J. Embryol. exp. Morph. 83, Supplement, 289-311 (1984)
Printed in Great Britain © The Company of Biologists Limited 1984
289
Models for positional signalling, the threefold
subdivision of segments and the pigmentation
pattern of molluscs
By HANS MEINHARDT
Max-Planck-Institut fiir Virusforschung, 7400 Tubingen, West Germany
TABLE OF CONTENTS
Summary
Introduction
Formation of primary organizing regions by autocatalysis and long-range inhibition
Interpretation of positional information
Mutual activation of cell states
Segmentation: the repetition of three cell states.
The SAP pattern and segment-polarity mutants in Drosophila
Formation of additional legs after removal of the segment border
Intercalary regeneration: different predictions of a positional information and a
mutual activation scheme
The pattern of shells of molluscs
Conclusion
References
SUMMARY
Models of biological pattern formation are discussed. The regulatory features expected
from the models are compared to those observed experimentally. It will be shown that:
(i) Stable gradients appropriate to supply positional information can be produced by local
autocatalysis and long-range inhibition.
(ii) Spatially ordered sequences of differentiated cell states can emerge if these cell states
mutually activate each other on long range but exclude each other locally. Segmentation
results from the repetition of three such cell states, S, A and P (and not of only two, as is usually
assumed). With a repetition of three states, each segment has a denned polarity. The confrontation of P cells and S cells lead to the formation of a segment border (... P/SAP/
SAP/S. ..) while the A-P confrontation is a prerequisite for appendage formation. Mutations
of Drosophila affecting larval segmentation are discussed in terms of this model.
(iii) The two models for the generation of sequences of structures in space (positional
information including interpretation versus mutual activation) lead to different predictions
with respect to intercalary regeneration. This allows a distinction between the two models on
the basis of experiments.
(iv) The pigmentation patterns of certain molluscs emerge from a coupled oscillation of
cells (that is, a lateral inhibition in time, instead of space). The oblique lines result from a chain
of triggering events.
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H. MEINHARDT
INTRODUCTION
Pattern formation during development of higher organisms is one of the challenging problems in contemporary biology. How can a particular group of cells
obtain a state of differentiation different from that of others despite the fact that
all cells contain, as a general rule, the same genetic information and are the
progeny of the same cell, the fertilized egg? Most information about this process
has been derived from experimental perturbations of normal development and
observations of how the systems respond. Of course, such observations do not
immediately reveal the molecular basis of development. However, one can
search for hypothetical molecular interactions which have the same regulatory
behaviour as that observed experimentally. To see whether such models are free
of internal contradictions and whether they account for the experiments in a
quantitative way, it is invaluable to formulate the hypothesis in a mathematically
precise way. In recent years we have elaborated several mathematically formulated models for the molecular basis of development at different stages (for
review see Gierer, 1981; Meinhardt, 1982). In this article, I will provide an
overview of some of these models. The formation of segments, intercalary
regeneration and the pattern on shells of molluscs will be discussed in more
detail.
FORMATION
OF PRIMARY ORGANIZING REGIONS BY
AND LONG-RANGE INHIBITION
AUTOCATALYSIS
The structure of the adult organism cannot be present in a latent form in the
fertilized egg since, after bisection of many different early embryos, both fragments often form a complete organism. Thus, in the early embryo, a selfregulating pattern must be generated which governs further cellular pathways.
A strongly patterned distribution of substances can be achieved if a local
autocatalysis is coupled to a long-range antagonistic effect (Gierer & Meinhardt,
1972; Gierer, 1981; Meinhardt, 1982). This leads to competition among the cells
which will be won by some cells which in turn suppress others. A simple
molecular realization would consist of a short-ranging autocatalytic activator
which catalyses in addition its long-ranging antagonist, the inhibitor (Fig. 1).
If, during the onset of pattern formation, the range of the activator is comparable to the size of the field (e.g. the size of the egg or the early embryo), space
is available only for one marginal activator maximum. The result is a polar
pattern. Thus, the model accounts for an essential feature: regions of the
developing organism become exposed to different concentrations of a substance
(morphogen). Different genetic information can be activated in different parts
of the embryo under the control of this morphogen gradient.
An activator maximum has many properties of a classical organizing region
such as the dorsal lip in amphibians (Spemann & Mangold, 1924). According to
Models for positional signalling
291
our model, small tissue fragments containing activator can regulate to a full
maximum due to autocatalysis. This will occur after a transplantation into a
region of low inhibitor concentration, i.e. at distance to an existing maximum.
The result would be a symmetric instead of a polar development. Further, a
fragment which does not contain the organizing region can regenerate a new one
since, according to the model, after removal of the activated region, inhibitor is
no longer produced in the fragment. After decay of the remnant inhibitor,
autocatalysis can start anew and, due to the selfregulating features of the patternforming process, a new maximum (organizing region) is generated in the frag­
ment. For the same reason, regulation is also possible after partial removal of the
organizing region.
A puzzling observation was the discovery that very unspecific perturbations
Fig. 1. Pattern formation by autocatalysis and lateral inhibition. (A) A possible
reaction scheme. An activator has a positive feedback on its own production as well
as on a rapidly diffusing inhibitor which slows down the activator autocatalysis
(Gierer & Meinhardt, 1972). ( B , C ) The instability: A local activator increase can
grow further since the additionally produced inhibitor (—) spreads out rapidly,
suppressing activator production in the surroundings. (D-F) Stages in the formation
of a stable gradient in a two-dimensional field of cells. Pattern formation is triggered
by random fluctuations. If the range of the activator is of the order of the field size,
only one marginal maximum can emerge. (A maximum in the centre would require
space for two activator slopes.) In the absence of strong polarizing influences, the
pattern will orient itself along the longest extension of the field. The polar pattern
can be maintained even after further substantial growth of the field. The gradient can
be used as positional information.
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H. MEINHARDT
can lead to the induction of a second organizing region. In our model, an unspecific inhibitor decrease in a non-activated region can lead to the onset of
autocatalysis. The inhibitor decrease may be caused by a leakage through an
injury or by reduced inhibitor production due to cell poisoning. After surpassing
a certain threshold, a second maximum (organizing region) develops in a selfregulatory fashion. For early insect development I have shown that the
activator-inhibitor model accounts for many experimental observations in a
quantitative manner. As we will discuss later, the same interaction but with
different time constants can lead to oscillating patterns. The pattern on shells of
molluscs will be discussed from this point of view.
Interpretation of positional information
The graded concentration profile generated by the mechanism discussed
above could be used to determine several structures in a well-defined sequence,
i.e. as positional information, using Wolpert's terms (1969,1971). From the data
available for early insect development (see Sander, 1976) we have deduced how
cells obtain different states of determination upon exposure to a morphogen
gradient: they are promoted step wise from more anterior (or in the leg from
more proximal) to more posterior (distal) determinations until the state of determination achieved corresponds to the local morphogen concentration (Meinhardt, 1977,1978). Each step is essentially irreversible. Thus, a cell remains in
a state of determination if the morphogen concentration decreases but it can be
promoted still further after an increase of the morphogen. This mode of interpretation seems to be at work in all developmental systems discovered so far for
which the experiments suggest the involvement of a morphogen gradient: the
body pattern of insects (Meinhardt, 1977) the segments of insect legs (Meinhardt, 1983a) and the digits of vertebrates (Tickle, Summerbell & Wolpert,
1975). It is remarkable that all these systems are segmented structures. The
stepwise and irreversible promotion is significant for the developing organism.
Usually, a field grows. However, the slope of a gradient remains unchanged as
long as the diffusion and the lifetime of the morphogen remains unchanged. This
means that in growing systems the morphogen gradient shrinks in relation to the
total field. Due to growth, a group of cells and their progeny increase their
distance to the source. They would be exposed to lower and lower morphogen
concentration. Only if determination is irreversible would the cells not forget
what they have learned. This idea of unidirectional promotion predicts rules for
the presence or absence of intercalary regeneration which will be discussed later
(see Fig. 6).
MUTUAL ACTIVATION OF CELL STATES
An alternative mechanism for the generation of sequences of structures in
space seems to be used to form non-segmented structures, for instance the pattern
Models for positional signalling
293
within a segment or the dorsoventral organization of organisms. In these systems
neighbourhood control appears to exist. Juxtaposition of normally non-adjacent
groups of cells lead to the intercalation of the missing intervening structures.
Depending on the type of juxtaposition, the intercalated structures may exhibit
polarity reversal. Polarity reversal of intercalated structures is very difficult to
understand by assuming a morphogen gradient with a source at one end and a
sink at the other since this would lead to the restoration of the normal slope of
the gradient.
We have proposed that this type of pattern formation is not based on the twostep mechanism mentioned above (in which first a morphogen gradient is
generated and second in which particular concentrations lead to particular cell
states), but that a direct mutual activation of cell states is involved (Meinhardt &
Gierer, 1980; Meinhardt, 1982). As discussed below in detail, this mechanism has
very good size-regulation properties, it accounts for the stripe-like arrangement
of differentiated cell states and it allows intercalation with reversed polarity.
Let us first ask how a pattern of two cell states - A and P - can be generated.
The states A and P must be locally exclusive: a particular cell should be either
in state A or in state P but not in a mixture of both states. This will occur if the
two cell states are created by two feedback loops which compete with each other.
A possible basis consists of two genes which each have a positive feedback on
their own activity but which compete with each other, for instance via a common
repressor. (Other mechanisms are also possible). The competition among the
alternative gene activities has the consequence that in a particular cell eventually
only one of the alternative genes remains active. To obtain a patterned
distribution (i.e. to avoid, for example, all cells in state A), we have to insure that
where gene A has won the competition, gene P must win in a neighbouring
region. For this to occur, over long ranges, cells in state A must activate gene P
and vice versa. Fig. 2 shows such an interaction and a simulation of a patternforming process based on it. A stripe-like distribution of cells in which either A
or P is turned on is especially stable since stripe formation is necessarily correlated with the formation of long boundaries between the two regions. This
enables an efficient mutual stabilization. Due to the narrowness of a stripe, each
A cell is not too distant from a P cell which is required for the stabilization of the
A state (and vice versa). Further, this mechanism has excellent size-regulating
features. For instance, if the A region is too large, the P state would be amplified
while the A state would obtain little help from the few P cells. This has the
consequence that some A cells switch into the P state until the correct ratio of
both types is obtained. Pattern regulation is possible after complete removal of
one cell type and the pattern can be maintained even if the field grows substantially (for details see Meinhardt & Gierer, 1980; Meinhardt, 1982). The mechanism of mutual activation is easily extended to more than two states (see Fig. 7).
A special feature is that such sequences can be cyclic. For instance, state 1 can
activate state 2, 2 can activate 3, and so on with state n activating state 1.
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H.
MEINHARDT
(long range help for P)
(long range help for A)
Fig. 2 . Pattern formation by long-range activation of two states. (A) A possible
reaction scheme: Two feedback loops, called A and P, compete with each other via
a common repressor but activate each other via the rapidly diffusing substances SA
and Sp. (B-D) Stages in the formation of a stripe-like pattern in a two-dimensional
field of cells. Pattern formation is initiated by a small A elevation at the right margin.
Alternating stable stripes of high A and high P concentration emerge. The long
borders between regions of high A and high P concentrations enable an efficient
mutual stabilization of the two feedback loops. (For the mutual activation mechan­
ism of more than two states see Fig. 7.)
Segmentation:
The repetition of three cell states
Segments are clearly periodic structures. The simplest periodic structure
would consist of the alternation of two states. Segments are usually assumed to
consist of a pair of two cell states (see, for instance Garcia-Bellido, 1975;
Lawrence & Morata, 1983). However, in an alternating sequence of two states,
e.g. A P A P A P . . . it is not obvious where in this sequence a segment border
should form; it could be A P / A P / A P or A / P A / P A / P , One could imagine that
a juxtaposition of two different segmental qualities is required to form a segment
border. However, from the phenotypes of mutations in the Bithorax gene com­
plex (Lewis, 1978), it is clear that this is not the case. The formation of segment
Models for positional signalling
(A)
295
(B)
N
Fig. 3. (A) A model for segmentation. It is proposed that segmentation results from
the reiteration of three subunits, to be called S, A and P. A segment border is formed
whenever P and S cells are juxtaposed. An A-P border is a prerequisite for the
formation of a leg (Fig. 5) or a wing. In Drosophila A and P should correspond to
the well-known anterior and posterior compartments (only these regions may
contribute to the cuticle of the adult fly). With the SAP pattern, each segment has
from the beginning an internal polarity. Transplantation of tissue from the anteriormost part (S) of a segment into a posterior location (P) (or vice versa) can also lead
to the formation of a segment border (Wright & Lawrence, 1981), in obvious agreement with our model. (B) It is not assumed (as in most current models of segmentation) that the primary event is the formation of a saw-tooth-like gradient which
controls in a second step the formation of the repetitive structures of the segments.
borders proceed normally even if all segments have the same identity. Therefore,
the formation of segment borders must be under the control of the periodic
pattern. The signal to form a segment border cannot be a border, for instance,
between meso- and metathorax, not even in connection with an AP border. For
this reason, I have proposed (Meinhardt, 1982) that segments consist of three
states, . . . P/SAP/SAP/S... A segment border is formed whenever P and S
cells are juxtaposed (Fig. 3), while the AP border, as will be discussed below, is
a prerequisite to initiate appendages. The grouping of the three elements is
unambiguous. With the iteration of three states, each segment necessarily has a
polarity.
The three states founding the segment enable a finer subdivision in a straightforward manner. Additional structures can be intercalated, leading, for instance,
to a sequence S1S2A1A2P1P2 etc. In contrast, any intercalation in an APAP
sequence would maintain the non-polar, symmetrical character of such a
sequence.
Nerves of the developing nervous system of insects show a predictable routing
in either the anterior or posterior direction (Bate, 1976; Bate & Grunewald,
1981). The assumption of the threefold subdivision of segments accounts for this
general feature. Imagine that a nerve is to be programmed to grow from an A
region towards the head, it would be sufficient to give the instruction: 'grow
towards S and then straight ahead'.
The threefold subdivision of segments is almost visible in leeches. As the rule,
each segment consist of three rings, the so-called annuli (Mann, 1953). In some
species, two of three annuli are fused. In other species, some annuli subdivide
further (A-» Ai A2 etc). The annuli play an essential role in the formation of the
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H. MEINHARDT
sensory nervous system and of the muscle pattern of the leech segments. It is
suggestive that each of the three annuli contain cells of a particular determination which forms the basic blocks of the segment.
The SAP pattern and segment-polarity mutants in Drosophila
The threefold subdivision of segments leads to the prediction of mutants which
correlate with phenotypes of mutations affecting the larval polarity of
Drosophila (Niisslein-Volhard & Wieschaus, 1980). If one of three feedback
loops (genes) is destroyed by mutation, a sequence of two would remain. An
alternation of two states would form always symmetrical patterns. We expect
three types, SASA..., S/P/S/P and APAP. To predict the phenotypes of such
sequences we have to correlate the pattern of the wild type with our hypothetical
SAP pattern. Since clonal analysis is not possible on the larval ectoderm, this can
be made only in a tentative way. The pattern of the larva consist of belts bearing
denticles separated by regions of naked cuticle (Lohs-Schardin, Crerrjer &
Niisslein-Volhard, 1979). The segment borders are located at the anterior side
of the denticle belts, or more precisely, between the first and the second row of
denticles (Niisslein-Volhard & Wieschaus, 1980). I will assume that the naked
cuticle contains the A and P region while the denticle band represents the S
region. Incidentally, the naked cuticle is quite precisely twice as large as the
denticle belt region posterior to the segment border (Fig. 4).
The denticle bands themselves have polarity. I propose that this polarity
results from the P confrontation on the one side and the A confrontation on the
other side of the S band: The P-S confrontation leads to the formation of the fine
denticles at the anterior margin of the denticle band (the segment border) while
the heavy denticles result from the S-A confrontation at the posterior side (Fig.
4). In the thoracic segments, only the denticles around the SA border seem to
remain. This assignment enables the prediction of phenotypes. If A is lost by
mutation, an S/P/S/P/S pattern remains. We expect the number of denticle
bands to remain normal but the number of segment borders to double. Due to
Fig. 4. Explanation of segment polarity mutations (Niisslein-Volhard & Wieschaus,
1980) in terms of changes in the SAP pattern. (A) Anterior portion of a Drosophila
wild-type larva. The denticle band is assumed to be the S region, the naked cuticle
to resemble the P and most of the A region. The precise location of the S-A border
is not yet known. Presumably the posterior portion of the denticle belt already
belongs to the A region (Jiirgens, personal communication). According to the
model, the segment borders (||) correspond to P/S juxtapositions. (B) In patch, it is
assumed that the A region is lost. The number of denticle bands remains the same
but the number of segment borders (P/S confrontation) is doubled. No heavy denticles (SA confrontation) are formed. (C) Another example for a mutation leading to
a symmetrical segment pattern: gooseberry. Presumably, here the P region is lost. In
terms of the model, an SASA pattern remains. This accounts for the symmetrical
pattern, the absence of segment borders and for the formation of bands with heavy
denticles (photographs kindly supplied by Ch. Niisslein-Volhard).
Models for positional signalling
297
the lack of the A region, a new S/P confrontation is created in each segment
which causes a new segment border, one at each flank of the denticle bands.
Since in a . . . P / S / P / S / P . . . arrangement, the S region is bordered on both sides
B
Fig. 4
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H. MEINHARDT
with P cells, no heavy denticles are formed. The naked cuticle should be of the
same size as the denticle bands (not twice as large as in the wild type). This is the
phenotype of the locus patched discovered by Nixsslein-Volhard & Wieschaus
(1980; figure 4).
Should the P region be deleted by mutation and an SASA... sequence remain,
symmetrical denticle bands would be expected but in this case with heavy denticles on both sides. No segment border should occur. The pattern of the mutation
gooseberry is close to this (despite the fact that the space between the bands is
not completely naked). The third case would be an S deletion and an APAP
pattern. This pattern is difficult to identify since the denticle bands as markers
of segmentation are expected to be absent. However, Jiirgens, Wieschaus,
Niisslein-Volhard & Kluding (1984) have identified a locus 'naked' in which,
depending on the strength of the allele, large portions of the belts are deleted.
At the blastoderm stage, the time at which the segment- and the A-P borders
are fixed, the anteroposterior extension of a segment is three to four cell
diameters (Lohs-Schardin et al. 1979). We have to conclude that each S, A or P
stripe consists of a single row of cells. This underlines the necessity of employing
a mechanism which can produce stripes: structures with a short extension in one
dimension and a long extension in the other.
Formation of additional legs after removal of the segment border
I have proposed that the intersection of the AP compartment border with the
DV compartment border forms the organizing region for appendages (Meinhardt, 1980,1983a). If the segments were to consist of... A P / A P / A P . . . pairs
only, there would be two AP borders per segment. The question would then arise
why two pairs of appendages are not formed per segment, one at each P/A
juxtaposition. In the SAP/SAP scheme, this problem does not exist because the
S region separates the P and A region. Strong support for this model comes from
Fig. 5. Explanation for the formation of a supernumerary leg after removal of a
piece of cuticle and ectoderm from an anterior portion of a thoracic cockroach
segment (experiments of Bohn, 1974a,b,c). (A) Formation of normal legs according
to the model. The repetitive /SAP/ pattern is shown. (B) Positional information for
leg formation is generated by cooperation of compartments (Meinhardt, 1980,
1983a) at the intersection of the A-P and D-V compartment border (assuming a
similar compartmentalization as in Drosophila). A cone-shaped gradient centred
over the intersection of compartment borders results which leads to a circular arrangement of leg primordia. (C) Removal of an anterior portion of a segment
(dashed in Fig. A) leads to the removal of the S region which separates the P and the
A region. After healing, a new PA/DV confrontation and thus an additional leg
results. It has the opposite handedness and orientation compared with the other legs
on the side of operation since the new A-P confrontation is inverted (P is anterior)
and AV, P, AD is clockwise instead of counterclockwise arranged. (D) Scanning
electron micrograph of a supernumerary limb (s) formed after such operation. The
left mesothoracic leg (msl) has been removed for photographic purpose; mtl:
metathoracic legs, specimen kindly supplied by H. Bohn, see Bohn, 1974a).
Models for positional signalling
Fig. 5
299
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H. MEINHARDT
experiments of Bohn (1974a-c). He removed small tissue fragments from the
anterior part of cockroach segments. (He has called this part the scleriteinducing membrane; for that reason I have termed the hypothetical anteriormost
part of a segment S). After wound healing an additional leg was observed to
form. According toour mo'del, S removal leads to an additional P-A confrontation at the same dorsoventral level. Thus, the condition for limb formation
obtains (Fig. 5). This experiment tells us also that the third cell state is located
at the anterior and not at the posterior margin of the segment, i.e. that an
. . . SAP/SAP... and not an APS/APS pattern exist. In Drosophila, the utilization of the cells around the A-P compartment border to form the structures of
the imaginal discs and thus of the adult fly may be the reason why an S region has
not been detected in experiments based on clonal analysis. The S region may be
used for larval structures only.
Other types of mutations (Niisslein-Volhard & Wieschaus, 1980) suggest that
the SAP pattern is not directly formed under the influence of the primary morphogen gradient but that first a subdivision into a few cardinal regions takes place
(as indicated by the gap mutations) and that a further intermediate subdivision
works on the level of double segments. The discussion of corresponding models
is outside the scope of the present paper.
INTERCALARY
REGENERATION:
DIFFERENT
PREDICTIONS
OF
POSITIONAL INFORMATION AND A MUTUAL ACTIVATION SCHEME
A
We have seen that two rather different mechanisms can produce ordered
sequences of structures in space: on the one hand positional information plus
interpretation and, on the other hand, mutual activation of cell states. It may be
helpful to discuss the different regulatory features expected from the two models
to provide clues about which mechanism may be involved in a particular developmental situation.
In a positional information scheme, the cells measure the local morphogen
concentration but they do not communicate directly with their neighbours to
check whether the neighbourhood is correct. The neighbourhood is normally
correct since the different cell states are activated under the influence of a
smoothly graded morphogen concentration. As mentioned, the available experiments indicate that the cells are unidirectionally promoted. This has important
consequences bearing on whether intercalation takes place or not (Fig. 6). If an
intervening portion is removed, after wound healing some cells will be located
closer to the source region (Fig. 6C). Thus they become exposed to a higher
morphogen concentration and, after corresponding promotion, the gap will be
repaired. This is possible, however, only if the system did not grow substantially
between determination and tissue removal (Fig. 6E).
An example of the presence and absence of intercalation in a positional information scheme is presumably given in the regulation of the proximodistal
Models for positional signalling
301
(B)
(A)
positional
information
(C)
(D)
/I
repair of a gap
i
T^T
(E)
no intercalation
after growth
3
1-
7 i
(F)
no intercalation
with polarity reversal
.6-
Fig. 6. Presence and absence of intercalation in a positional information scheme:
(A) The primary event is assumed to be the generation of a morphogen gradient
(either by autocatalysis-lateral inhibition, Fig. 1, or by cooperation, Fig. 5). (B) The
second step is a unidirectional promotion of cells from more anterior to more posterior or from more proximal to more distal determinations (distal transformation)
until their state of determination correspond to the local morphogen concentration.
The scheme leads to predictions of whether intercalation takes place after certain
graft experiments. (C,D) Removal of an intervening piece of tissue brings tissue
closer to the (shaded) source region. Some cells become exposed to a morphogen
concentration higher than they were originally exposed to. Thus these cells become
promoted (arrows) and the gap repaired. (E) If growth occurs between determination (B) and tissue removal, the distance of the tissue to the source may remain too
large. The local morphogen concentration is too small. No promotion can take place
and the gap will remain. (F) After grafting a large source-containing region to
another large fragment, the resulting discontinuity of cell states will remain. Since
the cells obtain their positional information only from the local morphogen concentration but not from a communication with their (unusual) neighbours, the
discontinuity cannot be repaired. Such regulatory behaviour can be seen in the
proximodistal axes of vertebrate limbs (Meinhardt, 19836).
pattern of amphibian limbs. Let us call the normal sequence of structures 1,
2 . . . 7. If a distal part of a blastema (67) is grafted on a proximal stump (123),
the intervening structures (45) are intercalated. In contrast, if a larger limb
blastema (/34567) is grafted on a distal stump (123456/) the pattern discontinuity is not repaired. This result appears surprising since after both operations cells
of the same proximodistal level are confronted (3/6 and 6/3). This shows
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H. MEINHARDT
unambiguously that intercalation in this case is not a strictly local process. However, this result corresponds precisely to the expectation from a positional information model (Fig. 6C,D,F; for detailed modelling and references see Meinhardt, 1982, 1983/?). Growth in a positional information scheme (Fig. 6E) is
presumably the reason for the absence of intercalation in the following graft
experiment. If a mid-tibia fragment of a cockroach leg is grafted onto a midfemur stump, the missing femur-tibia articulation is not replaced (Bohn, 1970).
According to the model, the source region - the intersection of compartments
at the limb tip - is too far away to enable any reprogramming of the stump cells.
This result is in contrast to intercalation within a segment (Fig. 7E) which is
assumed to be based on mutual activation.
In a positional information system, polarity reversal will never occur as long
as no new source region is introduced (Fig. 7E). If, however, polarity changes
occur, we expect that this is caused by a new source region and we expect
therefore that polarity reversal is connected with the formation of new terminal
structures. The formation of a second abdomen in insects or of a second digit 4
in the chicken wing (Tickle etal. 1975) is assumed to have this origin. In contrast,
no terminal structures are expected after a polarity reversal in a mutual activation scheme (see below).
An additional indication that a positional information scheme is involved is the
existence of a clear organizing regions, such as for instance the zone of polarizing
activity in the chicken wing (Tickle et al. 1975) or the posterior pole of insects
(see Meinhardt, 1977). In insects many examples are known where, after experimental interference, segments become neighbours which are usually determined
at large distances from each other. For instance, head and abdominal structures
can appear in close proximity. Such a pattern discontinuity will never be repaired
(see Meinhardt, 1982, for discussion), supporting the view that the cells do not
directly communicate with each other to detect gaps in the sequence of structures.
The situation is very different in a mutual activation scheme. The cells depend
on the correct neighbourhood, at least during certain phases of development.
Pattern discontinuities can be repaired even with a polarity reversal (Fig. 7). A
Fig. 7. Sequence formation by lateral activation of several cell states. (A) The
reaction scheme. Several feedback loops activate each other on long range but
exclude each other locally (see also Fig. 2). (B) Computer simulation of the formation of a sequence of cell states in a growing field. The density of dots indicate the
activity of the feedback loops ('genes'). New pattern elements are added whenever
sufficient space is available. (C) Gaps can be repaired even after substantial growth.
(D) Intercalation with polarity reversal is possible. (E,F) An experimental example
in cockroach leg segments (Bohn, 1970). The polarity reversal of the intercalate
(Reg) is indicated by the reversed orientation of the spines. (F) Confrontation of the
most extreme regions of a leg segment can also cause a new segment border (French,
1976), in analogy the mechanism of P/S confrontation proposed for the formation
of borders between body segments (Fig. 3).
(D) Intercalation with polarity reversal
(C) Intercalary regeneration
o
(B) Formation of a sequence
Fig. 7
Position
Position
Position
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H. MEINHARDT
polarity change is possible without the formation of the most terminal structures
since only a smoothing out of the pattern discontinuity takes place, in contrast
to a positional information scheme as mentioned above. Since there is no
unidirectional promotion under morphogen influence, the cells can change their
state of determination even in a distal-to-proximal direction, as observed in the
experiments of Bohn (1970; Fig. 7E). No special organizing region (morphogen
source) is expected; all regions have equal rights (in contrast to the dictatorship
of a local morphogen source and the long-ranging gradient).
Wolpert (1969, 1971) has pointed out that most embryonic fields are small,
about 1 mm or 100 cells across. The length of segments of cockroach legs can be
much larger and pattern regulation is still possible. According to our model, if,
let us say, 10 cell states are involved, also 10 diffusible substances are required
to generate the sequence. If the total field length is lcm, each diffusible substance would be responsible for only 1 mm, in agreement with the estimation of
Wolpert. Since the same signalling system is used repetitively in each segment,
this would not lead to an excess of required substances.
Combinations of a positional information mechanisms with its long-range
control and of a mutual activation mechanisms with its ability for size regulation
and intercalation can be envisaged too. Here, the morphogen gradient orients
the emerging sequence but does not determine particular structures by particular
thresholds. The sequence has by itself pattern-forming and selfregulatory
capabilities. The dorsoventral organization of vertebrates may be of this type
(Meinhardt, 1982).
THE PATTERN ON SHELLS OF MOLLUSCS
The pattern of pigmentation and of relief-like structures on the shells of
molluscs are presumably more a special case then a paradigm for pattern formation in general but the enormous diversity and their beauty invites the construction of models to understand their formation. The pattern emerges in most
species in a strictly linear fashion since new pattern elements are added only
Fig. 8. Formation of oblique lines of pigmentation by coupled oscillations. (A) Cells
are assumed to produce pigment in an autocatalytic way (activator) on the expense
of a precursor-molecule (an activator-inhibitor mechanism as shown in Fig. 1 would
work as well). By diffusion of the activator, neighbouring cells with high precursor
concentration can be 'infected'. With some delay, they enter into the activated
(pigment-producing) phase. This comes to rest due to a depletion of the precursor
molecules. Oblique lines result from propagating waves of such infections. If two
lines merge, they become extinct since no neighbouring cells with high precursor
concentrations are available. (B) If some cells oscillate somewhat faster, the characteristic W-like pattern emerges also all cells are to begin with in the same phase. (C)
Autocatalysis is assumed to be occasionally extinguished by external influences
(arrow); a more irregular pattern results. (D) The pattern on a bivalve shell for
comparison.
Models for positional signalling
Fig. 8
305
306
H. MEINHARDT
along the growing edge of the shell. The pattern on a shell is therefore a protocol
of what has happened at the growing edge during the life span of the animal. This
one dimensionality facilitates the design and testing of models since it makes
geometrical simplifications unnecessary. Formal models of the formation of the
tent-like pattern on shells have been proposed by Waddington & Cowe (1969)
and for bivalve shells by Lindsday (1982).
Most frequent are pigmentation lines which are oriented obliquely to the
direction of growth. These lines can be understood under the assumption that the
cells oscillate between a (short) pigment-producing phase and a (long) phase in
which no pigment deposition takes place (for a general discussion of oscillations
in biology see Winfree, 1980). Oblique lines emerge if a firing cell 'infects' its
neighbour which, after some delay, fires then too. The lines emerge as long
chains of such triggering events (Fig. 8). According to this model, the pattern
formation on mollusc shells has many similarities with the waves in slime moulds,
only in molluscs the pattern is strictly linear and a permanent record of the
temporary pattern remains in the second dimension.
The mechanism of autocatalysis and long-range inhibition discussed above can
be involved to generate the oscillations. In this case, the long range of the
inhibition belongs to the time coordinate and not to the space coordinate. Then,
in an activator-inhibitor mechanism, the inhibitor must not necessarily diffuse
more rapidly than the activator but it must have a slower turn-over (Meinhardt
& Gierer, 1974). High activator production (pigment production) can occur in
a burst-like manner since the antagonistic reaction follows too slowly. After such
a burst, the cells enter into a refractory period in which even an external addition
of activator would have no effect. Only after the decay of the inhibitor (or after
replenishing of precursor molecules necessary for pigment production) the cell
enters into a sensitive phase in which small amounts of activator, originating for
instance from a neighbouring, just-firing cell, suffices to trigger the next burst.
In the absence of such stimulus, the cell can fire spontaneously somewhat later.
Then, both neighbours can be in the sensitive period. Such a cell will be the
founder of two diverging lines (Fig. 8). If two lines merge, no neighbouring cell
is available which is in a sensitive period. The two lines become extinct.
In many shell patterns, branching of pigmentation lines occur. This indicates
that occasionally a 'reinfection' is possible. Fig. 9 shows a simulation in which
the pigment-producing time is due to statistical fluctuations occasionally larger
than the refractory period. The similarity to a real pattern is obvious. In the
Fig. 9. Branches in the pigmentation lines are formed by a 'reinfection' of a
previously active cell. This requires a short refractory period in comparison to the
activated phase. (A) A cell-automata model: each cell can be black for a certain
number of time units (4 ± 2), can be reinfected after a certain refractory period
(5 ± 3) from a black neighbour cell and can fire spontaneously after a long time
interval (33 ± 6). The resulting pattern resembles quite closely the shell pattern of
Cymbola (Aulicina) vespertilio (B).
Models for positional signalling
Position
Fig. 9
307
Fig. 10
X
I—I
w
u
o
oo
Models for positional signalling
309
autocatalysis-inhibition model, two ways are possible to get a longer firing
period in comparison to the refractory period. Either the refractory period must
be very short (in relation to the time interval required to get spontaneous firing)
or the duration of the firing time must be long. The first possibility can be realized
by a second inhibition which has a much shorter time constant. This rapidly
extinguishes the activation but, on the other hand, it disappears rapidly and in
this way enables a reinfection from the neighbouring, still pigment-producing
cells. A second, longer, time constant would determine the time after which the
next spontaneous firing will occur. The patterns shown in Fig. 10 are calculated
using this assumption.
The other possibility of forming branches would be that the duration of the
pigmentation period is elongated in some cells to such an extent that the neighbouring cell already has passed through its refractory period. This will occur if,
during the oscillation, a cell reaches a concentration which is close to the (semistable) steady state and remains there for a certain period.
An enormous diversity of pattern can arise by different time constants and
different couplings of the oscillating cells. Depending whether the antagonist
(inhibitor or depleted precursor) diffuses, transitions to a lateral inhibition
mechanism in space is possible. A dot-like pattern or stable stripes would result.
A detailed elaboration of these models is in progress.
CONCLUSION
Relatively simple interactions of few types of molecules are sufficient to
generate patterns with regulatory features which resemble closely those observed after experimental interference of a developing organism. The precise
mathematical formulation (for details see Meinhardt, 1982) has enabled us to see
how well a model fits. Usually, it has turned out that during development of a
model the initial assumptions have failed to give a satisfactory explanation for
the experiments. However, by computer simulations we learned at which point
mistakes in the intuitive thinking have been creeping in. In many cases this has
led to models very different from those originally envisioned. In addition these
models were often then able to account for more details of the experimental
observations than we originally intended. This makes us confident that the
models are close to what happens during pattern formation in real systems. We
hope that our models will help to unravel these processes.
Fig. 10. (A-C) Branching patterns generated by molecularly feasible interactions.
A long time constant (the replenishment of a precursor) determines the interval after
which spontaneous firing is possible. A rapidly accumulating inhibitor extinguishes
pigment production but it disappears also rapidly (short time constant) and enables
in this way a reinfection after a short recovery. (A-C) Patterns generated by simulation with somewhat different reaction parameters. For comparison, the shell pattern
of Conus aulicus (D) and of Conus perplexus (F).
310
H . MEINHARDT
I thank Prof. A. Gierer for a very fruitful collaboration, Chr. Niisslein-Volhard, R.
Lehmann and G. Jiirgens for many discussions concerning the segmentation of Drosophila,
Prof. A. Seilacher and W. Wommelsdorf for species of mollusc shells, and Prof. T. Cox for
a critical reading of the manuscript.
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