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## Sieve Elements - Comparative Structure, Induction and Development | H.-D. Behnke | Springer

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Seller Details View Store. Expand your business to millions of customers Sell this item on Snapdeal. Sold by. Sell on Snapdeal. Explore More Biology Books. More Biology Books From Books. In Same Price. Buy now Loading Easy Return Policy. The proposed method considers sieve plates with their pores Figs. When slices Fig. These images are comparable to images of transparent objects in an opaque matrix, which are sectioned by slices: their height Fig.

In the profile views, these images give lines of equal lengths to the previous height Fig. 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.

Images that are shorter than a limit which is named the lower image detection limit and labelled L S cannot be observed Fig. Because the pores are only approximately cylindrical, the image lengths are only approximately as it was described before. On views such as Figs. 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.

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 In this later method, D m and cv D are estimated by considering a fictive circle of diameter D l m. This diameter D l m 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.

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For given values of t and L S , D l m 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, in the first phase, D l m is estimated as follows. The surfaces are placed at w 0. S1, S2 and S3 on line. The surfaces were at distances that vary by 0. 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. Then, the mean image length L m of the circle of diameter D l m is calculated from D l m , 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 l m ; cv L is calculated from D l m , t and L S by Supplementary Dataset 1 on line. In principle, it would be estimated as previously with a number of surfaces with identical area as the circle of diameter D lm. 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 m using equations 18 and 19 in Supplementary Note on line.

Finally, the estimated number of pores per unit plate area, n A , is see Supplementary Note on line : 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. For this calculation, the frequencies of the images of circles with the normal diameter distribution D m , cv D , which are obtained from slices equally spaced by one hundredth of the largest diameter, are calculated.

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: 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 D m and cv D.

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.

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 each square, a circle of diameter d i,j is placed. The coordinates of the circle centre are: where a and b are random numbers within 0—1. The number of reconstructed circles is counted and the mean and the coefficient of variation of their diameters are calculated. A circle of diameter d i,j , which represents a pore cross-section dotted line , is shown in the small square between abscissa 2. Assume that the pore cross-section centres are uniformly dispersed over the plate. 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.

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. I selected four of the seven longitudinal microphotographs Fisher's Figs. The image lengths of the four plates and 23 pores in the four longitudinal microphotographs by Fisher 8 were measured, according to Fig. 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. 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. The minimal distances of the centres of the circles circumscribed to the plate or pore cross-sections were measured.

All data and calculations are provided in Supplementary Dataset 2 on line. Mullendore et al. 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. Then, whenever both sides of a strip section passed through a pore cross-section, the coordinates of the ends of the shortest side Fig. 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 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.

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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. The distribution of the pore image lengths is estimated from three normal equivalent-diameter distributions the values of D m and cv D are provided in Table 2 , which are estimated for three pore cross-section forms in Fig.

The pore cross-sections were successively approximated by one of the three forms observed in the transverse view Fig. These surfaces were placed at ten different distances, varied by 0. 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. 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.

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. 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.

Estimated relation between the equivalent-pore diameter and the minimal pore distance to the plate edge h.

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. 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 Under H1, these numbers were only slightly different 0.

All previous results were only slightly changed when 10, 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. The plate equivalent-diameter was 4. The pore cross-sections were almost uniformly distributed over the plate Fig. The pore equivalent-diameter distribution was roughly normal Fig. The values of D m , cv D and n A obtained from 53 pores were 0.

Each d i,j was equal to the sum of the value calculated by the regression in Fig. In the partial and almost transverse view of a sieve tube from Fisher 8 his Fig. 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. 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.

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. The reconstruction model was strongly validated by its application to the transverse view: i the pore centres were uniformly dispersed over the plate Fig.

When the reconstruction model was applied to the four longitudinal views, D m was smaller 0. Finally, the proposed method provided notably close estimates to the observed values in the transverse view 0. 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.

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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. 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. 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.