Estimating the number and size of phloem sieve plate pores using longitudinal views and geometric reconstruction

Because it is difficult to obtain transverse views of the plant phloem sieve plate pores, which are short tubes, to estimate their number and diameters, a method based on longitudinal views is proposed. This method uses recent methods to estimate the number and the sizes of approximately circular objects from their images, given by slices perpendicular to the objects. Moreover, because such longitudinal views are obtained from slices that are rather close to the plate centres whereas the pore size may vary with the pore distance from the plate edge, a sieve plate reconstruction model was developed and incorporated in the method to consider this bias. The method was successfully tested with published longitudinal views of phloem of Soybean and an exceptional entire transverse view from the same tissue. The method was also validated with simulated slices in two sieve plates from Cucurbita and Phaseolus. This method will likely be useful to estimate and to model the hydraulic conductivity and the architecture of the plant phloem, and it could have applications for other materials with approximately cylindrical structures.

T he pores of the sieve plates (Figs. 1, 2a and 3a) in the sieve tubes of plant phloem are shaped as short tubes. They are strongly suspected to largely control the transfers in the plant phloem because the total transfer section area at their level is small and the cross-sectional area of each pore is small, which results likely in high forces of viscosity. Because of their small size, they can be a barrier to microorganisms 1 . Their number and cross-sectional size can vary considerably according to species, organs and cell age, from less than one to several hundred per square micrometre of sieve plate and from hundredths of micrometres to micrometres [1][2][3][4][5][6] . Thus, it is important to be able to estimate these numbers and sizes, particularly to confirm that 95% of the hydraulic conductivity in the young tomato fruit pedicel 7 depends on the pores. More generally, modelling the architecture of the sieve plate could contribute to model fluxes in the tissue.
However, estimating the number and cross-sectional size of the pores is not easy. Under light microscopy, the pores are too small to be measurable and visible, so that even identifying the sieve tubes among the other phloem cells is difficult. In transmission electron microscopy, longitudinal views of the phloem can show the length of the sieve tubes and the pores (Fig. 1), and it is logical to expect that transverse views (Fig. 2a) give the area of the sieve plates and the number and cross-sectional sizes of their pores. However, obtaining transverse views with slices that are generally 0.07 mm thick on a level with a plate is notably improbable because of the small sieve plate thickness (only approximately 0.5-1 mm), whereas the cell length ranges from a few tens to hundreds of microns. Because there are at most only a few sieve tubes in one transverse view in transmission electron microscopy, only one view out of several tens or hundreds can be expected to show one sieve plate. In addition, because the plates are not often exactly perpendicular to the tubes, all plate pores cannot generally be observed in a slice. The literature 5,8 contains only a few transverse views on a level with a sieve plate where all pores are clearly visible. Fig. 2a shows the contours of the cross-sections of the plate and the pores from Soybean petiolar tissue, which were drawn from the main transverse view that was published by Fisher 8 .
Longitudinal views of phloem ( Fig. 1) are generally obtained from a tissue slice that does not pass exactly at the levels of the axes of the tube, plate and pores. However, they are near the tube and plate axes because it is easier to identify a cell such as a sieve tube if the pores are visible. Thus, short sections of plates are not observed. Tube or pore images depend on the tube or pore size, the position of the slice from their centre and the slice thickness. Elsewhere, greater tubes or pores have higher chances to be observed in the slice. Therefore, views such as Fig. 1 do not provide much directly useful information to estimate the sizes of the plate and pores and the number of pores.
Because these elements are only approximately cylindrical, it is desirable to use stereological methods 9 that are applicable regardless of the object shape. However, because these methods appear difficult to adapt to the present problem, whereas the objective is to study the mechanisms using classic flow laws, (e.g., Poiseuille's Law) which are based on perfect shapes such as cylinders, it is preferable to compare the tubes and pores to cylinders with cross-section areas equal to those of the tubes and pores, whose diameter is called equivalentdiameter.
I have recently proposed a method to estimate the numbers and diameter distributions of circles, cylinders and spheres in a matrix from their images, which are obtained by slices that section the matrix perpendicularly to a diameter, when the diameter distribution is perfectly or approximately symmetrical 10 . This method is applicable to opaque or transparent objects relative to matrix. It considers the slice thickness t and the lower image detection limit L S (when the shortest images cannot be observed). From this method, I developed a method 11 for planar objects that are only approximately circular with approximately known shapes. This method was successfully tested using simulated longitudinal slices, which were made in the previously reported transverse view from Fisher 8 .
In the here proposed method, these methods are used to estimate the equivalent-diameters of the plate and pores and the number of pores from longitudinal views. However, these methods, which assume representative samples of pores and plates, likely provide biased estimates when such views are obtained from slices that are near the plate centre, if the pore size varies with its distance from the plate edge. Therefore, the statistical relation between the The distance between the black arrows is 10.54 mm. The mean plate thickness is 0.77 mm. The image lengths of eight pores were measured as the lengths of the sides of the white rectangles along the plate image: from left to right: 0.52, 0.38, 0.61, 0.76, 0.76, 0.58, 0.41 and 0.31 mm. The image light contrast (minimum and maximum values) were modified using Image J to better show the pore edges, but this method does not change the pore image widths. The white arrow shows a site without clear pore image but with a long black zone of likely accumulation proteinaceous material. www.nature.com/scientificreports equivalent-diameters of the pores and their distance from the plate edge is estimated, and a plate reconstruction model is developed and introduced in the proposed method to consider this bias. The method was applied to a set of four longitudinal views, which were also published by Fisher 8 and taken from the same tissue as the previous transverse view from this author. The estimates were notably close to the values observed in Fisher's transverse view. The method was also successfully tested with images that were obtained from simulated slices in two sieve plates with their pores from two other species (Cucurbita maxima and Phaseolus vulgaris), which were published by Mullendore et al. 5 .

Results
The proposed method and its mathematical development. The proposed method considers sieve plates with their pores (Figs. 2a and 3a), which are sectioned by slices of thickness t perpendicular to the plates (Fig. 3b). When slices (Fig. 3c) are submitted to radiation, they produce 2D images (Figs. 1 and 3d) of the plate and pore sections existing in the slices. These images are comparable to images of transparent objects in an opaque matrix, which are sectioned by slices: their height (Fig. 3d), depends on the smallest object dimension, which delimits the radiation path. In the profile views, these images give lines of equal lengths to the previous height longitudinally sections the tube, it is perpendicular to the plate and pore cross-sections (b) and contains pore and plate sections (c). The smallest dimensions of these sections delimit the image size and particularly their height, which are obtained on planes parallel to the slices (d). In the profile view, the image length is equal to the previous height (e). The longest images of a circle with diameter D are provided by slices that pass through the circle centre (f), whereas the shorter images are provided by slices at distances shorter than D/2 2 t/2 (f) or much shorter if images smaller than L S cannot be observed (g).
www.nature.com/scientificreports SCIENTIFIC REPORTS | 4 : 4929 | DOI: 10.1038/srep04929 (Fig. 3e). These lines are thus also considered to be images of the pores and plates, and their length l is named the ''image length''. The image lengths vary with the plate or pore sizes, the slice thickness t and the position of the slice relatively to the plates and pores. For a perfectly cylindrical pore, the images are those of a transparent circle in an opaque matrix sectioned by slices perpendicular the circle. The longest image is obtained when the slice middle passes through the circle centre (Fig. 3f). Images that are shorter than a limit (which is named the lower image detection limit and labelled L S ) cannot be observed (Fig. 3g) because the resolving power of the observation system is too low or, in the case of plate images, only the slices that pass near the plate centre are considered. Because the pores are only approximately cylindrical, the image lengths are only approximately as it was described before.
On views such as Figs. 1 and 3d, the lengths of the plate and pore images and the distances of the pore images to the ends of the plate image can be measured.
The notation of the main variables of image length is provided in Table 1. The capital letter L marks the length of an image among the images that are obtained from a single circle or several circles with identical diameter, whereas the small letter l marks the length of an image among the images that are obtained from circles with different diameters. Similarly, the capital Greek letter L and the small greek letter l are for images of pores which are only approximately circular. L m , l m and l m are the means of the lengths L, l and l, respectively. D lm is the diameter of the circle of which all images that are obtained from slices spaced by small distances in all possible configurations have for mean length L m 5 l m . Similarly, D lm is the diameter of the circle of which the mean image length, L m , is equal to l m .
Estimation of the pore equivalent-diameter distribution and the number of pores per unit plate area, assuming that the pore images are obtained from a sample representative of the plate. First, assume that the cross-sections of all pores have identical form.
Let l m and cv l be the mean and the coefficient of variation of the pore image lengths.
From l m and cv l , the proposed method estimates, in a first phase, the mean l m and the coefficient of variation cv l of the image lengths that would be obtained if the cross-section of each pore was a perfect circle with identical area as the pore cross-section area, its diameter being called equivalent-diameter of the pore. In a second phase, the method estimates the mean D m and the coefficient of variation cv D of the diameters of these perfect circles from l m and cv l using the previously proposed method to estimate the diameter distribution of circles in a matrix which is sectioned by slices when the diameter distribution is perfectly or approximately symmetrical 10 .
In this later method, D m and cv D are estimated by considering a fictive circle of diameter D lm . This diameter D lm is calculated so that the mean image length L m of this circle over all possible configurations of the slice is equal to the mean length l m of the images of the circles provided by the slices. For given values of t and L S , D lm is calculated from l m by successive approximations, based on a number of relations (see Supplementary Note on line) using Supplementary Dataset 1 on line. For these calculations, the smallest and the largest measured image lengths, L min and L max , are assumed to be the central values of the smallest and the largest image length classes. Therefore, L S is approximated by L min 2 (L max 2 L min )/(m 2 1)/2, where m is the number of images.
Here, it is assumed that, if the pore cross-sections are sufficiently approximately circular, the cross-section area of a fictive pore that generates the mean image length l m is approximately equal to the area of a fictive circle of diameter D lm , that generates the mean image length L m equal to the mean length l m of the images of all circles with identical areas to the pore cross-sections.
Therefore, in the first phase, D lm is estimated as follows. D lm is assumed to be near the diameter D lm of a circle, which mean image length is l m . Therefore, an approximate value of D lm is calculated from l m , t and L S using Supplementary Dataset 1 on line. Then, a number w of surfaces (for example, one hundred surfaces) with the The surfaces were at distances that vary by 0.06 mm from the slice, which is 0.07 mm thick. Surfaces a9, b9, c9 and d9 were turned by an angle of 18u relative to surfaces a, b, c and d. The images of surfaces a, b, a9, b9 and c9 were the smallest sectioned sides of the slice (heavy lines). Surface c did not have an image because the smallest sectioned side of the slice was shorter than the lower image detection limit L S . Surfaces d and d9 did not have an image because they were not sectioned by the two slice sides.  form of the pore cross-sections and with the area of the circle of diameter D lm (i.e., p D lm 2 /4) are drawn at different positions relative to perpendicular slices, whose thickness is t. The surfaces are placed at w 0.5 (for example, ten) different distances, which vary by a slightly greater value than 1/(w 0.5 times D lm ) from the slice, and the surfaces are turned by w 0.5 (for example, ten) different angles, which vary by 180/w 0.5 degrees (Supplementary Figs. S1, S2 and S3 on line). The image lengths, which are labelled L 0 , of the w surfaces are measured as shown in Fig. 4, and the mean (L 0 ) m of the image lengths that are longer than L S is calculated. Note that the coefficient of variation cv L 0 of these lengths can be calculated for further reference.
Assuming that the mean image length of a circle is approximately proportional to the diameter when it slightly varies from D lm to D lm , D lm is estimated to be Eventually, more precise approximations of D lm can be successively made using surfaces with the area of the previously estimated diameter D lm (i.e., p D lm 2 /4) until (L 0 ) m becomes notably close to l m . Then, the mean image length L m of the circle of diameter D lm is calculated from D lm , t and L S using Supplementary Dataset 1 on line. Then, the mean length l m of the images of the circles with identical areas as the pore cross-sections is estimated: by definition, it is equal to L m .
The coefficient of variation cv l of these images is calculated as follows (see Supplementary Note on line): where cv L is the coefficient of variation of the image length of the circle with diameter D lm ; cv L is calculated from D lm , t and L S by Supplementary Dataset 1 on line. The coefficient cv L is the coefficient of variation of the image length L of the surface with identical area as the circle area with the estimated diameter D lm . In principle, it would be estimated as previously with a number of surfaces with identical area as the circle of diameter D lm . However, when D lm is close to D lm , this coefficient of variation is notably close to cv L 0 if t and L S are not too high. Therefore, cv L can be merely approximated using cv L 0 .
In the second phase, the parameters of the equivalent-diameter distribution are estimated as follows (see Supplementary Note on line): and D m is: where (c 1 ) Dlm and (c 2 ) Dlm are calculated with respect to D lm using equations (18) and (19) in Supplementary Note on line. Indications have been provided 10 about selecting the sign 6 in equation (3). Finally, the estimated number of pores per unit plate area, n A , is (see Supplementary Note on line): where L is the total slice length and (c 1 ) Dm , which is different from (c 1 ) Dlm , is calculated with respect to D m using equation (18) in Supplementary Note on line.
To verify that the sign 6 of equation (3) is correctly selected, the distribution of the pore image length, which is calculated from the normal equivalent-diameter distribution defined by D m and cv D , is compared to the distribution of the measured pore image lengths. For this calculation, the frequencies of the images of 100 circles with the normal diameter distribution (D m , cv D ), which are obtained from 100 slices equally spaced by one hundredth of the largest diameter, are calculated. Then, the limits of each class are multiplied by the ratio l m /l m . However, this verification may be impossible when different forms are distinguished (see below).
When q forms of pore cross-sections are distinguished, the previously described measurements and calculations are successively applied to each form. The forms are assumed present in all size classes. The calculated means of D m and n A are equal to the means of each form weighted by their proportions (respectively: of the form i, and i varies from 1 to q), and the coefficient of variation of the equivalent-diameter is calculated as follows (see Supplementary Note on line): To verify that the distribution of the pore image lengths which is calculated from the estimated normal equivalent-diameter distribution, which is defined by the previous means of D m and cv D , is comparable to the distribution of the measured pore image lengths, the procedure previously indicated to verify the selection of the root of equation (3), is applied using these means of D m and cv D . However, the limits of each class are calculated with the ratio l m /l m , where l m is equal to the sum of the estimated values of l m for each form, multiplied by the percentage of each form and the number of images of each population (calculated from equation (5)) and divided by the total number of images.
Estimation of the equivalent-diameter of the plates where the pores are observed. The plate image lengths are compared to image lengths of transparent circles in an opaque matrix, which are obtained from slices perpendicular to the circle plane and of thickness t.
With a notably small number of sieve plate images, the plate equivalent-diameter distribution is unlikely assessable and the pore cross-section form cannot be considered. Therefore, the plate images are assumed to be taken from perfectly cylindrical plates with identical diameter. This diameter, which is labelled P, is calculated using the mean and the coefficient of variation of the image length based on the relations in the circle (Supplementary Note on line) using Supplementary Dataset 1 on line.
With a greater number of sieve plate images, the value L S and the plate equivalent-diameter distribution can be estimated similarly to the previous case of pores from the plate image lengths.
Estimation of the relation between the pore equivalent-diameter and the minimal distance from the pore centre to the plate edge. When pore images are obtained from slices near the plate centre, the proportion of images obtained from the pores near the plate centre is higher than the proportion of these pores in the entire plate. Therefore, if, for example, the pores near the centre are greater, the mean pore size should be overestimated.
The possible variation of the pore equivalent-diameter with the minimal distance h from the pore centre to the plate edge is assessed as follows. First, the minimal distance h 0 from the pore image centre to the plate edge is calculated according to Fig. 5a using the value of P, the image lengths of the pores and plate, and the distance between their ends.
Let the statistic relation between the pore image length l and h 0 be and the part of the variance of l that is explained by the regression (7) be R 2 . Because the expected mean ratio of the pore equivalent-diameter to the image lengths is D m /l m , whereas the expected mean of the differences (h 0 2 h) is zero, the pore equivalent-diameter is expected vary with h according to the equation: Proposed geometric reconstruction model to estimate the pore equivalent-diameter distribution over the entire plate. If the previous  values of cv D and D m are estimated from pore images that are obtained by slices rather near the plate centre, and if D tends to vary with h, the following geometric reconstruction model of a plate view is used to re-estimate cv D and D m for the entire plate. On a plane (x, y), the coordinates of the centres of the circles with diameter d i,j (see below) which represent pore cross-sections within a circle of diameter P that represents a plate cross-section, are calculated as follows. In the square with centre (P/2, P/2) and side P (Fig. 6), n 2 squares are delimited. Each square has side P/n and two sides that are parallel to the y axis, with the abscissa: where i is an integer between 0 and n 2 1. The other two sides of the squares are parallel to the x axis, with the ordinates: y j~j P=n, and y jz1~j z1 ð ÞP=n ð10Þ where j is an integer between 0 and n 2 1. In each square, a circle of diameter d i,j is placed. This diameter d i,j is equal to the sum of the value provided by equation (8) and a random value taken from the normal distribution with zero for mean and (1 2 R 2 ) 3 (D m cv D ) 2 for variance, where (1 2 R 2 ) is the part of the variance of l that is not explained by h 0 . The coordinates of the circle centre are: where a and b are random numbers within 0-1. However, if d i,j is greater than P/n, then d i,j is equal to P/n. If the distance calculated by Pythagoras' theorem between the centre of a circle with diameter d i,j and the centre of the circle with diameter P is greater than P/2 2 d i,j / 2, the circle with diameter d i,j is not included in the plate. The number of reconstructed circles is counted and the mean and the coefficient of variation of their diameters are calculated.
Estimation of the distribution of the pore equivalent-diameter over the entire plate using the proposed geometric reconstruction model. Assume that the pore cross-section centres are uniformly dispersed over the plate. Then, the estimated number of pores n in the entire plate is equal to n A multiplied by the plate area, which is p P 2 /4: The previous proposed geometric reconstruction model is applied with some values of n close to n 0.5 (so that n 2 varies around n). The image lengths of the four plates and 23 pores in the four longitudinal microphotographs by Fisher 8 were measured, according to Fig. 1, including the callose, i.e., the white region, because it likely resulted from an artefact as reasoned by Fisher. The mean and the coefficient of variation of these lengths were calculated considering the magnification of each microphotograph. The distances of one end of the pore images to one edge of the plate images were measured.
In Fisher's Fig. 18, the contours of the cross-sections of the plate and its pores were drawn (Fig. 2a). The plate contour was delimited to the beginning of the dark cylindrical region surrounding the central part. The areas s of the plate or pore cross-sections were measured considering the magnification of each microphotograph. Their equivalent-diameters were calculated to be (4 s/p) 0.5 . The minimal distances of the centres of the circles circumscribed to the plate or pore cross-sections were measured.
In Fisher's Fig. 20, the area of the tube/plate section was measured, and the plate equivalent-diameter was calculated. Datasets from Mullendore et al. 5 and measurements. Mullendore et al. 5 published two cross-sectional views of phloem sieve plates of Cucurbita and Phaseolus (Figs. 3c-d from the authors), which were obtained using electron microscopy scanning. The outlines of the plate and pore cross-sections were drawn. Based on the indicated scale, the areas of the plate or pore cross-sections were measured, their equivalent-diameters were calculated and the minimal distances of the centres of the circles circumscribed to the pores to the plate edge were measured as for the dataset from Fisher 8 .
To simulate longitudinal views of these two sieve plates on an Excel Sheet, I superimposed these images with a grid of parallel strips, with 12 mm spacing for Cucurbita or 3 mm spacing for Phaseolus, to represent the intersections with slices of thickness t 5 0.07 mm. Then, whenever both sides of a strip section passed through a pore cross-section, the coordinates of the ends of the shortest side (Fig. 4) which was considered as the image length were noted. Similarly, the plate image length was obtained. The process was repeated for five other orientations of the grid, each of which was turned by 30 degrees with respect to the preceding position. The mean l m and the coefficient of variations cv l of the image lengths of the plate and pores were calculated.
The pore cross-sections were assumed to have identical forms as in the plate from Fisher 8 . Therefore, the method was applied with three previous pore forms 1, 2 and 3.
Estimates obtained by applying the method to the dataset from Fisher 8 and comparison with the true values. The estimates were firstly obtained without reconstructing the plate. Then, the equivalent-diameter of the plate and the relation between the equivalent-diameter and h were also estimated. The estimates were then obtained using the reconstruction model and compared to the observations. Pore equivalent-diameter distribution and number of pores estimated without reconstructing the plate. Based on 23 pore image lengths ( Fig. 7 and Supplementary Table S1 on line), L S , l m and cv l were equal to 0.093 mm, 0.359 mm and 0.413, respectively.
Based on L S , l m and t 5 0.07 mm, D lm was estimated to be 0.470 mm. The pore cross-sections were successively approximated by one of the three forms observed in the transverse view (Fig. 2a) and labelled 1 (rather rounded), 2 (rather oblong) or 3 (with some concavities), which were selected to be representative of this view. One hundred surfaces with identical area to the circle of diameter D lm and shaped like one of the forms 1, 2 and 3 were drawn at 100 different positions relative to perpendicular slices, which are 0.07 mm thick (Supplementary Figs S1, S2, and S3 on line). These surfaces  were placed at ten different distances, varied by 0.06 mm (which is slightly longer than one-tenth of D lm ) from the slice, and turned by ten different angles that varied by 18 degrees. The values of (L 0 ) m , D lm , l m , cv L and cv l are provided in Table 2 for the three forms. The estimates of cv D , D m and n A varied according to the pore cross-section form ( Table 2).
Assuming that the forms 1, 2 and 3 have equal proportions (hypothesis H1), the estimated mean values of D m , cv D and n A were 0.458 mm, 0.259 and 3.20 mm 22 , respectively. Assuming that these proportions are 42, 30 and 28% (hypothesis H2), for further reference, these values were 0.453 mm, 0.265 and 3.24 mm 22 . Under both H1 and H2, the estimated image length distributions were consistent with the distribution of the 23 pore image lengths, although the estimated images were slightly longer (Fig. 7).
Estimated equivalent-diameter of the plate. Four views were assumed to be taken from i) plates with identical size ii) at short distances from the plate cross-section centres because Fisher 8 likely selected views that showed a relatively large number of pores, which also contributed to identifying the sieve tubes. The image lengths of the four plates in these four views were compared to the image lengths of transparent circles with identical diameter P in an opaque matrix, which were obtained from perpendicular slices of thickness t 5 0.07 mm. From the mean and the coefficient of variation of these observed image lengths (Supplementary Table S1 on line), the values of L S and P were calculated to be 3.589 and 5.312 mm, respectively.
Estimated relation between the equivalent-pore diameter and the minimal pore distance to the plate edge (h). The pore image length l tended to increase with h 0 according to the regression in Fig. 5b. The low significance level was assumed to be largely because of the random distances of the slices from the pore cross-section centres, whereas the number of pores was small. Based on this regression and equation (8), the pore equivalent-diameter (mm) was estimated to vary as shown in Fig. 5c with respect to the minimal distance h between the pore cross-section centre and the plate edge (mm): Pore equivalent-diameter distribution and number of pores over the entire plate which were estimated using the proposed geometric reconstruction model. Because the number of pores in the entire plate that was estimated using equation (13) was 71.0 or 71.8 under H1 or H2, respectively, the proposed geometric reconstruction model was applied with n equal to 8, 9 and 10. One thousand replications of the plate reconstruction were made for each value of n under H1 and H2. Fig. 2b shows a replication with n 5 10 under H2. Under H2, the estimates of D m , cv D and n A which were extrapolated from the means that were obtained for n 5 9 and n 5 10 (Supplementary Table S2 on line), were 0.411 mm, 0.256 and 3.24 mm 22 . Fig. 7 shows the normal distribution of the pore equivalent-diameter based on these values of D m and cv D . Under H1, these numbers were only slightly different (0.415 mm, 0.251 and 3.20 mm 22 ). All previous results were only slightly changed when 10,000 replications were made.
Comparisons between the estimates obtained from the longitudinal views and the observations in the entire transverse plate view from Fisher 8 . There were 53 pores in the sieve plate in the main transverse view published by Fisher 8 (his Fig. 18) as previously reported and drawn in Fig. 2a. The plate equivalent-diameter was 4.647 mm. The pore cross-sections were almost uniformly distributed over the plate (Fig. 2a). The pore equivalent-diameter distribution was roughly normal (Fig. 7), and the equivalent-pore diameter tended to increase with the minimal distance from the pore cross-section centre to the plate edge according to the regression in Fig. 5c. The values of D m , cv D and n A obtained from 53 pores were 0.405 mm, 0.272 and 3.13 mm 22 , respectively.
The plate was reconstructed using the proposed geometric model with P 5 4.647 mm and different values of n: 8, 9 and 10. Each d i,j was equal to the sum of the value calculated by the regression in Fig. 5c and a random value from the reduced normal distribution with variance (1 2 R 2 ) 3 (D m cv D ) 2 , where R 2 is provided in Fig. 5c, and D m and cv D are 0.405 mm and 0.272, respectively. One thousand replications of the plate reconstruction were made for each value of n. Fig. 2c shows a replication with n 5 9. Because the number of pores over the entire plate is 53, the estimates of D m , cv D and n A , which were extrapolated from the means obtained for n 5 8 and 9 www.nature.com/scientificreports (Supplementary Table S2 on line), were 0.418 mm, 0.232 and 3.13 mm 22 , respectively. Fig. 7 shows the normal distribution based on these values of D m and cv D .
In the partial and almost transverse view of a sieve tube from Fisher 8 (his Fig. 20), the tube equivalent-diameter was estimated to be approximately 5.4 mm.
Estimates obtained by applying the method to the dataset from Mullendore et al. 5 and comparison with the true values. The values of P, D m , cv D and n A that were estimated from the measured lengths l of the pore and plate images obtained by the simulated slices in the two sieve plates of Cucurbita or Phaseolus phloem were notably near the true values ( Table 3). The pore equivalent-diameter (mm) that was calculated from the measured pore cross-section area tended to increase with its measured minimal distance h (mm) to the plate edge according to the regressions D 5 1.49 h 1 1.09 and D 5 0.82 h 1 2.03 in Cucurbita and Phaseolus, although the R 2 were low (0.27 and 0.04), respectively.
Because the grid was superimposed on the entire of the plate crosssection, the geometric reconstruction model was not applied.

Discussion
The estimated plate equivalent-diameter (5.312 mm) in the four longitudinal views from Fisher 8 was in the range (4.647-5.4 mm) of two plates that were observed entirely or partly in transverse views. This estimate was perhaps imprecise because based on only four plate images, but the consequences of this imprecision were likely limited to the number of plate pores and the diameters of the pores in the periphery.
Without reconstructing the plate, the estimates of D m , cv D and n A obtained from the longitudinal views were 0.453 mm, 0.265 and 3.24 pores per mm 2 , respectively, under hypothesis H2. Under H1, they were only slightly different. It was verified that the pore image length distribution that was predicted from the estimates of cv D and D m for the three pore forms by assuming normal equivalent-diameter distributions was largely consistent with the data. The observed deviation could be partly due to the equivalent-diameter distributions which were likely only approximately normal, like that observed in the transverse view (Fig. 7).
The reconstruction model was strongly validated by its application to the transverse view: i) the pore centres were uniformly dispersed over the plate (Fig. 2a); ii) the pore equivalent-diameter tended to increase with the pore distance from the plate edge; iii) the estimates of D m , cv D and n A obtained by reconstruction were close to their true values (0.418 vs 0.405 mm, 0.232 vs 0.272 and 3.13 vs 3.13 pores per mm 2 , respectively). When the reconstruction model was applied to the four longitudinal views, D m was smaller (0.411 vs 0.456 mm under H2) than the value which was estimated without reconstruction, which was consistent with the higher proportion of smaller pores in the plate periphery.
Finally, the proposed method provided notably close estimates to the observed values in the transverse view (0.411 vs 0.405 mm for D m , 0.256 vs 0.272 for cv D and 3.24 vs 3.13 pores per mm 2 for n A ), although the number of pore images was small (23). The small difference in diameter was consistent with a study 10 with bonbons using the Monte Carlo method with 100 simulations, where the slices were made at random distances from the bonbon centres: the deviation of the estimated bonbon diameter from its true value was lower than 5% in 95 cases of approximately 30 bonbon images. The estimates under H1 were only slightly different from those under H2.
Evidently, it is not certain that the pores were exactly comparable in the longitudinal views and the transverse view. However, the comparison heavily validates the proposed method because these views were taken from the same tissue (one to three adjacent vascular bundles of petiole). It is difficult to observe exactly identical plates in both longitudinal and transverse views.
This application to the dataset from Fisher 8 shows the method feasibility in real observation conditions of the phloem.
The proposed method assumes that the pore cross-section form is known. The examples from the datasets from Mullendore et al. 5 indicate that the identified forms in the plate from Fisher 8 are sufficiently common, so the method can provide useful approximate values of the size and number of pores for many species. Incomplete cross-sectional views can provide indications about the pore form.
The reconstruction model assumes that the pores are randomly distributed over the entire plate. This assumption was acceptable in the plate of the figure from Fisher 8 . In the plates of the figures from Mullendore et al. 5 , the pore density was higher in the periphery of the plates. Therefore, the total number of pores in the plate can be underestimated by the model. Thus, it is important to observe longitudinal views with short plate images.
In conclusion, the proposed method appears able to provide even with only a few dozen images, valuable estimates of the sizes and numbers of pores from longitudinal views when the pore form is approximately known. It is expected to be useful for many studies on phloem sieve tubes and modelling sap flow, particularly on the phloem hydraulic conductivity, which likely varies with the number and the diameter of the pores according to a previously proposed equation 7 , when the pores have identical size. The method is also expected to be applicable for other tissues and materials with approximately cylindrical structures when longitudinal slices can be obtained.