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.

One key aspect of cell division in multicellular organisms is the orientation of the division plane. Proper division plane establishment contributes to normal organization of the plant body. To determine the importance of cell geometry in division plane orientation, we designed a threedimensional probabilistic mathematical modeling approach to directly test the century-old hypothesis that cell divisions mimic “soap-film minima” or that daughter cells have equal volume and the resulting division plane is a local surface area minimum. Predicted division planes were compared 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 observed directly adjacent to the predicted division site, to avoid creating a potentially structurally unfavorable four-way junction. By comparing divisions of differently shaped plant and animal 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.

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We consider the mean dimension of some ridge functions of spherical Gaussian random vectors of dimension d. If the ridge function is Lipschitz continuous, then the mean dimension remains bounded as d → ∞. If instead, the ridge function is discontinuous, then the mean dimension depends on a measure of the ridge function's sparsity, and absent sparsity the mean dimension can grow proportionally to √ d. Preintegrating a ridge function yields a new, potentially much smoother ridge function. We include an example where, if one of the ridge coefficients is bounded away from zero as d → ∞, then preintegration can reduce the mean dimension from O( √ d) to O(1).