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Parameter degeneracy in the vertex model for tissues

Paulo C. Godolphim, Leonardo G. Brunnet, Rodrigo Soto

TL;DR

This work reveals a target-area degeneracy in heterogeneous vertex models, showing that the dynamics depend on individual cell targets and stiffnesses but are invariant under shifts that preserve the gauge pressure $P_g = \langle K_{Ac} A_{0c} \rangle$. It introduces the gauge-pressure interpretation and derives a symmetry of the parameter space, then proposes two practical routes to fix the gauge—a zero-mean internal pressure gauge (ZPG) and a fixed-gauge approach—so that observables like cell shape index and stress become comparable across parameter choices. The paper further analyzes how boundary conditions and curvature influence the degeneracy, demonstrating that open borders partially lift it while curved geometries (planar vs spherical) can either suppress or restore degeneracy depending on the force formulation used. The framework provides a general method to test for degeneracy in other physical models and has concrete implications for parameter inference from experimental data in tissue mechanics.

Abstract

The vertex model with homogeneous cell properties is known to exhibit a parameter degeneracy in which the system's dynamics is independent of the target area. Here, we show, for the heterogeneous vertex model where cells differ in size and stiffness, that degeneracy is also present with the average product of target areas and stiffness becoming dynamically irrelevant. Fixing this quantity is equivalent to fixing the global internal tissue pressure. Unless properly treated, this degeneracy undermines the physical relevance of key observables' numerical values, such as cell target shape index, cell pressure, and cell stress tensor. We present methods to resolve the degeneracy and to correctly set the gauge pressure via symmetry transformations applied to the cells' target areas. We further demonstrate that the degeneracy is removed under certain boundary conditions and partially lifted when spherical tissues are modeled using a locally planar approximation, leading to numerical consequences when fitting model parameters to experimental data. The approach extends beyond vertex models and provides a framework for testing whether the parameter spaces of other physical models are free from degeneracy.

Parameter degeneracy in the vertex model for tissues

TL;DR

This work reveals a target-area degeneracy in heterogeneous vertex models, showing that the dynamics depend on individual cell targets and stiffnesses but are invariant under shifts that preserve the gauge pressure . It introduces the gauge-pressure interpretation and derives a symmetry of the parameter space, then proposes two practical routes to fix the gauge—a zero-mean internal pressure gauge (ZPG) and a fixed-gauge approach—so that observables like cell shape index and stress become comparable across parameter choices. The paper further analyzes how boundary conditions and curvature influence the degeneracy, demonstrating that open borders partially lift it while curved geometries (planar vs spherical) can either suppress or restore degeneracy depending on the force formulation used. The framework provides a general method to test for degeneracy in other physical models and has concrete implications for parameter inference from experimental data in tissue mechanics.

Abstract

The vertex model with homogeneous cell properties is known to exhibit a parameter degeneracy in which the system's dynamics is independent of the target area. Here, we show, for the heterogeneous vertex model where cells differ in size and stiffness, that degeneracy is also present with the average product of target areas and stiffness becoming dynamically irrelevant. Fixing this quantity is equivalent to fixing the global internal tissue pressure. Unless properly treated, this degeneracy undermines the physical relevance of key observables' numerical values, such as cell target shape index, cell pressure, and cell stress tensor. We present methods to resolve the degeneracy and to correctly set the gauge pressure via symmetry transformations applied to the cells' target areas. We further demonstrate that the degeneracy is removed under certain boundary conditions and partially lifted when spherical tissues are modeled using a locally planar approximation, leading to numerical consequences when fitting model parameters to experimental data. The approach extends beyond vertex models and provides a framework for testing whether the parameter spaces of other physical models are free from degeneracy.

Paper Structure

This paper contains 12 sections, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Vertex model
  3. parameter space degeneracy
  4. Managing the degeneracy
  5. Gauge pressure
  6. Fixing the gauge pressure
  7. Parameter optimization procedures
  8. Boundary conditions and Curvature
  9. Finite tissues
  10. Curved tissues
  11. Conclusions and discussion
  12. Force expressions

Figures (1)

  • Figure 1: Tissue schematics showing the position $\mathbf{r}_i$ of a bulk vertex (top) and a border vertex (bottom), in their initial configurations (left column) and after a virtual displacement $\delta \mathbf{r}$ (right column). The displacement of the bulk vertex does not alter the net area of the cells it belongs to (highlighted in red), whereas the displacement of a border vertex does.