Untangling Complex Systems, page 24
104
Untangling Complex Systems
Pumping
FIGURE 4.7 Population inversion promoted by an efficient pumping.
Optical axis
Laser light
Active medium
Mirror
Mirror
Pumping
FIGURE 4.8 Sketch of an optical cavity.
optical axis and embedding the active medium emitting the radiation. One mirror is completely
reflecting, whereas the other is only partially reflecting. Only the radiation that propagates along the
optical axis of the cavity is reflected many times, and it is amplified by a lot of passages through the
active medium. The spatial separation d between the two mirrors and the phenomenon of construc-
tive interference select the wavelength ( λ) of the laser radiation. It is
λ = 2 d [4.13]
n
where n = 1, 2, … is a positive integer.
The generation of laser radiation may be described through the formulation of a system of two
differential equations: one regarding the number of photons, n, and the other regarding the number
of atoms or molecules in level 2: N . The system is
2
dn = B 21 N 2 n− Tn
dt
[4.14]
dN 2 = Pu − B 21 N 2 n− A 21 N
dt
2
In [4.14], B 21 represents the probability of the stimulated emission; T is the transmittance of the partially reflecting mirror of the optical cavity; A 21 represents the probability of the spontaneous emission; Pu is the pump strength. Let us suppose that N 2 relaxes much more quickly than n. Then, ( dN
)
2 dt = 0, and
P
N
u
2 =
[4.15]
B n
21 + A 21
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105
If we insert equation [4.15] into the first differential equation of the system [4.14], we obtain:
dn
B n
21 P
u
=
− Tn [4.16]
dt
B n
21 + A 21
We may rearrange equation [4.16] in the following form:
dn ( B
2
21 P
21 )
u − A T n − TB n
=
21
[4.17]
dt
B n
21 + A 21
In the latter form, the denominator is always positive, and the numerator looks like the differen-
tial equation originating a trans-critical bifurcation, i.e., [4.6]. We have two fixed points: ′
n 0 = 0
and ′′
n 0 = ( B 21 Pu − A 2 T
1 ) / TB 21. The solution ′
n 0 = 0 is stable when B 21 Pu < A T
21 . This condition corre-
sponds to the case wherein we do not have stimulated emission, and the device behaves like a lamp.
When B
n 0 ( B 21 Pu A 2 T
1 )
21 Pu becomes larger than A T
21 , the system crosses a bifurcation and ′′ =
−
/ TB 21
becomes the stable solution. The device becomes a laser. The critical value of pumping is given by
( P )
21 / 21 )
u
= ( A T B : the pumping must be particularly strong when A
c
21 and T are large.
4.2.3 PiTchfork bifurcaTion
If we determine the fixed points of equation [4.7] and we study their stability, we find a third type
of bifurcation, called pitchfork bifurcation. The equation x (λ − x 2 ) is null when either x′0 = 0 or x′′0 = + λ or x′′′0 = − λ . If we look just for real solutions of the variable x, when λ is negative or null, the fixed point is only the first: x′0 = 0. On the other hand, when λ is positive, all the three solutions are possible. We define the properties of the fixed points through the linear stability analysis (see
equations [4.8] and [4.9]).
dx
∫
f x
2
0
dt (λ 3 x 0 )
(
=
−
dt [4.18]
x − x 0 ) = ( )∫
∫
After integrating, we obtain:
( −3 2
λ
0
0 )
x x 0 e f ( x t) x 0 e
x t
=
+
=
+
[4.19]
From equation [4.19], it is evident that the fixed point x′0 = 0 is stable when λ is negative, whereas it is unstable when λ ≥ 0. When λ is positive, both x′′0 = + λ and x′′′0= − λ are stable solutions. The plot of all the solutions as a function of λ is shown in Figure 4.9. Due to its shape, the graph has been termed the pitchfork bifurcation diagram.
Figure 4.9 shows that if λ moves from negative to positive values, the solution x′0 = 0 changes from stable to unstable. At λ = 0, the system will evolve either towards the positive branch x′′0 = + λ
or the negative branch x′′′0 = − λ . It is not possible to predict the path that will be traced because
small, unpredictable fluctuations will push the system towards either one or the other route.
TRY EXERCISE 4.2
4.2.3.1 Chiral Symmetry Breaking
The concept of pitchfork bifurcation has been recently invoked to understand why chiral biomol-
ecules in living beings are present only as one enantiomer.
First of all, let us remember the meaning of chiral species. Any molecule whose structure is not
superimposable with that of its mirror image is defined chiral (the word chiral derives from the
ancient Greek “χείρ”, which means “hand”: our right hand is not superimposable to our left hand).
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Untangling Complex Systems
x
Stable
Stable
Unstable
λ
Stable
FIGURE 4.9 Pitchfork bifurcation diagram.
Mirror-image structures of a chiral molecule are called enantiomers. The two enantiomeric struc-
tures are usually distinguished by the sense of the rotation they give to a linearly polarized light
crossing their solutions. An enantiomer will be levogyre ( L or (–) as symbols), whereas the other
will be dextrogyre ( D or (+)). The absolute value of the rotation angle is the same for a couple of
enantiomers. When a chiral molecule is synthesized from achiral compounds and, in the absence
of any physical “force” having chiral character, for example, in the absence of any chiral interac-
tion (Kondepudi and Nelson 1984), it is obtained as racemate. We have a racemate when the two
enantiomers are in an equal amount (it is said that the enantiomeric excess is null).6 Two examples of natural chiral molecules are shown in Figure 4.10.
What is surprising is that all the chiral amino acids present in living beings exist only as L(−)
enantiomers, and the sugar molecules in DNA and RNA (2-deoxyribose in DNA and ribose in
RNA) are present just in the D(+) configuration. Wrong-handed amino acids would disrupt the
α-helix structure in proteins, and if just one wrong-handed sugar monomer were present, DNA
could not be stabilized in a helix. Therefore, a Really Big Question arises:
“How did the biomolecular asymmetry originate?”
A good model for the spontaneous generation of chiral asymmetry has been proposed and
confirmed experimentally by Kondepudi and Asakura (2001). It is based on a kinetic mechanism
CH3
CH3
C
C
H2N
COOH
HOOC
NH2
H
H
L-Alanine
D-Alanine
H
H
HO
O
CH
H
H
2OH
HOH2C
O
OH
H
H
H
H
H
OH
OH
H
2-deoxy-L-ribose
2-deoxy-D-ribose
FIGURE 4.10 Three-dimensional structures of the enantiomers of alanine and 2-deoxyribose.
6 If we want to produce a chiral compound in enantiomeric excess, it is necessary that the reaction takes place in an asymmetric chiral environment. The chirality of the environment may be due to either the reagents or the solvent or the presence of a chiral physical force (such as circularly polarized light).
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107
[4.20–4.24] involving a chiral autocatalytic species X, existing as a couple of enantiomers ( X and
L
X ), produced from achiral reactants, S and T.
D
k 1 L
S + T →
X L
←
[4.20]
k−1 L
k 1 D
S + T →
X
←
D [4.21]
k−1 D
k 2 L
S + T + X →
L
2 X L
←
[4.22]
k−2 L
k
S + T + X
2 D
→
D
2 X D
←
[4.23]
k−2 D
X
k
L + X D →
3
P [4.24]
In the first two steps of the mechanism ([4.20] and [4.21]), the two enantiomeric forms are produced
directly from the achiral substrates S and T, whereas in the third and fourth steps ([4.22] and [4.23]), they are produced autocatalytically. Finally ([4.24]), X and X combine to produce P. The chemical L
D
system can be maintained far from equilibrium with a constant inflow of S and T and a constant
outflow of P.
The dynamics of the system has a rather simple mathematical description in terms of the
variables α = ([ XL]−[ XD ]) / 2 and λ = [ S][ T ] in the vicinity of the critical point λ (Kondepudi c
and Nelson 1984) at which asymmetric states emerge. It is modeled by the following differential
equation:
dα
= − Aα3 + B(λ − λ )α [4.25]
dt
c
wherein A and B are functions of the kinetic rate constants of the mechanism [4.20–4.24]. Since
A and B are positive, when λ ≤ λ , the only stationary state solution is α = 0. When
, there are
c
λ > λc
three solutions of stationary state: α
1/2
[ B(
c )
]
1 = 0, α2 3
, = ±
λ − λ
A . The plot of the stationary state
solutions is shown in Figure 4.11.
By applying the linear stability analysis, we can get insight on the stability of the solutions.
If x = (α −α )
S represents a small perturbation to the stationary state α , its evolution over time is
S
given by:
x
f (α S ) ( t ti
− )
t = xt
[4.26]
i e
with
f(α
3
2
(
)
S ) = − Aα S + B λ − c
λ
[4.27]
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Untangling Complex Systems
α
0
0
λ − λc
FIGURE 4.11 Bifurcation plot for the system described by the equation [4.25]. All the solutions represent
stable stationary states except for those represented by the dotted line ( α = 0 when λ > λ ).
c
When λ ≤ λ , the solution
,
c
αS = 0 is stable because the perturbation decays exponentially. When λ > λc
the solution α
1/2
[ B(
c )
]
1 = 0 becomes unstable, whereas the solutions α2 3
, = ±
λ − λ / A is stable. This
result means that by changing the value of the parameter ( λ − λ ) from negative to positive, the
c
thermodynamic branch defined by the solutions α S= 0 changes its character from stable to unstable
at the point having coordinates α
. This point of the graph is a pitchfork bifurcation
S = 0 and λ = λc
point (see Figure 4.11). When the system is on the bifurcation point, small random thermal fluctuations can push it towards new stationary states. The system will move towards either the stationary
state described by the solution α or that associated with , unpredictably. The solutions and
2
α 3
α 2
α represent asymmetric states in which one enantiomer dominates over the other ([ ]
]).
3
XL ≠ [ XD
Such stationary states are said to have a broken symmetry.
When there is a chiral bias, the evolution of the system is different. The chiral interaction brings
about a small gap in the energies of X and X and those of the transition states, X # and X #, L
D
L
D
involved in the elementary steps producing the chiral species in mechanism [4.20–4.24]. If Δ E is
the energy gap between X # and X #, the kinetic constants k and k will be not anymore equal.
L
D
iL
iD
According to the Arrhenius equation, their ratio becomes
k
E
iL
eg
= where g = ∆ [4.28]
kiD
RT
The chiral interactions may be intrinsic, such as the electroweak force acting within the molecule,
or extrinsic, that is due to external electric, magnetic, gravitational and centrifugal fields. A few
calculations (Kondepudi and Nelson 1984 and 1985) show that g amounts to 10−11–10−15 for the
electroweak interaction, whereas it is less than 10−19 for ordinary extrinsic forces. In the presence of
the small asymmetric factor g, equation [4.25] becomes
dα
= − Aα3 + B(λ − )
c
λ α + Cg [4.29]
dt
Here C is a function of rate constants. The stationary state solutions for the system described by
equation [4.29] are shown in Figure 4.12.
When λ is below or equal to λ , there is only one solution that represents a stable stationary state
c
characterized by a small value of α, with small asymmetry. The chiral asymmetry suddenly grows
as λ passes through the critical point λ . As
c
λ increases, the system will tend to stay on the upper
branch: in other words, a small initial asymmetry due to a chiral bias is amplified because the chemi-
cal system has autocatalytic nature. When λ is larger than λ , three solutions representing three dif-c
ferent stationary states are possible (see Figure 4.12). Only the upper and lower solutions are stable
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109
α
D
λ − λc
FIGURE 4.12 Bifurcation plot for the system described by equation [4.29] when there is a chiral bias. The
points of the continuous lines represent solutions of stable stationary states, whereas the points of the dashed
curve represent unstable solutions. D represents the separation between two stable states in the proximity of the critical point λ . c
stationary states. The system evolves maintaining the upper branch unless it suffers a sufficiently
strong perturbation, which knocks it to the lower branch. These perturbations can be due to random
fluctuations, which are represented by the introduction of another term in the right part of equation
[4.29]: ε f ( t ) where f ( t ) is assumed to be a normalized Gaussian white noise (Kondepudi and Nelson 1984) with a root-mean-square value ε .7 The random fluctuations can play a role in the
proximity of the critical point when the separation D (see Figure 4.12) between the upper and lower branches is not large enough. Many different scenarios have been proposed for the possible origin of
biomolecular handedness (Cline 1996). Although we are not sure how the biomolecular asymmetry
originated, this case study offers the opportunity to make three important considerations.
First, a theory of hierarchical chiral asymmetry (Kondepudi 2003) can be formulated, based on
the sensitivity of chiral molecular symmetry breaking on small chiral biases. The breaking of chiral
symmetry at one spatial level is the source of a small chiral bias for the next higher spatial level.
For example, the parity-violating electroweak asymmetry influences a breaking asymmetry at the
molecular level. On the other hand, molecular asymmetry is the origin of morphological asymme-
tries, such as microtubules in cells.8
Second, the theory of bifurcation introduces the concept of “history” in physics and chemistry,
unequivocally (Prigogine 1968). If the system, whose bifurcation diagram is shown in Figure 4.13, was initially in state i and that at the end is in state f after a change in the contour conditions from λ to , then the knowledge of the bifurcation diagram allows us to track the evolution of the system
