Out-of-phase oscillations and traveling waves
with unusual properties: the use of
three-component systems in biology
Hans Meinhardt
Max-Planck-Institut für Entwicklungsbiologie, Spemannstraße 35,
D-72076 Tübingen, Germany
Abstract
Pattern formation requires the interaction of a self-enhancing component and its
long-raning antogonist. If once established maxima are quenched by a second localacting antagonist, three-component systems result that allow the generation of
highly dynamic patterns. Either out-of-phase oscillations in groups of cells, traveling
waves with a soliton-like behavior or a regular flashing up of signals at displaced positions are possible. By comparison with the pole-to-pole oscillations in E.coli, with
pigment pattern on tropical sea shells, with the orientation of chemotactic cells and
with the signaling for the initiation of new leaves on a growing shoot (phyllotaxis)
it is shown that three-component systems are appropriate to account for a wide
class of biological phenomena. Even when triggered by random fluctuations, these
pattern-forming systems obtain rapidly their characteristic properties although they
never reach a stable steady state.
Key words: Pattern formation / Oscillations / Solitons / Seashells / Penetrating
waves / Growth cones / Phyllotaxis / Chemotaxis
1
Introduction
As discovered by Turing in 1952 [1], dynamic systems with pattern-forming
capabilities can emerge by an interaction of two substances that spread with
different rates. These so-called reaction-diffusion systems are now well investigated [2-5] and chemically defined systems are known that display this behavior [6,7]. In a second part of his paper, Turing discusses interactions of three
Email address: e-mail: [email protected] (Hans Meinhardt).
Preprint submitted to Elsevier Preprint
24 May 2004
substances and showed that these can lead to traveling waves and to oscillations that are out-of-phase in adjacent regions. Turing mentioned ([1], page
67) that he is not aware of any biological example for such an out-of-phase
oscillation and that he did not make an effort to provide a molecularly feasible
interaction. This is presumably the reason why this part of his paper became
largely forgotten.
By searching for mechanisms that account for the pigment pattern on tropical sea shells we came across a reaction type that is able to generate highly
dynamic patterns that never reach a steady state [8, 9]. The basic idea was
that concentration maxima, generated by a conventional two-component system, become destabilized by an additional antagonist that locally quenches
the once established maxima. Two modes of system response are prevailing:
• Maxima disappear and thereafter reappear at a displaced position. At these
new positions the maxima will become quenched too. This leads either to a
regular out-of-phase oscillation between adjacent regions or to an irregular
appearance of new maxima.
• The local poisoning of maxima causes an ongoing displacement into adjacent
regions. This leads to traveling waves, which can have unusual properties.
In other words, our three-component systems have essentially the same properties as those discussed by Turing. Whether they are also mathematically
equivalent is not yet clear.
Meanwhile several biological pattern-forming systems are known that require
the presence of a local destabilization to understand their behavior. Here I
will review the general condition that leads either to out-of-phase oscillations
or to traveling waves. The properties of these systems will be illustrated by
comparison with biological processes that have overtly nothing in common:
(i) the pole-to-pole oscillation in E.coli bacteria that restricts initiation of cell
division to the center of the cell; (ii) the orientation of chemotactic cells and
neuronal growth cones by minute asymmetries imposed by external signals;
(iii) the flashing up of signals for leaf initiation behind the tip of a growing shoot that leads to the typical patterns of phyllotaxis; (iv) the pigment
patterns on tropical sea shells that preserve records of traveling waves that regularly penetrate each other. In contrast to conventional traveling waves with
their long extended wave front, these systems can produce moving spot-like
concentration maxima in two-dimensional fields. If the moving signals leave
behind traces of differentiated cells, long branching filamentous structures can
emerge.
2
(A )
(B )
A c tiv a to r
In h ib ito r
In h ib ito r
T im
T im
e
e
A c tiv a to r
Fig. 1. Periodic pattern formation by insertion of new or by splitting of existing
maxima. Assumed is an activator-inhibitor system [2,3] in a growing field. Growth
is simulated by the insertion of two new cells at random positions, one in each
half after a certain time interval. (A) Without saturation, new peaks are inserted
whenever the inhibitor concentration between the maxima becomes too low. (B) If
activator production saturates and, due to growth, more space becomes available
into which the inhibitor can escape, the broader maxima become even broader until
the centers become deactivated. Maxima split and shift to keep distance from one
another. Calculated with Eqs. 1, 2 and Da = 0.005; ra = 0.02; ba =0.03 ; sa = 0
(A) or 0.3 (B); s = .02 ± 1% random fluctuation; Db = 0.4; rb = 0.03.
2
Stable pattern formation and the possible shift of maxima
A very brief survey of the two-component systems has to be given and some
properties have to be described to make the special features of the threecomponent systems understandable. Not all reactions of two substances with
different diffusion rates will form patterns. To the contrary, we have shown that
only a very restricted set of reactions is able to do so. The crucial condition is
that the short-ranging substance has a positive non-linear feedback on its own
production while the long-ranging substance acts as an antagonist on this selfenhancement [2,3]. This mechanism is able to account for many observations in
early embryonic development [10], including the observed robustness against
perturbations [11]. In other ranges of parameter, the same type of reaction
can show a very different behavior. Synchronous oscillations are possible if
the antagonistic reaction has a longer time constant then the self-enhancing
reaction (see Fig. 2 E). The theory of oscillations in biology is well developed
[12,13]. Traveling waves in excitable media appear if in the oscillating mode
the autocatalytic reaction spreads moderately while the antagonistic reaction
remains local. The spread of a fire front in a forest fire or the spread of an
epidemic has the same origin. Thus, depending on the range and the time
constant of the antagonistic reaction either patterns in space or patterns in
3
time result.
A typical molecular realization of such a pattern-forming reaction consists
on an autocatalytic substance called the activator a that is antagonized by
a long-ranging substance, inhibitor b. The following set of equations describe
the change of the activator and inhibitor per time unit:
∂a
s(a2 + ba )
∂2a
=
−
r
a
+
D
a
a
∂t
b (1 + sa a2 )
∂x2
∂b
∂2b
= sa2 − rb b + Db 2 + bb
∂t
∂x
(1)
(2)
where t is time, x is the spatial coordinate, Da and Db are the diffusion coefficients, and ra and rb the decay rates of a and b. The factor s, the source
density, describes the ability of the cell to perform the reaction; s is assumed
to be modulated by minor fluctuations to allow initiation of pattern formation;
it remains unchanged during a simulation. ba is an activator-independent production of a that allows the initiation of the autocatalytic activator production
at low concentrations. For instance, if due to growth the distance between the
maxima surpasses a certain threshold, the inhibition may be so low that a
new activation becomes triggered via ba (Fig. 1 A). In contrast, a substantial
baseline inhibitor production bb leads to an additional non-patterned stable
steady state at low activator concentrations. Such a mode is required for traveling waves in excitable media where activation does only occur after a trigger
from an adjacent region. The factor (1 + sa a2 ) in the denominator of Eq. 1
leads to a saturation in the activator autocatalysis at high concentrations. A
substantial saturation causes an upper limit of the maximum a-concentration.
Since a maximum cannot grow in height, it extends in width until the balance between the self-activation and the antagonist is achieved. The system
obtains size-regulating properties in which activated to non-activated regions
obtain a certain ratio. Peak splitting is possible: when a peak becomes too
broad, the cells at the flanks are in a better position since they can more easily get rid of the inhibitor due to diffusion into the nearby non-activated cells.
In contrast, cells in the center of the maximum become de-activated due to
the accumulating inhibitor. Moreover, a broadened maximum can much easier shift towards a more favorable position since the activation can extend on
one side of the plateau-like profile while de-activation occurs simultaneously
at the other. The different behavior in a system without and with saturation
is illustrated in Fig. 1. Without saturation new peaks are inserted into the
growing interstices; with saturation, the enlarging maxima can split and shift.
In both cases periodic patterns emerge in a growing field. Other properties
of this reaction and their relevance to biology has been described elsewhere
[2,3,10].
4
3
Local poisoning of stable maxima: traveling waves by enforced
displacement
As mentioned, two-component systems can lead to traveling waves. They can
emerge if the autocatalytic reaction spreads moderately while the antagonistic reaction acts local and has a longer time constant than the activator. If
traveling waves can be generated by two substances, why to consider the interaction of three? The simplicity of wave formation by the two-component
system is, however, only apparent since additional conditions have to be met.
It must be specified where the wave should start, otherwise the system could
just oscillate in a synchronous way. For instance, the regular contraction waves
of the heart require a pacemaker region. In the sinus node the oscillation runs
somewhat faster, giving rise to the waves in an ordered fashion. Without a
pacemaker or after a severe perturbation, the oscillations may occur either at
random phases or in synchrony. Both situations would be disastrous. Thus,
a two-component system only appears to be simple since a complete system
would require at least four substances, two for the generation of the stable
pattern that defines the pacemaker region and two for the proper wave.
For wave formation by two-component systems the medium has to be excitable: a small activation derived from a neighboring cell triggers a full round
of the cyclic activation that triggers the subsequent cell. In a three-component
system wave formation works in a different way. Imagine that local high
concentrations have been generated by a short-ranging activator and a longranging inhibitor as shown in Fig. 1. Imagine further that a second inhibitor
exists, which does not spread and which accumulates slowly. When this second
inhibitor reaches a certain level, the autocatalysis breaks down. Consequently
also the long-ranging inhibition surrounding each maximum fades away. As
illustrated in Fig. 2, the activation can either reappear at a displaced position
or shift into an adjacent region. Biological examples for both modes will be
discussed further below.
4
Some prototype reactions
For the actual implementation of a second antagonist many interactions are
conceivable. It can consist of a necessary factor that is consumed during the
autocatalytic reaction or, as mentioned above, of a second inhibitor. Such
an inhibitor can either block the production or cause an elevated destruction
of the activator. Different non-linearities add further possibilities to the vast
amount of possible realizations. Pattern generation and pattern destruction
can even occur by separate systems (see Fig. 5). A systematic coverage and
analytical treatment of this reaction type is still missing. In the following only
5
C
B
A
D
T im e
T im e
P o s itio n
E
P o s itio n
P o s itio n
P o s itio n
P o s itio n
Fig. 2. Pattern formation modified by a second local and long-lasting inhibitor [8,9].
Simulation in a one-dimensional chain of cells; activation is plotted as function of
time. (A) Starting from a homogeneous activation, the second inhibitor leads first
to an overall breakdown. Due to the primary long-ranging inhibition the activation
reappears in a somewhat spotty way. Due to the second inhibitor, the activations
decline locally after some times and reappear in an adjacent position. Due to the
long-range inhibition, this is restricted to one side of a previous activation. Activations appear at regularly displaced positions. (B) With some saturation of the
autocatalysis (sa > 0) the maxima tend to be shifted instead of breaking down;
thus, traveling waves emerge. Note that these are not normal waves in an excitable
medium but are based on an enforced displacement. Such mechanism does not require a pacemaker region since local maxima form due to the long-range inhibition a feature that is foreign to traveling waves in excitable media. (C) If the long-ranging
inhibitor has also a long time constant, oscillations out of phase do occur. (D, E)
For comparison, without the second antagonist, either stable patterns (D) or synchronous oscillations (E) emerge, depending whether the half life of the inhibitor is
shorter (D) or longer (E) than that of the activator. (A-C) calculated with Eq. 3-5
and the following parameter: (A): Da = 0.003; ra = 0.005; ba = 0.01; sa = 0; sb =
1; sc = 0.8; s = 0.005 ± 1% random fluctuation; Db = 0.3; rb = 0.008; bb = 0; Dc
= 0; rc = 0.001. (B) as (A) except sa = 0.5 and sc = 0.5; (C) as (A) except rb =
0.002 [8,9].
some examples can be given with an attempt to provide some intuition for the
behavior of these systems.
In the following example, the self-enhancement of the activator a is antagonized by two inhibitors; b and c.
∂a
s(a2 + ba )
∂2a
=
−
r
a
+
D
a
a
∂t
b (1 + sa a2 )(1 + sc c)
∂x2
∂b
∂2b
= sa2 − rb b + Db 2 + bb
∂t
∂x
∂c
∂2c
= rc a − rc c + Dc 2
∂t
∂x
6
(3)
(4)
(5)
Fig. 2 shows simulations using this interaction. Depending on the parameter,
the activation either jumps to a new or is smoothly shifted to an adjacent
position. Alternatively, adjacent regions may oscillate out of phase.
The antagonistic effect in a pattern forming reaction can have its origin in the
depletion of a necessary precursor [2,3]. Such an activator-depleted substrate
interaction has inherently a saturation since the local self-enhancement comes
necessary to a rest if all the substrate is used up. Therefore, maxima that
are localized due to the depletion of a long-ranging precursor have a strong
tendency to shift into a region where still a high precursor concentration is
available [3]. In the following interaction, the substance b acts as a factor that
becomes depleted as described above, while c acts as inhibitor.
∂a
s b(a2 + ba )
∂2a
=
−
r
a
+
D
a
a
∂t
(sb + sc c)(1 + sa a2 )
∂x2
∂b
s b(a2 + ba )
∂2b
= bb −
−
r
b
+
D
b
b
∂t
(sb + sc c)(1 + sa a2 )
∂x2
∂c
∂2c
= rc (a − c) + Dc 2
∂t
∂x
(6)
(7)
(8)
New a-molecules appear with the same rate as b-molecules are used up. In
this interaction, the activator concentration (but not substrate concentration)
is to a large extend independent of c, since a reduction of c also leads to a
compensating decrease in substrate removal. If sb > 0 the additional inhibitor
c plays a role only at high c concentrations, while at low c concentrations this
system behaves like a standard activator-substrate system.
5
Active de-synchronization of coupled oscillators by a long-ranging
antagonist
There is a large body of literature stating that coupled non-linear oscillators
have a strong tendency to synchronize [14,15]. An example is the synchronization of the pulsing light emission by fireflies. In the two-component patterning
systems both a spreading activator or a spreading inhibitor causes synchronization (see Fig. 2E). However, there are many cases in biology where oscillations
occur with opposite phases in adjacent regions. A rich source of examples can
be found on the pigmentation pattern of tropical sea shells [8,9]. An example
is shown in Fig. 3.
The shells of mollusks consist of calcified material. The animals can increase
the size of their shells only by accretion of new material along a marginal
7
(C )
(B )
T im e
(A )
P o s itio n
P o s itio n
Fig. 3. Oscillations out of phase. A mollusk can enlarge its shell only at the growing
edge by accretion of new material. Most two-dimensional shell patterns are, therefore, time records of reactions that took place along the edge. (A) Chessboard-like
pigment pattern on the shell of the small sea snail Bankivia fasciata is a record of
an out-of-phase oscillations. (B) These patterns can be simulated by an activator two-antagonist interaction. Calculated with Eq. 6-8 and Da = 0.015; ra = 0.02; ba
= 0.003 ; sa = 3; sb = 0; sc = 1; s = ra ±5% random fluctuation; Db = 0.4; rb =
0; bb = 0.003; Dc = 0.002; rc = 0.004. The initially synchronous oscillation breaks
up into an oscillation in which adjacent regions oscillate in counter phase. (C) An
increase in the production rate of the substance b from bb = 0.003 to 0.004 causes
a transition from the chessboard pattern to oblique lines, i.e., from out-of-phase
oscillations to traveling waves. With the higher rate of substrate production, the
activation is stable enough not to collapse but to escape by shifting into an adjacent
region. Both modes resemble a simultaneous patterning in space and in time [8,9].
zone, the growing edge. In most species, pigment becomes incorporated during
growth at the edge. In these cases, pattern formation proceeds in a strictly
linear manner. The second dimension is a protocol of what happens as function
of time. In other words, shell patterns are natural space-time plots in which
the complete history of a highly dynamic process is preserved. The pattern on
the shell given in Fig. 3 is reminiscent to a chess board. Keeping in mind its
space-time character, it is evident that this pattern is a protocol of a pigment
production that oscillates out-of-phase.
The second and long-ranging antagonist, as introduced above, enforces a desynchronization. Imagine coupled oscillators that consists of an autocatalytic
and an antagonistic component. Initially all oscillators fire in synchrony. If
some cells become activated only a bit later than their neighbors, this phase
difference will increase during subsequent oscillations since the inhibitory influence that spreads from the advanced cells delays the retarded neighbors
even more. The phase difference will increase until it reaches 180o (Fig. 3 B).
In contrast, the diffusion of the activator tends to synchronize adjacent cells.
This has the consequence that groups of adjacent cells are in the same phase
and that an abrupt transition occurs to another group that oscillate exactly
8
in counter phase.
The patterns on the shell shown in Fig. 3 displays a transition from a chess
board into an oblique lines pattern, i.e., from an out-of-phase oscillation into
a traveling wave pattern. The common basis of these patterns is easy to understand. Both the chess board and the oblique line pattern are based on a
regular alternation of the pigmentation production along the space and along
the time coordinate. This dual pattern has its origin in the two antagonists:
a long-ranging antagonist has the tendency to generate pattern in space, a
long-lasting antagonist causes burst-like activations in time. Superimposed,
both antagonist create a periodic pattern along the space and along the time
coordinate, i.e., traveling waves and out-of-phase oscillations. Small parameter
changes can cause a transition from one pattern to the other (see also Fig. 2).
6
Soliton-like penetration of traveling waves
Modeling of traveling waves in excitable media is in an advanced state, especially due to the study of the Belusov-Zhabotinsky reaction [16,17,4]. However,
the traveling waves as preserved on shells display features that are otherwise
rarely observed in waves in excitable media. The shell of Tapes literatus (Fig.
4) shows oblique lines that regularly cross each other. Crossings are records
of waves that upon a collision do not annihilate, but penetrate each other.
This requires that upon a collision cells remain activated for a sufficiently
long period until the refractory period of the neighboring cells is over. Waves
generated by the enforced displacement as described above can show this behavior [8,9]. In the simulation Fig. 4, an activator-substrate mechanism (a, b in
Eq. 6-7) has been used with such parameters that a cell, once activated, would
just remain in a steady state (the condition for this is bb > ra , i.e., the supply
of the factor is higher than the removal rate of the activator). However, due
to the additional antagonistic action of the diffusive inhibitor c the activation
of the previously activated cells breaks down. Traveling waves result similar
to those discussed above. However, when two waves collide, the situation is
different (Fig. 4B). In this case no newly activated cells are available that produce sufficient inhibitor for down-regulation of the preceding cells. Therefore,
at the point of collision, both waves come temporarily to rest and cells remain
activated at a lower level. Neighboring cells can be re-infected again after their
refractory period is over. These newly activated cells extinguish via the newly
produced diffusible inhibitor the activation in those cells in which the activation escaped from annihilation, completing in this way the penetration of the
two waves.
The specimen showed in Fig. 4A shows an interesting perturbation. Many
oblique lines terminate at a particular growth line, presumably caused by an
9
A
B
C
T im e
P o s itio n
P o s itio n
Fig. 4. Wave penetration and the trigger of backwards-running waves. (A) Detail of a
shell of Tapes literatus. The crossing oblique lines document traveling waves that can
penetrate each other upon collision. (B) Penetration of waves in a three-component
system, Eqs. 6-8. Assumed is an activator - inhibitor system, a, c, (black area, gray
line). The autocatalysis of the activator depends on a non-diffusible substrate or
co-factor b (gray area). Its depletion leads to a local quenching of the signal and
to a shift of the activation into an adjacent region where still sufficient substrate
is available; traveling waves are the result. During collision, this shift is suspended.
Due to the rapidly declining long-ranging inhibitor the activation survives at a low
level. After the recovery of the factor, the waves start to move again into divergent
directions, completing the penetration. (C) The simulation in a larger field shows
that the model captures essential features: (i) waves are initiated in an initially
homogeneous situations; (ii) sometimes only one wave survive a collision, leading to
an amputated X (arrowheads). Occasionally cells are sufficiently long in a steady
state that backwards-running waves can be triggered (large arrows). (iv) After a
perturbation some lines bifurcate while others die out. The attempt to bifurcate
at this occasion is often visible (small arrows). This features are reproduced by
the assumption of a global lowering of the activator concentration. Calculated with
Eqs. 6-8 and Da = 0.01; ra = 0.08; ba = 0.0001 ; sa = 1; sb = 1; sc = 11; s = ra ±
5% random fluctuation; Db = 0; rb = 0.004; bb = 0.1; Dc = 0.4; rc = 0.02, i. e., b
acts as the local long-lasting and c as the long-ranging antagonist (from [8,9].
10
external event such as dryness or lack of food. Most remarkable, at the same
instance other lines bifurcate. It seems to be paradoxical that wave termination
and wave doubling occurs simultaneously. However, this phenomenon has a
straightforward explanation. A sudden lowering of the activator also causes
a decrease in the diffusible inhibitor, which, in turn, stabilizes the activator.
Thus, the situation during a perturbation is very similar as during a crossing:
two new diverging lines can emerge, in agreement with the natural pattern.
7
Out-of phase oscillations in E.coli for center finding during cell
division
How a bacterium finds its center in order to localize the division machinery
was a long-standing question. The preparation for division starts with the assembly of a polymeric tubulin-like FtsZ ring just underneath the cytoplasmic
membrane. Although more or less all molecules involved in this center-finding
process were known for long, it remained an open problem of how these components work together. In a review Shapiro and Losick [20] wrote: “we are
left with two topological mysteries: how does a bacterial cell knows where its
middle is and what is the medial mark that triggers polymerisation”.
The considerations given above were most helpful to integrate the known components into a molecularly feasible interaction, which was able to account for
the observation without assuming any pre-localized determinants [19]. Crucial
for center-finding is the highly dynamic behavior of the Min-proteins, MinC,
MinD, and MinE. High concentrations of membrane-bound MinD appear at
the poles in an alternating fashion (Fig. 5 A). A full cycle of this pole-to-pole
oscillation takes about 45s [18]. Thus, MinD shows an out-of-phase oscillation
as discussed above. MinD binds MinC that, in turn, inhibits FtsZ polymerization [21] and thus septum initiation. On time average the alternating appearance of MinD/C at the poles leaves only the non-inhibited center free of
the septum-repressing MinC.
For the out-of-phase oscillations a third player is indispensable: MinE. Without MinE, no oscillation takes place and MinD/C binds everywhere to the
membrane, abolishing any cell division. Initial visualization of the MinE protein has shown that it is more centrally localized. The question was then: How
can it be that a substance that accumulates in the center is responsible for an
alternating activation at the poles?
Although the similarities of the out-of-phase oscillation of sea shells and in
the center-finding mechanism of E.coli suggested an analogous mechanism,
a direct employment of the interactions developed for the sea shell patterns
turned out to be inappropriate. Both Eqs. 3-5 and 6-8 postulate the existence
11
(A )
(B )
T im e
P o s itio n
(C )
(D )
T im e
P o s itio n
Fig. 5. Out-of-phase oscillation in E. coli. (A) Binding of MinD protein to the membrane - visualized by GFP-labeling - resembles an oscillating polar pattern[18] (the
numbers indicate lapsed time in seconds, DIC is a phase contrast photograph of
the same bacterium). A full cycle occurs in ca. 45 seconds. MinD together with
other components inhibit initiation of cell division. In this way, cell division becomes restricted to the center. (B) Simulation: Assumed is that MinD (black) binds
everywhere to the membrane, MinE (gray) needs MinD for binding to the membrane, but after binding both molecules together detach from the membrane. In this
way, a MinE maximum destabilizes itself and enforces therewith its own shift into
a neighboring region where still sufficient MinD is present. Like the back and forth
movement of a windshield wiper, the MinE wave keeps the center free of MinD and
allows there the initiation of cell division. (C, D) If cell division is blocked, long cells
are formed in which MinD oscillates out-of phase in several patches. This is reproduced in the simulations. Note that the system finds very rapidly this mode without
any pre-localized components (for equations, parameters and animated simulations
see [19], photographs kindly provided by Piet de Boer).
of an inhibitor. However, no indication for such an inhibitor has been found in
the center-finding system in E.coli. On the other hand, in the models discussed
above there is no role for a more centrally localized component such as MinE.
A detailed modeling has revealed that the experiments can be described by
the following assumptions [19]: (i)MinD and MinE are molecules that can assembly at the membrane. (ii) This binding is the self-enhancing process as it is
required for any pattern-forming reaction. (iii) The antagonistic effect results
from the depletion of the corresponding precursor molecules that diffuse freely
within cytoplasm of the cell. (iv) In the absence of MinE, MinD accumulates
evenly along the membrane of the entire cell, in agreement with the observa12
tion [18] (in the model this occurs if the diffusion rate of non-attached MinD is
too low to allow pattern formation). (v)MinE association to the membrane depends on membrane-bound MinD and on diffusible unbound MinE molecules
in the cytoplasm. On its own, this would lead to a stable MinE maximum.
However, binding of MinE to MinD causes detachment of both molecules from
the membrane. Therefore, a MinE maximum enforces its own local destabilization, causing its shift into a neighboring region that is still rich in MinD,
and so on. The result is a moving ring of high MinE, concentration, which
‘peels’ MinD off the membrane. These MinE-waves have been directly visualized [22]. Such a wave comes to rest shortly before it reaches the pole due
to both a fading amount of membrane-bound MinD and a shortage of freely
diffusible MinE in the remaining portion next to the pole. Meanwhile MinD
re-assembles on the membrane in the opposite half of the cell. This attracts
the assembly of a new MinE-ring, which travels to the pole of this cell half as
well, etc. (Fig. 5). Thus, the out-of-phase oscillation of MinD does not result
from a direct self-destabilization of MinD, as in the simulations given above,
but from the back-and forth movement of a MinD-removing agent, MinE.
This special implementation in E.coli exemplifies the wide range of possible
realizations of the general mechanism.
The modeling shows that reliable patterning is possible without the need for
any pre-localized determinants. This is an important property of the model
because a requirement for such a factor would immediately raise the question
of how they themselves become localized, leading to an infinite regress. Starting with homogenous initial conditions, random fluctuations are sufficient to
initiate the patterning. In large cells resulting from a suppression of cell division, several such waves can be simultaneously at work, generating a periodic
out-of-phase oscillation, in agreement with the observation (Fig. 5 C,D [18]).
Two other models have been proposed for the MinD/MinE oscillation; both
depend on a recycling of the molecules involved. In the model of Kruse [23]
MinE also stimulates membrane detachment. MinD and maxima appear with
some phase shift. In Kruse’s model MinE does not form traveling waves (moving MinE rings), presumably because components causing lateral inhibition
effects do not play a role (in the model described above, the lateral inhibition results from a depletion of MinE momomers in the cytoplasm). In the
model of Howard et al. [24] the reaction consists only of an association to
and dissociation from the membrane. In their equations no direct autoregulatory components are involved. The instability has its origin presumably in
the mutual inhibition of two processes: MinD/MinE complexes, formed in the
cytoplasm, cause a dissociation of MinD from the membrane. Thus, MinEbinding lowers bound MinD. Conversely, MinE hinders the spontaneous association of MinD precursors to the membrane. Thus, MinD and MinE block
each other mutually, and an inhibition of an inhibition is in fact equivalent
of a self-enhancing reaction [3]. The diffusible precursor molecules play like13
wise an antagonistic role. In a more recent paper [25] it has been shown that
the number of molecules in the bacterial cell is high enough such that a reliable pattern formation is not abolished due to the corresponding unavoidable
fluctuations.
8
Chemotactic orientation of cell polarity
Many cells are able to migrate towards a source regions that secretes signaling
molecules. For instance, the growth cone at the tip of an axon enables path
finding [27] of an outgrowing nerve (Fig. 6A). Since the cells are small, the
concentration difference of the signaling substance between the front and rear
end of the cells is small. However, the cells are able to detect concentration
differences as low as 1-2% on their cell surface and orient their internal polarity
accordingly [28, 29]. This requires a very sensitive detection system. For an
oriented movement cells stretch out protrusions preferentially in the direction
to be moved. Many molecules involved in this process are known (for review
see [30-32]).
Earlier attempts to find models that account for the extreme sensitivity of
the cells in respect to external cues have revealed that the challenging problem is not the sensitivity itself. If a pattern forming system is in the instable
equilibrium, a minute asymmetry will orient the emerging pattern. However,
once a pattern is formed, the system is usually very stable and thus the position of activated regions on the cell surface can hardly be changed upon a
reorientation of the weak external asymmetric cue [3]. An important hint for
the nature of the underlying mechanism is the fact that this stretching out
and retraction also occurs in the absence of any external signal. This indicates
that a patterning mechanism is permanently active within a cell.
Three-component systems as outlined above provide a basis for an explication.
Due to the competition of the regions of the cell surface for activation, peaks
appear preferentially in the region that is favored by the external asymmetry. Subsequently, due to the second local antagonist, they become quenched
and disappear. Since the total activated area is maintained due to the longranging inhibition that covers the whole cell or growth cone, newly activated
regions emerge. The simulation shown in Fig. 6 is based on an activator - twoantagonist mechanism. One inhibitor is distributed rapidly within the cell,
which makes sure that only a certain fraction of the cell cortex becomes activated. It creates a competition that will be won by the side exposed to the
highest signal concentration. Using a saturating system, the activated region
obtains a certain extension. Together with some unavoidable random fluctuation and little or no activator diffusion, this leads to isolated maxima that
point in the direction of the guiding cue. These local activations on the cell
14
)
*
+
-
.
,
/
Fig. 6. Orientation of growth cones and chemotactic cells. (A) A growth cone of a
nerve growing in vitro. (B-G) Model [26]. Assumed is an internal pattern-forming
system in which the self-enhancing process saturates and in which the activator does
not diffuse; shown is only the activation. The distance from the inner circle is a measure for the local activation. The external orienting signal has a positive influence
on the internal patterning system of the cell. The concentration difference across
the cell is 2%; its orientation is indicated by the arrow. Assumed are max. 1% statistical variations in the cell cortex in the ability to perform the self-activation. (B-D)
Simulation: somewhat irregular active spots emerge that act as signals to stretch
out cell extensions towards the signaling source. Due to their limited half-life caused
by a local antagonistic process, they disappear subsequently and new ones emerge
instead. (E-G) After a change in the orientation of the external signal (arrow), the
locations of the temporary signals adapt rapidly to the new direction. Thus, the
system is able to detect permanently minute concentration differences (Photograph
kindly supplied by J. Löschinger).
surface are assumed to provide the signals for stretching out protrusions as
shown in Fig. 6A. The second and local antagonist is responsible for the finite
half-life of the protrusions. With the disappearance of an activated region the
global inhibition declines too and new spot-like activated centers will emerge
on the cell surface. They appear at the side pointing towards the highest signal
concentration even if the direction of the guiding signal has been changed. In
this way, the highly dynamic three-component systems provide the flexibility
to adapt to new situations. The dynamics has similarities in a human population: it is important that leading figures emerge. But it is also important that
these disappear after a while such that the next generation gets a chance to
respond to new challenges in a different way.
For the simulation in Fig. 6 the following interaction between an autocatalytic
activator a, the rapidly distributed inhibitor b and a local inhibitor c was
assumed (written here as a set of difference equations as they are used in the
computer simulations)
15
dai
si (ai 2 /b + ba )
=
− r a ai
dt
(sc + ci )(1 + sa ai 2 )
n
X
db
= rb
ai /n − rb b
dt
i=1
dci
= b c ai − r c c i
dt
(9)
(10)
(11)
i = 1...n denotes the surface elements (“cells”). The inhibitor b is assumed
to be redistributed so rapidly within the cell that its distribution becomes
uniform and its concentration can be calculated by averaging, i.e., its production rate is proportional to the sum of all activations on the cell surface.
The following constants have been used: activator a: decay rate ra = 0.02;
basic production required for initiation of the autocatalysis ba = 0.1; saturation of the autocatalysis to enable the coexistence of several maxima, sa =
0.005. The rapidly equilibrating inhibitor b: production and decay: rb =0.03.
Non-diffusible inhibitor c: production rate bc = 0.005; decay rate rc = 0.013.
Since the half life of the antagonist is longer then that of the activator (i.e.,
ra > rc ), oscillations do occur; Michaelis - Menten constant sc = 0.2; external
asymmetry and random fluctuations are integrated in the factor si ; it changes
by 2% over the circle and randomly by max. 1% from one space element to
the next. Further computational details for pattern forming reactions and PC
software for simulations can be found elsewhere [9].
A long-standing question is whether chemotactic orientation is achieved by
amplification of spatial (as in the model outlined above) or by temporal gradients. The model of Rappel et al. [33] is an example for the latter. These two
approaches are, however, not mutually exclusive. Both require a rapid spread
of an antagonistic effect in order to enhance the small differences imposed by
the external signal. In the model given above also a temporal signal will be
amplified since in such a mechanism the homogeneous situation is unstable
and any asymmetry, spatial or temporal, will be amplified. The observations
by Killich et al. [34] are very illuminating in this respect. They measured the
shape changes of isolated non-stimulated Dictyostelium cells and found that
the protrusions are not random. In one mode a rounded cell stretches to obtain
a more rod-like geometry and retracts later. The subsequent stretching occurs
perpendicular to the preceding elongation, and so on. In a second mode, the
cell keeps its elongation but protrusion and retractions of the cell leads to an
apparent windmill-like rotation. Both modes can be explained by the model
outlined above [26] according to which local signals become quenched shortly
after their generation. The first mode corresponds to an out of phase oscillation, the second to traveling waves on the cell surface. Both modes have been
shown above to occur in pigment patterns on shells (Fig. 3). Either the activity
collapses and reforms at the least poisoned position (mode 1) or it is shifted
into an adjacent position (mode 2). These experiments with non-stimulated
16
cells [34] reveal that pattern formation takes place permanently, whether the
cell is stimulated or not. In the absence of guiding signals the system is so
sensitive that asymmetries remaining from previous events in the cell are decisive for the determination where the next activation will take place. The
ever-changing cell shape demonstrates that this pattern formation never leads
to a stable steady state, a situation that requires three-component systems for
their description.
9
Phyllotaxis: initiation of leaves along spirals
The regular arrangement of leaves has fascinated people for centuries (see
[35,36] for a recent review and molecules involved). The tip of the shoot, the
apical meristem, contains undifferentiated cells that divide rapidly. Cells just
leaving this zone become competent to form new leaves. Thus, the leaf-forming
zone has the geometry of a ring. Similar as in shell patterning, the arrangement of leaves is a time record of the signal distribution in this leaf-forming
zone. Many different models have been proposed that have the assumption in
common that existing primordia have an inhibitory influence on the initiation
of the next leaf [36]. Implicit in this assumption is that the inhibition around
the circumference in the leaf forming zone and the separation along the axis,
i.e., the time span at which the next leaf can be formed, is based on the same
signaling. However, the existence of whorl-like leaf patterns suggests that the
spacing in space (i.e., around the circumference) and in time (e.g., when the
next whorl will come) is based on different mechanisms. This suggests an explanation of leaf spacing by two different inhibitions, one in space and one
in time [37], leading to models that are analogous to those proposed for shell
patterning (Fig. 7).
The formation of leaves at alternating opposite positions results from temporary signals displaced by 180o . This requires signals that jump from one side
of the leaf-forming zone to the other and back, analogous to the pole-to-pole
oscillation in E.coli discussed above. Since the leaf-forming zone has the geometry of a ring, and not of a rod as in E. coli, the displacement need not to
be 180o . In terms of the model, due to the local and long-lasting poisoning of
the signal, the signal could be hindered to jump back to the same position and
arises, therefore, at a displaced position (Fig. 7). A displacement by 137.5o ,
the golden angle, is a remarkable stable arrangement in the model [37], corresponding to a well-known pattern in phyllotaxis [36]. Fig. 7A, B illustrates
the formal similarity between pattern on a shell and of leaves.
17
)
,
*
+
.
!
"
#
$
%
Fig. 7. Flashing up of local signals at displaced positions in shell patterning and
phyllotaxis. (A) Oblique rows of spots on the shell of a mollusk, and (B, C) a helical
arrangement of leaves indicate the successive formation of spot-like signals at regularly displaced position. (D) Model: assumed is an autocatalytic activator (black)
that is antagonized by two inhibitors [8,9,37]. One has a short time constant and a
long range (dark pixels); it keeps the maximum localized. A second inhibitor (gray)
acts more locally and has a long time constant; it takes care that the leaf-initiating
signal disappears after a certain time interval. A new leaf-initiating signal can only
appear at a region where both inhibitions are below a threshold; calculation on a
ring, second dimension is time. (E) Calculation on a smaller ring; there is only space
for one helix. The most recent leaf initiation sites are numbered. (F) The same simulation in a plot as it is frequently used in botanical textbooks: youngest signals are
plotted close to the center, the older further to the margin. The two lines enclose
the golden angle of 137.5o , illustrating that the corresponding displacement of leaf
initiation sites are correctly described [37].
10
Traces behind moving spots: formation of branched filamentous
networks
While traveling waves in excitable media form long extended wave fronts, the
displacement of localized signals as described above can lead to spot-like regions of high activity that move over a field. If such signals cause permanent
changes like cell differentiation or cell extensions, the moving signals leave
behind trails of long extended filaments. The growth cone shown in Fig. 6A
that elongates a neuron was a first example. Many other systems are known
in which a moving local signal is involved in the elongation of branching filamentous structures. The formation of the lung, tracheae and blood vessels are
other examples (for review see [39-42]). The simulation in Fig. 8 is based on
18
+
)
*
,
-
.
Fig. 8. Moving signals as the driving force to generate net-like structures [38,3].
A local high activator concentration (black squares) is generated by an activator-inhibitor system. Cells exposed to the high signal concentration differentiate
to members of the net-like structure (open squares). Differentiated cells, in turn,
quench the local activation by removing a molecule (gray shading) that is necessary for the activator production. This causes a shift of the signal into an adjacent
position, inducing their cell differentiation too and thus the elongation of a filament. Branching can either occur by the generation of new signals along existing
filaments (A-C) or by splitting of signals whenever sufficient space became available
(D-F). The latter occurs if the autocatalysis in the signal formation saturates at
high concentrations (see Fig. 1).
a model proposed a long time ago [38]. Local signals cause cell differentiation
while, in turn, differentiated cells quench the signal, which causes a shift of
the signals into adjacent regions, and so on. Long filaments of differentiated
cells form behind the moving spot-like signals. The orientation of the shift is
controlled by a factor that is produced everywhere and which is removed by
the differentiated cells (possibly VEGF in the case of the blood vessel system,
see [42]). The substrate removal by the differentiated cells leads to graded
profiles around the filaments. The highest substrate concentration next to a
filament is in front of the tip. Therefore, in the absence of other constraints
such as nearby filaments, the extension of a filament is straight. The extension
occurs naturally towards regions that are not yet supplied by other filaments.
Branches can be generated in two different ways. In terms of the model, either new signals appear along existing filaments or the signal at the tip splits
causing a bifurcation of an extending filament at the elongation point. This
is related to the basic dichotomy between splitting of an existing signal or
the new trigger of a new one (Fig. 1). Branching in lung formation [39], for
instance, occurs typically by bifurcations at the growing tips.
19
11
Related mechanisms in non-biological systems
The mechanisms described above are not restricted to biological systems. Penetration of waves and wave splitting has been also observed in a catalytic reaction on platinum surfaces [43]. This phenomenon has been interpreted to
result from small imperfect regions on the crystal. The mechanism described
above suggest a different possibility: the involvement of a second, long ranging
antagonist that has a short time constant.
A simultaneous pattern formation in space and time has been observed with
temperature changes on catalytic ribbons [44]. Due to the catalytic reaction,
the temperature of an electrically heated wire increases, which has a positive
feedback on the actual reaction rate. With a thermo-sensitive camera, the
temperature has been recorded as function of time and position. The observed
space-time plots show traveling waves and oscillations with opposite phases in
neighboring regions that resemble closely the shell pattern shown in Fig. 3.
Electrical discharges become localized when the electrodes are plates of high
resistance [45, 46]. When compared with the three-component systems in biological pattern formation described above, the discharge resembles the selfenhancing process since an even higher current leads to a higher degree of
ionization and thus to an even higher current. The voltage drop around an
ongoing discharge functions as lateral inhibition, suppressing a second discharge next to an existing discharge. Accumulating space charges poisons in
the course of time the local discharge, forcing them to move to an adjacent
position [45,46]
12
Conclusion
Local destabilization of once established local high concentrations leads to
highly dynamic patterns that never reach a stable steady state. Such patterns account for diverse biological phenomena, from back-and-forth sweeping
waves that keeps the center of a bacterium ready to initiate cell division to
the continuous exploration of the environment by cells that are chemotactic
sensitive. Although a second antagonist provides only a moderate increase of
the complexity of a pattern-forming reaction, such three-component systems
provide a substantial enrichment of the toolbox.
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