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Physical configurations of a cell doublet with line tension, a theoretical study

Fabrice Delbary

TL;DR

The paper presents a theoretical, algebraic treatment of a cell doublet with line tension, extending the classical two-sphere-cap model to include a junction line tension $\kappa$ and deriving how this affects minimum-energy configurations under volume constraints. By formulating critical-point conditions as polynomial systems and employing Lagrange multipliers, the author characterizes interior and boundary configurations, uniqueness results, and the role of triangle (or quadrilateral) force-balance at the junction. Key findings include the existence of a unique interior minimum when tensions satisfy triangle inequalities, the emergence of boundary minima corresponding to internalization or separation, and the substantial qualitative and quantitative shifts induced by line tension—such as domain-boundedness, bulging near singularities, and non-unique tension inference. The work provides both analytical formulas and numerical strategies (including polynomial homotopy continuation) to map configuration regimes and offers insights into how line tension could influence early-embryo–scale foam-like tissue configurations.

Abstract

As a first approximation, early embryos may be modeled as foams whose shape depends on the surface tensions of each cell. However it has been remarked that exist line tensions at polarized exterior cellular interfaces (apical). In order to understand the changes it may imply on the usual foam model, a simple case study is considered: a double cell with line tension. Phase diagrams, bifurcations, possible new configurations are studied.

Physical configurations of a cell doublet with line tension, a theoretical study

TL;DR

The paper presents a theoretical, algebraic treatment of a cell doublet with line tension, extending the classical two-sphere-cap model to include a junction line tension and deriving how this affects minimum-energy configurations under volume constraints. By formulating critical-point conditions as polynomial systems and employing Lagrange multipliers, the author characterizes interior and boundary configurations, uniqueness results, and the role of triangle (or quadrilateral) force-balance at the junction. Key findings include the existence of a unique interior minimum when tensions satisfy triangle inequalities, the emergence of boundary minima corresponding to internalization or separation, and the substantial qualitative and quantitative shifts induced by line tension—such as domain-boundedness, bulging near singularities, and non-unique tension inference. The work provides both analytical formulas and numerical strategies (including polynomial homotopy continuation) to map configuration regimes and offers insights into how line tension could influence early-embryo–scale foam-like tissue configurations.

Abstract

As a first approximation, early embryos may be modeled as foams whose shape depends on the surface tensions of each cell. However it has been remarked that exist line tensions at polarized exterior cellular interfaces (apical). In order to understand the changes it may imply on the usual foam model, a simple case study is considered: a double cell with line tension. Phase diagrams, bifurcations, possible new configurations are studied.
Paper Structure (18 sections, 4 theorems, 137 equations, 20 figures)

This paper contains 18 sections, 4 theorems, 137 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Description
  3. Configuration at minimum energy under volume constraints
  4. Equations for the critical points of the constrained energy
  5. A more physical but equivalent way to proceed: the Lagrangian
  6. Separation and internalization
  7. Non-degenerate case
  8. Uniqueness of the local/global minimum
  9. A few words about inferring the tensions
  10. Configuration of the cell doublet for given pressures
  11. Cell doublet with line tension
  12. Equations for the critical points of the constrained energy
  13. Theoretical and numerical results, observations
  14. Inequalities for tensions and volumes
  15. Uniqueness of the interior minimum
  16. ...and 3 more sections

Key Result

Theorem 3.9

The polynomial $F$ defined in popol has a unique real root.

Figures (20)

  • Figure 1: Cell doublet configuration. Angles are taken in $(-\pi,\pi)$.
  • Figure 2: Tension forces and angles
  • Figure 6: Type of configurations depending on the tensions
  • Figure 7: Left: $w_1=0.5,w_2=0.5$. Right: $w_1=0.75,w_2=0.25$
  • Figure 8: $\kappa=0.1$, $t_3=1$, $w_1=w_2=0.5$.
  • ...and 15 more figures

Theorems & Definitions (20)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Remark 3.7
  • Remark 3.8
  • Theorem 3.9
  • ...and 10 more