Predicting Division Planes of Three-Dimensional Cells by Soap-Film Minimization

Abstract

One key aspect of cell division in multicellular organisms is the orientation of the division plane. Proper division plane establishment contributes to normal plant body organization. To determine the importance of cell geometry in division plane orientation, we designed a three-dimensional probabilistic mathematical model to directly test the century-old hypothesis that cell divisions mimic soap-film minima. According to this hypothesis, daughter cells have equal volume and the division plane occurs where the surface area is at a minimum. We compared predicted division planes to a plant microtubule array that marks the division site, the preprophase band (PPB). PPB location typically matched one of the predicted divisions. Predicted divisions offset from the PPB occurred when a neighboring cell wall or PPB was directly adjacent to the predicted division site to avoid creating a potentially structurally unfavorable four-way junction. By comparing divisions of differently shaped plant cells (maize [Zea mays] epidermal cells and developing ligule cells and Arabidopsis thaliana guard cells) and animal cells (Caenorhabditis elegans embryonic cells) to divisions simulated in silico, we demonstrate the generality of this model to accurately predict in vivo division. This powerful model can be used to separate the contribution of geometry from mechanical stresses or developmental regulation in predicting division plane orientation.

TANGLED1 mediates microtubule interactions that may promote division plane positioning in maize

“…Previous measurements of PPB placement were obtained from 2D micrographs, which might not accurately reflect the position of the PPB in 3D, particularly in cells with irregular shapes. To overcome this shortcoming, we used our recently developed mathematical modeling approach to accurately predict 3D division planes ( Martinez et al, 2018 ). This model generates soap-film minima from real, 3D cell shapes and allows us to compare purely geometric predictions to in vivo cell division sites ( Martinez et al, 2018 ).…”

“…Then, the gradient descent function in Surface Evolver was used to generate soap-film minima that divided the volume into two equal halves. These soap-film minima are division planes predictions (Martinez et al, 2018;Brakke, 1992). The predicted division planes were then compared with the in vivo PPB location (Fig.…”

Cell geometry determines symmetric and asymmetric division plane selection in Arabidopsis early embryos

“…While the revised version of the present paper was under review, Martinez et al [47] published a 3D model of symmetric cell divisions and Chakrabortty et al [48] reported how they attempted to predict the global orientation of division planes in the embryo based on the organization of cortical microtubules.…”

Plant cell divisions: variations from the shortest symmetric path

“…Another structure aligning with tension is the PPB whose orientation often follows the main orientation of cortical microtubules in interphase. PPB position is accurately predicted by the soap-film area minimization model run on real 3D segmented cell templates, in turn, the PPB position predicts the position of the future division plane [ 13 , 28 ]. However, PPB is not present in all plant species [ 29 ] and its function has recently been re-evaluated.…”

“…Divisions parallel to the longest axis of the cell, referred to as longitudinal divisions hereafter, are frequent during leaf development in maize [ 48 ]. However, they tend to occur in nearly isotropic cells, making the selected path not so distant from the shortest path [ 28 ]. Besides this situation that can be interpreted as an extreme case of the Besson and Dumais rule, some deviations from the shortest path cannot be explained by cell geometry-based rules alone.…”