss
The trace and the determinant of J are:
khET K
tr ( J ) = − k
M
d − (
0
+ [ ] ) <
2
KM
Y ss
kdkhET K
det ( J
M
) = (
0
+ [ ] ) >
2
KM
Y ss
2
2
khET K
The discriminant ∆ = ( tr( J )) − 4 det ( J )
M
kd
> 0
=
−
2
KM
Y
( +[ ] ss)
These conditions correspond to a steady-state solution that is a stable node.
The Emergence of
8 Temporal Order in a
Chemical Laboratory
Experiment is the interrogation of Nature.
Robert Boyle (1627–1691 AD)
8.1 INTRODUCTION
Oscillations are everywhere in nature: in the Universe, in our Solar System, in our planet, in ecosystems,
in the economy, and inside every living being, as we have learned in the previous chapters. It was
a great surprise to observe oscillations also in chemical laboratories because chemical oscillations
were imagined as a sort of perpetual motion machines in a beaker, in sharp contradiction with the
Second Law of Thermodynamics.
8.2 THE DISCOVERY OF OSCILLATING CHEMICAL REACTIONS
The first report of chemical oscillations in a lab was made by Robert Boyle in the late seventeenth
century. At that time, Boyle was investigating the luminescence of phosphorus in the gaseous phase,
in liquid solutions, and in the solid phase (Harvey 1957). He found with wonder that the oxidation
of phosphorous produces flashes of light. The second report didn’t occur until the beginning of the
nineteenth century when A. T. Fechner (1828) described an electrochemical cell producing an oscil-
lating current. About seventy years later, in the late 1890s, J. Liesegang (1896) discovered periodic
precipitation patterns in space and time (first mentioned by F. F. Runge in 1855), and Ostwald (1899)
observed that the rate of chromium dissolution in acid periodically increased and decreased. But all
these experiments, involving heterogeneous systems, did not undermine the widespread idea that
a chemical reaction proceeding towards its equilibrium state cannot show oscillations.
During the early decades of the twentieth century, Lotka (1910, 1920a, 1920b) published a hand-
ful of theoretical papers on chemical oscillations. His models inspired ecologists. The most famous
one is named as the Lotka-Volterra model, which is used to characterize the predator-prey interac-
tion in an ecosystem as we have seen in Chapter 5.
The report of the first homogeneous isothermal chemical oscillator is ascribed to Bray (1921),
and Bray and Liebhafsky (1931), who studied the catalytic decomposition of hydrogen peroxide by
the iodic acid-iodine couple. The mixture consisted of H O , KIO , and H SO and the reactions
2
2
3
2
4
were:
5H2O2 + I2 = 2HIO3 + 4H2O [8.1]
5H2O2 + 2HIO3 = 5O2 + I2 + 6H2O [8.2]
where hydrogen peroxide plays as both an oxidizing and reducing agent.
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Untangling Complex Systems
Bray serendipitously observed that the concentration of iodine and the rate of oxygen evolu-
tion vary almost periodically. He offered no explanation for the oscillation other than referring to
Lotka’s theoretical work. He suggested that autocatalysis was involved in the mechanism of its reac-
tion. Bray’s results triggered more works devoted to debunking them than to explaining it mecha-
nistically (Nicolis and Portnow 1973). Most chemists tried to prove that the cause of the oscillations
was some unknown heterogeneous impurity.
The next experimental observation of an oscillating reaction was made again accidentally by the
Russian biophysicist Boris Belousov. During the Second World War, he was working on projects in
radiation and chemical warfare protection. By investigating methods for removing toxic agents from
the human body, Belousov developed a keen interest in the role of biochemical metabolic processes
(Kiprijanov 2016). After the war, part of the Belousov’s research included the Krebs cycle. He was
looking for an inorganic analog of the citric acid cycle.1 In 1950, he studied the reaction of citric acid with bromate and ceric ions (Ce+4) in sulfuric acid. He expected to see the monotonic depletion of the yellow ceric ions into the colorless cerous (Ce+3) ions. Instead, he astonishingly observed
that the color of the solution repeatedly oscillated with a period of one minute or so between pale
yellow and colorless. Belousov also noted that if the same reaction was carried out unstirred in
a graduated cylinder, the solution exhibited awesome traveling waves of yellow. Belousov care-
fully characterized the phenomenon and submitted a manuscript containing the recipe in 1951 to
the Soviet periodical Journal of General Chemistry. Unfortunately, his paper was quickly rejected
because the phenomenon described by Belousov was in contradiction with the Second Law of
Thermodynamics. Belousov attempted a second submission to the Russian Kinetics and Catalysis
in 1955. But his manuscript was rejected again for the same reason—phenomenon like that could
not occur. Although Belousov furnished a recipe and photographs of the different stages of the
oscillations, the evidence was judged insufficient. Therefore, he decided to work more on that reac-
