A model generating the pattern of cartilage skeletal elements in the embryonic chick limb.

The development of the vertebrate limb involves the production of a specific external form arising from cell division and other growth processes at the cellular level, and the origin within it of specific patterns of tissues arising by cellular differentiation, of which the pattern of cartilages which pre-figure the limb skeleton is the most striking.In this paper we propose a model for the differentiation (or the preceding determination) process that, using only localized cell to cell interactions, can approximate the cartilage pattern in any limb shape. The model requires cells to modify their metabolism irreversibly at critical threshold levels of a diffusible morphogen which may be made or destroyed by these cells. Restrictions inherent in the successful development of a total limb pattern using this system lead to the prediction that the process is confined to a distal band which has no significant interaction with more proximal regions but within this band the characteristic features of the anterior-posterior axis of the limb develop without additional interactions. Cartilage elements are initiated as single “cells” and expand centrifugally to their final size; these elements developing sequentially along the anterior-posterior axis, showing a distinct polarity of size. The model also predicts that equivalent cartilage elements in all vertebrate limbs will be roughly the same relative size at determination, the extensive range of adult structures arising by differential growth and fusion, possibly controlled by global aspects of the model.It must be emphasized that this model only satisfactorily simulates the anterior-posterior patterning of cartilage elements, the disto-proximal pattern being externally imposed.The final cartilage pattern is shown to be a function of (1) the developing shape of the limb, (2) the position of an initiator region that starts the patterning process and (3) the rate of production of the diffusible morphogen. Using parameters selected with as much realism as possible the model gives a good approximation to the pattern of c cartilages found in the normal chick limb; modifying the shape of the limb to that of the talpid³ mutant produces the characteristics features of the cartilage pattern found in that mutant and modifying the rate of morphogen production simulates patterns resembling those found in some ancestral vertebrate fossil forms.     

Activator-inhibitor dynamics of vertebrate limb pattern formation

The development of the vertebrate limb depends on an interplay of cellular differentiation, pattern formation, and tissue morphogenesis on multiple spatial and temporal scales. While numerous gene products have been described that participate in, and influence, the generation of the limb skeletal pattern, an understanding of the most salient feature of the developing limb--its quasiperiodic arrangement of bones, requires additional organizational principles. We review several such principles, drawing on concepts of physics and chemical dynamics along with molecular genetics and cell biology. First, a "core mechanism" for precartilage mesenchymal condensation is described, based on positive autoregulation of the morphogen transforming growth factor (TGF)-beta, induction of the extracellular matrix (ECM) protein fibronectin, and focal accumulation of cells via haptotaxis. This core mechanism is shown to be part of a local autoactivation-lateral inhibition (LALI) system that ensures that the condensations will be regularly spaced. Next, a "bare-bones" model for limb development is described in which the LALI-core mechanism is placed in a growing geometric framework with predifferentiated "apical," differentiating "active," and irreversibly differentiated "frozen" zones defined by distance from an apical source of a fibroblast growth factor (FGF)-type morphogen. This model is shown to account for classic features of the developing limb, including the proximodistal (PD) emergence over time of increasing numbers of bones. We review earlier and recent work suggesting that the inhibitory component of the LALI system for condensation may not be a diffusible morphogen, and propose an alternative mechanism for lateral inhibition, based on synchronization of oscillations of a Hes mediator of the Notch signaling pathway. Finally, we discuss how viewing development as an interplay between molecular-genetic and dynamic physical processes can provide new insight into the origin of congenital anomalies.     

Generation and regeneration of sequence of structures during morphogenesis

Models for the generation and repair of sequences of structures in space are proposed. One possibility consists of the mutual or sequential induction of locally exclusive states. The general properties of such an interaction are demonstrated by a system of two components which mutually activate each other: the partition of a field into two parts with good size regulation is possible. Symmetric double gradients or periodic patterns can be formed. In a two-dimensional field, this type of interaction permits the formation of “stripes” of high concentrations of the components. In an extension to more than two components, sequences of structures are formed which show intercalary or terminal regeneration. Relatively simple molecular interactions can lead to such patterns.In some biological cases, experimental evidence suggests that the first step in the formation of a sequence is the determination of one or both terminal elements, followed by sequential filling in of the missing structures. The latter process can be mediated by a general signal formed at the discontinuity. If some monotonically increasing physical parameter r (positional value) occurs in the natural sequence, a discontinuity in the r-concentration is formed at the location of the gap. This discontinuity can be converted into a local maximum and/or minimum, serving as a gap sensing signal and leading to the induction of the missing structures.The other extreme type of model posits that the determination consists solely of the response of cells to the local concentration of a morphogen. In such a positional information scheme, a sequence of structures can be elongated by marginal growth if a feedback of the achieved states of determination orto the morphogen gradient is assumed. This permits the successive increase of the maximum morphogen concentration during the outgrowth enabling the accretion of new structures.The similarities in and differences between such models are discussed. Intermediate forms of these “pure” types are presumably involved in the control of development and some examples are given. Possible application to the developmental control of insects are discussed, in particular to the phenomena of intercalary regeneration and the duplication of excessive parts, as well as to the promimo-distal organisation of the vertebrate limb. Computer simulations are provided which demonstrate the feasibility of the models proposed.     

Butterfly Eyespot Patterns: Evidence for Specification by a Morphogen Diffusion Gradient

In this paper we describe a test for Nijhout's (1978, 1980a) hypothesis that the eyespot patterns on butterfly wings are the result of a threshold reaction of the epidermal cells to a concentration gradient of a diffusing degradable morphogen produced by focal cells at the centre of the future eyespot. The wings of the nymphalid butterfly, Bicyclus anynana, have a series of eyespots, each composed of a white pupil, a black disc and a gold outer ring. In earlier extirpation and transplantation experiments (Nijhout 1980a; French and Brakefield, 1995) it has been established that these eyespots are indeed organised around groups of signalling cells active during the first hours of pupal development. If these cells were to supply the positional information for eyespot formation in accordance with Nijhout's diffusion-degradation gradient model, then, when two foci are close together, the signals should sum, and this effect should be apparent in the detailed shape of the resulting pigment pattern. We give an equation for the form of the contours that would be obtained in this manner. We use this to test the morphogen gradient hypothesis on measurements of the outlines of fused eyespots obtained either by grafting focal cells close together, or by using a mutation (Spotty) that produces adjacent fused eyespots. The contours of the fused patterns were found to satisfy our equation, thus corroborating Nijhout's hypothesis to the extent possible with this particular type of experiment.     

Generation model of positional values as cell operation during the development of multicellular organisms

Many conventional models have used the positional information hypothesis to explain each elementary process of morphogenesis during the development of multicellular organisms. Their models assume that the steady concentration patterns of morphogens formed in an extracellular environment have an important property of positional information, so-called “robustness”. However, recent experiments reported that a steady morphogen pattern, the concentration gradient of the Bicoid protein, during early Drosophila embryonic development is not robust for embryo-to-embryo variability. These reports encourage a reconsideration of a long-standing problem in systematic cell differentiation: what is the entity of positional information for cells? And, what is the origin of the robust boundary of gene expression? To address these problems at a cellular level, in this article we pay attention to the re-generative phenomena that show another important property of positional information, “size invariance”. In view of regenerative phenomena, we propose a new mathematical model to describe the generation mechanism of a spatial pattern of positional values. In this model, the positional values are defined as the values into which differentiable cells transform a spatial pattern providing positional information. The model is mathematically described as an associative algebra composed of various terms, each of which is the multiplication of some fundamental operators under the assumption that the operators are derived from the remarkable properties of cell differentiation on an amputation surface in regenerative phenomena. We apply this model to the concentration pattern of the Bicoid protein during the anterior-posterior axis formation in Drosophila, and consider the conditions needed to establish the robust boundary of the expression of the hunchback gene.     

A model for generating aspects of zebra and other mammalian coat patterns

A model is put forward which is capable of generating chemical maps whose concentration contours are similar to the patterns seen on the flanks of zebras, cats and other mammals. The model derives from the reaction-diffusion kinetics invented by Turing (1952) and it is assumed that the necessary molecular apparatus is present in each cell of a two-dimensional array and that the cells are in diffusion contact. The model was expressed in differential equation form and solved digitally under a range of different initial, boundary and other conditions. The main forms of pattern that the model generated were spots of variable complexity, rings, and both vertical and horizontal stripes. If morphogen concentration levels are assumed to act as melanin-production switches, then a common basic mechanism is capable of generating a variety of skin patterns. Simple spots such as those found in the fallow deer or the serval, F. serval, are generated if the kinetics are initiated simultaneously in each cell and interpretation depends only on the presence or absence of morphogen, which is assumed for the deer to be an activator and for the cat a suppressor of pigment formation. The reticulated pattern of the giraffe is generated if there is a single high-value threshold. Complex spots typical of the leopards can be produced if there are different concentration thresholds for different colours. Rings of pattern typical of those found on cat tails are generated if the cellular array is a very narrow cylinder. Horizontal stripes are generated if the kinetics in each cell are initiated by a diffusion gradient whose source is the dorsal line of cells and these stripes may break up into spots to give a pattern very similar to that of, for example, the fishing cat, F. viverina. The vertical stripes of the caffre cat, F. caffra, or the zebras are formed if the kinetics are initiated by a vertically-moving constant-velocity wave which also allows morphogen diffusion between previously uncoupled cells. Thus far, the mechanism has generated neither the triradii that are commonly found on forelimbs nor the rings often observed on mammalian limbs. It does however incorporate the randomness that characterizes skin pattern, its operation is of the scale required in embryogenesis, it can be made stable to growth and it can explain certain degenerate patterns. Analysis of a spotted zebra in the light of the model provides evidence that zebra stripes arise from the inhibition rather than the stimulation of melanin; their pattern is thus of white stripes on a black background.     

The Morphostatic Limit for a Model of Skeletal Pattern Formation in the Vertebrate Limb

A recently proposed mathematical model of a “core” set of cellular and molecular interactions present in the developing vertebrate limb was shown to exhibit pattern-forming instabilities and limb skeleton-like patterns under certain restrictive conditions, suggesting that it may authentically represent the underlying embryonic process (Hentschel et al., Proc. R. Soc. B 271, 1713–1722, 2004). The model, an eight-equation system of partial differential equations, incorporates the behavior of mesenchymal cells as “reactors,” both participating in the generation of morphogen patterns and changing their state and position in response to them. The full system, which has smooth solutions that exist globally in time, is nonetheless highly complex and difficult to handle analytically or numerically. According to a recent classification of developmental mechanisms (Salazar-Ciudad et al., Development 130, 2027–2037, 2003), the limb model of Hentschel et al. is “morphodynamic,” since differentiation of new cell types occurs simultaneously with cell rearrangement. This contrasts with “morphostatic” mechanisms, in which cell identity becomes established independently of cell rearrangement. Under the hypothesis that development of some vertebrate limbs employs the core mechanism in a morphostatic fashion, we derive in an analytically rigorous fashion a pair of equations representing the spatiotemporal evolution of the morphogen fields under the assumption that cell differentiation relaxes faster than the evolution of the overall cell density (i.e., the morphostatic limit of the full system). This simple reaction–diffusion system is unique in having been derived analytically from a substantially more complex system involving multiple morphogens, extracellular matrix deposition, haptotaxis, and cell translocation. We identify regions in the parameter space of the reduced system where Turing-type pattern formation is possible, which we refer to as its “Turing space.” Obtained values of the parameters are used in numerical simulations of the reduced system, using a new Galerkin finite element method, in tissue domains with nonstandard geometry. The reduced system exhibits patterns of spots and stripes like those seen in developing limbs, indicating its potential utility in hybrid continuum-discrete stochastic modeling of limb development. Lastly, we discuss the possible role in limb evolution of selection for increasingly morphostatic developmental mechanisms.     

Space-dependent cell determination under the control of a morphogen gradient

A model is proposed for space-dependent cell determination under the influence of a morphogen gradient. It provides an explanation of how groups of cells can be programmed in a particular direction and how a jump from one determination stage to the next can occur between them even though the controlling signal is of a smoothly graded morphogen concentration. Together with an earlier proposed mechanism for pattern formation, these models offer a complete system for the generation and interpretation of positional information. Each member of a set of structure-controlling genes is assumed to feed back onto its own activation such that a gene, once activated, remains in the activated state. A repressor, however, is produced by any activated gene of this set. This assures that only one gene of this set is active in one cell at any one time. A selective activation of a particular gene is possible if (i) the morphogen competes with the gene-produced, non-diffusible repressor, (ii) the feedback loops have some overlap and (iii) a hierarchy exists among the structure-controlling genes. The kinetics of this determination have all the properties demanded earlier from a study of the early insect development: It proceeds stepwise from determination for more anterior to more posterior structures until the gene that is activated corresponds to the local gradient level. A more anterior structure will be formed if the gradient is destroyed before the final determination level is reached. A more posterior structure will be formed after an additional increase of the morphogen concentration. After completion of the determination, the repressor concentration in each cell depends on which gene has become activated and it can be made roughly proportional to the morphogen concentration which the cell has seen. Therefore, a stable parameter (positional value) becomes available which can be used for further developmental decisions.     

Pattern regulation in the embryonic chick limb: Supernumerary limb formation with anterior (non-ZPA) limb bud tissue

The formation of supernumerary limbs and limb structures was studied by juxtaposing normally nonadjacent embryonic chick limb bud tissue. A “wedge” (ectoderm and mesoderm) of anterior or mid donor right wing bud (stage 21) was inserted in a slit made in a host right limb bud (stage 21) at the same position as its position of origin or to a more posterior position. The AER of the donor tissue and host wing bud were aligned with each other. Donor tissue was grafted with its dorsalventral polarity the same as the host's limb bud or reversed to that of the host's. Depending on the position of origin of the donor limb bud tissue and the position to which it was transplanted in a host, supernumerary wings or wing structures formed. Furthermore, depending on the orientation of the graft in the host, supernumerary limbs with either left or right asymmetry developed. The results of experiments performed here are considered in light of two current models which have been used to describe supernumerary limb formation: one based on local, short-range, cell-cell interactions and the other based on long-range positional signaling via a diffusible morphogen.     

Reaction–Diffusion Systems and External Morphogen Gradients: The Two-Dimensional Case,with an Application to Skeletal Pattern Formation

We investigate a reaction–diffusion system consisting of an activator and an inhibitor in a two-dimensional domain. There is a morphogen gradient in the domain. The production of the activator depends on the concentration of the morphogen. Mathematically, this leads to reaction–diffusion equations with explicitly space-dependent terms. It is well known that in the absence of an external morphogen, the system can produce either spots or stripes via the Turing bifurcation. We derive first-order expansions for the possible patterns in the presence of an external morphogen and show how both stripes and spots are affected. This work generalizes previous one-dimensional results to two dimensions. Specifically, we consider the quasi-one-dimensional case of a thin rectangular domain and the case of a square domain. We apply the results to a model of skeletal pattern formation in vertebrate limbs. In the framework of reaction–diffusion models, our results suggest a simple explanation for some recent experimental findings in the mouse limb which are much harder to explain in positional-information-type models.     

A membrane-carrier mechanism for generating new sources and sinks within gradient-controlled tissues—Its application to regeneration in Oncopeltus

A model is presented that can, in principle, generate new sources and sinks within an existing gradient in the concentration of a morphogen. The novel and crucial feature of the model is that morphogens are transported between cells by membrane-based carrier molecules and not by diffusion. A further aspect of the model is the presence of a second substance within each cell whose concentration is uniform over the tissue; this molecule binds to but is not transported by the carrier and is therefore a competitive inhibitor of the morphogen. The concentration of free inhibitor in a cell determines its fate: if at any time it exceeds some threshold, that cell becomes a morphogen source; if it falls below a second threshold, the cell becomes a sink; in between them, the cell shows no special properties. Provided that differences in morphogen concentration between adjacent cells are not too great, the mechanism is indistinguishable from a normal, diffusion gradient. Examination of the kinetics of the system over a one-dimensional line of cells, however, shows that any stable morphogen difference leads to a carrier imbalance and to a change in the degree of inhibitor binding. If this difference is sufficiently great and if there is morphogen homostasis in each cell, then the free inhibitor concentration in the high morphogen cell may exceed the higher threshold causing it to become a source while the low morphogen cell becomes a sink.A numerical example of the mechanism is given and the results calculated for two-dimensional cellular arrays on either side of a morphogen discontinuity. The predictions match the observations of Wright & Lawrence (1981a, b) on Oncopeltus. These authors showed that, if pieces of epidermis from sufficiently different positions were grafted together in vivo, an ectopic boundary would form with regions of reversed polarity on either side of the join. The ability of the model to explain the regeneration and axial graft observations on hydra is also discussed and some experiments that might test the model are put forward. It is suggested that the significance of the membrane-carrier mechanism in vivo is twofold: first, to interpret the basic segmentation mechanism in embryogenesis by turning its morphogen discontinuities into source-sink pairs and so generating actual boundaries; second, to act as a homeostatic mechanism in later development, thus ensuring the maintenance of boundaries.     

A morphogen gradient model for pattern regulation. I. Formation of non-repetitive and repetitive structures

A model for pattern formation is proposed based on two concentration gradients S and Sigma. S is a local morphogen generated by a reaction-diffusion mechanism while Sigma is a by-product of the S-decomposition. Under certain conditions S is well approximated by S(x,L) = alpha(L)f(x L ), where alpha(L) is a scaling function of the length L and f(x L ) is a monotonie function of the relative distance x L from the origin. Sigma degradates and diffuses in the field, reaching a stable L-dependent homogeneous distribution. An allosteric protein P with several active sites reacts with S and Sigma and separates the field into segments. To every segment a corresponding active state of P is dominant. Pattern regulation is automatically achieved since the compartmerttal separation depends explicitly only on x L . For the case of repetitive patterns, a supplementary Gierer-Meinhardt mechanism is introduced for activator X and inhibitor Y. The level of Sigma can affect the decomposition rate of X or Y, e.g. via a second order degradation reaction, hence making the chemical wavelength lambda size-dependent. For a particular decay scheme of Y, a variation of L induces a change of lambda so that finally the number of repetitive structures becomes independent of the field size.     

Morphogen gradients in vertebrate limb development.

The developing limb is an excellent model for pattern formation in vertebrate embryos. Signalling by the polarizing region controls limb pattern across the antero-posterior axis of the chick limb. It was suggested first on theoretical grounds that signalling by the polarizing region could involve a morphogen gradient. Embryological manipulations provided evidence consistent with this model and, more recently, signalling molecules associated with the polarizing region have been identified and tested for their role as morphogens. It is still not clear whether any of the known molecules act directly as a morphogen. The extension of the morphogen model to patterning along the other axes of the limb has been proposed but this may not be applicable.     

Positional information and pattern formation

Spatial patterns of cellular differentiation may arise from cells first being assigned a position, as in a coordinate system, and then interpreting the positional value that they have acquired. This interpretation will depend on their genetic constitution and developmental history. Different patterns may thus arise from similar positional fields. The specification of positional value may involve a positional signal, such as the concentration of a diffusible morphogen, but can also depend on how long the cells remain in a particular region, such as a progress zone. Positional values may also be acquired by direct transfer from one cell layer to another, as in directed embryonic induction. Positional value, unlike a positional signal, involves long-term memory, and can be regarded as a type of cell determination. Cells of the same differentiation class may have different positional values and may thus be non-equivalent. Evidence is presented for a signal providing positional information along the antero-posterior axis during chick limb development. This signal has properties similar to those of a diffusible morphogen.     

Studies on the mechanism of retinoid-induced pattern duplications in the early chick limb bud: temporal and spatial aspects

All-trans-retinoic acid causes striking digit pattern changes when it is continuously released from a bead implanted in the anterior margin of an early chick wing bud. In addition to the normal set of digits (234), extra digits form in a mirror-symmetrical arrangement, creating digit patterns such as a 432234. These retinoic acid-induced pattern duplications closely mimic those found after grafts of polarizing region cells to the same positions with regard to dose-response, timing, and positional effects. To elucidate the mechanism by which retinoic acid induces these pattern duplications, we have studied the temporal and spatial distribution of all-trans-retinoic acid and its potent analogue TTNPB in these limb buds. We find that the induction process is biphasic: there is an 8-h lag phase followed by a 6-h duplication phase, during which additional digits are irreversibly specified in the sequence digit 2, digit 3, digit 4. On average, formation of each digit seems to require between 1 and 2 h. The tissue concentrations, metabolic pattern, and spatial distribution of all- trans-retinoic acid and TTNPB in the limb rapidly reach a steady state, in which the continuous release of the retinoid is balanced by loss from metabolism and blood circulation. Pulse-chase experiments reveal that the half-time of clearance from the bud is 20 min for all-trans- retinoic acid and 80 min for TTNPB. Manipulations that change the experimentally induced steep concentration gradient of TTNPB suggest that a graded distribution of retinoid concentrations across the limb is required during the duplication phase to induce changes in the digit pattern. The extensive similarities between results obtained with retinoids and with polarizing region grafts raise the possibility that retinoic acid serves as a natural "morphogen" in the limb.     

The possible role of “sibling neurite bias” in the coordination of neurite extension,branching, and survival

In this review we consider a novel mechanism, “sibling neurite bias,” which may explain aspects of the coordination of elongation, branching, and resorption among different neurites growing from the same neuronal cell body. In this model, growing neurites which incorporate structural precursors at higher rates would deplete the cellular pool of precursors available to their “sibling” neurites; neurites would compete for survival, but in addition they would bias each other's behavior during active growth. Evidence is reviewed that “sibling neurite bias” may contribute to the establishment and stabilization of specific neural connections. Specific examples examined include the loss of polyinnervation at the developing neuromuscular junction, contextual mapping in the retino-tectal system, and selective neurite growth patterns and synaptic connections in nerve tissue culture model systems.     

A Mechanochemical Model for Embryonic Pattern Formation: Coupling Tissue Mechanics and Morphogen Expression

Motivated by recent experimental findings, we propose a novel mechanism of embryonic pattern formation based on coupling of tissue curvature with diffusive signaling by a chemical factor. We derive a new mathematical model using energy minimization approach and show that the model generates a variety of morphogen and curvature patterns agreeing with experimentally observed structures. The mechanism proposed transcends the classical Turing concept which requires interactions between two morphogens with a significantly different diffusivity. Our studies show how biomechanical forces may replace the elusive long-range inhibitor and lead to formation of stable spatially heterogeneous structures without existence of chemical prepatterns. We propose new experimental approaches to decisively test our central hypothesis that tissue curvature and morphogen expression are coupled in a positive feedback loop.