Starting with a Single Cell
Models of Biological Pattern Formation Provide New Insights into a Fundamental Process
- Pigment patterns on a sea shell. A mollusc can only enlarge its shell along the growing edge. The two-dimensional pattern is therefore a faithful time record of a one-dimensional pattern-forming reaction. This feature makes sea shell patterns unique to decode patterning mechanisms that never reach a stable steady state. The pattern on Conus marmoreus can be explained by a bi-stable pigmentation reaction that spreads slowly (black triangles; time runs form top to bottom). Whenever the pigment reaction was active for a certain period, a second, faster-spreading reaction is triggered that extinguishes the first (red in the simulation, invisible on the natural pattern). Pigmentation survives at the margins and starts spreading again. In its logic, this reaction is close to that responsible for blood coagulation. After an injury, a wave spreads that initiates coagulation to stop bleeding. To avoid coagulation in the whole body, a second and faster wave starts with some delay that extinguishes the first. In this way large but spatially restricted blood clots are formed (from ).
- Fig. 1: Pattern formation by an activator – inhibitor interaction (A): Reaction scheme: the activator catalyses its own production. The production of its rapidly spreading antagonist, the inhibitor, is also under activator control [3–6]. (B): Simulations in a sheet of cells: if the activator has a range comparable to the field size, only a single maximum can appear at a marginal position. The initial, an intermediate and the final stable distribution is shown. Although all cells have the same genetic information, this mechanism allows activation of different genes in different parts of the developing embryo. (C) Biological example: the emergence of a local activation of Nodal in the sea urchin. In agreement with our prediction, Nodal has an autocatalytic feedback on its own production. (D) The inhibitor Antivin/lefty is under the same control as Nodal but inhibits its autoregulation. As expected, it is produced at the same position as the activating Nodal  (Photograph kindly supplied by Dr. Thierry Lepage).
- Fig. 2: (A): After a separation of a sea urchin embryo along the animal-vegetal axis, each fragment forms a complete embryo. Cells close to the area of separation were vitally stained (dots) to document the original orientation. The resulting patterning shows that the complete pattern is restored in both fragments and that a polarity reversal occurs in one. (B) Model: after separation, the remnant inhibitor decays in the non-activated fragment, and a new activation is triggered. The side with the originally lowest inhibitor level wins the competition and forms the new maximum. Both patterns are mirror-symmetric in relation to each other. In other organisms such as hydra intrinsic asymmetries cause regeneration with the original polarity (see [2, 6]).
- Formula 1A
- Formula 1B
- Prof. Hans Meinhardt, Max-Planck-Institute for Developmental Biology
In many branches of sciences, a precise mathematical description is a key to obtain a consistent understanding of the underlying principles. The complexity of the development of a higher organism seems to preclude such an approach in this discipline. However, recently it was shown that basic types of the molecular interaction allowing pattern formation can be described by sets of coupled partial equations. They allow computer simulations that mimic the observation rather precisely. Meanwhile these theories found direct support by observation on the molecular-genetic level.
The generation of the complex structure of a higher organism within each life cycle is certainly one of the most fascinating aspects of biology. Development starts, as the rule, with a single cell, the fertilized egg. At an early stage, many embryos can be fragmented and each fragment forms a complete organism. This indicates that a communication exists between different parts of the developing embryo. The removal of some parts is detected and the missing structures become replaced. This regulation makes development to a surprisingly robust process. However, from the reaction of the organism upon such an experimental perturbation one cannot directly deduce the molecular basis on which development is based.
Recently it was possible to clone genes and to isolate the substances involved. However, even if a particular gene is found to be expressed at a high level at a particular position and a mutation shows that this gene expression is crucial for a particular step, we have no information on how this local maximum is generated.
As in other branches of science, theories provide the bridges between observations and an understanding of the underlying mechanisms. The development of a higher organism may appear far too complex to allow such an approach in this discipline. However, this process can be separated into a number of elementary steps. A key process is the generation of local concentration maxima that act as signaling centers. The concentration of the signaling molecules declines with increasing distance from organizing regions, which allows a position-dependent activation of genes.
Such organizing regions where first found in classical transplantation experiments. Examples are the gastric opening of the freshwater polyp Hydra  and the organizer in amphibians . Upon transplantation, such organizing regions are able to instruct the surrounding cells. Meanwhile several genes are known that are expressed at high levels in such signaling centers.
A Theory of Biological Pattern Formation
There are good reasons to assume that every step in development is accomplished by molecules. Since many molecules interact, the description of development will involve many coupled differential equations that describe the production, spread, removal and, most importantly, the mutual regulation of molecules. Thus, the aim was to find hypothetical molecular interactions that reproduce the capability to generate structures in space, i.e., to accomplish pattern formation. Long before the molecular approach became feasible together with Alfred Gierer I have shown which type of molecular interaction allows the generation of such signaling centers and a concentration-dependent gene activation [3, 4]. These theories found meanwhile direct support by experimental observations [5, 6]. In particular, the theory accounts for the basic observations in the perturbation experiments mentioned above.
Local Self-enhancement and Long-range Antagonistic Effects as the Driving Force of Pattern Formation
Local concentration maxima can be generated if, and only if, strong (nonlinear) and local-acting positive feedback loops exist that are antagonized by a reaction that acts on a longer range [3-6]. In such an interaction the homogeneous distribution is instable. Small deviations from a homogeneous distribution further grow due to the local positive feedback while the long-ranging antagonistic effect keeps the emergent maxima localized and suppresses the onset of a similar patterning processes at larger distance.
Pattern formation starting from almost homogeneous initial conditions is also very common in non-animated systems; sand dunes, rivers, clouds, and lightning are examples. It is easy to see that these patterning processes are based on the same principle. For a sand dune a stone may provide a windshield that accelerates the local deposition of more sand. Erosion proceeds faster at an initial injury since more water collects there. Since the total amount of water or sand is limited, a local accumulation must be accompanied by an overall decrease elsewhere. The same type of interaction - self-re-enforcement and inhibition of a larger domain - is also the basis in many social interactions.
An Example: the Activator - Inhibitor Interaction
A biochemical realization of the general principle requires the interaction of at least two types of molecules (fig. 1). For instance, a short-ranging substance - to be called the activator - promotes its own production (autocatalysis) as well as that of its rapidly diffusing antagonist, the inhibitor. The concentrations of both substances can be in a steady state. A general increase of the activator concentration would be compensated by a corresponding increase of the inhibitor level that brings the activator back to the steady state concentration. However, any local elevation of the activator will further increase due to autocatalysis since the surplus of the inhibitor rapidly diffuses into the surroundings of this incipient maximum. It inhibits the activator production there while the local activator elevation further grows (fig. 1). Eventually a patterned stable steady state will be reached. Even random fluctuations are sufficient to initiate pattern formation. The following set of partial differential equations describes a possible interaction. It relates the change per time unit of the activator a and the inhibitor h as function of the actual concentrations.
Formula (1a) and (1b)
In this interaction t is time, x is the spatial coordinate, Da and Dh are the diffusion coefficients, µa and µh the decay rates of a and h. The source density ρa describes the ability of the cells to perform the autocatalysis. A small activator-independent activator production ρa can initiate the system at low activator concentrations. Such an interaction exhibits essential properties explaining basic steps in development. In small or growing fields, monotonic gradients are formed. The graded concentration profile can provide positional information for the formation of an embryonic axis. Although all cells carry the same genetic information, one side of the embryo develops different from the other in a reproducible way. The mechanism accounts for the regeneration of an organizing region in a straightforward manner. With the removal of an activated region, the region of inhibitor production is removed, too. The inhibitor decays and a new activation can be triggered from some base-line activator production (fig. 2). In fields that are large compared with the range of both substances, spatial periodic patterns can result, as required to initiate the formation of feathers and hairs. In other ranges of parameters, the same equation describes oscillations and traveling waves. Short computer programs that run on a PC can be found on our website.
The possibility to generate patterns by the interaction of two substances that have a different diffusion range was discovered by Alain Turing in 1952 . However, the crucial condition of local self-enhancement and long range inhibition is not inherent in his paper. The discovery of our condition enabled us to include different molecular realizations and non-linear interaction. This is required for the design of molecularly feasible interactions. At the time this mechanism was proposed (1972), the interaction was completely hypothetical. Meanwhile several activator-inhibitor systems have been found. An example is the nodal/lefty2 interaction that specifies the oral field in sea urchins (fig. 1). The same interaction is involved, e.g., in the left/ right patterning. Other examples and different realizations are described elsewhere .
Although in developmental biology the potential of mathematical modeling has been underrated, theoretical approaches are a powerful tool to provide essential new insights. They allow disentangling of the complex interaction on which development is based and reveal the minimum requirements for essential steps.
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