In the book “Models of Biological Pattern
formation” (Academic Press, 1982) I
described in chapter 14 models for
segmentation and somite formation.
In the following, an outline of these models is provided.
To see the animated simulation press F5
A TEX-remake of the book is available on our website:
http://www.eb.tuebingen.mpg.de/meinhardt/82-book
Hans Meinhardt Max-Planck-Institut für Entwicklungsbiologie
Tübingen / Germany
Digits, segments, somites…
A type of structure which is frequently encountered in higher organisms
consists of a sequence of similar but not identical substructures. Segments in
insects, the somites and the digits of vertebrates are examples. Usually their
total number is precisely regulated
As the basic mechanism we proposed
that cell states are involved that locally
exclude each other but activate each
other on long range
Meinhardt and Gierer (1980); J. theor.
Biol., 85, 429-450
Digits, segments, somites…
This mechanism found strong
support by the later discovered
logic in the engrailed-wingless
interaction: engrailed and wingless
are locally exclusive; engrailed
activates wingless in the adjacent
cell via hedgehog. In turn, wingless
molecules, transported in vesicles,
are absolutely required for the
engrailed activation in the
neighboring cell. As predicted,
wingless and engrailed activation is
autocatalytic. In engrailed, the selfenhancement is direct, that of
wingless involves sloppy paired.
Digits, segments, somites…
To illustrate the properties of such a type of interaction I used the following set
of equations:
2
a c s p a
2a

  a  Da 2  0
t
r
x
p c sa p 2
2 p

  p  Dp 2  0
t
r
x
r
 c s p a 2  c sa p 2   r
t
sa
 2 sa
  (a  sa )  Dsa 2
t
x
s p
2sp
  ( p  s p )  Dsp
t
x 2
This is Eq. 12.1 from the 82-book; only g1
and g2 is substituted by a and p as a label
for the compartmental specifications
a and p describe the local autocatalytic
feedback loops, sa and sp the mutual
long-ranging help. In this example it is
assumed that the mutual repression
occurs by a common repressor
r
which
is produced by both autocatalytic loops
and that acts on both. This mutual local
exclusion has the consequence that in
one cell only one of the feedback loop
can be active. Booth loops require the
help from the other cell state; a and p
expressing cells will appear next to
each other.
The mutual repression can also be
direct (see below)
Digits, segments, somites…
To illustrate the properties of such a type of interaction I used the following set
of equations:
2
a c s p a
2a

  a  Da 2  0
t
r
x
p c sa p 2
2 p

  p  Dp 2  0
t
r
x
r
 c s p a 2  c sa p 2   r
t
sa
 2 sa
  (a  sa )  Dsa 2
t
x
s p
2sp
  ( p  s p )  Dsp
t
x 2
In this simulation the homogeneous
distribution of a and p becomes instable,
high a and high p expression occur in
adjacent cells.
This mechanism show a good sizeregulation after partial removal of one cell
type.
Digits, segments, somites…
2
a c s p a
2a

  a  Da 2  0
t
r
x
p c sa p 2
2 p

  p  Dp 2  0
t
r
x
r
 c s p a 2  c sa p 2   r
t
sa
 2 sa
  (a  sa )  Dsa 2
t
x
s p
2sp
  ( p  s p )  Dsp
t
x 2
If the autocatalytic components are not
diffusible, the border between the two regions
will be absolutely sharp. This is the case in the
engrailed-wingless interaction for the
compartment formation in Drosophila: the clonal
borders are sharp and cannot be moved, as
observed.
However, if only one cell type remain, a partial
reprogramming may be possible (as observed
after fragmentation of of imaginal discs)
Digits, segments, somites…
2
a c s p a
2a

  a  Da 2  0
t
r
x
p c sa p 2
2 p

  p  Dp 2  0
t
r
x
r
 c s p a 2  c sa p 2   r
t
sa
 2 sa
  (a  sa )  Dsa 2
t
x
s p
2sp
  ( p  s p )  Dsp
t
x 2
An important feature of such a system is that
it can generate stripes. Since the different
cell types need each other for mutual
stabilization, a long common border leads to
a most stable situation. The ability to form
stripes is required for many such systems.
(In this simulation above pattern formation is
initiated by a slightly higher level of p (red) in
the right half of the field).
Digits, segments, somites…
a
c a2
2a
 2
  a  Da 2  0
2
t (a  p ) (sa   / p)
x
p
c p2
2 p
 2
  p  Dp 2  0
2
t (a  p ) s p
x
sa
 2 sa
  (a  sa )  Dsa 2
t
x
s p
t
  ( p  s p )  Dsp
(see Eq. 12.2 in the 82-book)
2sp
x 2
In this alternative example, the interaction
between the two feedback loops occurs
not by a long-ranging mutual activation but
by a self-inhibition (a2/sa). In competing
systems, a help for the other feedback
loop or a self-inhibition is equivalent. The
mutual exclusion is direct (a2 + p 2 ).
The term  / p in the first equation can
lead to a threshold decisive for a transition
for between an oscillating and an excitable
system (see below).
Segmentation: mutual long range activation
of locally exclusive cell states
In short germ insects segments are
sequentially added at a posterior elongation
zone.
In a most simple model using the
mechanism described above, a periodic
pattern is generated during posterior
outgrowth. Assumed is a doubling in the
right-most cell. Whenever a particular
activation (compartmental specification)
exceeds a certain extension, a flip to the
other activation will occur.
Note that the most-posterior cell
oscillates between the two cell states
Segmentation: mutual long range activation
of locally exclusive cell states
The simulation above shows the pattern at
successive stages. Note that at a particular
moment, a posterior terminal activation of
either the one or the other type is expected.
This fits with the observation of Damen, Weller
and Tautz (2000), PNAS 97,4515-4519 in the
spider
Prediction: at least three cell states are
required to generate a periodic structure with
an intrinsic polarity
A
P
A
P
A
A periodic pattern consisting of an
alternation of two cell states has no
intrinsic polarity
P
An alternation of three cell states has
an intrinsic polarity
Parasegment
A
P
S
A
Segment
P
S
A
P
S
For Drosophila it has been shown that
the A and the P cells resembling the
incipient anterior and posterior
compartment are indeed separated by
two other cells the are neither A or P.
(The separation of one somite from the
next is not yet clear and could be
different in different systems)
Oscillations and spatial pattern formation during
posterior outgrowth
Assumption: in the P-state
(red) the next HOX gene is
activated, but the transition is
blocked.
After transition to the A-state,
activation of the next HOX
gene is no longer blocked, but
no activation of the next Hoxgene: one full cycle for a next
gene
Top: periodic patterns that leads to segmentation; the periodic pattern has polarity.
Below: Specifying genes (HOX) genes
The segments formed during outgrowth resemble not only a periodic structure; they carry
a specificity. In the 82-book it was proposed that the terminal oscillation is also used to
activate the corresponding genes for specification (now known to be the HOX-genes).
According to the model, if the cells are in one cell state, an activation of the next (HOX)
gene is prepared but the full activation occurs only after switch to the other state. Thus,
for instance, with each P-to-A transition, one and only one new specifying gene was
assumed to become activated. The model describes correctly that both the sequential and
the periodic pattern are precisely in register with single cell precision (as observed).
Now we know that more than one cycle has to pass until the activation of the
next HOX gene occurs.
As segments in short germ insects, somites in the
ancestral Amphioxus are also formed at a posterior
elongation zone. However….
Remarkable: somites on the left and on the right side appear out of phase in an
alternative sequence
(see
Schubert, Holland, Stokes and Holland (2001) Dev. Biol. 240,262-273
…. in contrast, somite formation in vertebrates occurs
at a substantial distance from the posterior pole
Wolpert: Principles..
At the time I proposed my somite model, almost nothing was known about
the molecular basis of somite formation. An important piece of information
came from heat shock experiments in Amphibians. A short heat pulse lead
after a certain delay to some characteristic perturbations in the somite
patterning…
Such a perturbations lead frequently to a Y-shaped
split of a somite into two or to an incomplete border
between two somites.
Elsdale and Pearson (1979). Somitogenesis in amphibia. II. Origins in early embryogenesis
of two factors involved in somite specification. J. Embryol. exp Morphol. 53, 234-267
Such an irregularity is frequently followed by a second irregularity that
compensates partially the first, allowing that somite formation can proceed
normally. This I regard as a clear indication that a spatial component is involved
in the patterning.
Combining oscillations and spatial pattterning
To see how the oscillation derived for
insect segmentation and patterning in
space can be combined, it is essential to
see that the mechanism shown above can
act also as an oscillator. For instance, if
only A cells are present, the P-cell state
will get strong support, while the support
of the A state by the P cells is missing.
Thus, cells will switch from A to P and for
the same reason back to A, i.e, they will
oscillate between the two states. This
works in the same way if self-inhibition is
involved.
Now imagine that only one A cell exists
initially at the anterior end. The direct Pneighors will be be stabilized, while the
other will continue to oscillate. With each
complete cycle there would be one new pair
of A/P specifications (half-somites).
Conversion of an oscillating pattern into a
periodic pattern that is stable in time
In this way the oscillation gives rise to a
regular pattern in space. The non-trivial
prediction was made that a boundary
between a stable and an oscillating pattern
sweeps from anterior to posterior over the
field.
This raises, however, the question:
what makes the first border?
Position
Combining oscillations and spatial pattterning
The assumption was that at the
posterior terminal position a
gradient is generated and that a
certain concentration is required
to keep the oscillation going.
Cells below a threshold are
unable to oscillate.
Now it is clear that this prediction
was correct, the gradient shown
in the simulation in yellow has
been identified as FGF.
Palmeirim et al. (1997). Cell 91, 639-648
A comparison with the observation of Palmeirim et al., 1997 shows the striking
correspondence between the 82-model and the 1997 observation.
Combining oscillations and spatial pattterning

A switch between an oscillating or
an excitable system can be
accomplished by a baseline
inhibitor level or a MichaelisMenten type constant. If  is large
enough, the a activation can no
longer trigger spontaneously since
the denominator remains finite. The
positional information p (yellow)
lowers the influence of  . Thus, if p
is high enough (in the posterior),
the system oscillates, otherwise it is
arrested in the p state.
a
c a2
2a
 2
  a  Da 2  0
2
t (a  p ) (sa   / p)
x
(Hox) gene activation under the influence of the
oscillation
The aim was to also explain the
sequential activation of specifying
genes. The model proposed offered a
very convenient mechanism. The
number of the oscillations a cell has
been made corresponds
unambiguously to its position. Each
more posteriorly located pair of halfsomites proceeds exactly through one
more oscillation cycles. As shown
above for insects, this can be used to
activate specifying (HOX) genes
(lowest panel)
Although the precise mechanism of the coupling between the oscillation and Hox gene
activation is still unclear, there is some evidence for it:
Dubrulle, et al., (2001). Fgf signaling controls somite boundary position and regulates
segmentation clock control of spatiotemporal hox gene activation. Cell 106,219-232
Zakany et al., (2001). Localized and transient transcription of hox genes suggests a link
between patterning and the segmentation clock. Cell 106,207-217
The clock and wave front model of
Cooke and Zemann (1976)
Although Cooke and Zemann did not formulate their model in a mathematical way,
such a model can be easily provided by the elements of or models. In the
simulation above the wave is generated by an activator-inhibitor system with a nondiffusible inhibitor that has a longer half-life then the activator. A similar assumption
was made for the overall oscillation that takes place in the whole field (blue). The
wave triggers the activation of a gene for somite formation (brown) in a switch-like
manner. A burst of the oscillation blocks this activation, leading to a gap in the gene
activation thought to give rise to the somites.
It is easy to see that this model does not fit the observations, e.g., there are no
waves from the posterior that come to rest in the somite-forming zone, half-somites
do not play a role, the localization of oscillation does not play a role, the oscillation
is involved in the separation, not in their determination.
Summary
The model was the first fully mathematically formulated
model for somite formation. It correctly predicted that:
•
•
•
•
An out-of-phase oscillations occur in the posterior PSM
Activations spreads from posterior towards anterior
come to rest at the somite forming zone
Each full cycle adds one pair of A- and P-half-somites
The oscillation that leads to the periodic pattern can be
used to accomplish a very precise activation of (Hox)genes that specify the character of the periodic
elements. This issue is not yet solved.
Not yet clear: how are somites separated from each other
A final remark
The model was proposed in 1982 when nothing was
known about the molecular basis. Moreover,
computers were awfully slow at this time. Thus, it
was necessary to keep the model as simple as
possible. Therefore, in the model a single reaction
chain was used to describe:
•
•
•
The Oscillation
The transition into stable a pattern
Memorizing A and P specifications
 This is certainly too simple but shows the essence
As mentioned, a TEX-remake of the book is
available on our website:
http://www.eb.tuebingen.mpg.de/meinhardt/82-book
Essentially the same model was published in a proceeding volume Somites in Developmental
Biology (R.Bellairs, D.A.Edie, J.W. Lash, Edts), Nato ASI Series A, Vol 118, pp 179-189,
Plenum Press, New York as: Meinhardt, H. (1986b) Models of segmentation. (on our website)