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 EntwicklungsbiologieTübingen / 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 rangeMeinhardt 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 self-enhancement 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:

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)

2 2

02

pa

c s aa aa D

t r x

2 2

02a

p

c s pp pp D

t r x

2 2p a

rc s a c s p r

t

2

2( )a a

a sa

s sa s D

t x

2

2( )p p

p sp

s sp s D

t x

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

Digits, segments, somites…

To illustrate the properties of such a type of interaction I used the following set of equations:

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 size-regulation after partial removal of one cell type.

2 2

02

pa

c s aa aa D

t r x

2 2

02a

p

c s pp pp D

t r x

2 2p a

rc s a c s p r

t

2

2( )a a

a sa

s sa s D

t x

2

2( )p p

p sp

s sp s D

t x

Digits, segments, somites…

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)

2 2

02

pa

c s aa aa D

t r x

2 2

02a

p

c s pp pp D

t r x

2 2p a

rc s a c s p r

t

2

2( )a a

a sa

s sa s D

t x

2

2( )p p

p sp

s sp s D

t x

Digits, segments, somites…

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).

2 2

02

pa

c s aa aa D

t r x

2 2

02a

p

c s pp pp D

t r x

2 2p a

rc s a c s p r

t

2

2( )a a

a sa

s sa s D

t x

2

2( )p p

p sp

s sp s D

t x

Digits, segments, somites…

2 2

02 2 2( ) ( / ) aa

a c a aa D

t a p s p x

2 2

02 2 2( ) pp

p c p pp D

t a p s x

2

2( )a a

a sa

s sa s D

t x

2

2( )p p

p sp

s sp s D

t x

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 in the first equation can lead to a threshold decisive for a transition for between an oscillating and an excitable system (see below).

/ p

(see Eq. 12.2 in the 82-book)

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

Parasegment

A P S A P S A P S

Segment

A P A P A P A periodic pattern consisting of an alternation of two cell states has no intrinsic polarity

An alternation of three cell states has an intrinsic polarity

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

Top: periodic patterns that leads to segmentation; the periodic pattern has polarity.Below: Specifying genes (HOX) genes

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 Hox-gene: one full cycle for a next gene

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.

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.

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

Combining oscillations and spatial pattterning

Now imagine that only one A cell exists initially at the anterior end. The direct P-neighors 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).

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.

Conversion of an oscillating pattern into a periodic pattern that is stable in time

Tim

e

Position

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?

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.

A comparison with the observation of Palmeirim et al., 1997 shows the striking correspondence between the 82-model and the 1997 observation.

Palmeirim et al. (1997). Cell 91, 639-648

Combining oscillations and spatial pattterning

A switch between an oscillating or an excitable system can be accomplished by a baseline inhibitor level or a Michaelis-Menten 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.

2 2

02 2 2( ) ( )/ aa

a c a aa D

t a p ps 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 half-somites 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 non-diffusible 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)