“It’s a great paper,” said Linda Hirst, a physicist at the University of California, Merced, who was not involved in the work. “They are really describing the symmetry of the cell sheets in more detail than has been done before.”
Flow and Symmetry
Liquid crystals flow like fluids, but they still have a degree of crystalline order — a sort of inherent symmetry or directionality that’s a bit like the grain of wood. And just as a wood plank is strongest along its grain, a liquid crystal’s response to stimuli depends on its symmetry and orientation. This directionality, called anisotropy, is the optical magic behind modern liquid crystal displays, which refract light differently depending on their orientation.
Though we might be more familiar with the liquid crystals in TV screens, they are also common in cell biology, found inside cells and in cell membranes. Over the past few years, researchers have tried to show that tissues — organized groups of cells that act together — could be considered liquid crystals, too. If tissue could be accurately described as a liquid crystal, then the set of tools that physicists use to predict how crystals respond to forces could be put to work in biology, Hirst said.
However, these efforts hit a geometric roadblock. Experimentalists and theorists couldn’t agree on tissue’s symmetry — a liquid crystal’s most defining characteristic, and the key to predicting its behavior using fluid dynamics. In simulations of small groups of cells, theorists could describe tissues as liquid crystals with sixfold “hexatic” symmetry, a bit like tilings of hexagons. But in experiments, tissues instead acted like fluids made of bar-shaped particles with twofold “nematic” symmetry — a bit like what you’d see if you poured a barrel of toothpicks into a tube and watched them flow.
“There was a contradiction: Experiment says nematic; numerical experiments and models in general say hexatic,” said Livio Carenza, a computational physicist at Koç University in Istanbul. “How do these two things speak with each other?”
Preliminary simulations by Carenza — a former researcher in Giomi’s group — suggested that the disagreement could be resolved if both symmetries, sixfold and twofold, existed simultaneously in tissues. The idea was that if you zoomed in on a tissue with nematic symmetry, you’d find smaller-scale hexatic symmetry.
“But you cannot verify theory with theory,” Giomi said. “So we did the experiments.”
To do that, Giomi recruited Julia Eckert, then a doctoral student at Leiden University, to gather data from living tissue cultures.
“I pulled them to the microscope and showed them real cells, not only the cells they can see in the literature,” said Eckert, who is now a biophysicist at the University of Queensland. “I say, ‘Have you ever seen cells, you know, in real life?’ And it was like ‘No.’ No? OK, let’s go!”
A New Fluid Order
Eckert started by growing thin layers of epithelial tissue in the lab. Then she carefully marked out the boundaries of each individual cell in microscope images. Now Giomi and his team could get to work. They wanted to see whether the tissue’s symmetry differed between small scales — when they considered just a few cells and their neighbors — and zoomed-out, larger scales.
But to disentangle the nested symmetries in Eckert’s sheets of cells, the team needed a reliable way to distinguish nematic and hexatic orders in messy biological data.
The Leiden biophysicists devised a mathematical object called a shape tensor to capture information about the cells’ shapes and directions. Using it, Eckert measured the symmetries in the tissues at different scales, first treating individual cells as the crystal’s basic units and then doing the same for groups of cells.
At small scales, they found that the tissue had sixfold rotational symmetry and looked a bit like a tiling of smooshed hexagons. But when they examined groups larger than about 10 cells, twofold rotational symmetry emerged. The experimental results neatly agreed with Carenza’s simulations.
“It was pretty amazing how well the experimental data and numerical simulation matched,” Eckert said. In fact, it matched so closely that Carenza’s first response was that it must be wrong. The team jokingly worried that a peer reviewer might think they’d cheated. “It really was that beautiful,” Carenza said.
The observations answer a “long-standing question about the type of order present in tissues,” said Joshua Shaevitz, a physicist at Princeton University who reviewed the paper (and did not think they’d cheated). Science often “gets murky,” he said, when data points to seemingly conflicting truths — in this case, the nested symmetries. “Then someone points out or shows that, well, those things aren’t so distinct. They’re both right.”
Form, Force and Function
Accurately defining a liquid crystal’s symmetry isn’t just a mathematical exercise. Depending on its symmetry, a crystal’s stress tensor — a matrix that captures how a material deforms under stress — looks different. This tensor is the mathematical link to the fluid dynamics equations Giomi wanted to use to connect physical forces and biological functions.
Bringing the physics of liquid crystals to bear on tissues is a new way to understand the messy, complicated world of biology, Hirst said.
The precise implications of the handoff from hexatic to nematic order aren’t yet clear, but the team suspects that cells may exert a degree of control over that transition. There’s even evidence that the emergence of nematic order has something to do with cell adhesion, they said. Figuring out how and why tissues manifest these two interlaced symmetries is a project for the future — although Giomi is already working on using the results to understand how cancer cells flow through the body when they metastasize. And Shaevitz noted that a tissue’s multiscale liquid crystallinity could be related to embryogenesis — the process by which embryos mold themselves into organisms.
If there’s one central idea in tissue biophysics, Giomi said, it’s that structure gives rise to forces, and forces give rise to functions. In other words, controlling multiscale symmetry could be part of how tissues add up to more than the sum of their cells.
There’s “a triangle of form, force and function,” Giomi said. “Cells use their shape to regulate forces, and these in turn serve as the running engine of mechanical functionality.”
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