In overdamped vertex models of growing epithelial tissues, dissipation is conventionally represented by a single friction coefficient γ\gamma that is identical for every vertex. Guerrero and Perez-Carrasco (2024) showed that this supposedly innocuous choice deeply affects tissue growth, cell-size variability, division timing and topological transitions, because the friction-controlled relaxation time competes with the intrinsic timescale of cell-cycle progression. They call for (i) dissipative prescriptions in which the effective friction of each vertex reflects the amount of material it drags (for example, the sizes of the neighboring cells), and (ii) resolution of the parameter non-identifiability that prevents vertex models from serving as quantitative inference tools, supported by (iii) experimental measurement of full cell-cycle-length distributions rather than summary statistics. Both parts remain open: existing internal-dissipation vertex models address shear rheology and active flows, not the growth-coupled regime, and current inference methods recover quasi-static force parameters but not the dynamical ones, friction included.

Background

An epithelium is a sheet of tightly packed cells that covers surfaces in animals; during development such sheets grow and reshape through cell division, neighbor exchanges and cell removal. The dominant computational description of epithelial mechanics is the vertex model, which represents the tissue as a planar network of polygonal cells whose corners (vertices) carry the degrees of freedom. Mechanical behavior is encoded in an effective energy function EE with three standard terms, introduced in the classic study of Farhadifar et al. (2007): area elasticity, penalizing the deviation of each cell's area from a target area A0A_0 (modeling hydrostatic and osmotic pressure); line tension along each cell-cell junction (modeling cortical tension and adhesion); and perimeter contractility (modeling the contractile actomyosin belt at the cell cortex). Forces are the negative gradients of EE, and because inertial forces are negligible at cell scale, vertex motion is assumed overdamped: each vertex moves at a velocity proportional to the force acting on it, with the constant of proportionality being the friction coefficient γ\gamma. The network topology changes through T1 transitions (neighbor exchanges, when a junction shrinks below a threshold), T2 transitions (removal of a tiny cell), and cell division. Growth is coupled to mechanics because each cell's target area grows as it progresses through the cell cycle and the cell divides once its area reaches a critical threshold; Guerrero and Perez-Carrasco additionally prescribed an intrinsically stochastic cell-cycle duration with mean TcT_c and relative spread ε\varepsilon. If friction were the only timescale, one could rescale time by τ=t/γ\tau = t/\gamma and γ\gamma would drop out of all observables; the central observation of the source paper is that this rescaling symmetry is broken by the cell-cycle timescale, so the numerical value of γ\gamma controls the tissue growth rate, cell-size variability, the age of cells at division, and the rate of T2 transitions. Yet the standard literature practice sets γ\gamma to the same global constant for every vertex, which implicitly assumes every vertex dissipates as a point of identical mass regardless of how much material (membrane and cytoplasm of the adjoining cells) it drags. Compounding this, the parameters of vertex models are hard to infer from data: Kursawe, Baker and Fletcher (2018), using approximate Bayesian computation (a likelihood-free inference method that accepts simulation parameter sets whose simulated summary statistics match observed ones), found a substantial zone of non-identifiable parameter combinations compatible with a wide variety of biological scenarios, so that distinct mechanistic interpretations cannot be distinguished from available measurements. Together these issues obstruct the stated ultimate goal of vertex models: to infer mechanistic properties of real tissues from experimental data.

Problem Statement

For stochastic vertex models of growing epithelial tissues — overdamped dynamics γx˙i=−∇xiE\gamma \dot{\mathbf{x}}_i = -\nabla_{\mathbf{x}_i} E for each vertex ii, with a Farhadifar-type energy (area elasticity toward a cell-cycle-dependent target area, line tension, perimeter contractility), stochastic cell-cycle progression with prescribed mean and spread, division at a critical area, and topological transitions (T1 neighbor exchanges and T2 removal of tiny cells) — the problem, as posed by Guerrero & Perez-Carrasco (2024), is to replace the global constant friction coefficient γ\gamma with a physically grounded prescription in which the effective dissipation of each vertex depends on the amount of material it must displace, and to determine the consequences: (a) how vertex-dependent friction alters the competition between mechanical relaxation and cell-cycle timescales that the source identified as controlling tissue growth rate, cell-size variability, division-age distributions and T1/T2-transition rates; and (b) which parameter combinations of such a model — including the friction parameters themselves and the cell-cycle progression parameters — are identifiable from experimental data, and what inference strategy makes them identifiable, so that vertex models can be used reliably to infer mechanistic properties of real epithelia. The common objective uniting these parts is to give vertex models a dissipative structure faithful to the physics of the displaced material and a corresponding inference methodology, jointly enabling quantitative mechanistic inference for growing tissues; each part feeds the same end use and cannot be bypassed if that end use is the goal.

References

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