A variational method for image-based inference of internal stress in epithelial tissues.

Nicholas Noll,Boris I Shraiman,Sebastian J Streichan

doi:10.1103/physrevx.10.011072

Abstract

Cellular mechanics drives epithelial morphogenesis, the process wherein cells collectively rearrange to produce tissue-scale deformations that determine organismal shape. However, quantitative understanding of tissue mechanics is impaired by the difficulty of direct measurement of stress in-vivo. This difficulty has spurred the development of image-based inference algorithms that estimate stress from snapshots of epithelial geometry. Such methods are challenged by sensitivity to measurement error and thus require accurate geometric segmentation for practical use. We overcome this difficulty by introducing a novel approach - the Variational Method of Stress Inference (VMSI) - which exploits the fundamental duality between stress and geometry at equilibrium of discrete mechanical networks that model confluent cellular layers. We approximate the apical geometry of an epithelial tissue by a 2D tiling with Circular Arc Polygons (CAP) in which arcs represent intercellular interfaces defined by the balance of local line tension and pressure differentials between adjacent cells. The mechanical equilibrium of such networks imposes extensive local constraints on CAP geometry. These constraints provide the foundation of VMSI which, starting with images of epithelial monolayers, simultaneously approximates both tissue geometry and internal forces, subject to the constraint of equilibrium. We find VMSI to be more robust than previous methods. Specifically, the VMSI performance is validated by the comparison of the predicted cellular and mesoscopic scale stress with the measured myosin II patterns during early Drosophila embryogenesis. VMSI prediction of mesoscopic stress tensor correlates at the 80% level with the measured myosin distribution and reveals that most of the myosin activity in that case is involved in a static internal force balance within the epithelial layer. In addition to insight into cell mechanics, this study provides a practical method for non-destructive estimation of stress in live epithelial tissue.

Highlights

Cell and tissue mechanics is an important factor that both affects and regulates animal and plant development and is a subject of active study in developmental biology and biophysics, reviewed extensively in Refs. [1,2,3,4,5,6,7]
Applying variational method of stress inference (VMSI) to the embryonic epithelium images [36], we found that cell-array geometry observed over the first 60 min of convergent extension is quite well approximated by an equilibrium cell network, hEi ≈ 1, which means our best-fit equilibrium circular arc polygonal (CAP) geometry differs from the image segmentation by on average one pixel per edge
The VMSI algorithm described here was based on the model that assumed that (i) the 2D epithelial cell array is instantaneously in an approximate mechanical equilibrium and (ii) cell mechanics can be approximated by the balance of cytoskeletal tension localized at cell interfaces and the effective areal pressure

Summary

Introduction

Cell and tissue mechanics is an important factor that both affects and regulates animal and plant development and is a subject of active study in developmental biology and biophysics, reviewed extensively in Refs. [1,2,3,4,5,6,7]. For the purpose of tissue-scale mechanics, the full threedimensional force balance that shapes individual cells can be approximated by an effective two-dimensional model of the apical cytoskeletal network. In this simplified 2D view, the contractility of the junctional actomyosin “belts” balances against an effective two-dimensional pressure that prevents the collapse of the apical area under cortical tension: this type of an approximation underlies the widely used “vertex model” approach to epithelial cell mechanics [1,16,17,18]. Along with alternative models of epithelial mechanics, are reviewed in detail in Ref. [19]

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