Models for Positional Information and Positional Differentiation

Models for Positional Information and Positional DifferentiationAuthor(s): A. Babloyantz and J. HiernauxSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 71, No. 4 (Apr., 1974), pp. 1530-1533Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/63364 .

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Proc. Nat. Acad. Sci. USA Vol. 71, No. 4, pp. 1530-1533, April 1974

Models for Positional Information and P (gradient of morphogen/active transport/hydra graftin

A. BABLOYANTZ AND J. HIERNAUX

Universite Libre de Bruxelles, Faculte des Sciences, Brussels, Belgium

Communicated by I. Prigogine, January 2, 1974

ABSTRACT It is commonly thought that the formation of patterns in developing organisms is due to the existence of a gradient of a morphogen that determines the fate of cells as a function of position in the organism. A model is presented based on a molecular mechanism where the gradient is established by the active transport of a morphogen between source and sink. The cellular differentiation and the subsequent spatial pattern formation results from the interaction of this morphogen with the genetic regulatory mechanisms of cells. Some properties of the model are given and discussed in relation to grafting experiments in hydra.

One of the major problems related to pattern formation in biology is to understand how a given genome identical in all cells of a multicellular system can be expressed to give specific and varying spatial patterns of cellular differentiation.

Theories to explain pattern formation can be traced as far as 1895 when Driesch proposed the first version of what be- came later the concept of positional information (1). Driesch's basic idea was that some mechanism should provide a means whereby a given cell would know its position in an ensemble of cells forming the organism and differentiate accordingly. Later Child (2) suggested that the physiological gradients play an important role in embryological development. The relative position of cells with respect to their neighbors can be specified either through the action of two opposite gradients-their ratios specifying position-as suggested by Dalcq (3), or by a single gradient with fixed values of a dif- fusing substance of low molecular weight at boundaries as proposed by Crick (4). More recently, in the framework of the concept of positional information, Wolpert (1) postulates a universal two-step process in the course of development with two distinct time scales. In a first stage a coordinate system is established by a gradient-generating mechanism independently from differentiation processes, and positional information is assigned to the cells in this coordinate system. Wolpert assumes a linear function for positional information, which enables him to demonstrate size invariance. This assumption implies that the gradient is established by simple diffusion mechanism and the morphogen does not participate in any chemical reaction. In a second stage the positional information is translated in an unspecified manner into molecular differentiation according to the cell's genome. Thus, addi- tional rules must be given for interpretation of positional information into cellular differentiation.

Another type of theory such as the phase shift model of Goodwin and Cohen (5) is based on the phase angle difference between two periodic signals. Both signals are emitted by the pacemaker cell and travel with different velocities. There re-

1

ositional Differentiation g)

sults a phase angle difference which is thought to specify positional information.

All recent advances in this field again assume first the establishment by some mechanism and, in some way, of a primary pattern of some physico-chemical parameter such as the concentration of morphogens. The morphological pattern of the field is then the result of the primary pattern. As the process of cell differentiation is of all-or-none type with respect to the concentration of certain regulatory proteins, several models have been developed to show the existence of step distribution in the concentrations of morphogens which are responsible for the primary pattern. This kind of primary pattern is obtained by inhibitor-activator interaction coupled with diffusion in open systems. These patterns are typical example of dissipative structures (6, 7) introduced by Prigogine et al. Patterns exhibiting polarity can also be obtained via cell-to- cell contact (8).

On the other hand, there is a considerable literature on the control of molecular differentiation, mainly based on Jacob- Monod type control of gene transcription (9). Similar control mechanisms for eukaryotes are developed by several authors (10, 11). Cellular differentiation is also modeled in terms of genetic networks by considering the gene as an on-off binary device (12,13).

To our knowledge, there has been a single attempt, made by Edelstein (14), to obtain patterns of enzyme synthesis in terms of morphogen concentration based on empirical equations for induction of 2-galactosidase in Escherichia coli. Un- fortunately, the solutions he obtains are unstable towards random fluctuations that may appear in any physico-chemical system, and cannot account for stable patterns of developing systems.

Our aim is to propose models which interrelate the establishment of the gradient of positional information-the primary invisible pattern-and its subsequent expression by the cell's genome into positional differentiation, giving rise to the visible morphological pattern.

The elements used in our model are the well-known biologi- cal phenomena such as activation and repression of protein synthesis which can be thought to be universal mechanisms in living systems. We also use the idea that the morphogenetic fields are under the influence of a source and a sink of the morphogenetic substances. Two different mechanisms were used for the establishment of the gradient: diffusion of a morphogen, and an active or facilitated transport between source and sink as suggested by Webster and Wilby (15). A full discussion of models based on diffusion is reported elsewhere (16). Here we present a model where cell-to-cell trans-

530



Proc. Nat. Acad. Sci. USA 71 (1974)

port of morphogen is insured via active and facilitated transport.

Our model has the following properties: (a) Stable patterns are obtained simultaneously for the primary and morphological patterns. The patterns show polarity and are organized by source points. (b) The rules for interpretation of positional information are the result of gradient-gene interaction and need not be specified independently. (c) It can be shown that there is no need for the primary pattern to present step distribution in activator and inhibitor concentrations. Even a linear primary pattern of a single morphogen can give rise to an all-or-none type response in the concentration of regulatory proteins which characterize differentiation. (d) Size invariance can be demonstrated under certain conditions.

Model

Let us consider a linear array of N adjacent identical cells with the same genetic potentialities. The position of cells in the array is specified with respect to two fixed boundaries taken as source and sink of a substance I whose gradient determines positional information. This constitutes a morphogenetic field.

Each cell possesses a transport system for the uptake of the morphogen I from neighboring cells. It also can loose I to its neighbors. The substance I acts as a regulator at the genetic level and protein E is synthesized. The molecular mechanism whereby the action of I gives rise to a state of activity lead- ing to molecular differentiation need not be specified here; the important requirement for our model is that the E-I relationship be nonlinear of sigmoidal or S-shaped type as seen in Fig. 1. It is to be noticed that there is a threshold value of I for which a considerable jump in the value of E oc- curs. This kind of highly nonlinear relationship may be achieved through repression, activation, or a combination of those mechanisms known to exist in bacteria and probably also in eukaryotes.

One can construct models exhibiting Fig. I type of be- havior with elements proposed by Britten and Davidson (10), Georgiev (11), and others for genetic regulation in multicellular organisms (17). However, this would introduce into the model unnecessary assumptions and a great number of vari- ables. Instead, a simple model as described in Scheme 1 can

kl R' - Ri

k2

k3

Ri + 0O+ = ?O- k4

kg Ri + 2 1i E F,

k6

a + Oi+ - Oi+ + Ei + Mi ko

Mi + Ii-1 = Mi + Ii ko

k8 Mv + Ii+J = Mi + Ii

ko

klo

Mi--'F Ic1,

kl2

kl3

Scheme 1

be constructed, based on the bacterial induction mechanisms of the Jacob-Monod type, together with a very simplified

Model for Differentiation 1531

7

E5

1 d

Icl 2 3 4

FIG. 1. Variation of E per cell as a function of external concentration of I with R' = 0.75, a = 0.1, F, = 0.1, F = G = 0.05, kl = 2.5, k2 = 0.05, k3 = 7.0, k4 = 0.01, k, = 5.0, k6 = 1.0, k7 5.0, k8 = 2.5, kg = 0.2, klo = 0.05, k,l = 1.0, kl2 = 0.05, k13 = 1.0. The points c and d denote source and sink values.

model describing permease-mediated active and facilitated transport of the morphogen I. This model is inspired from a similar one proposed by Babloyantz and Sanglier (18) for the case of /-galactosidase induction in E. coli. The model, al- though simple, satisfies all requirements mentioned earlier and contains all important features needed for a morphogenetic field: an active transport mechanism providing cell-to- cell communication for transfer of the morphogen from cell to cell, and the production of a protein from DNA due to the action of this morphogen.

The Scheme I describes an induction mechanism of Jacob- Monod type: a repressor R is synthesized from its precursors and can repress the operator 0+ into 0-. The susbstance I may combine with repressor and inactivate it. The open operon can synthesize a protein E and a protein M, or permease, that mediates the entry of substance I into the cell. When this happens at a considerable rate we think of the cells as being differentiated, in contrast to nondifferentiated cells, which do not practically produce E and M. Ii-i and li+j are the concentrations of the substance I in the neighboring cells; M and E also might decay. ki. . .k13 are rate constants.

61 --

35

10 /

20 40 Cell Position

10.3 b

20 40 Cell Position

FIG. 2. (a) Gradient of morphogen in a source-sink active transport field of 40 cells. (b) Corresponding pattern of cell differentiation.



1532 Cell Biology: Babloyantz and Hiernaux

~10 ~~30 60 Cell Position

30 60

Cell Position

FIG. 3. (a) Gradient of morphogen in a source-source field of 60 cells. (b) Corresponding pattern of cell differentiation.

One has to consider here two entirely different situations according to the values of ks and kg, relative to the rate of entry of I into the cell. This point has been analyzed by Babloyantz and Sanglier (18), who have developed similar models for the case of a single cell. We briefly recall here some of their results. For ks > kg, the steady-state concentration of E as a function of the external concentration of I is repre- sented in Fig. 1. It is seen that there is a critical value Ic for which the concentration of E and of M jumps to a higher value. The cells then perform active transport, that is, they accumulate I inside the cell against a concentration gradient. For the same values of constants, if ks = k0, no active transport exists, but the amount of I entering the cell is greater than that obtained by a simple diffusion mechanism. This case corresponds to the phenomenon of facilitated transport.

For our morphogenetic field of N cells we choose ks > kg in order to permit cells to perform active transport. We take time variation of R, 0+, O- at quasi-steady state. The time variation of E, M, and I for this ensemble of N cells is given by:

dEi ak2k4k7 + a k4k5k7 (I ) 2 --= - -k12 Ei +Jr k13 G

dt klk3 R' + k3k6Fl + k2k4 + k4 k5 (Ii)2

dMi a k2k4k7 + a k4k5k7 (Ii)2

dt klk3R' + k3k6F + k2k4 + k4k5 (i)2 k M + k F

d-i kRI)5 ( + kF(1) + k6F1 dt k2+k l ()

+ ks Mi(Ii-1 + I+1) - 2 k9MiIi

- ks Ii (Mi-1 + Mi+x) + k9Mi- Ii-l + k9Mi+l3i+6

for i=2,N-l [1]

Moreover we have the obvious physical condition 0O + Oi = 1 where i+ and Oi- are respectively the probability that

the gene be open or closed. It must be noticed that the expressions giving the time

variation of Mi and Ei are quite similar. We assign to the boundary cells 1 and N, values of M, f,

and E corresponding to the points d on the lower and c on the upper branch of Fig. 1.


The steady-state solution of this set of equations determines the form of the gradient I and also the pattern of regulatory protein E. This set of equations can only be solved nu- merically. The steady-state solution is found by the Runge- Kutta method on a C.D.C. 6400 computer. The values of constants used are the same for all patterns presented in this paper and are given under Fig. 1. Fig. 2 represents the steady- state solution of Eqs. 1 for I and E starting from a state where the 40 cells are nondifferentiated. It is seen that al- though the variation in space of the concentration of I does not present an abrupt jump, the response curve for E is highly discontinuous and there is a net separation among the cells of the field into two categories. One part of cells remains undifferentiated and is unable to perform active transport. The other part, on the contrary, is differentiated and trans- ports I actively from regions of low I concentration to regions of high I concentration. We see that the presence of the protein E in a cell is determined by its position in the morphogenetic field.

It is important to notice that, by introduction of a small diffusion term approximated by Fick's law for substance I in the equations, with diffusion coefficient of order of 10 -, the gradient may become linear but the cells respond in an all-or- none fashion as in the precedent case.

Some properties of the model The patterns of Fig. 2 are stable towards random fluctuations that may appear spontaneously in each cell. If the concentration of a given substance suddenly changes in one or several cells, after a while the perturbed pattern returns to its original form.

An important feature of these patterns is that the value of I at the source organizes the pattern. Indeed, for a fixed source value the number of cells that are differentiated remains constant whether one adds or subtracts several cells from the morphogenetic field. Thus positional information "flows" from source to sink. This defines the polarity of the pattern.

Another striking feature of this morphogenetic field is the following: let n be the number of differentiated cells for a given source value. Let us assume also that the morphogenetic field comprises a number of cells N > 2 n. Now, if we take two sources as boundary values for the field, the patterns of Fig. 3 are found. Each source organizes its pattern independently from the other source and differentiated cells appear in the neighborhood of each source, whereas the remaining cells are undifferentiated. If, again in this pattern, the number of cells is increased or decreased without, however, reaching the critical 2n value, the pattern does not change and only the number of undifferentiated cells increases or decreases. On the other hand, if the number of cells in the field is less than 2 n, the pattern is completely altered. The same morphogenetic field with two boundaries taken as sink values gives no pattern.

All the above mentioned results can be repeated for the case of facilitated transport. However, the formed patterns are less pronounced.

In some cases, if a part of a developing organism is removed, the remaining part will regulate to give the original pattern in a manner such that the respective proportions of the different parts of the pattern are independent of the total size of the organism. This property is defined as "size invariance" or morphallaxis.




SC 1 1213141815161F

FIG. 4. Schematic representation of hydra.

In our model (see Fig. 2), we shall have size invariance if the number of induced cells is a constant fraction of the total number of cells. This can be achieved in our case if we assume that when part of the cells of a morphogenetic field is removed, a new source value is attributed to the boundary of the new field. On the other hand, if we do not modify the value of the source, when increasing or decreasing the number of cells of the field, we do not modify the number of induced cells. This corresponds to the phenomenon of epimorphosis in embryology.

Finally, an important and unexpected thermodynamic property can be shown for these patterns, namely, that the cell differentiation is followed by an increase in the rate of dissipation per unit mass of the system. It is important to notice that this property is in agreement with experimental observations in embryology. Details may be found elsewhere (19).

Axial grafts of hydra

In this section, we shall try to apply our results to the partic- ular problem of axial grafts in hydra. We feel that the lateral grafts and also a great number of other grafting experiments can only be explained with a two-dimensional and two-gradient model. Worl in this direction is in progress. Meanwhile the one-dimensional and one-gradient model can give some insight for the following grafting experiments.

Hydra is an animal a few mm in length, with a polar structure that can be described distoproximally as shown on Fig. 4.

Wilby and Webster (20) performed the following experiment on hydra. They removed segments H and 56 F from the animal and grafted the 56F part distally to the remaining part of the animal, obtaining F65/1234. They observed a change in the polarity of the field such as the appearance of an hypostome in the middle part of the animal. Let us suppose that in the pattern of Fig. 2 the source corresponds to the distal part of the hydra and the sink to the proximal part. Now, if we replace the source by a sink and follow the time evolution of the structure, we would simulate the above experiment. Our computer results show the formation of differentiated cells in the middle of the structure.

In another experiment, Wilby and Webster grafted to a 1234 gastric region a head at the proximal end and a penduncle

Model for Differentiation 1533

at the distal end to give F65/1234/H. They have obtained a middle hypostome in some experimental conditions. In our model, this would correspond to the change of boundary values of sink to source and source to sink. Again the time evolution of this pattern shows the appearance of a head structure in the middle of the pattern. Gierer et al. (21) performed reag- gregation experiments with cells produced from the head region H and the gastric region G. Then they centrifugated these aggregates on the top of each other in polyethylene tubing in such a way that the layer of G cells was formed in between two regions of H cells in an HGH sequence. They found that tentacles are produced at both ends and a head structure never appears in the middle. This kind of experiment can be accounted for by Fig. 3 type patterns when two sources at boundaries induce two heads with gastric regions in the middle part of the structure.

We are deeply indebted to Professors I. Prigogine for his constant interest and G. Nicolis for numerous and fruitful discussions. J.H. acknowledges financial support from the Institut pour l'En- couragement de la Recherche Scientifique dans l'Industrie et l'Agriculture.

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(The University of Chicago Press, Chicago). 3. Dalcq, A. M. (1938) Form and Causality in Early Develop-

ment (Cambridge University Press, Cambridge). 4. Crick, F. (1970) Nature 225, 420-422. 5. Goodwin, B. C. & Cohen, M. H. (1969) J. Theor. Biol. 25,

49-107. 6. Glansdorff, P. & Prigogine, I. (1971) Thermodynamics of

Structure, Stability and Fluctuations (Wiley-Interscience, New York).

7. Gierer, A. & Meinhardt, H. (1972) Kybernetik 12, 30-39. 8. McMahon, D. (1973) Proc. Nat. Acad. Sci. USA 70, 2396-

2400. 9. Babloyantz, A. & Nicolis, G. (1972) J. Theor. Biol. 34, 185-

192. 10. Britten, R. J. & Davidson, E. H. (1969) Science 165, 349-

357. 11. Georgiev, G. P. (1969) J. Theor. Biol. 25, 473-490. 12. Tsanev, R. & Sendov, Bl. (1971) J. Theor. Biol. 30, 337-

393. 13. Kauffman, S. A. (1971) J. Cybernet. 1, 71-82. 14. Edelstein, B. B. (1972) J. Theor. Biol. 37, 221-243. 15. Webster, G. (1971) Biol. Rev. 46, 1-46. 16. Babloyantz, A. & Hiernaux, J., submitted to J. Theor. Biol. 17. Hiernaux, J., unpublished results. 18. Babloyantz, A. & Sanglier, M. (1972) FEBS Lett. 23, 364-

366. 19. Hiernaux, J. & Babloyantz, A., to be published. 20. Wilby, 0. K. & Webster, G. (1970) J. Embryol. Exp. Mor-

phol. 24, 595-613. 21. Gierer, A., Berking, S., Bode, H., David, C. N., Flick, H.,

Hansmann, G., Schaller, H. & Trenkner, E. (1972) Nature New Bio6. 239, 98-101.



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