An allometric method for measuring leaf growth in eelgrass, Zostera marina, using leaf length data

Héctor Echavarría-Heras; Elena Solana-Arellano; Cecilia Leal-Ramírez; Ernesto Franco Vizcaino

doi:10.1515/bot-2012-0215

Article

An allometric method for measuring leaf growth in eelgrass, Zostera marina, using leaf length data

Héctor Echavarría-Heras , Elena Solana-Arellano , Cecilia Leal-Ramírez and Ernesto Franco Vizcaino

Published/Copyright: May 31, 2013

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From the journal Botanica Marina Volume 56 Issue 3

Abstract

The rate of production of leaf biomass in eelgrass (Zostera marina L.) is an indicator variable for environmental influences on the growth of this important seagrass species. The efforts to restore eelgrass meadows from the harmful human influences make the use of nondestructive evaluations essential. We present, here, an indirect procedure for the estimation of the growth rates of eelgrass leaves by using easily obtained measurements of leaf length and increases in leaf length, and allometric parameters linked to the scaling of leaf biomass and leaf length. This allometric method includes criteria that allow the estimation of leaf growth rates even when the sizes of some of the leaves cannot be determined because of herbivory or other environmental factors. To validate the proposed method, we performed simulation studies and analyzed data from two natural eelgrass populations in the East Pacific (México). These allometric projections of leaf growth rates displayed a high level of correspondence with observed values. We show that whenever the allometric parameters for the scaling of eelgrass leaf dry weight in terms of leaf length have been previously fitted, the method proposed here can provide an alternative for estimating biomass production that is both accurate and nondestructive and uses easily obtained data on leaf length and increases in leaf length between the sampling periods.

Keywords: allometric scaling; eelgrass; leaf growth assessments; marking interval

Corresponding author: Héctor Echavarría-Heras, Centro de Investigación Científica y de Educación Superior de Ensenada, carretera Ensenada-Tijuana No. 3918, Zona Playitas, Código Postal 22860. Apdo. Postal 360. Ensenada, B.C. Mexico

Appendix

In this appendix, we derive the results that sustain the method of the allometric approximation presented here. Let us consider a leaf in a retrieved shoot such that its length, l(t+Δt), from its base, at the top of the sheath, to its tip, can be plainly identified and measured. We also assume that the individual leaf biomass can be allometrically scaled (Duarte 1991, Harris et al. 2006) in terms of matching length, i.e.,

where α and β are the fixed parameters. Then, we can obtain an allometric surrogate, Δw_la(t, Δt), for the individual leaf dry weight increment, Δw_l(t, Δt), produced over the interval, [t, t+Δt]; this is,

Factoring βl(t+Δt)^α in equation (A2), we obtain

and by defining

we get

where λ_l(t, Δt) is the ratio of the leaf length increment to the total length,

Noticing that, at the marking time, the previous longitudinal growth aggregated by l(t) and the length increase, Δl, occurring over the interval [t, t+Δl] are formally related through

Hence, from equation (A3), we get,

then, factoring Δl and rearranging leads to

such that

and with

Now, as Δl is positive and bounded above by l(t+Δt), we get from equation (A6),

Then, equation (A10) shows that R_la(t, Δt) vanishes at both the minimum and maximum values attained by λ_l(t, Δt). In fact, for leaves emerging from the sheath during the interval [t, t+Δt], we have l(t+Δt)=Δl, this will set λ_l(t, Δt)=1, R_la(t, Δt)=0, and Δw_la(t, Δt)=β(Δl)^α. Hence, the term representing new tissue is the fundamental contribution to the growth for these leaves.

Because Solana-Arellano et al. (1997) demonstrated that the assumption of a restricted, asymptotic leaf growth in eelgrass is consistent with observations, the leaves having negligible sizes at marking time can be expected to grow most of their size measured at time t+Δt during the observation period [t, t+Δt]. That is, Δl would be expected to be similar to l(t+Δt), thus, making λ_l(t, Δt)≅1, and again rendering R_la(t, Δt); and therefore, the new leaf material, represented by β(Δl)^α, dominates the growth once more. For the older leaves that were almost fully grown at the marking time, the asymptotic growth assumption implies that l(t) and l(t+Δt) could be expected to be similar, thus, rendering the values of both Δl and λ_l(t, Δt) negligible, and correspondingly, R_la(t, Δt) will also become negligible making β(Δl)^α approximate Δw_l(t, Δt) in a reasonable way. Therefore, for leaves leading to values of λ_l(t, Δt) at the extremes of its variation range set by inequality (A12), R_la(t, Δt) will vanish, and we will have Δw_l(t, Δt)=β(Δl)^α. Meanwhile, for leaves leading to values close to the extreme values of λ_l(t, Δt), the continuity of R_la(t, Δt) implies that the bias term, R_la(t, Δt), will become negligible, making β(Δl)^α dominant in equation (A9).

Let us now assume that a marked leaf is retrieved undamaged, so its length, l(t+Δt), can be actually measured. Then, for such a leaf, the associated value of the ratio λ_l(t, Δt) will depend only on Δl, and correspondingly, the sign of R_da(t, Δt) depends only on the factor ø_la(λ, t, Δt). Furthermore, for fixed l(t+Δt), the value of βl(t+Δt)^α will also be fixed, and from equation (A10), we can see that the behavior of R_da(t, Δt) depends fundamentally on the variation of the factor ø_la(λ, t, Δt). Now, as we have

and

whenever α>1, R_da(t, Δt) is positive, its maximum value will be attained at λ_l(t, Δt)=0.5, and from equation (A10), we have,

As by fitting equation (A1) we have verified that α>1, then, R_la(t, Δt) remains positive for all the values of λ_l(t, Δt) in the interval (0, 1), and β(Δl)^α will systematically underestimate Δw_l(t, Δt). But for suitable values of λ_l(t, Δt), we can expect the value of the bias term, R_la(t, Δt), to diminish significantly, thus, making the approximation of Δw_l(t, Δt) through β(Δl)^α consistent.

In fact, letting θ(t, Δt)=λ_l(t, Δt)^-1, we have Δl=θ(t, Δt) l(t+Δt), then, from equation (A10), we have

where

Moreover, as

then, whenever 1≤α≤2 and θ satisfies 1≤θ<∞, the factor Ω_la(θ, t, Δt) increases.

Now, letting l_max=max{l(t+Δt)}, and Δl_min=min{Δl}, then, for

we have 1≤θ(t, Δt)≤θ_∗, and from equations (A9) and (A16), we have

Then, the smaller the value of Δl becomes, the lesser the difference between β(Δl)^α and Δw_la(t, Δt) will be. We have observed that most of the leaves that were damaged or had missing tips were to be found among the older and longer leaves. As the asymptotic growth assumption implies that Δl can be expected to be smaller for the older and larger leaves than for the newly produced ones, then inequality (A20) suggests the following criteria for calculating Δw_la(t, Δt),

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Received: 2012-10-16

Accepted: 2013-4-17

Published Online: 2013-05-31

Published in Print: 2013-06-01

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https://doi.org/10.1515/bot-2012-0215

Keywords for this article

allometric scaling; eelgrass; leaf growth assessments; marking interval