Some basic features of
pattern-forming reactions
From “The Algorithmic Beauty of Sea Shells”
© Hans Meinhardt and Springer Company
(TO SEE THE ANIMATIONS, PLEASE SWITCH TO THE FULL PRESENTATION MODE)
Pattern formation can start from an almost homogeneous
initial situation
The freshwater polyp hydra can be
dissociated into individual cells. After reaggregation, 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
a2
2a
 
 a a  Da 2  a
t
h
x
2
h

h
2
  a  h h  Dh 2  h
t
x
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
Pattern formation requires local self-enhancement
and long range inhibition (with Alfred Gierer, 1972)

Kybernetik
12, 30-39 (1972)
(on our web-site)
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.
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)
Regeneration
After removal of the activated region, the
remnant inhibitor decays and a new
maximum is triggered from a base-line
activator production.
Position
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
a
sa

 ....
2
t b (1  sa a )
2
(this is the crucial parameter)
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
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.
Activator (HetR)
Inhibitor (PatS)
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 nonlinear.
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.
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
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
Photo: Hülskamp
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,
a
a2
2a
 s
 ra a  Da 2  ba
t
h
x
b
 2b
2
 s a  rb b  Db 2  bb
t
x
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
a
a
2
  s a   a a  Da 2   0
t
x
2
s
 s
2
    s a   s s  Ds 2
t
x
2
An alternative realization: the long-range inhibition
may be accomplished indirectly by the depletion of
a cofactor that is derived from a larger area
a
a
2
  s a   a a  Da 2   0
t
x
2
s
 s
2
    s a   s s  Ds 2
t
x
2
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 forms
Strong 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
c
s
 2c

 rc c  Dc 2  bc
2
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rb
b
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
 rbb  Db 2  bb
2
2
t
sc  c / a
x
(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.
a
2a
 ra c  ra a  Da 2
t
x
(Chordin)
(BMP)
(ADMP)
Models for positional signalling… Development (Supplement 1989), 169-180.
Curr. Topic in Dev. Biol; (2008) 81, 1-63
This interaction also allows regeneration. Regeneration
of organizing regions is a well-known observation
BMP
Chordin
ADMP
c
s
 2c

 rc c  Dc 2  bc
2
t sc  b
x
rb
b
 2b

 rbb  Db 2  bb
2
2
t
sc  c / a
x
a
2a
 ra c  ra a  Da 2
t
x
(Chordin)
(BMP)
(ADMP)
Models for positional signalling… Development (Supplement 1989), 169-180.
Curr. Topic in Dev. Biol; (2008) 81, 1-63
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, 429-450
Models 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 nonoverlapping 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
After Patel et al. (1989).Development, 107, 201-212
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)
Compartmental specification
Hox-gene activation
With posterior growth, whenever one compartment 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
H.M., Models of Biological Pattern Formation, Academic Press, 1982 (available on our website)
Somite formation: a sequential conversion of a periodic
pattern in time into a periodic pattern in space
Chick embryos at the 5 and 12 somite stage
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.
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 activation
in the chick. Activity comes to rest in the next posterior half-somite
Palmeirim 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. Longrange 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 published
our basic theory, the right is a more recent picture (2006).
Academic Press (1982)
See also our website at
http://www.eb.tuebingen.mpg.de/meinhardt
available as PDF on our website