Figure 1.
Illustrations of typical Arabidopsis cell division and differentiation.
A Diagram of cell division in Arabidopsis thaliana showing meristemoids and stomata in blue, and pavement cells in green (or white). The cells labeled 1, 2 and 3 indicate oldest to youngest cells in that sequence of division. B Typical example of a pavement cell arrangement resulting from clockwise inward spiraling subdivision: first cell (1), second cell (2) and third cell (3). The axis of final pavement cell typically lies at 0°–10° to the axis of stomatal pore. The medial axis transform (MAT) lines of each cell are also shown. C This diagram represents a typical pavement cell in the early stages of division.
Figure 2.
Growth stages of Arabidopsis thaliana (Col.) leaf.
A Scanning Electron Microscope (SEM) images were taken from three different leaf ages, 10, 17 and 24 days after initial appearance of leaf. Day 10 leaf sizes ranged from 63 to 88 mm2 (average 75 mm2); day 17 leaf sizes ranged from 189 to 263 mm2 (average 237 mm2) and day 24 leaf sizes ranged from 559 to 809 mm2 (average 710 mm2). B Sample SEM images of Arabidopsis thaliana epidermal cells from three different ages as shown.
Figure 3.
Cell shape as described by Minimum Bounding Rectangle (MBR).
A Typical cell shape shown within its aligned MRB (short dashed line). The rhombus (long dashed line) represents half of the area of the MBR. Also shown, is the equivalent area circle (dotted line). B This graph plots MBR Length to MBR Width for each age group (days 10, 17 and 24). Mean length∶width ratios were 1.62 (95% CI 1.58–1.65), 1.65 (95% CI 1.62–1.68), and 1.60 (95% CI 1.57–1.64) for days 10, 17, and 24 respectively. Note: the golden ratio is 1.618034. C KS Density plots of MBR length∶width ratios for each day. Length∶width ratios were clustered in the range 1 to 3. Values rarely exceeded 3, and none were found to be greater than 6. D Plots the cell areas against half of the MBR. Interestingly, the linear fits lie almost exactly in line with the x = y plot, i.e. cell area = half of MBR.
Figure 4.
Cell Area analyses and Area versus Perimeter ratios.
A, B & C Show pavement cell area histograms for day 10, day 17 and day 24 from top to bottom. Day 10 pavement cell areas range from 20 to 786 µm2; average cell area = 120 µm2; median cell area = 102 µm2. Day 17 pavement cell areas range from 20 to 2289 µm2; average cell area = 165 µm2; median cell area = 119 µm2. Day 24 pavement cell areas range from 20 to 1914 µm2; average cell area = 178 µm2; median cell area = 138 µm2. D Shows a plot of cell area against cell perimeter for each day 10, 17 and 24. The grey points show the equivalent circle for day 24 data where area is the same and perimeter is found by . E KS Density plot of area∶perimeter ratio for days 10,17 and 24. The medians of the area∶perimeter ratios are 1.97 (95% CI 1.93–2.01), 2.23 (95% CI 2.16–2.30) and 2.38 (95% CI 2.30–2.38) at days 10, 17 and 24 respectively.
Figure 5.
Medial Axis Transform (MAT) angles and distribution frequency.
A Each cell outline was exported from AutoCAD into MATLAB using AutoLISP script. Here is the typical data extracted showing (a) stomatal pore axis; (b) cell boundary; (c) MAT internal segment lengths (red); and (d) external segment lengths (green). MAT angle measurements are also shown. B Graph shows the distribution curves of all angles along MAT at branch points for day 10 (blue), day 17 (green) and day 24 (red). The peaks all fall in the range 125–135°. C Possible angle subdivisions of 360°, e.g. golden angle plus two equal smaller angles would be 137.5°+111.25°+111.25°. Typical angle groupings found in our data set often exhibited such values, e.g. 144°+108°+108°. It is worth noting that the ratios of successive Fibonacci numbers converge on the golden ratio (137.5°) as n approaches infinity in the equation , where
. D Histograms of MAT angles for all cells (left) and only cells of area ≥500 µm2 (right) for Day 24. Generally the peak is around 130°, but when the giant cells are isolated, the increased frequency is seen around 100–110° and 130–140°.
Figure 6.
SEM image from day 24 data set: cell shapes and digit numbers per cell area.
A SEM image taken from the day 24 series. Shown overlaid (grey lines) is the Voronoi diagram calculated from stomatal centers. Three cells are selected to represent cells of different sizes. B Boundaries of the cells identified in A: blue cell has area of 88 µm2 and 5 digits; red cell has area of 519 µm2 and 7 digits; green cell has area of 1032 µm2 and 13 digits. The ratio of digits per 100 µm2 is 5.6, 1.3 and 1.2 respectively for the blue, red and green cell shown here. For the purposes of shape comparison, the three cells have been re-scaled so that each has roughly the same area. C Scatter plot showing number of digits versus cell area, for the three different age groups. The range of digit numbers is from 1 to 22 and rarely exceeds 15 irrespective of cell size. Also, we see in the younger samples that the digit to area ratio is higher than in the mature samples. Mean no of digits per 100 µm2 for all cells: 6.7 (SD 3.6), 5.6 (SD 3.9) and 5.1 (SD 3.7) for day 10, 17 & 24 respectively. Mean number of digits per 100 µm2 for cells ≥500 µm2: 2.3 (SD 0.3), 1.3 (SD 0.4) and 1.3 (SD 0.4) for day 10, 17 & 24 respectively. Note: the data shown here in C is based on MAT data pruned to remove any external segment <0.55 µm.
Figure 7.
Delaunay Triangulation (DT) and Voronoi analysis based on stomatal centers; KS density plots of largest DT angles and DT areas.
A Delaunay triangulation was conducted on each SEM image using the centroids of the stomata, i.e. stomatal pore and surrounding guard cells. B Voronoi diagram using centroids of stomata (centroids shown red) as seen in SEM images. C KS density distribution of the largest angles of the Delaunay triangulation: day 10 (blue), day 17 (green), and day 24 (red). Peaks are consistently around 80°–90°. D KS density distribution plots of DT areas for day 10 (blue) and day 24 (red). Peaks are consistent around 500 µm2. E Histogram with normal distribution fit overlay for all DT angles at day 10 (left) and day 24 (right). F Diagram exemplifies a golden spiral overlaying the golden triangle and golden gnomon.
Figure 8.
SEM image with giant pavement cell highlighted.
A SEM image taken from the day 24 series with a giant cell outlined in green. B Giant pavement cell as identified in A: cell boundary (green line); medial axis transform (MAT; black lines); Voronoi cell boundaries (red dashed lines) calculated using centroid of stomatal complex as input coordinates; stomatal pores and guard cells (shown in red and yellow). Cell area = 1575 µm2, cell perimeter = 333 µm; area∶perimeter ratio = 4.7; average MAT angle is 139.4° for the largest angles and 110.3° for the smaller two of three angles. C Example (left), from day 24 data, of giant cell sitting along the Voronoi boundaries. Cell area = 1032; cell perimeter = 242; area∶perimeter ratio = 4.3; length∶width ratio = 3.2. Example (right) from a young leaf.