Pattern formation from initially more or less homogeneous situations is not restricted to living systems. The formation of sand dunes, patterns of erosion, lightning, stars and galaxies are examples for pattern formation in non-animated systems. The possibility of generating patterns by the interaction of two substances with different diffusion rates was discovered by Turing [5]. However, a different spread of two interacting substances does not guarantee pattern-forming capabilities. In fact, only a very restricted class of interactions allows pattern formation. We have proposed that pattern formation is based at local self-enhancement and long-range inhibition [6–8]: small local elevations above the average grow further on the expense of the surroundings. It is easy to see that the examples of inorganic pattern formation mentioned above are based on the same principle.

For biological pattern formation, a possible molecular realization can consist of a substance that has an autocatalytic feedback on its own synthesis; we have called this substance the activator a(x). The production rate of the activator is slowed down by a long-ranging molecule, the inhibitor b(x), which is produced under the control of the activator. The following set of partial differential equations describe a possible interaction:

In contrast, oscillations will occur if the inhibitor has a longer half-life than the activator (Fig. 2B). A non-diffusible inhibitor can lead under this condition to travelling waves (Fig. 2C). Such behaviour is well known from waves in an epidemic. The epidemic can spread since the autocatalytic agent, the virus, can be transmitted from one individual to the next one, while the antagonistic reaction – the action of the immune system – remains confined to the individuum. Oscillations and travelling waves play a crucial role in many biological systems, for instance in the signalling of neurons or in formation of pigment patterns on seashells (see Fig. 3). The theory of oscillating systems in biology is well developed [4,11–16].

The two activator-independent production terms b_a and b_b can have important functions. b_a is required to initiate autocatalysis when necessary, e.g., during regeneration, insertion of new maxima (Fig. 2A) or during oscillations (Fig. 2B). In contrast, a non-zero b_b can suppress the spontaneous onset of activation. In this case, the system remains asleep until an inducing trigger occurs that elevates the activation above a threshold. In the case of travelling waves, this trigger is obtained from neighbouring cells (Fig. 2C).

The activator–inhibitor reaction is, of course, only an example. In most inorganic systems with pattern forming capabilities, the antagonistic effect results from a depletion of a prerequisite that is required for the self-enhancing process. Equations (2) describe a possible interaction between an activator a(x) and a substrate b(x) that is consumed during the autocatalysis [6]:

Patterns generated in this way obtain a stable steady state supposed that the antagonist has a shorter time constant than the self-enhancing component(s). As mentioned above, there are, however, situations indicating that the corresponding signals are only present for a short period. This will occur if a second antagonistic reaction, with a longer time constant, is at work, causing a breakdown of the maximum shortly after it has been generated [3,4]. While the primary antagonist has to spread rapidly, the second antagonist has to remain more local to accomplish the destabilisation. Two possible molecular interactions are given below. The mechanism described in equations (3) is an activator–substrate interaction (equations (2)) enriched by an additional inhibitor c(x).

The inhibitor c(x), produced under activator control, slows down the autocatalysis, and, therewith, also the removal rate of the substrate b(x); c_c accomplishes a Michaelis–Menten-type limitation of the reaction rate at low concentrations of the long-lasting inhibitor c; s_c describes its efficiency; s_a leads to an upper bound of the autocatalysis at high a concentrations. Either b or c can accomplish the long-ranging or long-lasting function respectively.

To give another example, an activator may not only catalyse a single inhibitor species but two. Again, one inhibitor has to have a long time constant and has to spread only slowly. Both inhibitors act in an additive way.

Depending on the parameters, the local de-stabilization by the second long-lasting antagonistic reaction can lead to two different patterns. Either the activation is permanently shifted into a neighbouring region, causing travelling waves. Alternatively, the activation disappears and reappears at a displaced position. Both types of behaviours have parallels in very different biological systems that should be discussed in the following sections. Transitions between both modes do occur within one system, indicating that both modes have indeed a common molecular basis (see Fig. 4).

If the autocatalytic component spreads somewhat faster than the long-lasting antagonistic reaction or if the maximum is somewhat flattened due to saturation, cells at the flank of the maximum can achieve a higher activator production than in the centre. Under this condition, the maximum starts to move (Fig. 3). At the new position, the maximum will also be ‘poised’ and thus forced to move further. Since the second antagonistic reaction is long lasting, the movement will continue into the same direction, i.e., away from the preceding activation. This leads to travelling waves.

Modelling of travelling waves is in an advanced state, especially due to the study of the Belusov–Zhabotinsky reaction [15,19,20] and wave formation in the slime mould Dictyostelium discoideum [21,22]. However, there are substantial differences between these classical travelling waves and those to be described. As mentioned, the classical waves in excitable media depend on a mechanism that is similar to the spread of an epidemic: a self-enhancing reaction leads to an avalanche-like increase, while a long-lasting antagonistic reaction brings the system back to the initial situation. These waves need two components only. With appropriate parameters, the interaction given in equations (1) and (2) can produce such waves. If two such waves collide, they annihilate each other, since a wave cannot enter a region that is refractory due to the remnants of the antagonistic component of the counter wave (Fig. 2C). This mutual annihilation is a typical behaviour of the Belusov–Zhabotinsky waves. The formation of such waves depends on pacemaker regions, i.e., on regions in which the oscillation occurs somewhat faster. Here, the next activation occurs always a bit earlier, giving rise to the new waves. Without such pacemaker region, all elements may oscillate in phase. The small spread of the self-enhancing component leads to a synchronization. There is no tendency for a de-synchronization.

The pattern on the shell shown in Fig. 3A document a different behaviour. Since the shells preserve a space-time record, the oblique lines are a protocol of travelling waves. Obviously, they do not need a pacemaker since there is no particular area from which the waves start. Their crossings result from mutual penetrations. These features are reproduced in the simulation using the interaction given by equations (3) (Fig. 3B and C). A pacemaker region is not required, since due to the long-ranging antagonist, the interaction has the capability to generate local maxima from which waves can start. Turing [5] already described in his well-known paper that the spontaneous generation of travelling waves in a homogeneous field requires the interaction of three substances, a discovery that is almost forgotten. Whether the three component interactions of Turing are mathematically equivalent to the activator–two-antagonist concept discussed here remains to be shown.

The reason why wave penetration can occur in systems employing two antagonists is easily understood (Fig. 3B). In a reaction according to equations (3) in the appropriate parameter range, the activator a together with the (non-diffusible) substrate b can form a steady-state activation. Due to diffusion, the activation can spread and infect neighbouring cells. Each newly activation occurs with some overshoot. The concomitantly produced surplus of the diffusible inhibitor c contributes to extinguish the activation of the preceding cells, which is more susceptible due to the reduced substrate concentration. Therefore, apparently normal travelling waves result. However, if two waves collide, the situation is different. At the point of collision, the spread of both waves comes temporarily to rest. Therefore, no subsequently activated cells are available to extinguish the activation of the cells at the point of collision. Since the diffusing inhibitor decays rapidly, the activation is maintained until the refractory period of the neighbouring cells is over. Cells next to the point of collision become activated again and extinguish via the diffusible inhibitor the activation of the cells at the point of collision. With this step, the penetration is completed. This penetration is therefore connected with a transient change in the wave velocity, a feature known from soliton waves in physics.

The shell shown in Fig. 3A displays an interesting perturbation. Presumably triggered by an external event, several waves terminate at a sudden moment. After a lag period, new waves emerge, that are in register with the terminated waves. This can be simulated under the assumption that the external event led to a dramatic reduction of the activator concentration. In terms of the model, after decay of the long-lasting antagonist, the remnant activation may be high enough to trigger a new wave. Initially these waves can spread to both sides, leading to a bifurcation shortly after the general breakdown. This feature is in full agreement with the natural pattern.

The wave formation described in the preceding section depends on the shift of the activation into a neighbouring region. As suggested by the examples given in Fig. 1A, in other systems the activation is not continuously shifted, but disappears and reappears at a displaced position.

If the long-lasting antagonistic reaction has a range somewhat larger than that of the self-enhancing component, such a shift is not possible, since the immediate neighbourhood is also poisoned. Under this condition, the maxima tend to collapse sometimes after their generation. Such behaviour is especially likely if the primary pattern forming reaction is also in an oscillating regime. In the case both antagonistic reactions are realized by inhibitors, both inhibitors have to decay until a new activation can be triggered from an activator-independent baseline production (b_a in equations (4)). The new activation has to appear at a displaced position, since the previous places are still forbidden due to remnants of the long-lasting inhibitor. This is clearly visible in the simulation in Fig. 1C. The new activation can only occur in regions where the concentration of both the far-spreading (red) and the long-lasting inhibitor (green) becomes sufficiently low. In the time record as given in sea shells, this leads to sequential appearance of spots that have a defined displacement from preceding spots. In the course of times oblique rows of dots emerge.

The question may arise why a simple reaction, for instance, that described by equations (2), is insufficient for activations that flash up at changing positions. The condition for pattern formation (D_b⪢D_a) and the condition for oscillation (b_b<r_a) can be satisfied simultaneously. However, in such a simple system, all cells would reach the threshold for triggering the next activation simultaneously, since the substrate distributes evenly due to its high diffusion rate. Only after this trigger, spatial pattern formation can proceed in which neighbouring cells compete with each other. Therefore, before the uniform activation can resolve into well-separated maxima, the activation will break down due to the depletion of the substrate (Fig. 4A). Thus, the generation of well-separated activations at positions with a predictable displacement requires the combined action of two antagonists that differ in their range and time constant.

The regular initiation of new leaf primordia behind the tip of a growing shoot [23], the so-called phyllotaxis, seems to have nothing in common with any pattern on seashells. This impression is, however, misleading. On growing shoots as in shells, the new pattern elements are initiated in a narrow zone. The tip of the shoot, the meristem, contains undifferentiated cells that divide rapidly. Only cells just leaving this zone are able to form new leaves. Thus, the leaf-forming zone has the geometry of a narrow ring (for recent review, see [24]). Common are leaf arrangements at which each new leaf appears at alternating opposing positions (distichous patterns). Also common is a pair-wise arrangement in which each subsequent pair is rotated by 90° (decussate arrangement). These patterns can be simulated with the basic model of equations (1) [25]. According to classical models, the initiation of a new leaf is inhibited by existing leaves [26]. Therefore, a new leaf can be initiated only at a certain distance from the last formed leaf. This long-range inhibition could be a part of the general pattern-forming mechanism we propose. Together with the self-enhancement, our model explains why a leaf can be formed although it is in the centre of this leaf-inhibiting zone [27].

Many flowering plants display, however, a helical patterns, indicating that that the new leaves are formed at regularly displaced positions, a situation similar to the displaced dots on the sea shells discussed above (Fig. 1A). The displacement between subsequent leaves is close to 137.5°, the golden angle. This helical arrangement requires that the positioning of a new leaf is not only influenced by the last but also by the earlier leaves. This is easily understandable. Imagine that a first leaf exists. The second can appear at 180°. If the third leaf to be formed is exposed to some inhibitory influence from the first, it cannot be formed at the same circumferential position as the first and will become displaced. By modelling it has turned out that this requires again two antagonists with different properties. Assuming only one rapidly diffusing inhibitor, the asymmetry of the inhibition originating from the penultimate leaf would be very low and no reliable positioning influence would be possible. The situation is different, however, if two antagonists are involved. The long-lasting but slowly spreading antagonist would signal that this angular position next to the meristem has been used recently (Fig. 4D and E; [8]). According to this model, the helical arrangement does not result from an inhibitory influence of the earlier leaves directly, but from the remnants of the long-lasting antagonist in the leaf-forming zone that a preceding leaf has left behind at this position. This view is supported by the recent finding that new leaf primordia become rapidly disconnected from the leaf-forming zone by closing the cellular junctions between the cells [28,29], a feature that would make a direct inhibitory influence of preceding leaves unlikely.

Recent observations have shown that the plant hormone auxin is required as prerequisites in leave initiation [30,31]. Suppression of auxin transport suppresses leaf initiation, a situation that can be rescued by a local supply of auxin. This suggests that one of the expected antagonistic reaction results from a depletion of auxin in a wider surrounding. For this reason, equations (3) have been used to simulate the phyllotactic pattern given in Fig. 4. The spread of auxin is assumed there to be accomplished by diffusion. A more realistic model has to include its active transport. From the model it is expected that the auxin concentration declines at a new leaf initiation site, in contrast to the expectation discussed in the literature [32]. If auxin is present but its spread is blocked, it is expected that the leaf initiation signals remain present but do not form a pattern. This is in agreement with the observation that blocking of auxin transport leads to a ring-shaped expression pattern of markers for organ identity [31], i.e., the normal phyllotactic pattern is lost.

In whorl-like patterns, the primordia are closely together around the circumference but their separation along apical-basal axis is large (for an example see, e.g., [33]). Such patterns support the proposed separation into a component responsible for the patterning around the circumference (long ranging antagonist) and a component for a patterning in time, i.e., for apical-basal patterning (long lasting antagonist). At a particular time, many primordia appear closely spaced in a ring-like arrangement. Only after substantial further outgrowth, the next ring appears.

So far for leaf initiation at changing positions around the meristem, but what generates the signal for the meristem itself? Molecular components found in meristem formation (for reviews, see [24,34]) are compatible with the mechanism for stable pattern formation outlined in the introduction. WUS, a homeobox protein, is regarded as the primary meristem organizer. STM is necessary to maintain WUS activation [35] and vice versa, suggesting that WUS/STM is part of the expected activating system. WUS controls the CLV3 gene, a ligand involved in WUS inhibition [36], suggesting that CLV3 together with CLV1 and CLV2 is part of the inhibitory system. As expected, CLV3 is produced only in the activated region, although its action is of longer range. Moreover, the CLV genes not only inhibit WUS, but also promote organ formation. Thus, meristem/leaf patterning seems to have much in common with patterning in hydra where a primary system controls hypostome formation, which, in turn, generates the precondition for a secondary system that initiates the surrounding tentacles [37].

Many cells are able to perform an oriented movement towards a target area that secrete signalling molecules. With a growth cone on an outgrowing axon, nerve cells posses a ‘search organ’ for path finding (Fig. 5) [38]. The cells are able to detect concentration differences of the signalling molecules as low as 1–2% on their cell surface and orient their internal polarity appropriately [1,2]. This requires a very sensitive detection system. For the oriented movement, cell stretch out protrusions preferentially in the direction to be moved. Many molecules involved in this process are known (see [39,40]).

An important hint for the nature of the underlying mechanism is the fact that this stretching out and retraction occurs also in the absence of external signals. This indicates a patterning system that is permanently active within a cell. In the absence of external signals, the protrusions are stretched out at more or less random orientations (Fig. 5, see also Fig. 7), while in the presence of a signal this dynamic patterning obtains a bias towards the target region. In the following, the ingredients of a mechanism are described, that are able to accomplish this task.

If a pattern-forming system is still in the instable homogeneous steady state, the system is very sensitive to external perturbations. Minute asymmetries are decisive at which position the maximum will emerge. In the absence of such an asymmetry, random fluctuations can be decisive. The situation may be compared with a pencil balancing on its tip. It will fall down and minute differences decide in which direction it will fall. However, a chemotactic cell has not only to decide once in which direction to go but has to adapt permanently to an optimum orientation. The pencil-analogy illustrates the problem. After it has dropped down, i.e. after transition from the instable to a stable equilibrium, a much higher external influence would be necessary to reorient the pencil. This illustrates that the problem is not so much to achieve the sensitivity but to maintain the sensitivity of a cell even after the internal amplification already led to a strong internal pattern.

The following model has turned out to accomplish a permanent sensitivity [41]. The autocatalysis required for the internal amplification saturates at high activator concentrations. This leads to an enlargement of the activated region at a lower absolute level (Fig. 6A and B). The inhibition spreads rapidly over the whole cell, leading to an overall competition. Only a certain fraction of the cell cortex can become activated. If the activation does not spread on the cell surface, the activated regions need not to be coherent. In the absence of an orienting signal, their positions are determined by fluctuations and appear at random. With an external signal the internal signals appear where the combined influence of the external signal and the random fluctuations are highest (Fig. 6C). A stronger external signal will lead to a more focussed appearance of the internal maxima.

A permanent sensitivity can be obtained if these internal signals have only a limited half-life. This is the case if a second local antagonistic reaction causes a breakdown of the individual internal signals after a certain time. The deactivation of one region leads to a lowering of the long-ranging antagonistic reaction and enforces therewith the formation of a new activation at another position on the cell surface. Which region will win this race depends on the external signal and the cell internal state of the pattern-forming mechanism. Fig. 6D–I shows snapshot of corresponding distributions. On time average, the maxima point in the direction where the signalling molecule is coming from. In the simulation, it is assumed tat the external signal has a positive influence on the internal patterning system. After a change in the direction of the external signal, the cell internal pattern follows rapidly. Thus, with two antagonistic reactions, the maintenance of a permanent sensitivity in a chemotactic cell requires a mechanism analogous to that discussed above for shell patterns and phyllotaxis. For the simulation Fig. 6D–I, the following equation has been used:

This model offers a simple mechanism for an important process that plays a role the in the patterning of the nervous system [38], in the movement of cells during development and during inflammation [42]. Calcium plays presumably a pivotal role in the formation of cell polarity [43], a fact that can be integrated into the model in a straightforward manner [41]. Not yet included into the model is the fact that the same molecules can work as attractants or repellents, depending on other molecules to which the growth cone is exposed [44].

Observations of isolated cells of the slime mould Dictyostelium discoideum have shown that even in the absence of external signals, the formation of protrusions occurs in a non-random way [45,46]. Two modes are especially frequent (Fig. 7A and C). In the first mode, protrusions stretch out at two opposing sides. After their retraction, a new pair results, with a 90° orientation in respect to the preceding one. The second mode leads to protrusions that move around the cells in a wave-like manner. In terms of the model, these observations show that in the absence of external signals the cells are so sensitive that remnants of the long-lasting antagonist are sufficient for orientation. As shown in Fig. 7B and D, the model describes both modes. According to the model, the oscillation in counter phase corresponds to the leaf initiation in 90° pairs (Fig. 4B and C), underlining the proposed inherent similarity of phyllotaxis and chemotactic orientation.

Highly dynamic pattern-forming processes can be described by one autoregulatory reaction that is balanced by two antagonistic reactions. Such systems allow a simultaneous patterning in space and in time and may never reach a stable steady state. Patterning in diverse biological situation is proposed to depend on such processes. The position of the newly formed high concentrations can either depend on minute external signal (as in chemotaxis) or on the remnants of the antagonistic effects (as in leaf formation and shell patterning). The examples given are certainly not exhaustive. The centre finding in E. coli for the localization of the division plane is based on a wave that oscillates around the centre, keeping in this way the centre free of any division inhibitor [47]. The known molecules suggest that this wave formation is based on a similar mechanism [48].

The destabilization of a once-generated pattern has an antipode in a mechanism in which a pattern leads on the long term to a stabilization of the pattern, not to a destabilization, as discussed above. The competence of cells to perform the pattern forming reaction is then preferentially maintained in regions near to an activated (organizing) region. In this way, the formation of multiple maxima during growth as shown in Fig. 2A can be avoided. This is an important prerequisite to generate extended embryonic axes in a reliable way [7,10]. Thus, a positive or a negative feedback of a pattern on its own adds new features that play important roles in development.