The origin of universal cell shape variability in a confluent epithelial monolayer
Abstract
Cell shape is fundamental in biology. The average cell shape can influence crucial biological functions, such as cell fate and division orientation. But celltocell shape variability is often regarded as noise. In contrast, recent works reveal that shape variability in diverse epithelial monolayers follows a nearly universal distribution. However, the origin and implications of this universality are unclear. Here, assuming contractility and adhesion are crucial for cell shape, characterized via aspect ratio (AR), we develop a meanfield analytical theory for shape variability. We find that a single parameter, α , containing all the systemspecific details, describes the probability distribution function (PDF) of AR; this leads to a universal relation between the standard deviation and the average of AR. The PDF for the scaled AR is not strictly but almost universal. The functional form is not related to jamming, contrary to common beliefs, but a consequence of a mathematical property. In addition, we obtain the scaled area distribution, described by the parameter µ . We show that α and µ together can distinguish the effects of changing physical conditions, such as maturation, on different system properties. The theory is verified in simulations of two distinct models of epithelial monolayers and agrees well with existing experiments. We demonstrate that in a confluent monolayer, average shape determines both the shape variability and dynamics. Our results imply the cell shape variability is inevitable, where a single parameter describes both statics and dynamics and provides a framework to analyze and compare diverse epithelial systems.
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Evaluation Summary:
By theoretically analysing the energy of a confluent epithelial tissue, the authors unveil the reason for nearly universal shape fluctuations that have been reported earlier. With a better justification of some of the underlying approximations used by the authors, the manuscript would be relevant for all people with an interest in tissue structure and dynamics.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #2 agreed to share their name with the authors.)

Reviewer #1 (Public Review):
Sadhukhan and Nandi study theoretically the variation of cell shapes in an epithelial layer. Specifically, they consider the aspect ratio of the cell surface area and the surface area distribution. The authors use an effective equilibrium theory, where they restrict themselves to a regime, where the cell areas make a negligible contribution to the monolayer energy, which only depends on the cell perimeters. The energy is governed by the target perimeter P_0 and the perimeter elastic constant \lambda_P. The authors compute the distributions for the aspect ratio and the area. Each distribution depends on a single parameter, respectively called \alpha and \mu. A priori, neither of the two distributions is universal, but if the average aspect ratio shows a certain relation to \alpha, then the distribution of the …
Reviewer #1 (Public Review):
Sadhukhan and Nandi study theoretically the variation of cell shapes in an epithelial layer. Specifically, they consider the aspect ratio of the cell surface area and the surface area distribution. The authors use an effective equilibrium theory, where they restrict themselves to a regime, where the cell areas make a negligible contribution to the monolayer energy, which only depends on the cell perimeters. The energy is governed by the target perimeter P_0 and the perimeter elastic constant \lambda_P. The authors compute the distributions for the aspect ratio and the area. Each distribution depends on a single parameter, respectively called \alpha and \mu. A priori, neither of the two distributions is universal, but if the average aspect ratio shows a certain relation to \alpha, then the distribution of the scaled aspect ratio is universal. The deviation between the dependence of the mean aspect ratio on \alpha required for universality and the actual relation are small, such that shape fluctuations are nearly universal. The authors' derivation puts earlier experimental findings on a solid ground and very importantly shows that the distribution is NOT a consequence of jamming.
The authors also find that the relation between the standard deviation and the mean aspect ratio is universal. They check their analytical results by simulations of epithelia in terms of a cellular Potts model and a vertex model for which they explicitly verify their assumptions. Due to the success of these approaches had in the past to describe salient features of epithelial tissues, these comparisons strongly support the relevance of the authors' calculations for real epithelia.
The results obtained by the authors clarify the origin of the very intriguing near universality of aspect ratio fluctuations found in epithelial monolayers and should be of interest to all researchers with an interest in tissue properties. Unfortunately, the presentation is not at par with the quality of the results the authors obtain. Most importantly they often make use of jargon that is hard to understand for readers without a formal training in physics.

Reviewer #2 (Public Review):
The manuscript of Sadhukhan and Nandi presents a theoretical study of the shape fluctuations in a confluent epithelial monolayer. The theory which is following the lines of what has been done for 2 dimensional foams is knowledge new and original and the derivation of the results looks sound. It leads to the surprising result that the fluctuations in shape are "almost" universal and the distribution of the rescaled area depends only on a single parameter. The results are obtained with a series of approximations that would need to be discussed more extensively.
