Algorithms of Beauty: The Mathematical Genetics of Flower Pattern

Nature is full of striking patterns. Zebra stripes, leopard spots, and spiral seashells are just a few of the endless examples.

Have you ever wondered how these patterns form? It's a question that's fascinated researchers for a long time. The search for the answer spans several fields, including biology, chemistry, and math.

Turing Patterns

reaction-diffusion model

Examples of reaction-diffusion equations (top). Variables include the amount of morphogen present, its diffusion rate, and how strongly the morphogens affect one another. Many patterns can be modeled with the equations (bottom). Patterns based on Kondo 2010.

You can watch computer simulations of Turing patternshere.

In 1952, the mathmatician Alan Turing proposed a model for how biological patterns form. Using mathematical equations, he described how substances spread through a space and react with one another to create patterns.

He called the substances “morphogens”, an intentionally vague term. In the context of pattern formation, it can refer to chemicals, genes, proteins, or even groups of cells. The important thing is that one morphogen — an activator — makes a change that leads to a trait. A second morphogen — an inhibitor — stops the change. The activator increases its own activity, and it turns on the inhibitor too. The activator and inhibitor diffuse, or spread, at different speeds.

Turing’s model could explain the spots on a lady bug, for example. Let’s say “A” is an activating morphogen. It causes black pigment. “I” is an inhibitor that prevents the trait.

Where “A” is produced, it makes more of itself. It diffuses slowly, so a lot of “A” builds up in a small area.

“A” also turns on “I”. But “I” diffuses quickly. There’s a small amount overlapping with “A,” but most moves further away.

As a result, “A” is high and “I” is low in a small area. Black pigment is made. Further away, “A” is low, but there’s plenty of “I”. No black pigment is made.

Turing’s work is now known as the reaction-diffusion model. An important point is that it only requires two morphogens to establish a periodic pattern. In computer simulations, changing a small number of variables causes a dazzling variety of patterns. Many resemble those found in nature.

Formation of Petal Spots

Many real-life patterns could form from the reaction-diffusion model. Proving they actually do is more difficult.

The hunt’s been in progress for a long time, and verified examples of Turing patterns are still scarce. The few identified so far include zebrafish stripes, chick feather buds, mouse hair follicles, and flower petal spots.

Two key pieces of evidence are needed to prove a pattern forms by reaction-diffusion. They are the identity of the activator, and the identity of the inhibitor. In the monkeyflower Mimulus lewisii, researchers first solved this puzzle. Then, they tested if the activator-inhibitor pair they found fit other predictions of the model.

Monkeyflowers have nectar guides, spots of dark pink anthocyanin pigment on a yellow background, that attract pollinators. The guides form from the activator-inhibitor pair NECTAR GUIDE ANTHOCYANIN (NEGAN) and RED TONGUE (RTO). Both NEGAN and RTO are transcription factors, proteins whose job is to turn gene activity up or down.

NEGAN and RTO fit the reaction-diffusion model well. NEGAN turns on genes needed to make anthocyanin pigment. It also activates itself and RTO. Once on, RTO inhibits NEGAN by competing for a specific partner protein they both need to do their jobs. RTO diffuses from where it’s made into neighboring cells, restricting NEGAN’s activity.

monkeyflower petalsmonkeyflower petalsmonkeyflower petals

LEFT: Activator and Inhibitor, Normal Spots
MIDDLE: No Activator, No Spots
RIGHT: No Inhibitor, Expanded Spots

The pattern of anthocyanin spots on monkeyflower petals is a real-life example of the reaction-diffusion model. The dissected monkeyflower petals above are from genetic experiments to test and expand the model.

The activator-inhibitor pair, NEGAN and RTO, fit predictions made by the reaction-diffusion model.

Patterns Influence Pollinator Choice

patterns on monkeyflowers
patterns on monkeyflowers

Examples of pattern in different species of monkeyflower. M. rupicola (top left), M. varigatus (top right), M. luteus (bottom left), and M. pictus (bottom right).

Flower patterns don’t just look nice, they’re also really important. Patterns help attract pollinators to a flower. The more pollinators visit a flower, the more likely it is to reproduce.

In monkeyflowers, petal patterns affect pollinator choice. M. luteus is a species of monkeyflower that grows in the Andes Mountains. Even within the species, patterns differ. Some flowers have broad patterns and are favored by hummingbirds. Others have narrow patterns, and are favored by bees.

Because patterns influence pollinator choice, it’s possible reaction-diffusion played a role in flower evolution. Small changes in reaction-diffusion networks have a big impact on pattern. This could be part of the reason we see so many diverse flower patterns today.



Ding, B., Patterson, E. L., Holalu, S., Li, J., Johnson, G. A., Stanley, L. E., ... & Yuan, Y. W. (2018). Formation of periodic pigment spots by the reaction-diffusion mechanism. bioRxiv, 403600.

Jung, H. S., Francis-West, P. H., Widelitz, R. B., Jiang, T. X., Ting-Berreth, S., Tickle, C., ... & Chuong, C. M. (1998). Local inhibitory action of BMPs and their relationships with activators in feather formation: implications for periodic patterning. Developmental Biology, 196(1), 11-23.

Kondo, S., & Miura, T. (2010). Reaction-diffusion model as a framework for understanding biological pattern formation. Science, 329(5999), 1616-1620.

Medel, R., Botto-Mahan, C., & Kalin-Arroyo, M. (2003). Pollinator‐mediated selection on the nectar guide phenotype in the Andean monkey flower, Mimulus luteus. Ecology, 84(7), 1721-1732.

Sick, S., Reinker, S., Timmer, J., & Schlake, T. (2006). WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism. Science, 314(5804), 1447-1450.

Torii, K. U. (2012). Two-dimensional spatial patterning in developmental systems. Trends in Cell Biology, 22(8), 438-446.

Turing, A. M. (1990). The chemical basis of morphogenesis. Bulletin of Mathematical Biology, 52(1-2), 153-197.

Yamaguchi, M., Yoshimoto, E., & Kondo, S. (2007). Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism. Proceedings of the National Academy of Sciences, 104(12), 4790-4793.

Yuan, Y. W., Sagawa, J. M., Frost, L., Vela, J. P., & Bradshaw Jr, H. D. (2014). Transcriptional control of floral anthocyanin pigmentation in monkeyflowers (Mimulus). New Phytologist, 204(4), 1013-1027.