# Periodic pattern formation in reaction-diffusion systems: an introduction for numerical simulation.

## Summary (4 min read)

### Introduction

- Mathematical modeling, numerical simulation, pattern formation, Turing, also known as Key words.
- As the activator peaks grow, inhibitor peaks should also grow in response because the activator promotes the production of inhibitor.

### What is a differential equation?

- These equations are called differential equations (more specifically, partial differential equations) and describe the rate of change of an internal state of a certain physical system in both time and space.
- To fully specify the problem requires three factors: (i) initial conditions; (ii) governing equations; and (iii) boundary conditions.
- The governing equations define the rules on how these values will change in time and space.
- Boundary conditions define how the system behaves at its boundary; for example, the system may be confined within a certain domain, so there would be no flux out of the boundaries.
- The authors will explain what these terms actually mean by using the simplest example.

### Initial conditions

- In that case, the authors have 20 pieces of tissue (1/0.05 = 20) that contain both activator and inhibitor, so they define the initial distribution of activator and inhibitor by assigning 20 random numbers to each tissue piece.
- This set of random numbers represents the small noise that should exist in the actual system and, as the authors will see later, any set of random numbers as the initial condition will eventually lead to a similar periodic pattern in this simulation.

### Discretization

- Because it is quite difficult to think directly about the time-course of spatial distribution of the molecules, at first the authors divide this rod-like structure into small pieces that have horizontal length dx (dx << 1, meaning dx is much smaller than 1) and suppose the spatial distribution of activator and inhibitor in these small pieces to be homogeneous.
- Then, the authors will think about the concentration change of activator and inhibitor in these small elements.
- There are two factors that affect the concentration of activator and inhibitor in these small pieces: (i) the interaction of activator and inhibitor within each element; and (ii) the transfer of activator and inhibitor between each element and its two nearest neighbors.
- Basically, time should be continuous, but the authors consider updating the system in discrete time steps, dt.

### Reaction term

- At first the authors suppose the spatial distributions of activator and inhibitor at time m × dt are known and think about what will happen in each small tissue element after the short time period dt.
- Then, the authors will consider the events that occur solely inside each tissue piece.
- In the activator–inhibitor type reaction–diffusion model (Fig. 3), the activator promotes its own production and promotes the production of inhibitor and the inhibitor inhibits the activator production and decays with time.
- For mathematical simplicity, the authors allow negative values for p and q and set their initial values to 0.
- The dt term arises from observing that the actual increase in chemical concentration is obtained by multiplying the net rate of production by time.

### Diffusion term

- Next, the authors will consider the interaction between a tissue element and its two nearest neighboring tissue elements during the time interval (m × dt, (m + 1) × dt).
- In the biological context, there are many ways to transmit signals spatially but, for simplicity, here the authors consider that both activator and inhibitor diffuse passively between tissue elements.
- Therefore, the amount of activator that is transferred from the right neighboring piece (arrow in Fig. 4) is: [4] where dp represents the diffusion coefficient of the activator.
- Similarly, the concentration change of inhibitor is: [10 ] in which dq is the diffusion coefficient of the inhibitor.
- This part, which represents the effect of diffusion, is called the ‘diffusion term’ in the reaction–diffusion system.

### Boundary conditions

- As the authors have seen above, the tissues at the boundary should be treated separately.
- By evaluating these cells, the authors can obtain the complete spatial distribution of activator and inhibitor at time dt.

### Zero-flux boundary condition

- The authors assume that the boundary is impermeable; that is, no material is transferred across it.
- This condition is also called the Newmann boundary condition.

### Transformation to continuous equation

- The authors can obtain the continuous differential equations corresponding to the discrete equations above by making dt and dx infinitely small.
- At first, the authors define the concentration of activator and inhibitor as u(x, t) and v(x, t), which are now continuous functions (x and t are now real values instead of integers).
- This is a ‘partial derivative’ and is defined as a derivative of a function of several variables when all but one variable (the variable of interest) are held fixed during the differentiation.
- The authors describe the continuous equations only for deciphering equation 1, but actually an analytical treatment of the system is possible for the continuous case (see Murray, 2003).
- Numerical calculation of the reaction– diffusion system Parameters and equations used in the numerical calculations.

### Governing equation

- First, the authors will calculate the concentration of activator (p) and inhibitor (q) at a certain tissue piece after time dt has passed.
- Let us calculate the activator and inhibitor concentrations for a certain column (C) and write the values to C4 and C5.
- The authors can obtain values in other cells by applying the same equation (Fig. 6).
- Fortunately, Excel automatically converts these equations to appropriate forms by simply copying and pasting cells C4 and C5 to other cells.

### Numerical calculation

- Fortunately Excel again automatically converts the equations to appropriate forms by simply copying and pasting the whole row to the rows below.
- By doing this 100–200 times repeatedly, you can observe that the concentrations gradually form a periodic structure, as shown in Fig.
- Numerical calculation with Mathematica Obviously, it is too labor consuming to undertake the above procedure, so the authors generally use Mathematica (Wolfram Research, Champaign, IL, USA) to calculate the results.
- The details of the program can be provided electronically.
- (All the calculations in the present paper are performed by Mathematica and the source code (with additional instructions on linear stability analysis) is freely available on request from the authors.).

### Properties of Turing reaction–diffusion systems

- Relationship between domain size and number of structures.
- One characteristic of reaction–diffusion systems is that they tend to form structures of similar size , so if the domain size is changed the number of structures should change, not the size of each structure.

### Changing initial conditions

- One characteristic of the Turing reaction–diffusion system is that it has the ability to form de novo stable periodic patterns; that is, without any prepattern.
- As you can see in Fig. 10, if the initial condition is not homogeneous, pattern formation can occur sequentially.
- The final periodic structure is more or less the same.
- Detailed analysis on the pattern appearance speed has been recently undertaken in Miura and Maini (2004).

### Changing diffusion coefficients

- As you can see from the above discussion, the reaction–diffusion system has the ability to make a periodic pattern from an almost homogeneous initial state and the wavelength of the pattern is decided by the parameters in the reaction and diffusion terms.
- This is experimentally assayed by Miura and Shiota (2000a) in a limb bud mesenchyme cell culture system.
- Actually, the result is easy to understand without numerical calculation.
- The result is confirmed by numerical calculation (Fig. 12).

### Cross-type reaction–diffusion model

- The activator–inhibitor scheme is well known among developmental biologists, but actually there is another type of reaction–diffusion model that has the ability to generate periodic pattern.
- This system, usually called the substrate-depletion system, also consists of two hypothetical molecules, the substrate and the enzyme.
- Numerical simulations are shown in Fig. 13.
- In the activator–inhibitor system, the inhibitor should be expressed at chondrogenic sites and have an ability to inhibit chondrogenesis.

### Reaggregated tissue experiment

- As the authors saw in the previous subsection, one important characteristic of the Turing reaction–diffusion system is an ability to form a periodic pattern from various initial states and the process is quite robust.
- Here, the authors try to emulate the most extreme case, where the tissue is dissociated into single cells and reaggregated.
- In contrast, if the number of structures is quite stable, the authors have to couple the reaction–diffusion system with other mechanisms to explain the stability.
- The reliability of the pattern formation mechanism was first discussed by Bard and Lauder (1974) and a possible scenario for robust pattern formation is proposed by Crampin et al. (1999).
- The authors introduce the latter in the following subsection.

### Growing domain

- In actual biological systems (especially embryonic tissue), the size and shape of the pattern formation field is usually not constant, but grows.
- The Turing reaction–diffusion system has the property that it retains the periodic structure of fixed wavelength, so if the tissue grows the authors can expect that the pattern will change according to the growth.
- Next, the authors slightly modify the reaction term of the system (Fig. 16b).
- In the previous simulation, additional peaks are inserted between pre-existing peaks, but this time each peak is split into two peaks.
- This distinction of peak increase (called ‘mode doubling’) is investigated analytically by Crampin et al. (2002).

### Two dimensions: stripe–spot selection

- The authors can easily modify the above program for simulation on a two-dimensional spatial domain (a flat surface).
- One finds stripes, spots or more complex patterns.
- One can see, in Fig. 17, a stripe- and spot-like pattern.the authors.
- Stripe–spot selection has been studied in a special case (Ermentrout, 1991; Lyons & Harrison, 1992), but the mechanism, in general, is not fully understood.
- Numerical calculations on two-dimensional curved surfaces (Varea et al., 1999) and three dimensions (Leppänen et al., 2002) have been done recently and some interesting features are observed concerning the connectivity of the periodic pattern.

### Future prospects

- There are several groups of researchers who are trying to understand biological pattern formation using mathematical models.
- The field is expanding rapidly and a considerable number of researchers (although much less than in developmental biology) is involved.
- As far as the authors know, there are only two to three groups in the world that can deal with this kind of problem both theoretically and experimentally.
- It seems that the two cultural differences described above can be an energy barrier for experimental people to enter this field.

### Acknowledgments

- The authors thank Professor Stuart Newman (New York Medical College) and Dr Chad Perlin (Department of Human Anatomy and Genetics, University of Oxford) for their critical reading of the manuscript.
- This work was supported by the Japan Society for the Promotion of Science.

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### Cites background from "Periodic pattern formation in react..."

...In the isotropic case, the boundary conditions in equation (4.7a) simplify to: vui vn̂ Z 0; (4.7b) The initial conditions are: uðx; 0ÞZ u initðxÞ: (4.8) For biological applications of RD see, among others, Meinhardt (1982), Murray (1993) and Miura & Maini (2004a)....

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...Turing (1952) introduced the idea that interactions of reacting and diffusing chemicals (usually of two species) could form self-organizing instabilities that provide the basis for biological spatial patterning (e.g. animal coat patterning; see Murray 1993; Miura & Maini 2004a for reviews)....

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...Several models attempt to account for RD of morphogenetic signalling during chondrogenesis in the limb (Newman & Frisch 1979; Hentschel et al. 2004) and its isolated mesenchymal tissue (Miura & Shiota 2000a,b; Miura et al. 2000; Miura & Maini 2004b)....

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##### References

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### "Periodic pattern formation in react..." refers methods in this paper

...This model was originally proposed by the British mathematician Alan Turing (Turing, 1952) and a significant amount of work has been done using this idea in the field of mathematical biology (for reviews, see Bard, 1990; Meinhardt, 1995; Murray, 2003)....

[...]

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### "Periodic pattern formation in react..." refers background in this paper

...Examples include the skin pigment pattern in zebra (Bard, 1981; Murray, 2003), angelfish (Kondo & Asai, 1995; Shoji et al., 2003), zebrafish (Asai et al., 1999) and sea shells (Meinhardt, 1995), feather follicle formation (Jung et al., 1998), tooth development (SalazarCiudad & Jernvall, 2002) and…...

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### "Periodic pattern formation in react..." refers background or methods in this paper

...This model was originally proposed by the British mathematician Alan Turing (Turing, 1952) and a significant amount of work has been done using this idea in the field of mathematical biology (for reviews, see Bard, 1990; Meinhardt, 1995; Murray, 2003)....

[...]

...…pattern in zebra (Bard, 1981; Murray, 2003), angelfish (Kondo & Asai, 1995; Shoji et al., 2003), zebrafish (Asai et al., 1999) and sea shells (Meinhardt, 1995), feather follicle formation (Jung et al., 1998), tooth development (SalazarCiudad & Jernvall, 2002) and digit formation during limb…...

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...This results in a periodic pattern of activator and inhibitor peaks (Meinhardt, 1995; Kondo, 2002)....

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##### Frequently Asked Questions (2)

###### Q2. What future works have the authors mentioned in the paper "Periodic pattern formation in reaction–diffusion systems: an introduction for numerical simulation" ?

In summary, there is no established course to study pattern formation during development and the number of researchers who take this approach is quite limited. It seems that the experimental application of mathematical models to biological pattern formation is quite a promising area because the lack of progress is more or less due to technical reasons described above, not the lack of importance. As far as the authors know, there are only two to three groups in the world that can deal with this kind of problem both theoretically and experimentally. It seems that the two cultural differences described above can be an energy barrier for experimental people to enter this field.