Quantifying leaf venation patterns: two-dimensional maps


(fax +1 613 562 5486; e-mail arolland@uottawa.ca).


The leaf vasculature plays crucial roles in transport and mechanical support. Understanding how vein patterns develop and what underlies pattern variation between species has many implications from both physiological and evolutionary perspectives. We developed a method for extracting spatial vein pattern data from leaf images, such as vein densities and also the sizes and shapes of the vein reticulations. We used this method to quantify leaf venation patterns of the first rosette leaf of Arabidopsis thaliana throughout a series of developmental stages. In particular, we characterized the size and shape of vein network areoles (loops), which enlarge and are split by new veins as a leaf develops. Pattern parameters varied in time and space. In particular, we observed a distal to proximal gradient in loop shape (length/width ratio) which varied over time, and a margin-to-center gradient in loop sizes. Quantitative analyses of vein patterns at the tissue level provide a two-way link between theoretical models of patterning and molecular experimental work to further explore patterning mechanisms during development. Such analyses could also be used to investigate the effect of environmental factors on vein patterns, or to compare venation patterns from different species for evolutionary studies. The method also provides a framework for gathering and overlaying two-dimensional maps of point, line and surface morphological data.


Plant vascular networks play crucial roles in transport and mechanical support. Leaf venation patterns vary between species. Recent work has shown that leaf vein architecture limits photosynthesis via its effect on hydraulic efficiency (Brodribb et al., 2007), and that different types of venation architectures display different tolerances to hydraulic disruption through wounding (Sack et al., 2008). Understanding how vein patterns develop therefore has many implications from both physiological and evolutionary perspectives.

How vascular network patterns arise from non-differentiated cells within a growing leaf is not well understood, but much progress has been made in recent years using the model plant Arabidopsis. During leaf development in Arabidopsis, veins form reticulations, and the pattern becomes more intricate as the leaf develops (Candela et al., 1999; Scarpella et al., 2006; Wenzel et al., 2007). The main approaches used to investigate leaf vein patterning have included genetic analyses, pharmacological approaches and theoretical modeling.

Many mutants have been shown to display altered leaf venation patterns (Cnops et al., 2006; Gonzalez-Bayon et al., 2006; Petricka and Nelson, 2007; reviews by Scarpella and Meijer, 2004; Sieburth and Deyholos, 2006). In most cases, corresponding genes have been identified. Moreover, reporter constructs have been generated (e.g. Athb8::GUS, PIN1:GFP) that allow vein patterning events to be described in great detail (Scarpella et al., 2004, 2006; Wenzel et al., 2007). The effect of pharmacological treatments on vein formation has been studied. For example, auxin transport has been linked to vein pattern formation, as treatment with auxin transport inhibitors results in altered vasculature (Mattsson et al., 1999, 2003; Scarpella et al., 2006; Sieburth, 1999).

Using such experimental approaches, it has been shown that expression of the MONOPTEROS (MP) gene, which is involved in auxin signaling and vein patterning (Hardtke and Berleth, 1998), is up-regulated by auxin and that the location of its expression progressively narrows from broad domains to future sites of vein formation (Wenzel et al., 2007). Expression of MP overlaps with that of PIN1, a protein involved in auxin efflux that ‘predicts’ the location of vein formation (Scarpella et al., 2006; Wenzel et al., 2007). Together, these results support the canalization hypothesis (Sachs, 1969, 1981, 1989), which states that a signal, probably involving auxin, enhances its own transport to create channels of increased signal flux that later become veins. However, the molecular mechanisms involved are unclear.

The aim of theoretical models is to determine what mechanism underlies vein pattern formation and to simulate how the patterns self-organize. Models of leaf vein patterning are loosely based on current experimental knowledge and were recently reviewed by Merks et al. (2007). At present, models are not sufficiently constrained (i.e. lack of experimental data means that many models with different sets of assumptions are plausible) and are not quantitative. Therefore, they are difficult to validate or disprove.

An alternative approach is needed to provide a link between current experimental and theoretical studies. We propose that a useful alternative approach is to quantify vein patterns spatially at the tissue level throughout leaf development. Veins develop within the leaf tissue, and, in the case of reticulate venation, the areas enclosed by veins, called areoles or sometimes loops (Scarpella et al., 2006), enlarge and are split by new veins as the leaf grows (Figure 1). This suggests a spacing mechanism that allows new veins to form within a loop once it reaches a certain area. Therefore, quantifying vein patterns spatially at the tissue level, not only in terms of vein densities but also in terms of loop shapes and sizes, should provide new insights into patterning mechanisms. Possible applications would be the comparison of mutant phenotypes, plants treated with substances affecting vein patterning, or plants grown under different environmental conditions. Quantitative pattern data could also be extracted from modeled vasculature, providing a way to test the outcome of pattern formation models against experimental data, thereby providing a link between current experimental and theoretical studies.

Figure 1.

 Example of leaf vein pattern.
The diagram is based on a first rosette leaf of Arabidopsis harvested 9 days after sowing (DAS). The arrow indicates a loop (an area enclosed by veins) that is being split by a vein segment.

So far, vein patterning defects in mutants have mostly been assessed qualitatively using image data (Candela et al., 1999; Scarpella et al., 2004, 2006; Schuetz et al., 2007; Sieburth, 1999; Wenzel et al., 2007), making it difficult to compare phenotypes and investigate patterning mechanisms. In some cases, venation patterns have been quantified to some extent by measuring characteristics such as vein density, angles between vein segments, vein thickness or radius, and the number of branching points, free vein endings and loops (Alonso-Peral et al., 2006; Bohn et al., 2002; Candela et al., 1999; Hill, 1980; Kang and Dengler, 2004; Scarpella et al., 2004; Steynen and Schultz, 2003; Ueno et al., 2006). One study estimated loop areas in grasses, by multiplying longitudinal and transversal vein segment length (Ueno et al., 2006). Current methods do not provide the necessary spatial resolution to investigate patterning processes occurring across the leaf.

In this paper, we present a method for quantifying vein patterns in space that can be used to investigate any visible network pattern, such as xylem elements or gene expression markers used to visualize vascular precursors (Bauby et al., 2007; Konishi and Yanagisawa, 2007; Sawchuk et al., 2007; Scarpella et al., 2004, 2006). We illustrate the method by a quantitative analysis of vein pattern formation in Arabidopsis leaves as shown by differentiated xylem elements. We discuss applications of spatial quantitative analyses of vein patterns for developmental, physiological and evolutionary studies. The programs that we developed to quantify vein patterns as described below are available from the authors on request.


We developed a method for quantifying leaf venation patterns in two dimensions, and applied it to the analysis of xylem pattern formation in the first rosette leaf of Arabidopsis. We collected data as patterns of differentiated xylem elements for 15 leaves each day for 9 days: from 8–16 days after sowing (DAS) or 5–13 days after germination. In the following, we refer to DAS (Candela et al., 1999) rather than days after germination due to slight heterogeneities in germination time between plants.

Extracting vein pattern data from leaf images

Software was developed in the lab to extract venation patterns from leaf images, and the process can be summarized as three steps (see Experimental procedures for details). First, leaf parameters were measured. The resolution of each image was recorded. The area, length and width were calculated for the leaf petiole and the leaf blade. The center point of the blade was recorded. The total leaf area and outline were recorded, as well as the leaf orientation on the image. Second, vein segments were digitized using a user interface: points along each segment were recorded, leading to a matrix representing all vein segments on the leaf. Third, the topology of the network was calculated from the extracted series of vein segments, and, using the known topology together with the coordinates of points along each vein segment, each vein segment and loop could be identified. We recorded each loop’s position on the image, its area, major axis length and minor axis length (i.e. length and width), its orientation along the major axis (relative to the proximal–distal axis of the leaf), and its ‘sub-division level’ (set to 1 if free-ending segments were present within the loop, 0 otherwise). We also recorded the number and position of branching points and free vein endings. The entire venation pattern and leaf outline for each sample was stored in matrix form.

An example of a leaf harvested on DAS12 with extracted vein pattern information is given in Figure 2. In Figure 2(a), vein branching points and free vein endings are identified and labeled. In Figure 2(b), vein loops are identified and color-coded according to their area (darker for larger loop areas). In Figure 2(c), the main orientation of each loop is identified and displayed as a line.

Figure 2.

 Example of pattern data for a single leaf harvested at DAS12.
(a) Branching points (in black) and free vein endings (in gray) are identified. The base of the primary vein is counted as a free vein ending by the algorithm, but is removed from the count of free vein endings for later analyses.
(b) Loops color-coded with shades of gray according to their area. Small loops are shown in light gray and large loops in darker gray.
(c) Loop orientations are shown as lines centered on each loop and oriented in the direction of loop elongation.

Temporal analysis of vein patterns in the first rosette leaf of Arabidopsis

Once pattern data had been extracted from leaf images and saved, parameters such as overall vein density or overall number of loops or segments across the leaf could easily be calculated.

The first rosette leaves of Arabidopsis from our data set grew throughout the period analyzed, with a decrease in growth rate after DAS14 (Figure 3a). To assess the overall variation in vein patterns throughout development, we calculated mean vein densities, loop numbers, segment numbers, and densities of free vein endings and branching points for each time point.

Figure 3.

 Mean leaf size and pattern data from DAS8–16.
Mean leaf data are shown as a function of developmental stage in DAS (= 15 leaves per stage). Error bars indicate the standard error of the mean.
(a) Mean blade area (mm2) (logarithmic scale).
(b) Mean vein density (mm−1).
(c) Mean loop number per leaf.
(d) Mean segment number per leaf.
(e) Mean density of free vein endings (mm−2).
(f) Mean density of branching points (mm−2).

The vein density for each leaf was calculated as the total vein length across the leaf divided by the whole leaf area. The mean leaf vein density for each day is shown in Figure 3(b), which shows that venation decreases from around 5.6 to 2.6 mm−1 over the period analyzed. A non-parametric analysis of variance (Kruskal–Wallis test) on vein density with day as a factor was highly significant (χ2 = 122.705, df = 8, P < 0.001), but was not significant if only DAS8–10 were considered (χ2 = 0.856, df = 2, = 0.652). This suggests that vein density remained constant from DAS8–10, then decreased from DAS10–16 (Figure 3b).

The mean loop number per leaf and the mean segment number per leaf for each day are shown in Figure 3(c,d). Again, a Kruskal–Wallis test on each variable with day as a factor was highly significant (χ2 = 109.97, df = 8, < 0.0001 for loop number; χ2 = 113.82, df = 8, P < 0.0001 for segment number). Loop numbers from DAS12–16 were not significantly different (χ2 = 7.010, df = 4, = 0.135), and segment numbers from DAS13–16 were not significantly different (χ2 = 5.598, df = 3, = 0.133). This suggests that the number of loops per leaf increased until DAS12, and the number of vein segments increased until DAS13.

The densities of free vein endings and branching points were calculated as the number of free vein endings and branching points per unit leaf area (mm−2) (Figure 3e,f). A Kruskal–Wallis test (χ2 = 14.92, df = 8, = 0.061) did not show a significant difference in mean densities of free vein endings between days, but we observed a significant difference in branching point densities between days (Kruskal–Wallis test, χ2 = 113.44, df = 8, < 0.001). The observed increase in branching point densities between DAS8 and 10 (Figure 3f) is consistent with the constant vein density at these stages as two branching points are created for each new segment linked on both sides to previous venation.

If the vein density were kept constant over time, we would expect vein loops to roughly double in size before being split or sub-divided into new loops. We therefore compared mean loop areas for each day for loops that had started sub-dividing and those that had not. To do so, loop areas from all samples for each day were pooled and split into two groups: loops with free vein endings within their perimeters were considered as currently sub-dividing, whereas those without free vein endings within their perimeter were considered as not sub-dividing. As expected, mean loop areas were consistently higher for loops that were sub-dividing. The ratios of mean areas of sub-dividing loops to mean areas of non-sub-dividing loops for each day are given in Table 1 and ranged from 1.378 to 2.502 between DAS8 and 16.

Table 1.   Loop areas in sub-dividing loops versus non-sub-dividing loops
DayNon-sub-dividing loopsSub-dividing loops
CountMean (1)SEMCountMean (2)SEMRatio (2)/(1)
  1. SEM, standard error of the mean.


Obtaining spatial data on venation patterns

Vein pattern parameters not only varied with the time point analyzed, but also with the position on the leaf, as shown in Figure 4. Spatial heterogeneities may reflect patterning and/or growth heterogeneities and are therefore of special interest.

Figure 4.

 Example of variation in loop areas in space and time.
One leaf for each day from DAS8–16 was selected as an example. Colors reflect loop areas. The color bar legend indicates loop area values (mm2).

Vein patterns typically vary between leaves of the same genetic background, even if these leaves are at similar developmental stages. Accordingly, our dataset of 15 leaves per time point showed variation in leaf size, shape and vein patterns between leaves harvested on the same day (see Figure S1).

In order to assess spatial variation in pattern parameters, mean pattern data across several samples were calculated. In order to do this, for each group of leaves harvested on the same day, we first calculated the average leaf shape, and warped all leaf shapes and venation patterns for that day to the average shape (Figure 5a; see Experimental procedures). New positions (Figure 5b) could then be assigned to all vein pattern elements measured (segments and loops positions), and the original calculated parameters (e.g. loop areas) were retained for further analyses.

Figure 5.

 Methodology used to obtain mean spatial data.
(a) Average leaf shapes for DAS8–16 (from left to right).
(b) Average leaf shape for DAS13, with all vein patterns from 15 samples superimposed.
(c) All free vein endings for DAS13.
(d) Grid of regions superimposed onto average leaf shape and all vein patterns from DAS13.
(e) Mean vein density per grid region for DAS13 (mm−1). Colors reflect values (yellow represents larger values than green).
(f) Method to calculate mean loop data for a given time point: for each point (x,y) of the average leaf surface, all loops overlapping (x,y) are gathered. The mean loop area or average loop shape can therefore be calculated at each (x,y).
(g) Contour map of loop areas for DAS13.
(h) Color code for (g); numbers indicate that values equal to or greater than the label value are displayed in the color to the right of the label (e.g. light green represents values equal to or greater than 0.5). Units are mm2.

We extracted spatial information in various ways, depending on the variables of interest. For instance, the spatial distribution of free vein endings for a given time point was readily visualized by overlapping data from all samples for that day (Figure 5c). Vein density in various parts of the leaf was calculated after splitting the leaf shape into various regions (Figure 5d,e). The mean loop area could be calculated at each point of the leaf surface as the mean area of loops overlapping that point (Figure 5f). Contour maps of mean loop areas could then be generated (Figure 5g,h). The same method was used to monitor other loop-related parameters, such as loop shape.

Spatial analysis of vein patterns in the first rosette leaf of Arabidopsis

Leaf vein patterns during development of the first rosette leaf of Arabidopsis varied in time and space (Figure 4). We therefore quantified spatial variations in venation patterns throughout the period analyzed. In particular, we focused on the densities of free vein endings, vein densities, loop areas and loop shapes.

The densities of free vein endings did not show a clear spatial pattern (Figure 6a,b). Vein densities decreased progressively in distal parts first, then in proximal parts, between DAS10 and 14 (Figure 6c,d). From DAS14 onwards, spatial variations in vein density decreased, with similar vein densities in all parts of the leaf at DAS16. Because the results may depend on the size of sampling regions, we repeated the analysis with other grid sizes (Figure S2) and obtained similar results.

Figure 6.

 Spatial variation in vein pattern parameters from DAS8–16.
(a) Maps showing the densities of free vein endings;
(b) legend for the maps in (a). Units are mm−1.
(c) Maps showing vein density; (d) legend for the maps in (c). Units are mm−1.
(e) Maps showing the spatial variation in loop areas; (f) common color code for the whole period analyzed. Units are mm2.
(g–i) Maps showing the spatial variation in loop areas, with color codes adjusted to reveal spatial heterogeneities at each stage.
(j), (k) and (l) are color codes for (g), (h) and (i), respectively. Units are mm2.
(m) Maps showing the spatial variation in loop shape, measured as the ratio of the length of the main axis of elongation to the length of the axis perpendicular to the main axis, i.e. loop length/loop width.
(n) Corresponding color code.
The labels on color bars in (f), (i–k) and (n) may overlap one or several boundaries between color boxes. The label indicates that the values equal to or greater than the label value will appear in the color that is to the right of the leftmost boundary overlapped by the label. For instance, the arrow in (f) indicates the color that represents values equal to or greater than 0.4, and the arrow in (l) indicates the color that represents values equal to or greater than 0.55.

Contour map data revealed clear spatial heterogeneities in loop areas and shapes. Loop areas increased from marginal to central regions (Figure 6e–l), and this could be detected from DAS10 onwards. The gradient in loop sizes from the centre to the edges appeared to be consistent throughout development from DAS10 onwards (Figure 6g). To confirm these visual results, we performed a linear regression of loop area as a function of the distance from the centre of the blade for each time point, using the loop centroid as a marker of loop position on the blade. The regression was significant for all days from DAS8 (= 32 loops, df1 = 1, df2 = 30, = 14.585, = 0.001), and especially from DAS10 (= 207 loops, df1 = 1, df2 = 205, = 43.119, < 0.001) onwards, confirming the map results. Regression lines for sample days are shown in Figure 7(a). The slopes of the regression lines were similar for all developmental stages.

Figure 7.

 Relationship between loop area and shape and loop position.
(a) Regression lines showing the relationship between loop area and the distance from the leaf blade center. Each line is labeled with the corresponding DAS. For clarity, only results for even DAS numbers are shown.
(b) Regression lines showing the relationship between loop shape and position along the proximal–distal axis of the leaf (the leaf blade center has ordinate 0). Only the results for even DAS numbers with statistically significant regression are shown.

Loop shapes varied in space as well, with a distal to proximal decrease in the ratio of loop length to loop width (Figure 6m,n). A regression of loop shape as a function of loop centroid position along the proximal–distal axis of the leaf was not significant for DAS8 (= 32 loops, df1 = 1, df2 = 30, = 1.073, = 0.309), but was significant for later days, in particular from DAS10 (= 207 loops, df1 = 1, df2 = 30, = 27.676, < 0.001) onwards. Regression lines for sample days are shown in Figure 7(b). The slopes of regression lines decreased in absolute value as DAS increased.


We developed a method to quantify leaf vein patterns in two dimensions throughout the leaf surface, and applied it to the analysis of vein pattern formation during leaf development in the first rosette leaf of Arabidopsis, as shown by differentiated xylem elements. Quantitative descriptions of vascular patterns at the tissue level may be used to explore vein patterning mechanisms as well as the basis for pattern variation between species or the effect of environmental factors. Temporal variations in venation patterns reflect the interplay between vein patterning and growth, and spatial variation may reflect both growth heterogeneities and/or heterogeneities in patterning across the leaf.

Timing of patterning with respect to growth

Temporal analysis data showed that vein patterning could be separated into three different phases. Until DAS10, the vein density remained constant, suggesting that patterning proceeds with growth (phase 1). After DAS10, the vein density decreased, but veins carried on forming (phase 2), with loop numbers increasing until DAS12 and the number of vein segments increasing until DAS13. This delay between the arrest of loop formation and the arrest of vein segment formation is consistent with the logical final step of the slowdown in vein formation: new vein segments are still formed between DAS12 and 13, but do not reconnect to other veins and therefore do not form loops. These results are consistent with previous descriptions of vein patterns (Scarpella et al., 2004). After DAS 13 (phase 3), there was no significant increase in vein segment numbers, indicating that vein pattern formation was complete, and the pattern just enlarged as the leaf carried on growing.

The fact that vein density decreased between DAS10 and 13 (phase 2) but new veins were still forming indicates that patterning was slowing down overall with respect to growth during that period. A possible cause for this slowing down of patterning in phase 2 is that vein patterning may stop at different times in different parts of the leaf as the carries on growing. Spatial data on vein density show that vein density starts to decrease in the distal part of the leaf first (Figure 6c,d), suggesting that vein patterning stops in the distal part of the leaf first. These results are consistent with the distal to proximal wave of vein differentiation proposed by Scarpella et al. (2004).

Our results are also consistent with a previous analysis (Candela et al., 1999), in which vein densities and branching point densities were estimated from DAS11 onwards in the first rosette leaf of Arabidopsis (Ler ecotype), that showed a decrease in both variables throughout the period analyzed. In our dataset, we found overall lower venation density and branching point densities, which could be due to different environmental conditions and/or the use of different ecotypes.

Spatial analysis of vein patterns and the use of two-dimensional maps

Throughout developmental stages, we observed a distal to proximal gradient in loop shapes, with more isodiametric loop shapes at the distal end of the leaf and more elongated shapes at the proximal end. This could indicate the presence of a proximal–distal gradient specifying growth rate and/or anisotropy. The spatial pattern of loop shapes varied over time, with the more isodiametric areas progressively becoming more confined to the distal parts of the leaf. This temporal variation in spatial patterns may be due to growth heterogeneities, as tissue proliferation in Arabidopsis follows a longitudinal gradient with cell-cycling arrest occurring first in the distal part of the leaf (Donnelly et al., 1999). The spatial distribution of loop orientations (data not shown) may reflect directions of growth throughout the leaf.

A center to margin gradient in loop sizes was apparent from DAS10 onwards, and could reflect spatial variations in factors that affect patterning. For instance, a margin to center gradient of a substance affecting the size at which loops sub-divide could be responsible for loops sub-dividing at a smaller size near the leaf margin. While vein patterning is occurring, growth heterogeneities may not have an impact on loop sizes because loops that are enlarging faster just sub-divide faster. This would explain why loop shapes reflect the proximal–distal growth gradient but loop areas do not. Data on leaf growth patterns will be necessary to investigate further what underlies the observed spatial heterogeneities in loop areas and shapes.

The spatial analyses described here can be used not only to compare venation patterns between developmental stages, but also to compare mutants or plants under different growth conditions to investigate the effects of mutations or growth conditions on vein patterning. Such analyses may also be used be used to compare species for evolutionary or systematics studies. The method illustrates how average spatial maps can be generated from point (free vein endings), line (vein segments) and surface (loops) data such that many layers of morphological and/or developmental information can potentially be combined.

Bridging the gap between modeling studies and experimental molecular data

There is an ongoing effort in the scientific community to reconcile theoretical pattern formation studies with molecular biology. For instance, in plants, reaction–diffusion models may explain trichome pattern formation and root hair pattern formation (Pesch and Hulskamp, 2004). With the recent accumulation of molecular data on the mode of action and transport of auxin, computer models involving auxin have been proposed in order to simulate various patterning mechanisms in plants (Berleth et al., 2007; Grieneisen et al., 2007; Kramer, 2008), including vein pattern formation (Merks et al., 2007).

Models may be used as tools to complement experimental studies and uncover pattern formation mechanisms, provided they can be tested quantitatively using experimental data. The proposed framework for investigating patterning at the tissue level may be used to guide experiments and modeling, and to evaluate computer-generated patterns. Quantitative analyses at the tissue level therefore provide a promising approach to link theoretical and experimental work and explore patterning mechanisms.

Limitations and future improvements

The time-consuming step of the analysis so far is vein-digitizing. Although methods have been published to automatically digitize pattern data (Bohn et al., 2002; Houle et al., 2003), the software we tried (Lilian Blot, University College London, unpublished data) to automatically digitize veins was not sufficiently accurate to replace computer-assisted digitization. Automated vein digitizing would speed up the analyses and should allow large-scale comparative analyses of vein patterning. Both image quality and software will need to be optimized. It should also be noted that averaging of spatial data through warping distorts individual leaf shapes and vein patterns (see Experimental procedures), thus, if this procedure is to be used, the group of leaves to be warped to the group average needs to be reasonably homogeneous in size and shape.

Experimental procedures

Data collection

For the time-series analysis, Arabidopsis plants of ecotype Col-0 (provided by E. Coen, John Innes Centre, Norwich, UK) were grown under greenhouse conditions. Germination occurred between 3 and 4 days after sowing (DAS). We harvested 30 plants each day from DAS8–16 (approximately 5–13 days after germination). Harvested plants were cleared according to the protocol described by Steynen and Schultz (2003). Plants were immersed in 3:1 ethanol acetic acid for 2 h, and dehydrated in 70% ethanol for 1 h and 95% ethanol overnight. The next morning, plants were immersed in 5% NaOH at 60°C for 1 h. The first pair of leaves from each plant was then dissected in 50% glycerol and placed on a glass slide. Leaves were photographed under dark-field microscopy using a digital camera. For further analysis, one leaf was selected randomly from each pair of leaves, as we could not distinguish between the first and second leaves. This leaf was then referred to as the ‘first leaf’. Fifteen leaves per day were randomly selected and analyzed further.

Although it has been reported that xylem can differentiate in a discontinuous way, all our images showed continuous xylem. A possible reason is that xylem appeared bright in dark-field microscopy even before full differentiation.

Image analysis and pattern extraction

Custom programs were written in the lab using Matlab (The Mathworks Inc., http://www.mathworks.com/) to obtain quantitative pattern data from leaf images. Here we describe the general steps performed by the programs.

  • (i)A user interface was developed to measure and record leaf parameters.
  • (ii) A user interface was developed to digitize vein segments: points along each segment were recorded, leading to a matrix S representing all vein segments on the leaf. For instance:


    where xi represents the ith x coordinate, yi represents the ith y coordinate, and j represents the jth segment. In this example, segment 1 is made up of three points. n point coordinates specify a total of m segments.

  • (iii)The topology of the network was automatically calculated from the extracted series of vein segments. To do so, each segment was identified by its starting and ending point (e.g. in S above, (x1,y1) and (x3,y3) are the end points for segment 1), as well as a mid-point on the segment. We then identified which points had the same x and y coordinates to determine which segments were linked together. For instance, in S above, if x3 = x4 and y3 = y4, segments 1 and 2 are connected. Nodes (branching points) and free vein endings could therefore be identified. An adjacency matrix was calculated, identifying which segment ends were connected to each other. In order to identify reticulations, free-ending segments were removed from the network iteratively until no free endings remained. Loops were then identified by taking a starting node at random and ‘walking’ along the network until getting back to a visited node.
  • (iv)Using the known topology of the network, together with the coordinates of points along each vein segment, we were able to record segment and loop parameters. We also recorded the number and position of branching points and free vein endings. For each leaf image, a folder was generated containing information on the leaf measurements, the segments (matrix S), positions of free vein endings, a list of loop data, and the list of coordinates for each segment and loop. These data could then be used to display pattern data as maps for each leaf. Knowing the image resolution and the orientation of the leaf on the image, we were able to display all leaves analyzed in a common orientation and at the same scale if so desired.

The recorded data were also used to automatically generate excel files (Microsoft Corporation, http://www.microsoft.com/) containing data to be analyzed using statistical analysis software (SPSS Inc., http://www.spss.com/). More details on the methodology are given in Figure S3. A list of parameters calculated is given in Appendix S1.

Obtaining spatial data from leaf samples

Leaves harvested on the same day showed variations in venation patterns (Figure S1). To analyze variations in pattern parameters, leaves harvested on the same day were therefore grouped, and the average leaf shape for that day was computed. Each leaf and its vein pattern was then warped to the average shape using a second-order polynomial transformation (Wolberg, 1992). The position of each vein pattern feature on the average shape could then be determined, e.g. position of loop centroids, free vein endings and branching nodes (Figure 5). Whilst warping was necessary to map all data to a common shape, it slightly altered individual leaf patterns, therefore warped positions were recorded whilst retaining pre-warping parameter values such as loop areas and segment lengths. New excel files were generated containing original calculated parameters with adjusted loops and vein segment positions (Appendix S1). These data were further analyzed using SPSS.

Programs were written using Matlab to calculate and display parameters such as mean vein densities and loop areas or shapes as they varied in space for a given time point (Figure 5). Vein densities were calculated from warped vein patterns in various regions of a grid superimposed to the average leaf shape. Because loop data represent surfaces and covered most of the leaf, we used a grid-free approach, as shown in Figure 5. The Matlab ‘contourf’ function was used to generate contour maps. Due to pattern variability between leaves, at some points on the surface means were calculated from fewer than 15 samples (Figure S4). However, the mean was calculated from 15 samples for most of the leaf blade for most stages, with DAS8 and 9 having larger areas for which the mean was calculated from a smaller number of samples.


We thank Kirill Yankelevich and Julie Gupta (University of Ottawa, Canada) for help with vein digitization, and Enrico Coen (John Innes Centre, UK) and Przemyslaw Prusinkiewicz (University of Canada, Calgary) for providing facilities during the early stages of this work (lab space and computer facilities, respectively). We thank an anonymous reviewer for helpful comments on an early version of the manuscript. This work was funded by an NSERC Discovery grant to A.-G.R.-L., and early stages of the work were supported by an Alberta Ingenuity fellowship and a Pacific Institute for the Mathematical Sciences fellowship (to A.-G.R.-L.).