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Sci Rep. 2014; 4: 4929.

Published online 2014 May 12. doi: 10.1038/srep04929

PMCID: PMC5381363

Received 2013 August 23; Accepted 2014 April 23.

Copyright © 2014, Macmillan Publishers Limited. All rights reserved

This work is licensed under a Creative Commons Attribution 3.0 Unported License. The images in this article are included in the article's Creative Commons license, unless indicated otherwise in the image credit; if the image is not included under the Creative Commons license, users will need to obtain permission from the license holder in order to reproduce the image. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/

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.

The pores of the sieve plates (Figs. 1, ,2a2a and and3a)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 μm thick on a level with a plate is notably improbable because of the small sieve plate thickness (only approximately 0.5–1 μm), 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 equivalent-diameter.

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 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}.

The proposed method considers sieve plates with their pores (Figs. 2a and and3a),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 and3d)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 (Fig. 3e). These lines are thus also considered to be images of the pores and plates, and their length *λ* 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 and3d,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 *Λ* and the small greek letter *λ* are for images of pores which are only approximately circular. *L*_{m}, *l*_{m} and *λ*_{m} are the means of the lengths *L*, *l* and *λ*, respectively. *D _{l}*

First, assume that the cross-sections of all pores have identical form.

Let *λ*_{m} and *cv _{λ}* be the mean and the coefficient of variation of the pore image lengths.

From *λ*_{m} and *cv _{λ}*, the proposed method estimates, in a first phase, the mean

In this later method, *D*_{m} and *cv _{D}* are estimated by considering a fictive circle of diameter

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 *λ*_{m} is approximately equal to the area of a fictive circle of diameter *D _{l}*

Therefore, in the first phase, *D _{l}*

Assuming that the mean image length of a circle is approximately proportional to the diameter when it slightly varies from *D _{λ}*

Eventually, more precise approximations of *D _{l}*

Then, the mean image length *L*_{m} of the circle of diameter *D _{l}*

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

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})_{Dl}_{m} and (*c*_{2})_{Dl}_{m} are calculated with respect to *D _{l}*

where is the total slice length and (*c*_{1})_{D}_{m}, which is different from (*c*_{1})_{Dl}_{m}, is calculated with respect to *D*_{m} using equation (18) in Supplementary Note on line.

To verify that the sign ± 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 (

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: labelled and where *G*_{i} is the proportion 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

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.

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 *λ* and *h*_{0} be

and the part of the variance of *λ* 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}/*λ*_{m}, whereas the expected mean of the differences (*h*_{0} − *h*) is zero, the pore equivalent-diameter is expected vary with *h* according to the equation:

If the previous values of *cv _{D}* and

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), *ν*^{2} squares are delimited. Each square has side *P*/*ν* and two sides that are parallel to the *y* axis, with the abscissa:

where i is an integer between 0 and *ν* − 1. The other two sides of the squares are parallel to the *x* axis, with the ordinates:

where j is an integer between 0 and *ν* − 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 − *R*^{2}) × (*D*_{m}
*cv _{D}*)

where *a* and *b* are random numbers within 0–1. However, if *d*_{i,j} is greater than *P*/*ν*, then *d*_{i,j} is equal to *P*/*ν*. 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 − *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.

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 π *P*^{2}/4:

The previous proposed geometric reconstruction model is applied with some values of *ν* close to *n*^{0.5} (so that *ν*^{2} varies around *n*). A number (for example, 1000) of replications of plate reconstruction are made for each value of *ν*. The values of *D*_{m}, *cv _{D}* and

The method was tested with three datasets: one from Fisher^{8} with true longitudinal views and one transverse view from the same tissue and two from Mullendore et al.^{5} with transverse views, from which I simulated longitudinal views.

Based on a set of electron microscopic photographs of soybean leaf petiole phloem, Fisher^{8} showed that the callose formations, which may occur in the sieve plate pores, could result from artefacts because of sample preparation. This set included seven longitudinal views with an entire sieve plate section, one transverse view (Fisher's Fig. 18) with an entire sieve plate with its pores and another transverse view with a sieve tube, which is at the level of an incomplete plate with only a few visible pores (Fisher's Fig. 20). I selected four of the seven longitudinal microphotographs (Fisher's Figs. 11, 12, 21 and 24) because in the other three longitudinal views, the limits of some pores were not clearly visible or the tubes showed signs of immaturity (Fisher's Figs. 15, 22 and 23).

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*/π)^{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.

All data and calculations are provided in Supplementary Dataset 2 on line.

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 μm spacing for Cucurbita or 3 μm spacing for Phaseolus, to represent the intersections with slices of thickness *t* = 0.07 μm. 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 *λ*_{m} and the coefficient of variations *cv _{λ}* 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.

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.

Based on 23 pore image lengths (Fig. 7 and Supplementary Table S1 on line), *L _{S}*,

Based on *L*_{S}, *λ*_{m} and *t* = 0.07 μm, *D _{λ}*

The estimates of *cv _{D}*,

Assuming that the forms 1, 2 and 3 have equal proportions (hypothesis H1), the estimated mean values of *D*_{m}, *cv _{D}* and

*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* = 0.07 μm. 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 μm, respectively.

*Estimated relation between the equivalent-pore diameter and the minimal pore distance to the plate edge (*h*).* The pore image length *λ* 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 (μm) 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 (μm):

Although the regression in Fig. 5b explained only a notably small part (*R*^{2} = 0.075) of the pore image length variations, equation (14) suggested possible large variations of *D* (between 0.313 μm for pores notably near the edge and 0.509 μm for pore cross-sections near the plate centre under H2).

*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 *ν* equal to 8, 9 and 10.

One thousand replications of the plate reconstruction were made for each value of *ν* under H1 and H2. Fig. 2b shows a replication with *ν* = 10 under H2. Under H2, the estimates of *D*_{m}, *cv _{D}* and

*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 μm. 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

The plate was reconstructed using the proposed geometric model with *P* = 4.647 μm and different values of *ν*: 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 −

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 μm.

The values of *P*, *D*_{m}, *cv _{D}* and

Because the grid was superimposed on the entire of the plate cross-section, the geometric reconstruction model was not applied.

The estimated plate equivalent-diameter (5.312 μm) in the four longitudinal views from Fisher^{8} was in the range (4.647–5.4 μm) 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

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

When the reconstruction model was applied to the four longitudinal views, *D*_{m} was smaller (0.411 *vs* 0.456 μm 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 μm for *D*_{m}, 0.256 *vs* 0.272 for *cv _{D}* and 3.24

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.

I greatly thank B. Delécolle (INRA, France) who guided me through problems raised in electron microscopy and who made observations of tomato pedicel tissue, particularly the photo in Fig. 1, and N. Bertin for notably helpful suggestions on a previous version of the manuscript.

The authors declare no competing financial interests.

**Author Contributions** P.B. contributed extensively to the work presented in this paper.

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