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Inspired by biological systems, materials scientists have long sought to harness self-assembly to build nanomaterials. The challenge: the process seemed random and notoriously difficult to predict. Now, researchers from the Institute of Science and Technology Austria (ISTA) and Brandeis University have uncovered geometric rules that act as a master control panel for self-assembling particles. The results, which could find applications ranging from protein design to synthetic nanomachines, were published in Nature Physics.
Life is the ultimate nanotechnologist. Biology has long fascinated physicists with its ability to build complex molecular machines and structures from molecules that snap into place like magnets. But what governs this phenomenon that frequently occurs in nature?
To find out, PhD student Maximilian Hübl and Assistant Professor Carl Goodrich from the Institute of Science and Technology Austria (ISTA) teamed up with scientists Daichi Hayakawa and Thomas Videbaek from W. Benjamin Rogers’ group at Brandeis University in the United States. Together, they set out to crack the code of molecular self-assembly and apply it to nanotechnology. “For decades, scientists have dreamed of harnessing the power of molecular self-assembly to build custom-made nanomaterials,” says Goodrich. “But a major challenge has been predicting exactly what shapes will form when thousands of tiny pieces are set in motion.” In a dual approach that combines theoretical and experimental methods, the team developed and validated a tool that distinguishes ‘designable’ or viable structures from those that cannot be assembled. As it turns out, the outcomes of self-assembly are governed by geometry.
Light in the darkness
Self-assembling particles are not exactly new to Goodrich’s research interests. However, he was only convinced to tackle the topic head-on after developing a concrete approach to address it. His initial strategy encompassed numerical calculations, including automatic differentiation and differentiable programming. When Hübl joined the Goodrich group at ISTA, he started examining the project using this same approach. However, he quickly identified a more general and effective method. “With our initial strategy, we looked at the problem as if we were inside an unknown room, in pitch darkness, searching around with a flashlight. Eventually, we realized the room had a light switch. Turning the lights on allowed us to view all the possibilities that self-assembly can achieve, as well as the areas it cannot access.” Thus, it turns out that self-assembly is far from a random process in a vast ocean of mathematical possibilities. By finding the right approach, the team was able to clarify the limits between feasible and infeasible self-assembly configurations.
Rulebook of a hidden geometric shape
With Hübl’s method, the team focused on the effects of tuning the concentrations and binding energies of assembling particles. Eventually, this approach would help them determine which sets of structures are ‘designable.’
“We used the binding energies as input for the calculation. As an output, we determined what structures will be formed by the particles, and in what quantities,” Hübl explains. “This allowed us to identify constraints that prevent certain outcomes from ever occurring in particles.” An example of such constraints is that obtaining a specific structure with 100 percent yield may be plainly impossible. In such a case, an additional structure might be what the scientists call a ‘necessary chimera,’ meaning an inevitable byproduct that is thermodynamically unavoidable under these conditions. Goodrich underlines, “Our method could explain why some attempts at designing specific nanomaterials are especially challenging.”
But what theoretical framework did the scientists identify, precisely? Together, the computed thermodynamic constraints form a hidden mathematical shape that captures the range of possible assembly outcomes: a ‘high-dimensional convex polyhedron.’ This geometric shape would serve as the ‘theoretical rulebook’ of self-assembly in equilibrium. “The polyhedral structure demonstrates that equilibrium assemblies follow rules that could serve as tools for nanotechnologists and molecular designers,” Goodrich explains. “This underlying physics tells us whether a given target structure is possible at all.”
DNA origami
To test the practical utility of this geometric shape that governs self-assembly, the ISTA scientists teamed up with researchers from the Rogers group at Brandeis, who use techniques from biological physics and soft matter physics to understand self-assembly. They designed and synthesized triangular DNA origami building blocks and devised experiments to validate the theory. By extending single-stranded DNA from the triangles’ sides and adjusting their sequences to program specific interactions, they implemented a set of theoretical binding rules experimentally. “We found striking quantitative agreement between the theory and experiment, confirming that we had indeed uncovered some of the fundamental rules of assembly,” says Rogers.
According to the authors, the experimental results are clear evidence of the theory’s real-world applicability. “Essentially, we used our geometric ‘rulebook’ to predict the experimental outcomes without modeling the details of the interactions. The experiments closely matched the predicted results, without us having to review any part of the theory or adjust any factors,” says Hübl. Consequently, besides identifying ‘designable’ structures, the ‘theoretical rulebook’ has proven its practical worth.
Nature’s playground & its blueprints
By uncovering the underlying geometry that draws the line between what is possible to build and what is impossible to create, the team demonstrated the limits of self-assembly. “Self-assembly is that grand, crazy thing that nature does. But Max’s theory now explains why some attempts to replicate this don't work, and how they could be done better. It’s like having a blueprint that traces the boundaries of nature’s playground. Ultimately, this model could serve as an architect’s tool, a master control panel for designing nanostructures,” says Goodrich.
According to the team, applications will likely include inverse design in a wide range of experimental settings, such as de novo protein assembly from smaller building blocks, DNA nanoparticles, and synthetic nanomachines.
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Funding information
This project was supported by funding from the Gesellschaft für Forschungsförderung Niederösterreich under project FTI23-G-011, the Brandeis University Materials Research Science and Engineering Center (MRSEC) under grant number NSFDMR-2011846, and the Smith Family Foundation.
Maximilian C. Hübl, Thomas E. Videbæk, Daichi Hayakawa, W. Benjamin Rogers, and Carl P. Goodrich. 2026. A polyhedral structure controls programmable self-assembly. Nature Physics. DOI: 10.1038/s41567-025-03120-3
https://www.nature.com/articles/s41567-025-03120-3
https://ista.ac.at/en/research/goodrich-group/ Research group "Theoretical and Computational Soft Matter" at ISTA
https://www.brandeis.edu/physics/people/profiles/rogers-benjamin.html W. Benjamin Rogers at Brandeis University
Biology has long fascinated physicists with its ability to build complex molecular machines from sel ...
Copyright: © ISTA
ISTA physicists created a ‘theoretical rulebook’ of self-assembly. The study’s first author, PhD stu ...
Copyright: © ISTA
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