Mathematical Background of Pattern Formation

The Mathematics of Reaction-Diffusion Mechanisms

Probably the best introduction to the mathematics of pattern formation in developmental biology is on Hans Meinhardt’s website, a remarkable and beautiful resource.

The classic paper Gierer, A. and Meinhardt, H. 1972. A theory of biological pattern formation, Kybernetic 12: 30-39 is a critically important paper in modeling developmental phenomena. It is in Acrobat PDF format.

A biography of Alan Turing.

I. The Basic Activator-Inhibitor System

(Based on Meinhardt, H. 1998. The Algorithmic Beauty of Sea Shells, Second ed., Springer-Verlag, New York.)

The mathematics of the reaction-diffusion system involves the interactions between an autocatalytic activator (P or a) and its antagonist, the inhibitor (S or b). The partial differential equations relate the concentration change per time of both these substances as a function of their concentrations:

Activator:

Inhibitor:

Where:

t is the time, x is the spatial coordinate, D_a and D_b are the diffusion coefficients for the activator and the inhibitor, respectively (and the diffusor inhibits faster than the activator).

The terms can then be understood as such:

sa²/b is the production rate of the activator. The activator has an autocatalytic rate such that the more a is present, the more a is made. The product formation is slowed down by the presence of b. The source density s describes the ability of the agents to perform autocatalysis.

-r_aa is the rate at which the activator is removed. As a general rule, the rate at which molecules disappear is proportional to the number of molecules present.

D_a(Δ²a/Δx²) is the exchange by diffusion. It is proportional to the second derivative for the following reason: The net exchange of molecules by diffusion is 0 if all cells have the same concentration. The net exchange is also 0 if a constant concentration difference exists between neighboring cells (as in a linear concentration gradient). In that case, each cell loses and receives the same amount of substance. Thus, this term represents not the change of concentrations, but the change in the change of concentrations in space.

b_a is the basic activator production. A small activator-independent production can initiate the system at low activator concentration levels. This is the term that was altered in the equations to produce Figure 1.22 in the textbook. It is required for sustained oscillations during growth.

b_b is the basic inhibitor production. This function can change the stable state at which the system arrives. It is also important in producing "travelling waves."

II. More Detailed Mathematics of the Reaction-Diffusion System

From Gene Networks Database, Olga Kirrilova

One of the most interesting and widely studied topics in modern theoretical biology is the structure formation from a more or less homogeneous egg. Here some global field models of such processes are represented. A characteristic feature of these models is that they consider autonomous growth (autocatalysis) coupled to dissipative processes, such as diffusion.

In some cases one can consider these models as Turing's systems of the first kind, that is represented by differential equation:

More rarely Turing's systems of second kind are used:

As was noticed, before the stable pattern can be generated two conditions have to be fulfilled: a local deviation from an average concentration of a pattern forming substance should further increase (1), and this increase should not go to infinity (2). In supposition that increase in one part of the field is necessarily connected with a decrease in another part of it, (i.e. that the total amount of substances is roughly conserved), the process of the pattern formation should reach the stable steady state.

Thus the mechanism underlying pattern formation should be similar to local autocatalysis with strong positive feedback and lateral inhibition. We separately consider the model with diffusion (class 1 models) and without it (class 2 models). Here we will only discuss class 1 models.

Models with diffusion (Class 1 models)

Here we represent the application of principles of autocatalysis and lateral inhibition to biochemical reactions with diffusion. Let us consider a substance a, called an activator, which stimulates its own production (autocatalysis) and the production of its antagonist i, called an inhibitor. To carry out the necessary long-range inhibition, the inhibitor must diffuse more rapidly. In an extended field of cells, a homogeneous distribution of these substances is unstable, since any small local elevation of the activator concentration, resulting perhaps from random fluctuations, will be amplified by the activator autocatalysis. The inhibitor, which is produced in response to the increase in activator production, cannot halt the locally increased activator production, since it diffuses quickly into the surrounding tissue and suppresses activator production outside the activated center. Thus, the locally increased activator concentration will increase further, and with increasing concentration, the maximum becomes narrower and narrower until some limiting factor comes into play. For instance, the loss of activator from the narrow peak by diffusion become sequel to the net production.

A stable activator and inhibitor profile is ultimately obtained, although both the substances continue to be made, to diffuse, and to be broken down. Such a simple system of two interacting substances is, therefore, able to produce a stable, strongly patterned distribution from a nearly homogeneous initial distribution, as it occurs in biological pattern formation. The general representation of such a system is:

The first term in the right part represents the diffusion process, where D_a and D_i are diffusion constants for a and i correspondingly; the second term represents other processes such as production and decay, they may be written as:

here if c₁=0, then c₂=1 and vice versa. In these equations, production is represented by the first and third terms, the second term represents the decay. One of the most known and frequently used types of systems of such a kind can be derived if to set c₁=1 (correspondingly c₂=0), k=r₁=r₂=0, namely

Resulting distribution may be monotonous or periodical and changes as a function of parameters. The pattern formed in this way may be stable or oscillating with the time. The positioning of high concentrations is produced by small internal or external asymmetries or by local disturbance. This local high concentration can serve as a signaling system, for instance, to initiate head formation in hydra. A pattern formed in this way has strong self-regulatory properties. Let us also consider a special case of such a systems with a feedback loop.

To do this set c₂=0, then

The small basic (activator-independent) activator production r₁ can initiate the autocatalysis in areas of low activator concentration. As we will see, this term is important if new centers have to arise during growth or regeneration. In contrast, the basic inhibitor production can suppress the appearance of secondary maxima, a feature which is important if an ordered sequence of structures is to be specified by positional information. If the activator production saturates at a high concentration due to term 1/(1+k × a × a), the activated area is self-regulated.

Other molecular realization of autocatalysis and lateral inhibition principle may be the processes in which the inhibition effect is realized by depletion of a substrate consumed in autocatalysis. From a mathematical point of view this means:

Also one has to remember Sel'kov's model in developmental biology:

where n is known as Hill's number.

In this model, one assumes that the activator reproduction is compensated by self-regulated reproduction of the inhibitor. The stable pattern formation is also possible in this case.

For a deeper mathematical understanding, see Oscillating Reaction-Diffusion Spots.

III. Reaction-Diffusion in Dictyostelium

As documented on pages 39-44 of the textbook (and in Vade Mecum²), the slime mold Dictyostelium discoideum is a part-time multicellular organism. Most of the time, Dictyostelium is an amoeba, crawling around the forest floors of North America, eating bacteria. However, when the food supply is insufficient, they aggregate by the thousands. The molecule signaling the aggregation is cyclic AMP (cAMP). This molecule appears to be autocatalytic, in that it tells the cell receiving it to make more of it. In fact, when cAMP is sensed by the responding cell, it takes only 30 seconds for that cell to start making more cAMP (Gerisch 1968; Devreotes 1989; Goldbeter 1996). However, that same autocatalysis also signals inhibition as the receptor molecules become phosphorylated. Once phosphorylated, the receptor cannot bind cAMP. The refractor period lasts about five minutes and then the receptor can function again. If Dictyostelium amoebae are packed together densely enough, the activation spreads in a wave-like motion (see Vade Mecum²). Those cells that initiated the release of cAMP will become the centers of aggregation. Cells will move towards these centers since one side of their cytoplasm receives the signal before the other. In this way, the wave is propagated from the center to the periphery of the aggregate. The movement of the slug also seems to be under the regulation of the waves of cAMP. To see movies of this wave going across the aggregation, look at the darkfield microscopy of the waves traversing the multicellular aggregate.

Literature Cited

Devreotes, P. N. 1989. Dictyostelium discoideum: A model system for cell-cell interactions in development. Science 245: 1054-1058.

Gerisch, G. 1968. Cell aggregation and differentiation in Dictyostelium. Curr. Top. Devel. Biol. 3: 157-232.

Goldbeter, A. 1996. Biochemical Oscillations and Cellular Rhythm: The Molecular Bases of Periodic and Chaotic Behavior. Cambridge University Press, Cambridge.

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