The smallest ever fractal in the natural world discovered

From the jagged coastline to the delicate fern leaf or the romesco broccoli, fractals are mesmerising shapes that defy conventional geometry and are present at the macroscale in our natural world. They possess infinite detail, generated through iterative processes or mathematical equations.

This is a story about fractals, but it doesn’t start with fractals. Georg Hochberg, evolutionary biologist at Max-Planck Institut, had never worked in fractals but instead was studying the arrangement of protein subunits that form a functional enzyme complex (called quaternary assembly) to understand in what way -if any- such arrangements contribute to function.

Romesco broccoli is a typical example of fractals in nature at the macroscale.

“We picked a microbial enzyme called citrate synthase from the cyanobacterium, which is a very simple system because it always does the same thing”, explains Hochberg, corresponding author of the paper. The enzyme, present in most organisms catalyses the first step of the Krebs cycle, a metabolic pathway that most organisms use to break down nutrients such as sugars.

Normally citrate synthases form relatively small and symmetrical assemblies. However, when Hochberg and his colleagues studied this protein using the technique of mass photometry, which weighs molecules with light, they realised that the molecule was unusually big.

They subsequently used Cryo-EM and discovered that it self-assembled into a pattern that resembled a famous fractal form – the Sierpinski triangle.  This triangle is a geometric pattern that starts with an equilateral triangle and recursively removes smaller triangles from its centre.

From the Cryo-EM data, they could observe that the molecule could go up to the second level of the Sierpinski triangle, but they couldn’t see any further than that due to the limitations of the technique. In addition, when trying to put together the high resolution structure using Artificial Intelligence (AI), they found themselves stuck because AI classifies and finds repeating patterns in the data. And so AI gets really confused by fractals because it finds smaller structures and bigger structures, so the team had to manually assemble the data.

SAXS to see beyond Cryo-EM

In order to overcome the limitations of Cryo-EM, the team turned to Small Angle X-ray Scattering at the ESRF’s beamline BM29. “This technique doesn’t solve the structure, but it tells you about the sizes, so we could show that these assemblies could replicate further in a pattern of continuous growth, at least part of which we strongly suspect they are Sierpinski triangles”, says Hochberg. “The ESRF was very important because it helped us show that these assemblies can keep growing infinitely”, he adds. For Mark Tully, beamline scientist at BM29, the experiment on fractals was a first: ”I was intrigued when Georg discussed his experiment, having never measured a fractal molecule before. Then when I saw the SAXS patterns of the fractal shapes and how they assembled, I was amazed.”

Harmless evolutionary accident

Do these fractals accomplish any function? “The very large assemblies we discovered with ESRF would have trouble fitting into a bacterium, so we think they are probably non-physiological, but we initially thought the smaller fractals might have a use”, says Hochberg. This was because the scientists found that when the enzyme is assembled into these fractal forms, it can’t really catalyze the reaction it needs to catalyze, suggesting fractal assembly might regulate activity. But the scientists could not find any evidence that this regulation was in anyway important to the bacteria. Instead, the fractals may simply be a harmless quirk.   In cyanobacteria citrate synthase is only needed during the day when they are photosynthesizing, but the fractals assemble only at night. During the day, the solution conditions in the cytoplasm of this organism are such that these assemblies fall apart, and the enzyme can dissociate into its active form. “It is plausible that the fractals could be a kind of harmless evolutionary accident”, adds Hochberg.

Hochberg is enthusiastic about the follow-up of this research: “Protein engineers have tried making protein Sierpinski fractals and declared them unofficially as unengineerable, but now that we know that this protein broke some rules that we thought were universal for how proteins assemble, there’s a question as to what else you might be able to build, what kind of 2D materials you can make if you're allowed to break these rules”. This could potentially lead to synthesis of systems that might have interesting material properties.

Reference:

Sendker, F.L., Lo, Y.K., Heimerl, T. et al. Emergence of fractal geometries in the evolution of a metabolic enzyme. Nature (2024). https://doi.org/10.1038/s41586-024-07287-2

Text by Montserrat Capellas Espuny.

Top image: Citrate synthase spontaneously assembles into a fractal pattern known as the Sierpinski triangle. Credits: MPI f. Terrestrial Microbiology/ Hochberg