Untangling Complex Systems, page 55
in paragraph 9.7.
9.6.1 ProPagaTor-conTroller model
An easy mathematical description of how chemical waves form is the Propagator-Controller model
(Fife 1984). Let us assume that our chemical system consists of two variables, x and y, which react
according to the kinetic laws f ( x, y) and g( x, y), and diffuse along the spatial coordinate r. The system of partial differential equations is:
∂ x 1
2 x
=
f ( x, y) +
∂
Dx
[9.42]
2
∂ t η
∂
r
∂ y
2 y
= g( x, y) +
∂
Dy
[9.43]
2
∂ t
∂
r
In equations [9.42] and [9.43], Dx and Dy are the diffusion coefficients of x and y; the parameter η
that is assumed to be much smaller than one is introduced to highlight the difference in the rates
of x and y transformations: the variable x changes much more quickly than y. Moreover, we assume that the kinetic laws f and g avoid that the concentrations of the variables grow to infinity. In particular, the nullcline f ( x, y) = 0 has the S shape we found in paragraph 8.3 (Figure 9.16). Finally, the time scale of the chemical transformations is much shorter than that associated with the diffusion
processes. Therefore, the system of two partial differential equations [9.42 and 9.43] can be reduced
nearly everywhere in time and space to the following equations:
f ( x, y) = 0
[9.44]
y 2
y
xR( y)
xL( y)
y 1
x
FIGURE 9.16 The S-shape of the f ( x, y) = 0 nullcline.
The Emergence of Order in Space
271
dy = g( x( y), y)
[9.45]
dt
In [9.45], the variable x, which changes rapidly, assumes instantaneous values that depend on that
of y.
As we learned in chapter 8, the points on the left-hand and right-hand branches of the f ( x, y) = 0
nullcline (labeled as x ( y) and x ( y) respectively, in Figure 9.16) represent stable solutions, whereas L
R
the points on the middle branch represent unstable solutions.
9.6.1.1 Phase Waves
Let us suppose that the second nullcline g( x, y) = 0 intersects the middle branch. This condition means the chemical system is in oscillatory regime (remember paragraph 8.3). Since the system is
not under stirring, the variable x may be discontinuous in r. There will be regions where x assumes a value of the right-hand branch and regions where it assumes a value of the left-hand branch.
The area that separates the two regions is called boundary layer. For η → 0, the boundary layer is
a point in a one-dimensional medium, whereas it is a very thin slice for a bi-dimensional medium.
In the boundary layer, x assumes a value belonging to the middle branch whereas y assumes a value
between y
. The layer will move with a velocity that depends on the “Controller”
1 and y 2
y and in
its movement the “Propagator” variable x jumps from x to x or vice versa. If we fix our atten-L
R
tion to a specific point in space, the frequency of the appearance of the boundary layer is equal
to the frequency of oscillations in that location. Such kind of chemical wave is called phase wave.
Nothing moves as a phase wave propagates. In other words, the propagation of a phase wave does
not rely upon the physical interchange of materials between adjacent points. In fact, in the case of
the BZ reaction it has been demonstrated (Kopell and Howard 1973) that phase waves, run in a
cylinder, pass through a barrier impenetrable to diffusion.12 If we define the phase φ of an oscillation as the ratio between the time delay ( t) and the period ( τ) of an oscillation (see the upper plot of Figure 9.17), then it derives that the velocity of a phase wave depends inversely on the phase gradient, Δ φ, between adjacent points in space. For example, in Figure 9.17, it is shown the situation where adjacent points along the spatial coordinate r have a phase difference ∆ϕ = 0 1
. . At time t , in
0
the five points, labeled as r , , , , , ϕ is different from both 0 and 1. Hence, in all five points,
1 r 2 r 3 r 4 r 5
[ X] is low. An instant later, at t , ϕ =
, and just in , [
, the condition
1
1 in r 1
r 1 X] becomes large. At time t 2
ϕ =1 is verified in the adjacent point r , and the maximum of [
. In , ϕ has become
2
X] is now in r 2
r 1
1.1, and [ X] has dropped, again. At time t , the condition ϕ =
. And so on. In
3
1 is verified only in r 3
this way, the wave front propagates as shown in Figure 9.17.
9.6.1.2 Trigger Waves
When the phase gradient is large, the phase wave moves slowly. In this case, the diffusion of reagents
is not anymore negligible, and the front can propagate faster than a phase wave (Reusser and Field
1979) because another type of chemical wave propagates, called trigger wave. The velocity of a
trigger wave is controlled by the rates of chemical reactions and diffusion; it cannot go through an
impenetrable barrier as a phase wave does. Trigger waves do not require necessarily an oscillatory
medium to appear. They can be originated even in an excitable medium. We learned the definition of
excitability in the previous chapter (see paragraph 8.3.1). Our system is excitable when the nullcline
g( x, y) = 0 intersects the first nullcline, f ( x, y) = 0, in any point of either the left-handed x or the L
right-handed x branch (in Figure 9.18, g( x, y) = 0 intersects f ( x, y) = 0 on the left-hand branch).
R
Before any perturbation, the system stays indefinitely in its stable steady state that is the intersection
point, labeled as 0, in Figure 9.18. If the system feels a sufficiently strong perturbation that pushes it
12 It is possible to observe phase waves between physically isolated oscillators as described in Ross et al. (1988). Such phase waves are also called kinematic waves. They do not involve mass transfer from one oscillator to another.
272
Untangling Complex Systems
[ X]
φ = tτ
τ
t
φ = 0,9 0,8 0,7 0,6 0,5
t 0
r
r 1
r 2 r 3 r 4 r 5
φ = 1 0,9 0,8 0,7 0,6
t 1
r
r 1
r 2 r 3 r 4 r 5
φ = 1,1 1 0,9 0,8 0,7
t 2
r
r 1
r 2 r 3 r 4 r 5
φ = 1,2 1,1 1 0,9 0,8
t 3
r
r 1
r 2 r 3 r 4 r 5
FIGURE 9.17 The upper plot illustrates the definition of phase φ for an oscillating reaction. In the lower part, the sketch, which must be read from t up to , helps to understand how a phase wave propagates along
0
t 3
the spatial coordinate r.
2
Back
x − x
y
1
2
4
3
Back 3
Front
y − y 1
y 1
1
0
1 Front
2
0
0
r
4
(a)
x 1
x 2
(b)
FIGURE 9.18 Propagating pulse in an excitable chemical system. Graph (a) shows the variations of the
Propagator x and the Controller y in their phase plane, with the front and the back of the pulse depicted as dashed traces. Graph (b) shows the same pulse along the spatial coordinate r.
up to reach point 1, then it initiates a large excursion especially in the x variable that follows the cycle
0-1-2-3-4-0 in graph a of Figure 9.18. A pulse of a trigger wave propagates (see graph b of the same figure). The leading edge of the wave is at 0-2, whereas its trailing edge is at 3-4.
The perturbations that promote trigger waves can be induced by the experimenter or be spontaneous.
A chemical system can be pushed deliberately beyond its threshold of excitability in several ways.
The Emergence of Order in Space
273
For example, chemically, by sinking a silver wire in a solution containing the BZ reagents in a non-
oscillatory regime, or by adding drops of acid into a pH-sensitive bistable reaction. Otherwise, it is
possible to induce a trigger wave thermally, by dipping a hot wire in the solution. In one case, it has
been demonstrated that waves can be electrochemically initiated in an unstirred thin film of a solution
containing iodate and arsenous acid (Hanna et al. 1982). Waves in photosensitive systems can be
initiated by irradiation at the appropriate wavelength and intensity. A remarkable example in litera-
ture is offered by the light-sensitive version of the BZ reaction that involves the ruthenium bipyridyl
complex [Ru(bpy) ]2+ as catalyst (Kuhnert et al. 1989). The spontaneous generation of trigger waves
3
may be due to concentration fluctuations at a particular point in the solution or to the presence of an
artificial pacemaker point such as a dust particle. The point where there is the dust particle may be
in oscillatory regime, while around it the system is excitable. The pacemaker point is the source of
phase waves that ultimately turn in trigger waves when they propagate in the excitable bulk medium.
9.6.2 shaPes of chemical waves
Chemical waves can assume different shapes depending on the geometry of the medium hosting them.
9.6.2.1 Mono- and Bi-Dimensional Waves
One-dimensional waves are formed in a narrow medium like a test tube, and they may consist of
a single pulse or a train of fronts (Tyson and Keener 1988). Two-dimensional waves occur more
frequently in nature and can be easily observed in the laboratory by using thin layers (1 mm deep)
of a solution (with proper chemicals) held in a Petri dish. In a two-dimensional medium, a wave
originating in a point produces a circular front when the wave propagates at the same velocity in
all directions. When the system gives rise to repeated waves, we observe a pattern of concentric
waves, called target pattern (like that shown in picture b of Figure 9.15). When the fronts of two waves belonging to two distinct target patterns collide, they annihilate and lead to cusp-like structures in the vicinity of the area of collision. When an expanding circular wave front is disrupted,
spiral waves are formed. In the case of the BZ reaction, the disruption of a circular wave front can
be performed by pushing a pipette through the solution or with a gentle blast of air from a pipette
onto the surface of the reacting solution (Ross et al. 1988). The wave curls in and forms a pair of
counter-rotating spirals as shown in picture c of Figures 9.15 and 9.19.
TRY EXERCISE 9.15
It is possible to obtain multi-armed spirals if a drop of KCl (the chloride anion interferes with the
oxidation-reduction chemistry of the BZ reaction) is added to the center of rotation of a forming spi-
ral wave (Agladze and Krinsky 1982). More recently, it has been shown that when the BZ reaction
is carried out inside the nanoreactors of a water-in-oil microemulsion with sodium bis(2-ethylhexyl)
sulfosuccinate (AOT) as the surfactant, antispiral waves are formed (Vanag and Epstein 2001b).
Antispiral waves are spirals traveling from the periphery toward their centers. In the same AOT-
microemulsions, packets of waves traveling coherently either toward or away from their centers of
curvature have been observed (Vanag and Epstein 2002). They range from nearly plane waves to
target-like patterns. Under certain conditions (Vanag and Epstein 2003), packet, target, and spiral
Time
FIGURE 9.19 Simulation of the propagation of a spiral wave.
274
Untangling Complex Systems
waves may break into small segments. These short segments, or dashes, propagate coherently in the
direction perpendicular to the breaks.13
9.6.2.2 Three-Dimensional Waves
When we increase the thickness of the chemical system, we expect to observe three-dimensional
chemical waves. Experimentally, it is difficult to detect them because there are many disturbing
phenomena, like convection and bubble formation. However, some results have been achieved in
sealed vessels or using gels that hinder convective motions. In three-dimensions, we can observe
either spherical waves or scroll waves. Spherical and scroll waves are 3D extensions of target pat-
terns and spiral waves, respectively. A scroll wave is made of a two-dimensional surface that rotates
around a one-dimensional filament or axis (Figure 9.20) that terminates at the boundary of the
three-dimensional spatial domain. The filament may assume a vertical orientation, or it may curve
or twist. Sometimes, it joins itself into a ring within the spatial domain. Several types of scroll
waves have been observed experimentally and numerically. A list and more-in-depth analysis can
be found in the papers by Tyson and Keener (1988), and Winfree and Strogatz (1984).
9.6.2.3 Effect of Curvature
In two and three dimensions the wave fronts may be curved. Any curvature is quantitatively char-
acterized by specifying its radius of curvature that is the radius of the circle best fitting the front.
If the radius is r, the curvature is c = (1 r
/ ). By convention, the curvature c is taken with a positive
sign when the curved front propagates towards the center of the circle; on the other hand, c is nega-
tive when the curved front propagates far from the center. It has been proved theoretically (Tyson
and Keener 1988) and experimentally (Foerster et al. 1988) that the velocity of a curved wave front
depends on its curvature, according to the eikonal equation
vc = vp + c * D
[9.46]
In [9.46], vp is the velocity of the plane wave ruled by the concentration of the controller species
(i.e., y in equations [9.42 and 9.43]) at the front; D is the diffusion coefficient of the propagator (i.e., species x in equations [9.42 and 9.43]) and c is the curvature. It is clear that for a curved front FIGURE 9.20 Sketch that depicts the two essential elements of a scroll wave embedded in a medium represented by the rectangle: (I) a surface that rotates around (II) a filament that is the vertical segment.
13 Wave segments can also be achieved by generating waves in an excitable medium and propagating in a sub-excitable medium. A medium is sub-excitable when the threshold of excitation is sufficiently large that a chemical wave cannot be maintained. The sub-excitable regime has been easily obtained by using a light of the proper intensity and the photosensitive version of BZ reaction where the photocatalyst ruthenium(II)-bipyridyl produces the inhibitor bromide (Kádár et al.
1998).
The Emergence of Order in Space
275
propagating far from the center of its curvature, the velocity of the curved wave is smaller than that
of a planar wave front considered at the same chemical composition. When the curved wave propa-
gates toward the center of the circle, the opposite is true.
TRY EXERCISE 9.16
Note that in the case of three-dimensional waves, the eikonal equation becomes
v
( 1 2)
c = v p + c + c
* D
[9.47]
where c and are the principal curvatures of the wave front surface (Tyson and Keener 1988).
1
c 2
9.7 “CHEMICAL” WAVES IN BIOLOGY
The phenomenon of traveling chemical waves is widespread in biology. The reason is that it is an
effective means of transmitting chemical information. In fact, it can be much faster than pure diffu-
sion. For example, imagine that a protein has the task of transporting a message through the axon of
a neuron. A neuron is a cell specialized in receiving, integrating, processing and transmitting infor-
mation. The axon of a neuron is a cable-like structure that carries signals (see Figure 9.21). Let us consider an axon that is 1 cm long.14 The protein diffuses through the axon with D ≈ 5 μm2/s.15 The time the protein spends to cross the axon only by diffusion is enormously long: almost eight months!
It is evident that neurons must exploit other strategies to transfer information. In fact, information
crosses an axon of 1 cm in just a few milliseconds! How is it possible? Such fast transfer is made
possible by the involvement of electrochemical waves. What are electrochemical waves?
9.7.1 waves in a neuron
Any neuron within our brain receives chemical signals through the dendrites (Figure 9.21). Such
information is transduced in transmembrane electrochemical potential in the soma and, finally,
Axon hillock
Axon
Nucleus
Synapsis
+ + + + + + + + + + + +
Dendrites
− − − − − − − − − − − −
− − − − − − − − − − − −
