Biological Near-symmetries Explain the Similarity and Diversity of Life

Despite the wide variety of shapes that organisms exhibit, their forms are not random. Instead, much of life possesses relatively clear structural patterns; humans and other mammals are bilaterally symmetric, sea stars have five-fold symmetry, and flowers display a wide variety of symmetrical patterns.

However, none of these biological symmetries are exact in a mathematical sense. The near-symmetries in living things are nevertheless real — to the extent that evolutionary, developmental, and environmental pressures must play a part. Yet when symmetries are not mathematically precise, mathematicians must find new ways to express them and determine their meanings.

Mathematical biologists Punit Gandhi (Virginia Commonwealth University), Veronica Ciocanel (Duke University), Adriana Dawes (Ohio State University), and Karl Niklas (Cornell University) teamed up to develop novel processes for identifying biological symmetries to glean a deeper understanding of their origins. They presented their research during a session that was moderated by Niklas at the 2022 American Association for the Advancement of Science (AAAS) Annual Meeting, which took place virtually in February. “If [an organism] has bilateral symmetry but you try to flip its image, it will simply not align in the way that mathematics would quantify it,” Ciocanel said. “We were inspired by the idea that biology will never have that kind of perfect symmetry.”

To tackle this problem, the researchers examined multiple species of flowering plants, algae, and animals with embryos that are easy to study. They used algorithms that minimize human choice—since people miss subtle patterns and are prone to bias when identifying symmetries—to analyze both computer-generated images and real-life photos. To quantify near-symmetries in organisms, the team employed a measure called transformation information (TI). “One way to interpret TI is by how much information is lost when assuming that the object is exactly symmetric under the transformation,” Gandhi said.

Although researchers have developed other methods for the quantification of inexact symmetries, these approaches generally require that one pick a particular type of transformation (e.g., rotation by a fixed angle) or select invariant points or axes before proceeding. By contrast, TI allows scientists to compare multiple potential symmetries and algorithmically determine invariants. “I think that TI is a really nice hands-off quantification of symmetries in an unbiased way,” Dawes said. “It requires less time, there’s less opportunity for bias (intentional or not) to be introduced into the measure, and it also gives you a way of then saying, ‘How different is this flower from that flower?’”

In more general terms, TI helps extract symmetry information from biological systems where—for instance—a single plant might have numerous non-identical flowers, or several specimens from one species might exhibit variations in the closeness with which they hew to a particular symmetry. For these cases, TI can in principle identify the underlying symmetry separately from random variation. “Most daisies are going to have rotational symmetry,” Dawes said. “You’ve got your round center and you’ve got your white petals around the outside. We identify a daisy by that rotational symmetry and the color properties” (see Figure 1).

While daisies may deviate from this Platonic ideal, Dawes argued that investigating representative specimens can directly inspire the identification of general symmetries. “If you want to know the common symmetry features among all daisies, it would make sense to do TI on a bunch of different samples and look at the common features in the TI function,” she said.

Perfect and Imperfect Symmetries

G.V. Vstovsky first introduced TI in 1997 to apply basic concepts of information theory to broken symmetries in physics-based applications [4]. Approximate symmetries play important roles across scientific fields, from frustrated lattices (solids that do not form perfect crystal structures) to computer vision problems.

To quantify asymmetry, one must first define a function \(\mu: D \rightarrow \mathbb{R}^+\) that maps points on the object \(D\) to positive real values. For a two-dimensional (2D) object,

\[\textrm{TI}=\frac{1}{|{\tilde D}|}\int_{\tilde D} \mu(x)\ln\left[\frac{\mu(x)}{T_a\mu(x)}\right]dA.\]

\(T_a\) is a given symmetry transformation and \(\tilde D\) is the region of overlap between the original domain \(D\) and transformed domain, while

\[|\tilde D|=\int_{\tilde D} dA.\]

If a symmetry transformation is mathematically exact, its TI is zero because the transformation will map the domain onto a perfect copy. For real biological systems, near-symmetries correspond to minimum (albeit typically nonzero) TI values. The group tested their technique on computer-generated patterns from a reaction-diffusion model that was developed by Alan Turing [3]. Such models are useful for this kind of test because they contain exact symmetries and do not require complicated pre-processing procedures; the center of the image is also the center of the pattern, and the algorithm does not need to detect backgrounds or edges.

The AAAS presenters then examined bay leaves and desmids (a type of green algae) for bilateral symmetry, along with several flowers for rotational symmetry (see Figure 2). These common organisms have basic symmetries that one can identify by sight; Pachypodium blossoms—which have a distinctive pinwheel curl to their petals that breaks reflection symmetry but exhibits an approximate five-fold rotational symmetry—are another example [2]. The TI algorithm identified the center of various flowers as well as appropriate near-symmetries.¹

However, flat images of three-dimensional (3D) objects create their usual set of problems. “Projection issues are definitely something we worried about,” Gandhi said. For instance, he and his collaborators compared two photos of the same flower that were taken from both a face-on and flattened view. “We could actually see differences in the symmetries just by the process of how we image it.”

Dawes elaborated on this concept. “[This is] where a little bit of user discrimination comes into play,” she said. “Karl [Niklas] and I went through pictures and decided which ones were appropriate for the analysis because of differences in viewpoints that weren’t capturing the symmetry characteristics we were interested in.” She pointed out that one could use TI in three dimensions in principle, though doing so raises its own set of challenges. “How do you do a meaningful comparison of TI with 3D data compared to 2D data?” Dawes asked. Of course, there is the question of how to obtain usable 3D images in the first place.

Endless Forms Most Wonderful

Because the group also examined development within single cells and embryos, Ciocanel’s AAAS talk addressed the challenge of symmetry that breaks as cells divide and develop. “We were interested in taking a spherical cell embryo and understanding how the symmetry of that system would break through rules that are somewhat conserved across model organisms,” she said.

Similar to how physicists look for broken symmetries to explain different particle properties, mathematical biologists can seek out meaningful underlying principles beneath near-symmetries. Such asymmetries might arise from external influences—like light, gravity, environmental chemistry, and so forth—or from developmental or genetic factors. “One of the things that we looked into with these models of cell polarity was symmetry breaking, driven by different initial conditions: different protein localizations around the boundary of the cell,” Ciocanel said. “We found that the measure can capture those things really well but also shows robustness.”

Dawes presented a TI-driven principal component analysis for C. elegans—a type of nematode worm—as it grows within an egg. “When you watch the worm, you can see that it’s undergoing these tremendous changes in body shape,” she said. “But it’s all occurring within the constraint of the egg shell. What happens is that the TI really picks up on that ellipsoidal constraint from the egg shell.”

These biological examples pertain to the question of why many organisms have near symmetry at all, as opposed to perfect forms or complete amorphousness (in fact, exact symmetry can trigger the “uncanny valley” response because we are psychologically primed to expect imperfections). Gandhi and his colleagues suspect that there are reasons for variations in basic symmetry; otherwise, these patterns would be undesirable from an evolutionary standpoint. The data back up this supposition. “There’s evidence [of] some advantage to having slight asymmetries in terms of resilience,” he said.

In the concluding paragraph of On the Origin of Species, Charles Darwin waxed eloquent about “endless forms most beautiful and most wonderful” and indicated that diversity of form evolved from simpler ancestors in the distant past [1]. Near-symmetries may help explain why—despite the huge diversity of life—organisms have symmetries in the first place.

¹ A sample transformation information (TI) algorithm for MATLAB is available online. 

References

[1] Darwin, C. (1859). On the origin of species by means of natural selection. London, England: John Murray.
[2] Gandhi, P., Ciocanel, M.-V., Niklas, K., & Dawes, A.T. (2021). Identification of approximate symmetries in biological development. Philos. Trans. R. Soc. A, 379(2213).
[3] Turing, A.M. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. B, 237(641), 37-72.
[4] Vstovsky, G.V. (1997). Transform information: A symmetry breaking measure. Found. Phys., 27, 1413-1444.