By Matthew R. Francis
Cyanobacteria are a diverse group of single-celled organisms whose lineage dates back to the earliest days of life on Earth. Sometimes known as blue-green algae, these bacteria predate true algae (which are classified as plants) by billions of years and are the earliest known organisms to harvest sunlight for energy via photosynthesis. While some cyanobacteria produce toxins that are harmful to animal life, they are also responsible for a significant amount of nitrogen fixation within food webs.
Modern cyanobacteria subsist in a wide range of environments, from soil, streams, and freshwater lakes to extreme locales like hot springs, Antarctic lakes, and rocky bacterial reefs called stromatolites. Although cyanobacteria are prokaryotes—single-celled organisms without nuclei or other distinct internal structures—that form seemingly undifferentiated green slimes called microbial mats on a macroscopic scale, some species can self-organize into microscopic networks (see Figure 1). And they have always done so.
Figure 1. A fluorescent microscopic image of cyanobacteria that form a distinct web-like pattern, where individual bacterial fibers align with each other rather than randomly distribute. Figure courtesy of [1].
“You get the same patterns, same structure, and same kind of activity in billion-year-old fossils and nowadays,” physicist Lucas Goehring of Nottingham Trent University said. “Basically, there’s something very important [about] how these structures provide some sort of fitness for the cyanobacteria to the landscape.”
In other words, understanding how and why self-organization arose in this ancient lineage can provide clues about the history of evolution as a whole. Goehring and Mixon Faluweki of the Malawi University of Science and Technology recently collaborated with Jan Cammann and Marco Mazza of Loughborough University to construct a big-picture view of these organisms’ behavior. The model organisms in the resulting study—published in Physical Review Letters [1]—were Oscillatoria lutea and Kamptonema animale: cyanobacteria that form long filaments of many individual cells. These strands organize themselves into the distinctive web-like networks that appear in both fossils and modern environments.
“We didn’t really have an idea [as to] how these patterns emerge, so we set out to model their motion,” Cammann said. “After some attempts, we finally arrived at [a model] that we were happy with, which reproduces these patterns exactly.”
Chain-Chain-Chain, Chain of Cells
Like all bacteria, filamentous cyanobacteria reproduce by cell division. However, the daughter cells remain stuck together and divide exclusively along a single axis — yielding long, non-branching chains that are only one bacterium wide. So from a self-organization standpoint, the primary unit is not a cell but a filament, which can be nearly a thousand times longer \((L \approx 1.5 \,\textrm{mm})\) than it is wide \((\sigma \approx 4 \,\mu\textrm{m})\). The researchers thus refer to cyanobacterial tangles as “active spaghetti.” “This [chaining] represents an early transition from single-celled to multicellular life,” Goehring said, though the cells do not perform specialized functions in the way that many other organisms would later evolve to do.
Mathematically, the process of filamentation means that we cannot treat these organisms as point particles. Though they do not have heads or tails, the strands move preferentially in one direction in an almost worm-like way. A single filament can interact with several others at different points along its length, and these other filaments may link up with separate chains. “The filamentous nature turns out to be the key in the pattern formation that we observe,” Mazza said. “The [microbial mat] is not just an amorphous green slime. [It] can react to external perturbation, be it a rapid change in temperature, light intensity, and so on.”
The collaborators focused their attention on one particular property: density. At low bacterial concentrations, the filaments lack any measurable order and can be treated as a fluid; at higher densities, however, they spontaneously organize into distinctive web patterns. Therefore, the challenge was to build a mathematical model that describes this behavior and test it against experimental data.
From Liquid Crystals to Bacterial Webs
Cammann led the theoretical portion of this study as part of his doctoral research, drawing inspiration from well-established models of nematic liquid crystals. These filamentous yet nonbiological materials also self-organize under certain conditions and undergo clear phase transitions, which enable myriad technological applications. “The filamentous nature is the minimal extension of point particle models that is required to quantitatively reproduce what we see in nature,” Cammann said.
He treated the bacteria as links in a flexible chain that is confined to a two-dimensional surface, such as a stromatolite or rock face. The model describes the orientation angle \(\theta_i\) of each filament head relative to the lab coordinate system, as well as its angular speed \(\omega_i\)—i.e., how quickly the filaments twist—via coupled differential equations that are modified from condensed matter physics:
\[\frac{d\theta_i}{dt}=\omega_i-J\mathcal{F}(\theta_i) \\ \frac{d\omega_i}{dt}=-\frac{1}{\tau}[\omega_i-J\mathcal{F}(\theta_i)]+\sqrt{2D_{\omega}}\xi_i(t). \]
Here, \(J\) is the interaction strength, \(\tau\) is the time scale for curvature fluctuations (measured experimentally), \(D_\omega\) is a diffusion coefficient, and \(\xi_i(t)\) is a Gaussian random noise term with zero mean and unit variance. The force term
\[\mathcal{F}(\theta_i)=-(N_{i})^{-1}\sum_{j\sim i}\frac{\partial}{\partial\theta_i}\cos[2(\theta_i-\phi_j)]\]
connects the head of chain angle \(\theta_i\) to the orientation angle \(\phi_j\) of the nearest link in each chain, which goes to zero when the filaments are aligned. The summation is only over the \(N_i\) bacteria that are in physical contact with each other, so the force term is set to zero outside of a distance \(d\) that is equal to the mean bacterial diameter. The simulations also account for experimentally determined filament velocities and curvatures, thereby allowing the researchers to obtain the diffusion and interaction parameters.
“Everything in the model is designed around what we see observationally,” Goehring said. The key quantity of interest for both the model and laboratory work was the critical density of bacteria, which was on the order of 50 strands per square millimeter (see Figure 2). “You go from gas-like behavior at low density to a collective response phase at higher density, which we can predict from back-of-the-envelope calculations, more detailed modeling, and experiments.” Goehring continued. “They all agree with each other quantitatively.”
Figure 2. Snapshots of cyanobacteria with increasing density. 2a–2d. Microscopic images of the bacteria. 2e–2h. Images generated by a mathematical model. The transition to self-organization occurs at the same density in both instances. Figure courtesy of [1].
The Four Fs of Mathematical Biology
This quantitative match is essential because humans frequently spot patterns that do not actually exist. Although bacterial webs may look strikingly like neuron structures in the brain or galaxy clusters on cosmic scales, the physical interactions are radically different. “Similarity of form can be incredibly deceptive,” Goehring said. “You have to compare numbers; otherwise you can easily fool yourself.”
“Biologists talk about three Fs: form, function, and force,” Mazza said. “This trinity of Fs plays a role here because the form is the filaments and the fluctuations in their shapes. If you were to condense our paper to one statement, it would be that ‘the fluctuations in shape lead to this pattern, which gives rise to important biological function.’”
Mazza then acknowledged the importance of another F: the feedback loop between theoretical modeling and experimental evidence. Cammann agreed. “The parameters that we put in our model are all based on [laboratory] measurements of individual filaments or pairwise interactions of filaments,” he said. “With all of these measured parameters on individual behavior, we get an emergent pattern that matches the emergent pattern that also appears in experiments with many filaments.”
Fossil stromatolites are some of the oldest traces of life on Earth, and the filamentary webs of long-dead cyanobacteria indicate that this type of self-organization significantly predates multicellular plant or animal life. In other words, organisms were organizing since the very beginning. Understanding this organization validates the importance of collective action as an evolutionary strategy since the dawn of life. And despite the mathematical model’s complexity, the present-day ubiquity of cyanobacteria also helps to explain the relevance of this work.
“We have a very pretty fountain at Loughborough,” Cammann said. “Whenever someone asked me, ‘What do you do for your Ph.D.?’ I pointed to the fountain and said, ‘The green stuff in there, this is what I do.’”
References
[1] Faluweki, M.K., Cammann, J., Mazza, M.G., & Goehring, L. (2023). Active spaghetti: Collective organization in cyanobacteria. Phys. Rev. Lett., 131(15), 158303.
Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is BowlerHatScience.org.