Hans Meinhardt

A theory of biological pattern formation.

One of the elementary processes in morphogenesis is the formation of a spatial pattern of tissue structures, starting from almost homogeneous tissue. It will be shown that relatively simple molecular mechanisms based on auto- and cross catalysis can account for a primary pattern of morphogens to determine pattern formation of the tissue. The theory is based on short range activation, long range inhibition, and a distinction between activator and inhibitor concentrations on one hand, and the densities of their sources on the other. While source density is expected to change slowly, e.g. as an effect of cell differentiation, the concentration of activators and inhibitors can change rapidly to establish the primary pattern; this results from auto- and cross catalytic effects on the sources, spreading by diffusion or other mechanisms, and degradation.Employing an approximative equation, a criterium is derived for models, which lead to a striking pattern, starting from an even distribution of morphogens, and assuming a shallow source gradient. The polarity of the pattern depends on the direction of the source gradient, but can be rather independent of other features of source distribution. Models are proposed which explain size regulation (constant proportion of the parts of the pattern irrespective of total size). Depending on the choice of constants, aperiodic patterns, implying a one-to-one correlation between morphogen concentration and position in the tissue, or nearly periodic patterns can be obtained. The theory can be applied not only to multicellular tissues, but also to intracellular differentiation, e.g. of polar cells.The theory permits various molecular interpretations. One of the simplest models involves bimolecular activation and monomolecular inhibition. Source gradients may be substituted by, or added to, sink gradients, e.g. of degrading enzymes. Inhibitors can be substituted by substances required for, and depleted by activation.Sources may be either synthesizing systems or particulate structures releasing activators and inhibitors.Calculations by computer are presented to exemplify the main features of the theory proposed. The theory is applied to quantitative data on hydra — a suitable one-dimensional model for pattern formation — and is shown to account for activation and inhibition of secondary head formation.

Pattern formation by local self-activation and lateral inhibition.

In 1972, we proposed a theory of biological pattern formation in which concentration maxima of pattern forming substances are generated through local self-enhancement in conjunction with long range inhibition. Since then, much evidence in various developmental systems has confirmed the importance of autocatalytic feedback loops combined with inhibitory interaction. Examples are found in the formation of embryonal organizing regions, in segmentation, in the polarization of individual cells, and in gene activation. By computer simulations, we have shown that the theory accounts for much of the regulatory phenomena observed, including signalling to regenerate removed parts. These self-regulatory features contribute to making development robust and error-tolerant. Furthermore, the resulting pattern is, to a large extent, independent of the details provided by initial conditions and inducing signals.

Pattern formation in Escherichia coli: a model for the pole-to-pole oscillations of Min proteins and the localization of the division site.

Proper cell division requires an accurate definition of the division plane. In bacteria, this plane is determined by a polymeric ring of the FtsZ protein. The site of Z ring assembly in turn is controlled by the Min system, which suppresses FtsZ polymerization at noncentral membrane sites. The Min proteins in Escherichia coli undergo a highly dynamic localization cycle, during which they oscillate between the membrane of both cell halves. By using computer simulations we show that Min protein dynamics can be described accurately by using the following assumptions: (i) the MinD ATPase self-assembles on the membrane and recruits both MinC, an inhibitor of Z ring formation, and MinE, a protein required for MinC/MinD oscillation, (ii) a local accumulation of MinE is generated by a pattern formation reaction that is based on local self-enhancement and a long range antagonistic effect, and (iii) it displaces MinD from the membrane causing its own local destabilization and shift toward higher MinD concentrations. This local destabilization results in a wave of high MinE concentration traveling from the cell center to a pole, where it disappears. MinD reassembles on the membrane of the other cell half and attracts a new accumulation of MinE, causing a wave-like disassembly of MinD again. The result is a pole-to-pole oscillation of MinC/D. On time average, MinC concentration is highest at the poles, forcing FtsZ assembly to the center. The mechanism is self-organizing and does not require any other hypothetical topological determinant.

Cell determination boundaries as organizing regions for secondary embryonic fields.

A model is proposed for pattern formation in secondary embryonic fields. It is stipulated that the boundaries, resulting from the primary embryonic organization of a developing organism, act as organizing regions for secondary embryonic fields, e.g., imaginal discs in insects. This boundary mechanism would allow very reliable pattern formation in the course of development: Primary positional information leads to cells of different determination, separated by sharp borders. At these borders, in turn, positional information would be generated for the next finer subdivision, and so on. This occurs if two or more differently determined cell types (e.g., compartments) cooperate for the production of a morphogenetic substance. A high concentration of the morphogen would appear at the common boundary of the cell types involved. Many experiments reported in the literature, for instance, the formation of duplicated and triplicated insect legs and the regeneration-duplication phenomenon of imaginal disc fragments can be explained under this assumption. The proposed boundary mechanism provides a molecularly feasible basis for the polar coordinate model.

Dynamic localization cycle of the cell division regulator MinE in Escherichia coli

The MinC protein directs placement of the division septum to the middle of Escherichia coli cells by blocking assembly of the division apparatus at other sites. MinD and MinE regulate MinC activity by modulating its cellular location in a unique fashion. MinD recruits MinC to the membrane, and MinE induces MinC/MinD to oscillate rapidly between the membrane of opposite cell halves. Using fixed cells, we previously found that a MinE–green fluorescent protein fusion accumulated in an annular structure at or near the midcell, as well as along the membrane on only one side of the ring. Here we show that in living cells, MinE undergoes a rapid localization cycle that appears coupled to MinD oscillation. The results show that MinE is not a fixed marker for septal ring assembly. Rather, they support a model in which MinE stimulates the removal of MinD from the membrane in a wave‐like fashion. These waves run from a midcell position towards the poles in an alternating sequence such that the time‐averaged concentration of division inhibitor is lowest at midcell.

Models of Biological Pattern Formation: From Elementary Steps to the Organization of Embryonic Axes

An inroad into an understanding of the complex molecular interactions on which development is based can be achieved by uncovering the minimum requirements that describe elementary steps and their linkage. Organizing regions and other signaling centers can be generated by reactions that involve local self-enhancement coupled to antagonistic reactions of longer range. More complex patterns result from a chaining of such reactions in which one pattern generates the prerequisites for the next. Patterning along the single axis of radial symmetric animals including the small freshwater polyp hydra can be explained in this way. The body pattern of such ancestral organisms evolved into the brain of higher organisms, while trunk and midline formation are later evolutionary additions. The equivalent of the hydra organizer is the blastopore, for instance, the marginal zone in amphibians. It organizes the anteroposterior axis. The Spemann organizer, located on this primary organizer, initiates and elongates the midline, which is responsible for the dorsoventral pattern. In contrast, midline formation in insects is achieved by an inhibitory signal from a dorsal organizer that restricts the midline to the ventral side. Thus, different modes of midline formation are proposed to be the points of no return in the separation of phyla. The conversion of the transient patterns of morphogenetic signaling into patterns of stable gene activation can be achieved by genes whose gene products have a positive feedback on the activity of their own gene. If several such autoregulatory genes mutually exclude each other, a cell has to make an unequivocal decision to take a particular pathway. Under the influence of a gradient, sharply confined regions with particular determinations can emerge. Borders between regions of different gene activities, and the areas of intersection of two such borders, become the new signaling centers that initiate secondary embryonic fields. As required for leg and wing formation, these new fields emerge in pairs at defined positions, with defined orientation and left-right handedness. Recent molecular-genetic results provide strong support for theoretically predicted interactions. By computer simulations it is shown that the regulatory properties of these models correspond closely to experimental observations (animated simulations are available at www.eb.tuebingen.mpg.de/meinhardt).

A model for pattern formation on the shells of molluscs

Models based on reaction-diffusion mechanisms are proposed for the generation of pigmentation and relief-like patterns on mollusc shells. They extend an earlier model to describe the formation of more complex patterns. Shell patterns are time records of a one-dimensional pattern forming process along the growing edge. Oblique lines result from travelling waves of activation (pigment production). Branches and crossings result from a temporary shift from an oscillatory into a steady state mode of pigment production. Checkerboard or meshwork-like patterns require systems with at least three components, one autocatalytic substance antagonized by two inhibitions, a diffusible inhibiting substance which generates the pattern in space and a non-diffusible one which is responsible for the pattern in time. Wavy lines, rows of dots and fish-bone like patterns can result from the superposition of two patterns: a pattern which is stable in time controls the oscillation frequency of the pigment-producing process. By computer simulations it is shown that the models reproduce not only fine details of the natural patterns but account also for pattern regulation such as observed on some species after injury.

Generation and regeneration of sequence of structures during morphogenesis

Abstract 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 or to 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.

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

Abstract 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.

Modeling seashells

This paper presents a method for modeling seashelfs, suitable for image synthesis purposes. It combines a geometric description of shelf shapes with an activator-inhibitor model of pigmentation patterns on shell surfaces. The technique is illustrated using models of selected sheUs found in nature. CR Categorks: 1.3.5 [Computer Graphka]: Computational Geometry and Object Modeling: Curve, siufue, solid and object represenfutwn. 1.3.7 [Computer Graphka]: l%ree-f)imensionaf Graphics and Reafism. J.3 Life and Medkal Sciences]: Biology. Keyworda: realistic image synthesis, modeling of natural phenomenq seashell, logarithmic helico-spiral, sweep representation, reactiondfision pattern model.