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

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.

On the best possible remaining term in the Hardy inequality.

We give a necessary and sufficient condition on a radially symmetric potential V on a bounded domain Omega of (n) that makes it an admissible candidate for an improved Hardy inequality of the following type. For every element in H(1)(0)(Omega) integral(Omega) |vector differential u|2 dx - ((n - 2)/2)2 integral(Omega) |u|2/|x|2 dx > or = c integral(Omega) V(x)|u|2 dx. A characterization of the best possible constant c(V) is also given. This result yields easily the improved Hardy's inequalities of Brezis-Vázquez [Brezis H, Vázquez JL (1997) Blow up solutions of some nonlinear elliptic problems. Revista Mat Univ Complutense Madrid 10:443-469], Adimurthi et al. [Adimurthi, Chaudhuri N, Ramaswamy N (2002) An improved Hardy Sobolev inequality and its applications. Proc Am Math Soc 130:489-505], and Filippas-Tertikas [Filippas S, Tertikas A (2002) Optimizing improved Hardy inequalities. J Funct Anal 192:186-233] as well as the corresponding best constants. Our approach clarifies the issue behind the lack of an optimal improvement while yielding the following sharpening of known integrability criteria: If a positive radial function V satisfies lim inf(r-->o) ln(r) integral(r)(o),sV(s) ds > -infinity, then there exists rho: = rho(Omega) > 0 such that the above inequality holds for the scaled potential v(rho)(x) = v((|x|)(rho)). On the other hand, if lim (r-->0) ln(r) integral(r)(o),sV(s) ds = -infinity, then there is no rho > 0 for which the inequality holds for V(rho).