Zebra stripes are the result of genetic and environmental factors. Scientists are still debating the exact reasons for the evolution of zebra stripes, but some hypotheses include camouflage, temperature regulation, and the deterrence of biting flies. Nature’s patterns, such as stripes and spots, are thought to be the result of chemical interactions. According to this new study, the mathematical basis of these patterns also governs how the sperm tail moves.

The findings, published in Nature Communications, show that flagella movement in sperm tails and cilia, for example, follows the same pattern formation template discovered by the famous mathematician Alan Turing. Flagellar undulations generate waves that travel along the tail to propel sperm and microbes forward.

Alan Turing is best known for his role in breaking the Enigma code during WWII. He did, however, develop a pattern formation theory that predicted that chemical patterns could appear spontaneously with just two ingredients: chemicals spreading out (diffusing) and reacting together. Turing was the first to propose the reaction-diffusion theory for pattern formation.

Live spontaneous motion of flagella and cilia is observed everywhere in nature, but little is known about how they are orchestrated. They are critical in the health and disease, reproduction, evolution, and survivorship of almost every aquatic microorganism on Earth.

Dr. Hermes Gadêlha

Turing paved the way for a completely new type of investigation that used reaction-diffusion mathematics to understand natural patterns. Turing patterns are the chemical patterns that Turing first imagined. Although not yet proven by experimental evidence, these patterns are thought to govern many patterns in nature, including leopard spots, the whorl of seeds in a sunflower’s head, and sand patterns on the beach. Turing’s theory has applications ranging from biology and robotics to astrophysics.

Mathematician Dr Hermes Gadêlha, head of the Polymaths Lab, and his Ph.D. student James Cass conducted this research in the School of Engineering Mathematics and Technology at the University of Bristol. Gadêlha explained: “Live spontaneous motion of flagella and cilia is observed everywhere in nature, but little is known about how they are orchestrated. They are critical in health and disease, reproduction, evolution, and survivorship of almost every aquatic microorganism on earth.”

The team was inspired by recent observations in low-viscosity fluids that the surrounding environment plays a minor role in the flagellum. They used mathematical modeling, simulations, and data fitting to show that flagellar undulations can arise spontaneously without the influence of their fluid environment.

Mathematically this is equivalent to Turing’s reaction-diffusion system that was first proposed for chemical patterns. In the case of sperm swimming, chemical reactions of molecular motors power the flagellum, and bending movement diffuses along the tail in waves. The level of generality between visual patterns and patterns of movement is striking and unexpected and shows that only two simple ingredients are needed to achieve highly complex motion.

Dr. Gadêlha added: “We show that this mathematical ‘recipe’ is followed by two very distant species — bull sperm and Chlamydomonas (a green algae that is used as a model organism across science), suggesting that nature replicates similar solutions. Traveling waves emerge spontaneously even when the flagellum is uninfluenced by the surrounding fluid. This means that the flagellum has a fool-proof mechanism to enable swimming in low viscosity environments, which would otherwise be impossible for aquatic species.”

“This is the first time that model simulations have performed well in comparison to experimental data.” We are grateful to the researchers who made their data freely available to us, as we would not have been able to conduct this mathematical study otherwise.”

These findings could be used to better understand fertility issues associated with abnormal flagellar motion and other ciliopathies (diseases caused by ineffective cilia in human bodies) in the future. As the team discovered a simple ‘mathematical recipe’ for creating movement patterns, this could be explored further for robotic applications, artificial muscles, and animated materials.

Dr. Gadêlha also works in the SoftLab at Bristol Robotics Laboratory (BRL), where he uses pattern formation mathematics to develop the next generation of soft robots.

“In 1952, Turing unlocked the reaction-diffusion basis of chemical patterns,” according to Dr. Gadêlha. “We show that the flagellum, the ‘atom’ of motion in the cellular world, uses Turing’s template to shape patterns of movement driving tail motion that propels sperm forward.”

“While this is an important step toward mathematically decoding spontaneous animation in nature, our reaction-diffusion model is far too simple to fully capture all complexity.” Other models may exist in the space of models with equal, or even better, fits with experiments that we are simply unaware of, and thus much more research is required!”