Some basic features of pattern-forming reactions

From “The Algorithmic Beauty of Sea Shells” © Hans Meinhardt and Springer Company

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Pattern formation can start from an almost homogeneous initial situation

The freshwater polyp hydra can be dissociated into individual cells. After re-aggregation, de-novo pattern formation takes place in these clumps of cells. In this example, two heads form at opposite position and a single foot at the center. Later a separation leads to two viable animals.

(Gierer et al., 1972; Photograph kindly by Thomas Holstein

Pattern formation requires local self-enhancement and long range inhibition (with Alfred Gierer, 1972)

Kybernetik 12, 30-39 (1972) ;(on our web-site)

A simple realization: an activator has a positive feedback on its own production and catalyzes also the production of a rapidly diffusing inhibitor, which blocks the self-enhancement

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Pattern formation requires local self-enhancement and long range inhibition (with Alfred Gierer, 1972)

Kybernetik 12, 30-39 (1972)(on our web-site)

Dubocet al., (2004). Nodal and BMP2/4 signaling organizes the oral-aboral axis of the sea urchin embryo. Dev. Cell, 6,397-410. (Photograph kindly supplied by Dr. Thierry-Lepage)

The formation of the organizing region for the oral field of sea urchins by Nodal/lefty2(antivin) is an example for the employment of an activator-inhibitor interaction for the generation of an embryonic axis. As expected, the inhibitor (bottom right) is produced in the same cells that produce also the activator.

Regeneration

Position

After removal of the activated region, the remnant inhibitor decays and a new maximum is triggered from a base-line activator production.

If some remnants of the original maximum remains, the polarity may be maintained

Polarity reversal in a non-activated fragment

Left: after separation of a sea urchin embryo, both fragments regenerate; in one fragment the polarity reverses (Hörstadius 1939).

Model: after separation, the remaining inhibitor gradient in the non-activated fragment imposes an asymmetry. It is lowest at the side that was most distant to the originally activated region (blue arrow). This side will win the competition. As mentioned, the oral field of sea urchins is generated by a Nodal/Lefty2 interaction – an activator inhibitor system. It is the non-Nodal-expressing fragment (V) that reverses polarity, as expected by the model.

Pattern regeneration in fragments

If some activator-producing cells are included in the non-activated fragment (blue arrow), the non-activated fragment will regenerate with the original polarity. In the fragments the final maximum activator concentration will be lower since there is less space into which the inhibitor can escape.

In other systems such as hydra, the system generates an intrinsic asymmetry such that regeneration will always occur with the original polarity

Gradient formation

It is a property of these pattern-forming systems that a certain minimum extension has to be achieved until pattern formation can take place. During growth the first pattern that can emerge is a high concentration at one and a low concentration at the opposite side. In other words, even if initiated by random fluctuation, only a polar pattern can emerge. This is a most important step, e.g.,in the generation of embryonic axes.

Gradient formation

This, of course, works also in a two-dimensional field.

In this simulation, an attempt to generate a second maximum at opposite position is visible. However, eventually, due to the mutual competition, only one maximum survives. In larger fields two maxima or symmetrical distributions can emerge.

Saturation of the autocatalysis: stripe formation

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Due to saturation, the activated region enlarges, although the maximum concentration remains lower. Due to the lateral inhibition, the activation of cells is favored if they have non-activated neighbors into which the inhibitor can be dumped. Both requirements are satisfied in stripe formation

An example for an activator-inhibitor system: Heterocyst formation in Anabaena

Inhibitor (PatS)

Activator (HetR)

Activator: HetR is a DNA-binding molecule that activates its own gene. HetR forms dimers, in agreement with the expectation from the theory that the reaction must be non-linear.

Inhibitor: PatS, a 13-17 AA polypeptide, binds to HetR, which abolishes HetR binding to the DNA. In this way PatS blocks the autoregulation of HetR. It is small enough to be exchanged between the cells

For experiments see Zhang et al. (2006). Mol. Microbiol. 59: 367-375

In the blue-green alga Anabaena under nitrogen deprivation, nitrogen-fixating cells are inserted during growth. Whenever the distances between two such heterocysts become larger then ca. 12 cells, a normal cell differentiates into a heterocyst cell. The signaling is based on an activator-inhibitor mechanism. In the model, if the inhibitor concentration is too low, a new activation is triggered.

An additional inhibitor (blue) can be prevent the onset of this pattern formation. It fades away under nitrogen deprivation, causing pattern initiation at the right condition.

Saturation and no diffusion of the activator:salt-and-pepper distributions

Saturation of the autocatalysis restricts the mutual competition; activated cell can coexist close to each other. However, the ratio of activated / non-activated cells is regulated. The initial activation of prestalk- and prespore cells in Dictyostelium is of this type. The collection of the prestalk cells at the future tip of the slug is a later and separate process.

If the range of the inhibitor is smaller than the field size,periodic patterns are formed

If initiated in a field that is large compared to the range of the inhibitor, several peaks form. The spacing is somewhat irregular but a certain maximum and minimum spacing is maintained.

For the formation of the hairs in leaves (trichomes), the observation that the inhibitory components (e.g., Tryptichon, try) are expressed only in those cells that form the trichomes was regarded as counterintuitive. However, this is exactly the expectation of the model: the inhibitor is only produced by the cells that produces the activator and thus the hair-forming signal (see equation).

For observations see Hülskamp (2004). Nat Rev Mol Cell Biol 5, 471-480.Esch et al, (2004). Plant J. 40, 860,

Photo: Hülskamp

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Insertion of new structures

Bristles on the bug Rodnius. Newly inserted bristles during the last moult are shaded. (Wigglesworth, 1940)

During growth, the inhibitor concentration can drop in the interstices to such low levels that a base-line activator production can lead to a trigger of a new activator maximum. Due to the concomitantly produced inhibitor, the new peak will obtain the same size and height as the others. Insertion of new signaling centers is typical for the patterning of hairs and bristles (top right).

An alternative realization: the long-range inhibition may be accomplished indirectly by the depletion of

a cofactor that is derived from a larger area

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An alternative realization: the long-range inhibition may be accomplished indirectly by the depletion of

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In this reaction, the antagonistic component (red) has at the position of the maximum its lowest concentration – in contrast for the situation in activator-inhibitor systems.

Local accumulation that depends on a depletion in the surroundings is frequent in pattern formation in the non-living world

The activator-depletion model is convenient to account for intracellular patterning: the self-enhancing process takes

place at the cell cortex

For intra-cellular patterning, the self-enhancing effect can consist of a cooperative binding to the cell cortex (green). The local accumulation goes on expense of the molecules diffusing in cytoplasm (red). The different diffusion rates of molecules at the membrane and in the cytoplasm satisfy our general pattern-forming condition: short-range for the activation and long-range for the antagonistic reaction. In this case long range means: across the cell.

An asymmetry can enforce the formation of a single organizer

Only a single maximum can be formed in a small field….

In a larger field multiple maxima can appear (which could be a disaster for the developing embryo)…

However, if there is a graded competence imposed (for instance due to maternal determinants), only a single organizer formsStrong asymmetries are the rule in large eggs/embryos such as in amphibians.

Spemann organizer: Autocatalysis by an inhibition of an inhibition (BMP/Chordin)

Candidate for the inhibitor: ADMP

BMP Chordin

ADMP

Models for positional signalling… Development (Supplement 1989), 169-180.Curr. Topic in Dev. Biol; (2008) 81, 1-63

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(ADMP undermines the inhibitory action of Chordin on BMP and blocks in this way the indirect autocatalysis. Since ADMP acts as antagonist, the region of Chordin/ADMP expression is much smaller than that of BMP expression.

This interaction also allows regeneration. Regeneration of organizing regions is a well-known observation

BMP Chordin

ADMP

Models for positional signalling… Development (Supplement 1989), 169-180.Curr. Topic in Dev. Biol; (2008) 81, 1-63

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The lateral inhibition may be realized by a lateral activation of another feedback loop that locally excludes

the first. This long-range activation can be mutual

Meinhardt and Gierer (1980) J. theor. Biol., 85, 429-450Models of Biological Pattern formation (1982), on our website

For segmentation we proposed that cell states activate each other on long range but exclude each other locally. This mutual activation allows the formation of stripes since a long common border enables an efficient mutual stabilization.

The lateral inhibition may be realized by a lateral activation of another feedback loop the locally excludes

the first. This long-range activation can be mutual

Meinhardt and Gierer (1980) J. theor. Biol., 85, 429-450

This prediction found full experimental support. The gene engrailed (en), the key gene for posterior compartmental specification, is autocatalytically activated. Via the diffusible molecule hedgehog (hh), en activates in addition the gene wingless (wg) that is crucial for the anterior compartment. The gene sloppy paired is involved in the wg-autoregulation. The wg protein can reach adjacent cells via vesicle transport and is required there to stabilize en. As expected from the theory, the activity of the en gene in a cell requires an active wg gene in an adjacent cell and vice versa, although both genes are transcribed in non-overlapping regions. The prediction of such a complex molecular interaction by a theory could hardly be more precise.

Oscillations and spatial pattern formation during posterior outgrowth

H.M., Models of Biological Pattern Formation, Academic Press, 1982 (available on our website)

In short germ insects new segments are formed at the posterior pole. These segment represent not only a periodic structure, they are also different from each other. Only the thoracic segments form legs (arrows)

With posterior growth, whenever one compart-ment becomes too large, the mutual activation will be insufficient and some cells will flip into the alternative specification. Thus, the most posterior cells oscillate. This switching can be used to activate segment-specifying (HOX-) genes with a single-cell precision. In the model, the activation of a new gene occurs with the transition from posterior (red) to anterior (green) specification

Hox-gene activation

Compartmental specification

After Patel et al. (1989).Development, 107, 201-212

Somite formation: a sequential conversion of a periodic pattern in time into a periodic pattern in space

Somites are the primary segmented structure in vertebrates. They give rise to many structures, including the vertebrae. In contrast to segments in short germ insects, the somites are formed not at the posterior pole but at a more anterior position.

Chick embryos at the 5 and 12 somite stage

Somite formation: a sequential conversion of a periodic pattern in time into a periodic pattern in space

Models of Biological Pattern Formation, Academic Press, 1982 (on our website)

Oscillation and anterior spread of c-hairy1 activationin the chick. Activity comes to rest in the next posterior half-somitePalmeirim et al. (1997). Cell 91, 639-648

Starting from the posterior oscillation expected for short-germ insects I proposed 1982 that somites are formed by an oscillation in the posterior body part. This oscillation spreads in a wave-like manner towards anterior and comes there to rest at low levels of a gradient (yellow, now known as FGF). Each full cycle in the oscillation was assumed to add one pair of anterior/posterior half somites. 15 years later evidence for this predicted oscillation has been found by the Pourqui-group.

Conclusion:

The mechanism of pattern formation by local self-enhancement and long range inhibition provides a rich toolbox to account for essential steps in development. Graded profiles, periodic structures, stripes and oscillations are possible outcomes. The patterns are self-organizing, very robust and allow, e.g., regeneration. Different realizations are compatible with the general scheme. Long-range inhibition can depend on a depletion of a substrate or co-factor in a larger surroundings. Autocatalysis can be realized by an inhibition of an inhibition. The long range inhibition can be realized by a long range activation of a second feedback loop that locally excludes the first – a mechanism that enables a controlled neighborhood of structures. Highly complex patterns can be realized in a combinatorial fashion. The pigment patterns on tropical sea shell provide a rich set of examples.

I am most grateful to Alfred Gierer for a wonderful collaboration over many years. The left picture were taken about at the time we publishedour basic theory, the right is a more recent picture (2006).

Academic Press (1982)

available as PDF on our websiteSee also our website at http://www.eb.tuebingen.mpg.de/meinhardt