# Stochastic terminal dynamics in Epithelial cell intercalation

Epithelial cell rearrangement is important for many processes in morphogenesis [1]. During germband extension in early gastrulation of Drosophila embryos, exchange of neighbors is achieved by junction remodeling that follows a topological T1 process [2, 3]. Its first step is the constriction of dorsal-ventral junctions and fusion of two 3x vertices into a 4x vertex, a process believed to be junction autonomous. We established a high throughput imaging pipeline, by which we recorded, segmented and analysed more than 1000 neighbor exchanges in Drosophila embryos. Characterizing the dynamics of junction lengths we find that the constriction of cell contacts follows intriguingly simple quantitative laws. (1) The mean contact length follows , where is the finite collapse time. (2) The time dependent variance of contact lengths is proportional to the square of the mean. (3) The time dependent probability density of the contact lengths remains close to Gaussian during the entire process. These observations are sufficient to derive a stochastic differential equation for contact length analytically tractable in small noise approximation. To find the universal laws of the constriction dynamics the data is analyzed by aligning the stochastic trajectories of the junction length to their collapse point. To account for this alignment, we model the collapse by a stochastic process with a well-defined final condition that evolves backwards in time. For this we use the theory of time-reversed stochastic differential equations. The model provides an effective description of the non-equilibrium statistical mechanics of contact collapse. All model parameters are fixed by measurements of time dependent mean and variance of contact lengths. The model predicts the contact length covariance function that we obtain in closed form. The contact length covariance function closely matches experimental observations suggesting that the model well captures the dynamics of contact collapse on a time scale of minutes.

[1] C.-P. Heisenberg, Y. Bellaiche, Cell 153, 948 (2013).

[2] K. D. Irvine, E. Wieschaus, Development 120, 827 (1994).

[3] Y. Zhang, D. Kong, L. Reichl, N. Vogt, F. Wolf, J. Grosshans, Developmental Biology 390, 208 (2014).

### Members working within this Project:

Stephan EuleNicolas Lenner

Fred Wolf