Geometric Modeling and Scientific Visualization Centre, King Abdullah University of Science and Technology, Thuwal, Kingdom of Saudi Arabia.
PLoS One. 2012;7(9):e43546. doi: 10.1371/journal.pone.0043546. Epub 2012 Sep 11.
The quest for the discovery of mathematical principles that underlie biological phenomena is ancient and ongoing. We present a geometric analysis of the complex interdigitated pavement cells in the Arabidopsis thaliana (Col.) adaxial epidermis with a view to discovering some geometric characteristics that may govern the formation of this tissue. More than 2,400 pavement cells from 10, 17 and 24 day old leaves were analyzed. These interdigitated cells revealed a number of geometric properties that remained constant across the three age groups. In particular, the number of digits per cell rarely exceeded 15, irrespective of cell area. Digit numbers per 100 µm(2) cell area reduce with age and as cell area increases, suggesting early developmental programming of digits. Cell shape proportions as defined by length:width ratios were highly conserved over time independent of the size and, interestingly, both the mean and the medians were close to the golden ratio 1.618034. With maturity, the cell area:perimeter ratios increased from a mean of 2.0 to 2.4. Shape properties as defined by the medial axis transform (MAT) were calculated and revealed that branch points along the MAT typically comprise one large and two small angles. These showed consistency across the developmental stages considered here at 140° (± 5°) for the largest angles and 110° (± 5°) for the smaller angles. Voronoi diagram analyses of stomatal center coordinates revealed that giant pavement cells (≥ 500 µm(2)) tend to be arranged along Voronoi boundaries suggesting that they could function as a scaffold of the epidermis. In addition, we propose that pavement cells have a role in spacing and positioning of the stomata in the growing leaf and that they do so by growing within the limits of a set of 'geometrical rules'.
探寻生物学现象背后的数学原理是一个古老而持续的课题。我们对拟南芥（Col.）上表皮的复杂交错的 pavement 细胞进行了几何分析，以期发现可能控制这种组织形成的一些几何特征。分析了来自 10、17 和 24 天大叶片的超过 2400 个 pavement 细胞。这些交错的细胞揭示了许多在三个年龄组中保持不变的几何特性。特别是，每个细胞的 digit 数很少超过 15，与细胞面积无关。每个 100 µm(2)细胞面积的 digit 数随年龄的增加而减少，并且随着细胞面积的增加而减少，表明 digit 是早期发育编程的。由长度与宽度之比定义的细胞形状比例随时间高度保守，与细胞大小无关，有趣的是，均值和中位数都接近黄金比例 1.618034。随着成熟，细胞面积与周长的比值从平均值 2.0 增加到 2.4。通过中轴变换（MAT）定义的形状特性进行了计算，结果表明，MAT 上的分支点通常由一个大角度和两个小角度组成。在考虑到的发育阶段中，这些角度保持一致，最大角度为 140°（± 5°），较小角度为 110°（± 5°）。气孔中心坐标的 Voronoi 图分析表明，大型 pavement 细胞（≥ 500 µm(2)) 倾向于沿着 Voronoi 边界排列，这表明它们可以作为表皮的支架。此外，我们提出 pavement 细胞在生长叶片中气孔的间距和定位中起作用，并且它们通过在一组“几何规则”的限制内生长来实现这一点。
