Geometric Limits of Mitotic Pressure Under Confinement

Abstract

Cells often divide under mechanical confinement, where surrounding structures restrict shape changes during cytokinesis. Although forces generated during confined division have been measured experimentally, it remains unclear how confinement geometry and mechanics determine the transmitted force. Here we develop a minimal mechanical theory of cell division under confinement. Modeling the cell as an incompressible volume bounded by an interface with effective isotropic tension, we show that confinement restricts the set of mechanically admissible furrow shapes. As the furrow radius decreases, it reaches it reaches a confinement-induced minimum. Beyond this point, further ingression does not alter the interface shape, and both pressure and axial force saturate. We analyze force and pressure in rigid, soft, and strong three-dimensional confinement and demonstrate that a single geometric mechanism underlies these distinct cases. After rescaling force and length by the appropriate geometric scale, cells of different size and surface tension collapse onto a single universal curve. The relevant length scale is the cell size for rigid and soft confinement, and the confinement size in fully enclosing three-dimensional confinement. In soft confinement, environmental stiffness and spindle-generated axial forces determine the operating force and pressure, while the geometric constraint fixes the maximal attainable levels. In summary, our results show that mitotic force transmission and mitotic pressure during cytokinesis are bounded by confinement geometry, with material properties and active forces selecting the operating point within these geometry-imposed limits.

Figures (3)

• Figure 1: Cell division under rigid axial confinement. (a–c) Schematic of cell division without confinement (free) and between rigid parallel walls (confined). The cell of fixed volume $2V_\mathrm{cell}$ is characterized by the geometric furrow radius $R_\mathrm{div}$ at the division plane and the maximum radius $R_{\max}$. Under confinement, axial elongation is restricted by the wall separation $4L$. (d) Geometric furrow radius $R_\mathrm{div}$ as a function of cytokinetic ring radius $R_{\mathrm{cyto}}$. In the absence of confinement, $R_\mathrm{div} \simeq R_{\mathrm{cyto}}$. Under confinement, $R_\mathrm{div}$ saturates at the confinement-induced minimum $R_\mathrm{div}^{\min}$. (e) Dimensionless axial force $F/F_0$ as a function of furrow radius. The force increases during ingression and saturates once $R_\mathrm{div}^{\min}$ is reached. (f) Axial force in physical units for different cell radii and surface tensions. Increasing cell size at fixed tension changes the minimum furrow radius but not the saturating force, whereas increasing tension at fixed cell size rescales the force without affecting the minimum furrow radius. (g) Scaled saturation force $F/F_0$ as a function of dimensionless wall separation $L/R_\mathrm{cell}$ for fixed cell volume. (h) Confinement-induced minimum furrow radius $R_\mathrm{div}/R_\mathrm{cell}$ as a function of $L/R_\mathrm{cell}$. (i) Scaled saturation pressure difference $\Delta P/\Delta P_0$ as a function of $L/R_\mathrm{cell}$. (j) Radius of curvature at saturation $R_\mathrm{H}/R_\mathrm{cell}$ as a function of $L/R_\mathrm{cell}$. Here $F_0 = 2\pi \gamma R_\mathrm{cell}$ is the characteristic force scale, and $\Delta P_0 = 2\gamma / R_\mathrm{cell}$ is the corresponding pressure scale.
• Figure 2: Cell division under soft confinement. (a–b) Schematic of soft axial confinement modeled as an elastic element of stiffness $k$. During cytokinesis, furrow ingression elongates the cell along the division axis and compresses the surrounding medium. (c) Graphical construction of mechanical equilibrium. The axial force generated by the cell under rigid confinement, $F_{\mathrm{cell}}(L)$, is shown together with the elastic force–length relation of the environment, $F_{\mathrm{env}}(L)$. The operating point $(L^\ast, F^\ast)$ is given by the intersection of the two curves. Increasing stiffness $\kappa$ steepens the environmental response and shifts the equilibrium toward smaller separations and larger forces. In the limit $\kappa \to \infty$, the solution approaches the rigid-wall limit. (d) Equilibrium force $F^\ast$ as a function of stiffness $k$. (e) Equilibrium cell half-length $L^\ast$ as a function of stiffness. (f) Equilibrium force $F^\ast$ as a function of spindle-generated axial force, showing shifts of the operating point along the force–geometry relation.
• Figure 3: Cell division under cylindrical confinement. (a) Geometry of a cell confined within a cylindrical chamber of radius $R_{\mathrm{cyl}}$ and length $L_{\mathrm{cyl}}$, with hemispherical caps. Two regimes are shown: for excess free volume $\Delta V > \Delta V_{\mathrm{th}}$ the cell accommodates division by axial elongation without transmitting axial force; for $\Delta V < \Delta V_{\mathrm{th}}$, ingression requires pushing against the caps, leading to nonzero axial force. (b) Dimensionless axial force at saturation $F/F_0$ as a function of reduced free volume $\Delta V/V_0$. The inset shows the same relation on logarithmic scales. (c) Furrow radius at saturation $R_\mathrm{div}/R_\mathrm{cyl}$ as a function of $\Delta V/V_0$, illustrating pinning at a confinement-induced minimum. (d) Scaled saturation pressure $\Delta P/\Delta P_0$ as a function of $\Delta V/V_0$, showing saturation in the confinement-limited regime. (e) Radius of curvature at saturation $R_\mathrm{H}/R_\mathrm{cyl}$ as a function of $\Delta V/V_0$. Here $V_0 \equiv 2\pi R_\mathrm{cyl}^3/3$ is the reference volume, $F_0 = 2\pi\gamma R_{\mathrm{cyl}}$ is the characteristic force scale under cylindrical confinement, and $\Delta P_0 = 2\gamma/R_{\mathrm{cyl}}$ is the corresponding pressure scale.