Ann Druyan, an American author and producer specializing in cosmology and popular science, once said

For most of the history of our species we were helpless to understand how nature works. We took every storm, drought, illness, and comet personally. We created myths and spirits in an attempt to explain the patterns of nature.

As disordered as nature appears, a number of natural phenomena adhere to consistent patterns. One such phenomenon observed in nature is that of phyllotaxis, a term that refers to the arrangement of leaves around a stem. Although different species of plants may look very different, leaves across all plant species have a highly conserved arrangement: the angle formed between adjacent leaves, as observed when looking down at the stem, is approximately 137.5 degrees (Figure 1A-B). Also, the formation of petals and branches [1] is commonly observed to be in accordance with the Fibonacci sequence, a mathematical sequence of numbers in which each subsequent number is the sum of the previous two; thus, the sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

The Fibonacci sequence is often found in patterns in nature. For example, daisies are often found with 13, 21 (Figure 1C), 34, 55 or 89 petals.

Figure 1. Examples of phyllotaxis. (A) Schematic diagram of leaf arrangement, with numbers of successive leaves numbered. (B) Example of Aloe polyphylla displaying an angle of 137.5 degrees between adjacent leaves, as indicated by the red lines (Image credit: Flickr user Genista) (C) Daisy with 21 petals (21 is a Fibonacci number) (Image credit: Melissa Wright).

Turing model

Intrigued by the natural phenomenon of phyllotaxis, Alan Turing strove to use mathematical equations to model biological processes. In 1952, he published the landmark paper, “The Chemical Basis of Morphogenesis”, in which he presented a mathematical model that attempted to explain the formation of self-regulated patterns during the development of plants and animals (morphogenesis literally means “the formation of the body’s shape”). His reaction-diffusion model demonstrates how patterns can form anew from simple molecules, and thus, proposes a process by which complex spatial structures in tissues and organs may arise. [2]

Figure 2. Hypothetical system in the reaction-diffusion model. Each cell produces, and is capable of responding to, two morphogens (red-colored triangles and blue-colored diamonds). The morphogens diffuse into the space around the cells at different rates, bind to their respective receptors (same colors) and either stimulate or inhibit their own production. The final distribution of morphogens results in a specific pattern. (Image based on [3]).

The concept behind the reaction-diffusion model hinges on the presence of two chemical substances (termed “morphogens”) within plant or animal tissues. Both morphogens can spread away (or “diffuse”) from where they are originally produced in the tissue. One morphogen diffuses slowly, and serves as an activator of development at nearby sites. The second morphogen is an inhibitor of the first, and spreads more rapidly from where it originated, thus acting at longer ranges. It is the interaction (or “reaction”) of these two morphogens that allows for the autonomous generation of spatial patterns during development. The model, whose novelty is derived from the idea that morphogens can interact with each other (Figure 2), provides an explanation for how pattern formation can occur independently of the initial conditions of the system (Figure 3). Turing proposed that the interaction of the morphogens is able to produce six different scenarios upon reaching a stable, constant state [2–4], as listed below (with analogies given to help visualize the processes, which may not be actual examples of what the model refers to during development):

• Case I: Uniform, stationary state. All cells are the same, behaving as if each were in isolation. Eg, mixing two food colorings to a cup of water changes the water to a single hue of color.
• Case II: Uniform, oscillating state. Similar to case I, except that the cells depart from equilibrium in an oscillatory manner. Eg, cycles of flowers closing and opening in plants.
• Case III: Salt-and-pepper formation. Cells that have changed or “differentiated” inhibit neighboring cells from differentiating. Eg, garden plot of sunflowers where spacing between adjacent plants is determined by the ability of the plants to inhibit growth of one another due to factors such as competition for nutrients and water.
• Case IV: Oscillation of salt-and-pepper state.
• Case V: Traveling wave formation. This scenario requires at least three morphogens, as two morphogens may only result in a stationary pattern (Figure 3B-C). Eg, movement of a slinky at one end propagates itself until it reaches the opposite end.
• Case VI: Stationary wave formation (Turing patterns). A wave pattern is maintained by a system that has reached dynamic equilibrium (where the morphogens continue to move and react but there is no net change in the overall state). The pattern’s wavelength (the distance between the “peaks” of the wave) is determined by how the molecules react and rates at which they diffuse (Figure 4).

Figure 3. Morphogen-gradient models and reaction-diffusion model. (A) A single morphogen provides positional information to cells based on the concentration of the molecule it “sees”, depending on how far the cells are from the morphogen source (the dotted lines represent different concentration thresholds that provide different instructions to the cells). (B) Two morphogens allow for the formation of more complex patterns that depend on the concentrations of both, however, the system is entirely dependent on the initial positional information and cannot regulate itself. (C) Interaction between two morphogens allow for the system to be self-regulated, and consequently, capable of forming a plethora of patterns that are independent of the initial conditions of the system. (Image based on [3]).

Figure 4. Generation of spontaneous Turing patterns (Case VI) based on reaction-diffusion mechanism. (A) The conditions of the model yield a peak of the slowly diffusing activator (P) and a lower peak of the rapidly diffusing inhibitor (S) at the same location. (B) The distribution of the morphogens is initially random. As the concentration of activator (P) increases locally, it stimulates the production of more of itself (called “autocatalysis”), as well as of the inhibitor (S) which diffuses away quickly to inhibit more peaks of P from forming, resulting in a series of P peaks (red arrows) at regular intervals whose positions don’t change with time (a “standing wave”). (Image based on [5]).

Fish stripe formation resembles the reaction-diffusion mechanism

Though the Turing model was proposed in the 1950s, it is only now starting to become accepted among experimental biologists, due to a gradual increase in evidence demonstrating that computer simulations based on the Turing model can accurately predict the dynamic properties of pattern formation during animal development.

In one such example, it was observed that the changes in the pigmentation pattern on the skin of the marine angelfish (Pomacanthus imperator) can be predicted by the Turing model (Figure 5) [6]. To understand the reaction-diffusion mechanism, scientists studied the zebrafish, a well-established model organism. Although development of stripes occurs in a predictable and conserved manner, artificially disrupting development leads to changes in stripe pattern that are consistent with what scientists would expect based on the reaction-diffusion model.

Figure 5. Changes in the pigmentation pattern of the angelfish and computer simulation predicting these changes. (A) Adult P. imperator (Image credit: Flickr user Al@in76). (B-D) Schematic of computer-based simulation of the pigmentation pattern at (B) 10 months, (C) 12 months and (D) 13 months, based on the reaction-diffusion model. (Images based on [6]).

In zebrafish, skin pigmentation comes from three cell types – those that contribute to the black stripes, yellow stripes, and light-reflecting stripes [7-8]. After using a laser light to kill the pigment cells of two black stripes on the upper back side of young zebrafish (40 to 80 days old), it was observed that the black stripe from the lower abdominal area moved upward to fill the vacant space. This dynamic movement of the stripes is characteristic of the stationary wave in the Turing model (case VI) and may be modeled with a computer simulation [9]. Furthermore, it has been shown that pigment formation is established and maintained in zebrafish by a process similar to that predicted by the reaction-diffusion model. Specifically, pigmentation patterns result from interactions between black pigment cells and yellow pigment cells [10]. Though an exciting discovery, much remains to be understood about zebrafish pigmentation.

Other areas of research where the Turing mechanism is being investigated include hair patterning in mammals, feather patterning in birds, and left-right asymmetry in vertebrates. Despite the ability of the Turing mechanism to model various developmental processes, there are limitations to what the model can explain. One such limitation is that the Turing model does not explain how patterns can be scaled according to the size of an organism. For example, the morphogens involved in patterning the head-tail axis of insects have been well-characterized, and are conserved across different species of insects. However, it is not understood how the morphogens are regulated to scale accordingly so that the size of the different segments along the head-tail axis is proportional to the overall size of the insects. Consequently, while the Turing model provides a new approach to deciphering biological mysteries, it is not a panacea for understanding all developmental processes.

Jessica W. Chen is a PhD student in Biological and Biomedical Sciences at Harvard Medical School.

References:

[1] Tung, K.K. Topics in Mathematical Modeling. Princeton: Princeton University Press. 2007.

[2] Turing, Alan M. The Chemical Basis of Morphogenesis. 1952. Philosophical Transactions of the Royal Society; 237: 37-72. <www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf>

[3] Kondo, Shigeru and Miura, Takashi. Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation. 2010. Science; 329: 1616-1620.

[4] Roth, Siegfried. Mathematics and biology: a Kantian view on the history of pattern formation theory. 2011. Dev Genes Evol; 221: 255-279.

[5] Gilbert, Scott F. “Mathematical Modeling of Development.” Developmental Biology. 6th ed. Sunderland, MA: Sinauer Associates, 2000. <http://www.ncbi.nlm.nih.gov/bookshelf/br.fcgi?book=dbio&part=A79>.

[6] Kondo, Shigeru and Asai, Rihito. A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. 1995. Nature; 376: 765-768.

[7] Hirata, M., Nakamura, K., Kanemaru, T., Shibata, Y., and Kondo, S. Pigment cell organization in the hypodermis of zebrafish. 2003. Dev Dyn; 227: 497-503.

[8] Kelsh, R.N., et. al. Zebrafish pigmentation mutations and the processes of neural crest development. 1996. Development; 123: 369-389.

[9] Yamaguchi et. al. Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism. 2007. PNAS; 104: 4790-4793.

[10] Nakamasu, A., Takahashi, G., Kanbe, A., and Kondo, S. Interactions between zebrafish pigment cells responsible for the generation of Turing patterns. 2009. PNAS; 106(21): 8429-8434.

One thought on “Understanding pattern formation during morphogenesis”

1. James Harvey-Brown says:

Hi Jessica,

Thanks for an interesting article. However, I feel that your assertion that “leaves across all plant species have a highly conserved arrangement: the angle formed between adjacent leaves, as observed when looking down at the stem, is approximately 137.5 degrees” may be an overstatement. It seems that this rule may only apply to leaves that form in a spiral around the stem. The alternate leaves of a walnut tree, for example, are a constant 180 degrees in relation to each other.