- Open Access
Using allometric procedures to substantiate the plastochrone method for eelgrass leaf growth assessments
© Echavarria-Heras et al.; licensee BioMed Central Ltd. 2013
- Received: 2 November 2012
- Accepted: 14 April 2013
- Published: 16 May 2013
Estimation of leaf productivity in eelgrass (Zostera marina L.) is crucial for evaluating the ecological role of this important seagrass species. Although leaf marking techniques are widely used to obtain estimates of leaf productivity, the accuracy of these assessments, has been questioned mainly because these fail to account for leaf growth bellow the reference mark and also because they apparently disregard the contribution of mature leaf tissues to the growth rate of leaves. On the other hand, the plastochrone method is a simpler technique that has been considered to effectively capture growth in a more realistic way, thereby providing more accurate assessments of both above- and below-ground productivities. But since the actual values of eelgrass growth rates are difficult to obtain, the worth of the plastochrone method has been largely vindicated because it produces assessments that overestimate productivity as compared to estimates obtained by leaf marking. Additionally, whenever eelgrass leaf biomass can be allometrically scaled in terms of matching leaf length in a consistent way, the associated leaf growth rates can be also projected allometrically. In this contribution, we used that approach to derive an authentication of the plastochrone method and formally demonstrate that, as has been claimed to occur for leaf marking approaches, the plastochrone method itself underestimates actual values of eelgrass leaf growth rates. We also show that this unavoidable bias is mainly due to the inadequacy of single-leaf biomass assessments in providing a proxy for the growth of all leaf tissue in a shoot over a given interval. Moreover, the derived formulae give conditions under which assessments of leaf growth rates using the plastochrone method would systematically underestimate matching values obtained by leaf marking procedures. And, assessments of leaf growth rates obtained by using the present data show that plastochrone method estimations underestimated corresponding proxies obtained allometrically (27%), or through leaf marking (35%). Allometric projection is recommended as a simpler and more effective procedure to reduce the bias in eelgrass leaf productivity estimations that associates to the use of plastochrone methods.
- Allometric scaling
- Leaf growth
- Plastochrone method
- Formal validation
Eelgrass (Zostera marina) is a cosmopolitan seagrass species that plays an important role in shallow and nearshore ecosystems. This temperate macrophyte is distributed in Northern Hemisphere habitats from the Arctic Circle to the Tropic of Cancer , where it provides a nursery for fish and as a substrate for attached algae and epifauna [2, 3]. By fixing large amounts of carbon through photosynthesis, eelgrass plays an important trophic role, sustaining detrital food chains and other secondary producers . Eelgrass also helps in the remediation of contaminated sediments  by filtering and retaining nutrients from the water column  and contributing to the stabilization of sediments . Moreover, eelgrass meadows reduce erosional forces by stumping wave energy, thus promoting the stabilization of adjacent shorelines [8, 9].
The variability in eelgrass biomass constitutes a dynamic link between its structural and trophic roles, because changes in the amount of organic carbon that can be fixed modulate the structure of the habitat for the associated biota. These organisms are affected in different ways when changes in biomass occur seasonally or unpredictably . Therefore, accurate measurements of the standing crop and productivity of eelgrass constitute an important input for evaluating the ecological functions and values of this significant seagrass species .
Several methods have been developed to measure seagrass growth and productivity [12–14]. Procedures for measuring productivity in aquatic macrophytes have included the assessment of changes in biomass over a growing season, as well as the quantification of oxygen production or incorporation of 14C during photosynthesis. But the accuracy of these methodologies was questioned early on [15–18]. Growth in seagrasses occurs through the expansion of modules formed by rhizome segments, which have bundles of attached leaves and roots. Because every leaf produced corresponds to the production of a rhizome node, it is reasonable to assume that seagrass growth and leaf formation are equivalent processes . This conspicuous feature has encouraged efforts to estimate the growth of eelgrass, as well as that of other seagrasses with ribbon-like leaves, by measuring leaf growth. Leaf growth in seagrasses can be estimated by using the leaf marking technique which was originally proposed as a way of avoiding the underestimation of growth when using oxygen metabolism measurements [16, 18]. Leaves were marked above the sheath with a small staple at the height of a reference frame placed above the sediment, and then recovered after a period of 2–4 weeks. The new growth in each leaf between the reference frame and the staple was then weighed. The ratio of this weight to the number of days of growth determined the leaf growth rate. Because this procedure is considered to give consistent estimations [19, 20] it has been used and modified by several authors (e.g. [21–23]).
For assessment of growth in Zoostera marina in particular, workers have marked blades using felt-tip pens, staples and hypodermic needles [4, 24–26], with the reference level being either the leaf tip [25, 27] or a point at the top of the sheath [12, 24, 26, 28]. However, Brouns  asserted that this technique only estimates blade production and is not suitable for measuring total production. The reliability and accuracy of the leaf-marking method has been also questioned by others; for example, because it does not account for new leaf growth within the sheath below the reference mark , or it might inflict damage to tissues , or fail to account for leaf maturation processes [11, 13, 30], or that it requires destructive dry-weight measurements.
Based on the observation that the production of every seagrass leaf is associated with the production of a rhizome node, Patriquin  envisioned that seagrass root and rhizome production can be also estimated by determining the time interval for the formation of new leaves and by counting the associated leaf scars. The time elapsed between the appearance of two consecutive leaves has been termed a plastochrone interval or plastochrone index [32, 33]. For eelgrass, Jacobs  used the average weight of the third leaf on sampled shoots as a substitute for the leaf biomass gained by a shoot over a plastochrone interval and divided this leaf biomass surrogate by the value of the plastochrone interval to assess growth rates of leaves. Gaeckle and Short  endorsed the method of Jacobs  and used the weight of a mature leaf to represent all growing leaf tissue in a shoot over a given a plastochrone interval to calculate eelgrass growth, calling the leaf-biomass to plastochrone ratio the plastochrone method for eelgrass leaf-growth assessments. And Gaeckle and Short  concluded that leaf marking results in lower estimates of leaf growth than the plastochrone-based method, as asserted by Brouns , and they considered that this approach both fully captures growth and is also simpler and non-destructive.
But despite the advantages of plastochrone methods, their use has not been yet formally substantiated. In the present research we attempt to fill this gap by using allometric models and a discrete mathematical formulation for the increment in biomass gained by an eelgrass leaf over a time interval to derive an equation which expresses the mean shoot leaf-growth rates in terms of a factor of the leaf biomass to plastochrone ratio plus a remainder, and explore their analytical implications. And, we extend aforementioned result to provide a formal device aimed at obtaining the absolute deviations between leaf growth-rate assessments estimated by using the conventional leaf-marking technique and those obtained by using the plasthochrone method. We also corroborate the derived formulae using both simulation and real data. Finally, we stress the advantages of using allometric methods in eelgrass research and discuss the findings of this study.
Where N(t + ∆t) is the number of shoots collected at a time t + ∆t. In what follows the calculated values for the Lg(t, ∆t) rates will be referred to as in situ mean leaf-growth rates.
where , is a residual term.
where is the involved estimation error.
where the term Ba(t, ∆t) stand for the pertinent approximation bias. Again, only in the case in which both equations (10) and (11a) hold we will have that the Lg(t, ∆t) rates of equation (4) and their allometric representations Lga(t, ∆t) are equivalent, this setting Ba(t, ∆t) = 0. In what follows the result of equation (12a) will be called “allometric method for the projection of eelgrass leaf growth rates” or simply as “allometric method”.
which taking into account inequality (9) could be only satisfied whenever the order relationship ∆t ≤ p(t, ∆t) holds.
Again, since varies continuously and Lgp s (t, ∆t) does not vanish and depends continuously on both 3s (t, ∆t) and p(t, ∆t) from equation (23) we conclude that Cap(t, ∆t) varies continuously and takes all the positive values within the range set by inequality (26).
This result shows that the minimum possible value that Bp(t, ∆t) can attain is BLap(t, ∆t) and that whenever the inequality p(t, ∆t) ≥ ∆t holds the upper bound for its variation range will increase from the value Bap(t,∆t). Moreover this order relationship will induce wider variation ranges for Bp(t, ∆t) relative to the complementary case p(t, ∆t) < ∆t, therefore substantiating the claim by Short and Duarte  who stated that for an efficient application of the plastochrone method the length of the observation period ∆t must exceed the value p(t, ∆t) of the leaf plastochrone interval.
Derivation of a formal relationship between leaf marking and plastochrone method assessments of eelgrass leaf growth rates
holds, the result of equation (44) implies that the Lgm(t, ∆t) assessments will be underestimated by the Lgp(t, ∆t) values.
Using the present leaf data, we calculated the values for in situ leaf growth rates Lg(t, ∆t) by means of equation (4). The concomitant Lgm(t, ∆t) estimations produced by the leaf-marking technique were also obtained. Using the formula in Jacobs , we produced biweekly estimations of plastochrone-interval values. Plastochrone method proxies Lgp(t, ∆t) for the in situ leaf growth rates Lg(t, ∆t) were also calculated. The model of equation (5) was identified as consistent with the present data set, producing a coefficient of determination of 0.82. The values of the fitted parameters were α = 0.00002 and β = 1.3 similar to those previously reported for this site [34, 35]. We then used these parameter values and associated leaf data to produce allometric estimations Lga(t, ∆t) for the in situ leaf-growth rates using equation (12a). An analysis of variance found significant differences between mean annual leaf-growth rates among methods (F = 17.95 df : 3, 112, p < 0.001) and an a posteriori Tukey test found non-significant differences among in situ values and those produced by the leaf marking method (p = 0.89), among in situ values and those obtained through the allometric method (p = 0.08), and among leaf marking values and those produced by using the allometric method p = 0.29. The Tukey test also showed that the mean annual leaf-growth rate obtained through the plastochrone method is statistically different from that linked to the in situ values, that calculated from leaf-marking data, and the one obtained from allometric projections (p < 0.001).
Root mean squared error (RMSE) and concordance correlation coefficient ( ) values for comparison of observed leaf growth rates and corresponding proxies calculated by the leaf- marking, allometric or plastochrone methods
95% Confidence interval
Moreover, an analysis of Lgm(t, ∆t) versus Lg(t, ∆t) values found a statistically significant regression through the origin; Lg(t, Δt) = 1.040 Lgm(t, Δt), r2 = 0.98, p < 0.001 (Figure 1b), also when we analyzed Lga(t, ∆t) versus Lg(t, ∆t) the same regression model was found highly consistent; Lg(t, Δt) = 1.163 Lga(t, Δt), r2 = 0.96, p < 0.001 (Figure 1c). Meanwhile, when we compared Lgp(t, ∆t) versus Lg(t, ∆t) no regression through the origin could be fitted. Instead only a poor fit; Lg(t, Δt) = 0.779 Lgp(t, Δt) + 0.002, r2 = 0.26, p < 0.003 could be produced (Figure 1d). Additionally, values of the Lin  Concordance Correlation Coefficient ( for reproducibility of the Lg(t, ∆t) rates by the methods addressed here (Table 1), reveal that both the leaf marking ( and allometric ( methods performed better than the plastochrone method (. Indeed we can observe that the bias Ba(t, ∆t), produced by the allometric method of equation (12a) is relatively smaller than that linked to plastocrone method assessments. Since the allometric approach is a simpler technique than leaf marking procedures [34, 35], allometric projection is recommended as an effective method to reduce the bias associated to plastochrone method assessments.
Allometric scaling models of the form Y = αX β , where Y is a response and X is an explanatory variable, are commonly used in biological research. In this formulation, α is positive and is known as the normalization constant, while β is called the scaling exponent. From an empirical standpoint these allometic models have been fitted to many pairs of biologically traits X and Y in a highly consistent way, providing reliable methods for estimating variables that are difficult to quantify by using other variables that can be easily and directly measured. Empirical scaling models can be viewed as static relationships in which both the allometric exponent and the normalization constant take values that pertain to a particular data set. But beyond a purely empirical advantage, allometric approaches have demonstrated that prediction is not an unachievable aim in ecology. Indeed, allometric methods can be successfully used to predict roles and characteristics of organisms [37, 38]. The predictive realm of allometric laws, moreover, is not circumscribed only to individual properties of organisms but can also apply to energy flows in whole ecosystems . For example, the consistency of allometric relationships between body size and metabolism, which span 27 orders of magnitude in body mass for living organisms in both aquatic and terrestrial ecosystems, has provided a basis for the development of the metabolic theory of ecology .
The benefits of allometric methodologies in seagrass research were stressed by Duarte , who exemplified the use of static allometric models to grasp the implications of differences in plant size for productivity. For eelgrass in particular, Hamburg and Homann  and Echavarria-Heras et al. used empirical allometric methods for the assessment of aboveground biomass. But allometric methodologies in eelgrass research are not limited only to empirical approaches for nondestructive appraisals of leaf biomass, but have also been used to formally substantiate empirical paradigms employed in eelgrass research . They demonstrated that allometric scaling can provide criteria for evaluating the suitability of the leaf-biomass-to length ratio as a paradigm for nondestructive leaf biomass evaluations. And from a theoretical standpoint, it has also been established that allometric models can be linked to a paradigm for the indirect assessment of leaf growth rates in terms of simple measurements of leaf length or area .
It has been considered that conventional leaf-marking techniques produce biased estimations of eelgrass leaf-growth rates . One fundamental objection states that, since these methods measure leaf growth by accounting only for the biomass of all the pieces of leaf material that have developed during the period between the marking and recovery of shoots, any leaf material produced below the reference point is not captured, and thus that portion of leaf production is missed. But even though the marking point could be placed on the sheath itself, as a way to avoid this inconvenience, the modified method still relies on newly produced leaf tissue, which is considered to be immature and flaccid. These tissues have a lower weight-to-length ratio than mature leaves , resulting in an underestimation of leaf growth.
The plastochrone method for eelgrass leaf growth rate estimations [4, 11] was proposed in order to avoid the lack of representativeness of the leaf marking techniques. Unlike the leaf-marking techniques, the plastochrone method has been considered to produce simple, direct, nondestructive and unbiased estimations of eelgrass growth. Nonetheless, the reliability of this method has been largely vindicated at an empirical level, and practitioners have justified its application by asserting that this paradigm fully captures the growth pattern of eelgrass, in which new plant parts are incrementally produced while young tissue is simultaneously maturing.
The implementation of the plastochrone method is centered on the idea that the biomass of a single leaf representing mature tissue on a shoot, divided by the leaf plastochrone interval, provides a reliable substitute for the contribution of the shoot to the mean growth rate of leaves. But so far, a mechanistic explanation is still missing as to how a single leaf can capture the contribution to growth produced by all the leaves in a shoot over a given observation period. This is where allometric approaches can contribute to the formal substantiation of the method. Indeed, equation (15) was obtained by starting from the basic balance equation (3), then representing the contributions to growth of all the individual leaves on a shoot using the allometric representation of equation (6); and finally, by averaging over the number of retrieved shoots, we obtained equation (22). This equation relates the observed values of eelgrass leaf-growth rate Lg(t, ∆t) in a direct dynamical way to their proxy plastochrone method values Lgp(t, ∆t), thereby providing a formal authentication for this paradigm.
The balance equation (3) shows, moreover, that any fixed value of Lg s (t, ∆t) links all individual contributions to leaf growth in an essential way, with the biomass gained by a particular leaf being lost by the remaining ones. Hence, a single leaf would capture all the leaf biomass gained by a shoot over a given growing period, if and only if the biomass of the residual leaves in the shoot remained static, which is unsupported by the observed growth mode of eelgrass. This is indeed what inequality (19) properly expresses, and its validity formally explains that we can expect a downhill bias when plastochrone method calculations are used as proxies for observed leaf growth rates. This was in fact corroborated with the present data; the values obtained for the derived formulae confirmed that the plastochrone-method proxies underestimated the leaf-growth rates observed.
Likewise, the gains obtained from the use of allometric approaches are also shown by equation (37), which provides a device for the estimation of the contribution of leaf maturation processes to growth. But using the present theoretical settings, we can also address the controversy between the leaf marking and plastochrone methods for leaf-growth assessments. Indeed, equation (44) relates leaf-growth rate estimations obtained by means of the leaf marking technique to those obtained through the plastochrone method. And, the derived formulae set conditions under which it could be expected that plastochrone method approximations of eelgrass leaf-growth rates would systematically underestimate matching values obtained by using leaf-marking procedures. Indeed the condition set by inequality (46) was verified for the present data; then as a consequence, the plastochrone method proxies underestimated similar values obtained by means of leaf-marking methods. That is the inequality Bp(t, Δt) > Bm(t, Δt) holds through time, as shown in Figure 1. Further, using the present allometric framework, we formalized the claim of Short and Duarte  who stated that, for an efficient application of the plastochrone method, the length of the observation period ∆t must exceed the leaf plastochrone interval p(t, ∆t). And, although this condition was satisfied for the present data, we obtained Ba(t, Δt) < Bp(t, Δt) when approximating observed values. That is, the bias produced by the allometric method of equation (12a) was found to be smaller than that linked to plastocrone-method estimates. Furthermore, for this data the order relationship Bm(t, Δt) < Ba(t, Δt) < Bp(t, Δt) was maintained through time. It can be also observed from Figure 1 and from the high value of the associated Lin  Concordance Correlation Coefficient ( for reproducibility of the Lg(t, ∆t) rates through Lgm(t, ∆t) values (Table 1), that the contributions of leaves within the sheath, and of leaf maturation processes to growth, were not significant enough to reduce the effectiveness of the leaf marking technique.
And although a larger bias is associated with allometric projections than with leaf-marking techniques, the allometric approach is recommended over the plastochrone method because it is simpler and more precise (Table 1). Moreover our results support the conjecture that, taking into account the contribution to growth of all the leaves on an eelgrass shoot, as is done in both the leaf-marking technique and the allometric method provides a better proxy than the weight of a single leaf, as traditionally done in plastochrone method assessments. To this, we add that the lack of representativeness of this single-leaf-weight surrogate could explain the deviations shown in Figure 1.
Most methods used to assess leaf productivity in eelgrass have been vindicated at an empirical level. In particular, the plastochrone method has been favored over leaf-marking techniques because it is supposed to eliminate the alleged downhill bias linked to these assessments. This study confirms that a formal interpretation of assessment procedures is a crucial step in appraising their reliability, and we have shown how allometric methods can pave the way to achieve this important step. Formally, the plastochrone method will unavoidably produce biased estimations of the actual eelgrass leaf growth rate values. Empirical corroboration validated this inference, endorsing the view that allometric scaling relationships are not circumscribed only to empirical descriptions of the linkage between variables. Rather, they can be used to produce theoretical tools aimed towards the clarification of relevant issues in eelgrass research.
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