While the biology of how an entire organism develops from a single cell has long been a source of fascination, recent research has increasingly highlighted the role of mechanical forces. “If we want to have rigorous predictive models of morphogenesis, of tissues and cells forming organs of an animal,” says Konstantin Doubrovinski at the University of Texas Southwestern Medical Center, “it is absolutely critical that we have a clear understanding of material properties of these tissues.”
Now Doubrovinski and his colleagues report a rheological study explaining why the developing fruit fly (Drosophila melanogaster) epithelial tissue stretches as it does over time to allow the embryo to change shape.
Previous studies had shown that under a constant force, tissue extension was proportional to the time the force had been applied to the power of one half. This had puzzled the researchers, since it did not fit a simple model in which epithelial tissues behave like linear springs. In such a model, the extension obeys Hooke’s law and is proportional to the force applied alone, such that the exponent of time in the relation would be zero.
They and other groups had tried to explain this observation of an exponent equal to 0.5 as due to the viscosity of the medium surrounding the cells, which would lead to deformation near the point of pulling that then gradually spreads. However, their subsequent experiments ruled out viscosity as a cause of the non-zero exponent.
For their measurements, the researchers had exploited a convenient feature of Drosophila epithelial cells – a small hole, through which they could manipulate a droplet of ferrofluid to enter using a permanent magnet. Once inside the cell, a magnet acting on this droplet could exert forces on the cell to stretch the surrounding tissue.
For the current study, the researchers first tested the observed scaling law over longer periods of time. A power law gives a straight line on a log–log plot but as Doubrovinski points out, curves also look like straight lines over short sections. However, even when they increased the time scales probed in their experiments to cover three orders of magnitude – from fractions of a second to several minutes – the observed power law still held.
Understanding the results
One of the post docs on the team – Mohamad Ibrahim Cheikh – stumbled upon the actual relation giving the power law with an exponent of 0.5 while working on a largely unrelated problem. He had been modelling ellipsoids in a hexagonal meshwork on a surface, in what Doubrovinski describes as a “large” and “relatively complex” simulation. He decided to examine what would happen if he allowed the mesh to relax in its stretched position, which would model the process of actin turnover in cells.
Cheikh’s simulation gave the power law observed in the epithelial cells. “We totally didn’t expect it,” says Doubrovinski. “We pursued it and thought, why are we getting it? What’s going on here?”
Although this simulation yielded the power law with an exponent of 0.5, because the simulation was so complex, it was hard to get a handle on why. “There are all these different physical effects that we took into account that we thought were relevant,” he tells Physics World.
To get a more intuitive understanding of the system, the researchers attempted to simplify the model into a lattice of springs in one dimension, keeping only some of the physical effects from the simulations, until they identified the effects required to give the exponent value of 0.5. They could then scale this simplified one-dimensional model back up to three dimensions and test how it behaved.
According to their model, if they changed the magnitude of various parameters, they should be able to rescale the curves so that they essentially collapse onto a single curve. “This makes our prediction falsifiable,” says Doubrovinski, and in fact the experimental curves could be rescaled in this way.
When the researchers used measured values for the relaxation constant based on the actin turnover rate, along with other known parameters such as the size of the force and the size of the extension, they were able to calculate the force constant of the epithelial cell. This value also agreed with their previous estimates.
Doubrovinski explains how the ferrofluid droplet engages with individual “springs” of the lattice as it moves through the mesh. “The further it moves, the more springs it catches on,” he says. “So the rapid increase of one turns into a slow increase with an exponent of 0.5.” Against this model, all the pieces fit into place.
“I find it inspiring that the authors, first motivated by in vivo mechanical measurements, could develop a simple theory capturing a new phenomenological law of tissue rheology,” says Pierre Françoise Lenne, group leader at the Institut de Biologie du Development de Marseille at L’Universite d’Aix-Marseille. Lenne specializes in the morphogenesis of multicellular systems but was not involved in the current research.
Next, Doubrovinski and his team are keen to see where else their results might apply, such as other developmental stages and other types of organisms, such as mammals, for example.
The research is reported in Physical Review Letters and bioRxiv.
The post A model stretch: explaining the rheology of developing tissue appeared first on Physics World.