A mathematical model to study the dynamics of epithelial cellular networks

Abstract

Epithelia are sheets of connected cells that are essential across the animal kingdom. Experimental observations suggest that the dynamical behavior of many single-layered epithelial tissues has strong analogies with that of specific mechanical systems, namely large networks consisting of point masses connected through spring-damper elements and undergoing the influence of active and dissipating forces. Based on this analogy, this work develops a modeling framework to enable the study of the mechanical properties and of the dynamic behavior of large epithelial cellular networks. The model is built first by creating a network topology that is extracted from the actual cellular geometry as obtained from experiments, then by associating a mechanical structure and dynamics to the network via spring-damper elements. This scalable approach enables running simulations of large network dynamics: the derived modeling framework in particular is predisposed to be tailored to study general dynamics (for example, morphogenesis) of various classes of single-layered epithelial cellular networks. In this contribution, we test the model on a case study of the dorsal epithelium of the Drosophila melanogaster embryo during early dorsal closure (and, less conspicuously, germband retraction).

Fig. 1

Pictorial top-view of a single cell in an epithelial network.

Fig. 2

Two-dimensional pictorial top-view of two adjacent epithelial cells: actin accumulation (blue) along cell boundaries (black outer lines). Adherens junctions (green) connect different circumferential actin bundles between the cytoskeleta of adjacent cells along their boundaries.

Fig. 3

Nonlinearity in cell elasticity (stress-stiffening).

Fig. 4

A small graph . Loose numbers denote vertices, whereas circled numbers denote edges.

Fig. 5

A spring-damper model of a single cell. Discrete points (large magenta dots) are adapted to the actual cell shape extracted from experiments (see Section 5.1). Black lines denote cell boundaries. Blue spring-damper elements are associated to boundaries, whereas green ones to intermediate pairs of non-adjacent points.

Fig. 6

A single point mass element m_i connected to three point mass elements (all denoted as m_j ) through spring-damper elements.

Fig. 7

One-dimensional spring model used to explain the applied prestress.

Fig. 8

Isolated cellular network at starting time, used for model building.

Fig. 9

Pictorial model of the embryo, depicting a segment of the epithelium (blue). The dorsal closure forces acting on the epithelium over the amnioserosa (pink region) are indicated by the blue arrows.

Fig. 10

Top: A pictorial longitudinal view of the embryo with indication of calibration, back projection, and folding steps, which together yield the folded, three-dimensional model of the epithelium. Bottom: A three-dimensional view of the morphing procedure.

Fig. 11

Three-dimensional model of the epithelium of the embryo (symmetric representation). The green cylinders denote the central nervous system. Notice that part of the ventral epithelium has not been captured by the microscope and hence has not been modeled.

Fig. 12

The Laplace-Young Law relates the pressure difference Δp over a surface to the surface tension T.

Fig. 13

trajectory imposed on the leading edge vertices. Top down displacement profile p^LE(t), velocity profile v^LE(t), and acceleration profile a^LE(t). The time axis has been normalized.

Fig. 14

Three-dimensional simulation outcomes. The cartesian axes are x (red), y (yellow), and z (green).

Fig. 15

Experimental cellular network at final time, used as a benchmark for the simulation outcomes.

Fig. 16

Two dimensional projection of the lateral-view simulation output in Figure 14(c).