Coral geometry and why it matters

Introduction

The colony morphology of clonal organisms varies widely between and within taxa and has many physiological and ecological implications associated with adaptation to environmental conditions. For reef building corals, colony morphology has been associated with adaption to light, hydrodynamic regime, sedimentation, and subaerial exposure (Jackson, 1979; Chappell, 1980). While many coral species exhibit morphological phenotypic plasticity in response to environmental conditions, their general colony shape is often conserved and can be characteristic within a taxon (Veron, 2000; Todd, 2008; Zawada, Dornelas & Madin, 2019). Coral colony morphology can dictate a coral’s internal light regime (e.g., Anthony, Hoogenboom & Connolly, 2005; Kaniewska, Anthony & Hoegh-Guldberg, 2008; Kaniewska et al., 2014), hydrodynamics and mass transfer (e.g., Hossain & Staples, 2019; Hossain & Staples, 2020), and exposure to peripheral processes (Meesters, Wesseling & Bak, 1996). The general geometric shape of a colony can also dictate and constrain the quantitative relationships between various growth parameters (Barnes, 1973) which can have additional ecological implications. For corals, these common growth measurements include linear extension rates (length per unit time), areal growth rate (area per unit time), and calcification rate (mass per unit area per unit time) where calcification (G) is related to volume growth rate via average density (M =ρV, where M is mass, V is volume, and ρ is avg density) (Pratchett et al., 2015).

For all calcifying organisms including reef building corals, there are two distinct components of growth/production: (1) organic or tissue biomass/biovolume and (2) inorganic or calcium carbonate skeleton (although there is a small fractional component of organic matter in coral skeletons) (DeCarlo, Ren & Farfan, 2018). Each aspect of growth requires their own set of elemental resources (e.g., nutrients vs carbonate alkalinity) which have characteristically different levels of availability depending on the habitat. For organisms like reef building corals (and coralline algae and sclerosponges), the growth in biomass (i.e., production) is proportional to growth in surface area (Anthony, Connolly & Willis, 2002) while the growth in skeleton mass (i.e., calcification) is proportional to volume (via density). Therefore, biomass production is directly related to surface area growth. In contrast, skeletal mass is roughly proportional to volume (assuming a constant mean density), so a constant calcification rate leads to a constant volume growth rate (per unit surface area). The relationship between these two aspects of growth can be influenced by coral geometry which may cause one aspect of growth to “geometrically” constrain the other which has important implications for life strategies.

For some geometric shapes, the surface area to volume ratio changes with size potentially causing ontogenetic changes in the ratio of biomass production to calcification and the relationship between various growth parameters causing them to size-dependent. One of the most common metrics of coral growth across all morphologies is linear extension rate due in part to the relative ease in which it can be measured and used for comparative purposes (reviewed in Pratchett et al., 2015). Linear extensions rates are generally assumed to be characteristics of a species, independent of size (e.g., Buddemeier & Kinzie, 1976; Hughes & Jackson, 1985; Kinzie & Sarmiento, 1986), and often used as a foundation of the census-based approach to calculate calcium carbonate reef budgets (e.g., Januchowski-Hartley et al., 2017; Perry et al., 2018; Perry & Alvarez-Filip, 2019; Molina-Hernández et al., 2020; Cornwall, Diaz-Pulido & Comeau, 2019). Changes in linear extensions rates are often interpreted as an environmental signal (Carricart-Ganivet, 2011). However, even with a constant rate of calcification, linear extension rates can be colony size-dependent due to the dynamics of surface area to volume ratio which is dictated by the geometry of coral colony (e.g., Kahng et al., 2023). Therefore, the relationship between linear extension rate and colony size for various geometries is important to quantify and understand.

Recent reports have investigated the question whether coral growth is isometric by virtue of the colonial modular design or subject to allometric constraints like non-clonal metazoans (Edmunds, 2006; Dornelas et al., 2017; Madin et al., 2020; Carlot et al., 2022; Medellín-Maldonado et al., 2022). If growth is allometric with an ontogenetic decrease in growth rate, there may be important implications for ecosystem function based on the reduction of age/size distributions of coral communities from mass mortality events during the Anthropocene (Dietzel et al., 2020; Carlot et al., 2021). While environmental factors in the form of spatial constraints or partial mortality due to stress or senescence obviously influence these growth parameters (Barnes, 1973; Edmunds, 2008; Medellín-Maldonado et al., 2022), the inherent geometric relationships between growth parameters must be clarified to be able to quantitatively reconcile the different metrics of coral growth which are regularly measured using different techniques under a variety of conditions (Pratchett et al., 2015).

In this study, four common reef building coral morphologies are modeled using geometric shapes to illustrate the inherent mathematical relationships between various growth parameters and demonstrate how colony geometry can characteristically alter whether relationships are size-dependent or size-independent. Using previously published empirical values, the geometry-dependent relationships between organic (biomass production) and inorganic (calcification) growth are investigated, and the associated ecological implications are discussed.

Materials and Methods

To demonstrate the inherent mathematical relationships between various coral growth parameters, four geometric shapes were used to model how calcification rate per unit area (G) affects linear extension rate (C) across time, change in surface area (ΔS), change in planar area (ΔP), and change in volume (ΔV): hemispheroids across the full range of eccentricity values (oblate, circular, prolate), flat discs with constant thickness, branching corals with constant branch diameters (cylinder model), and a single conical branch/colony with constant aspect ratio (α = height/basal radius). The accuracy of using simple geometric models has been explicitly tested and confirmed for several diverse coral taxa using 3D scanning and photogrammetry techniques (Courtney et al., 2007; Naumann et al., 2009). Surface area productivity (SAP = ΔS/S) and planar area productivity (PAP = ΔP/S) were calculated to demonstrate how quickly a coral generates area (per unit area) across time.

Size dependent relationships were illustrated by calculating parameters as a function of size (e.g., radius, thickness, length, or height). Average or common empirical values for G and density (ρ) were used from the literature for the various species modeled. To illustrate the effects of geometry independent of G and ρ values, SAP was calculated for all shapes using the same values for G and ρ. For this modeling exercise, favorable growth conditions were assumed with no partial mortality or spatial constraints on growth.

A hemispheroid shape was used to model the relationships between these parameters for massive corals (e.g., Porites spp.). The effect of the eccentricity (e) was calculated across the full range of possible values from oblate (flattened) to circular to prolate (tall) hemispheroids. For a circular hemisphere (e = 0), the calcification rate $\left(\mathrm{G}=\frac{△\mathrm{V}\rho }{{\mathrm{S}}_{\mathrm{t}}}\right)$ and volume ($\mathrm{V}=\frac{2}{3}\pi {\mathrm{r}}^3$) equations were used to solve for extension rate (C_t = r_t+1 − r_t). To calculate SAP $\left(\frac{\Delta \mathrm{S}}{{\mathrm{S}}_{\mathrm{t}}}=\frac{{\mathrm{S}}_{\mathrm{t}+1}-{\mathrm{S}}_{\mathrm{t}}}{{\mathrm{S}}_{\mathrm{t}}}\right)$, the calcification equations was solved for r _t+1 and substituted into the equation where surface area S_t = 2πr². PAP $\left(\frac{\Delta \mathrm{P}}{{\mathrm{S}}_{\mathrm{t}}}=\frac{{\mathrm{P}}_{\mathrm{t}+1}-{\mathrm{P}}_{\mathrm{t}}}{{\mathrm{S}}_{\mathrm{t}}}\right)$was solved using the equation for planar area $\left(\mathrm{P}=\pi {\mathrm{a}}^2\right)$.

For a prolate hemispheroid with a circular base (a=b), the polar radius (c>a), where c is the polar radius and a and b are the equatorial radii in the two horizontal dimensions. The surface area $\left(\mathrm{S}=\pi {\mathrm{a}}^2+\frac{\pi \mathrm{ac}}{\mathrm{e}}{sin}^{-1}\mathrm{e}\right)$ and volume $\left(\mathrm{V}=\frac{2}{3}\pi {\mathrm{a}}^2\mathrm{c}\right)$ equations were used to solve for SAP using the definition for eccentricity $\left(\mathrm{e}=\sqrt{1-\frac{{\mathrm{a}}^2}{{\mathrm{c}}^2}}\right)$, where c>a. PAP $\left(\frac{\Delta \mathrm{P}}{{\mathrm{S}}_1}\right)$was solved using the equation for planar area $\left(\mathrm{P}=\pi {\mathrm{a}}^2\right)$. For an oblate hemispheroid with a circular base (a =b), the polar radius (c<a), where c is the polar radius and a and b are the equatorial radii in the two horizontal dimensions. The surface area $\left(\mathrm{S}=\pi {\mathrm{a}}^2+\frac{\pi {\mathrm{c}}^2}{2\mathrm{e}}ln\left(\frac{1+\mathrm{e}}{1-\mathrm{e}}\right)\right)$ and volume $\left(\mathrm{V}=\frac{2}{3}\pi {\mathrm{a}}^2\mathrm{c}\right)$ equations were used to solve for SAP using the definition for eccentricity $\left(\mathrm{e}=\sqrt{1-\frac{{\mathrm{c}}^2}{{\mathrm{a}}^2}}\right)$, where c<a. PAP $\left(\frac{\Delta \mathrm{P}}{{\mathrm{S}}_{\mathrm{t}}}\right)$was solved using the equation for planar area $\left(\mathrm{P}=\pi {\mathrm{a}}^2\right)$. For graphing purposes, oblate eccentricity was denoted as negative values to distinguish it from prolate eccentricity. The effects of eccentricity and size (equatorial radius) on SAP and PAP were calculated.

A flat circular disk with radius (r) and height/thickness (h) was used to model plate-like corals such as those common in the lower photic zone (i.e., Leptoseris spp. and Agaricia spp.) (Kahng et al., 2019; Tamir et al., 2019; Gijsbers et al., 2022). The calcification rate $\left(\mathrm{G}=\frac{△\mathrm{V}\rho }{{\mathrm{S}}_{\mathrm{t}}}\right)$ and volume (V  = hπr²) equations were used to solve for extension rate (C_t = r_t+1 − r_t). To calculate surface area productivity or SAP $\left(\frac{\Delta \mathrm{S}}{{\mathrm{S}}_{\mathrm{t}}}=\frac{{\mathrm{S}}_{\mathrm{t}+1}-{\mathrm{S}}_{\mathrm{t}}}{{\mathrm{S}}_{\mathrm{t}}}\right)$, the calcification equations was solved for r _t+1 and substituted into the equation where surface area S = πr². The effect of skeletal thickness (h) on SAP was calculated within the range of empirical values reported (Kahng et al., 2020).

A cylinder was used to model branching corals with constant average branch radius (r) (e.g., some branching Acropora spp.). As the colony grows, the sum of all branch lengths or total branch length (h_t) can be used to calculate skeletal volume, independent of the number of branches (β_t) or average branch length (h_t/ β_t). The surface area (S_t = 2πrh_t + πr²) can be calculated from the side of the cylinder using total branch length. Since each new branch base covers the side of its parent branch, there is no net increase in surface area from each new branch tip, with the exception of the area of the very first branch tip (πr²). In this model, the ratios between growth in volume (△V = πr²[h _t+1-h _t]), surface area (△S = 2 πr[h _t+1-h _t]), and total linear extension rate (C _t =[h _t+1-h _t]) remain constant which is consistent with empirical growth data for Acropora cervicornis (Million et al., 2021). The calcification rate $\left(\mathrm{G}=\frac{△\mathrm{V}\rho }{{\mathrm{S}}_{\mathrm{t}}}\right)$ and volume (V  = hπr²) equations were used to solve for average branch extension rate $\left(\frac{{\mathrm{C}}_{\mathrm{t}}}{{\beta }_{\mathrm{t}}}\right)$, where C_t = h_t+1 − h_t. To calculate surface area productivity or SAP $\left(\frac{\Delta \mathrm{S}}{{\mathrm{S}}_{\mathrm{t}}}=\frac{{\mathrm{S}}_{\mathrm{t}+1}-{\mathrm{S}}_{\mathrm{t}}}{{\mathrm{S}}_{\mathrm{t}}}\right)$, the calcification equations was solved for h_t+1 and substituted into the equation where surface area. The effect of branch diameter (2r) on SAP was calculated within the range of empirical values reported in the literature.

A cone with a constant aspect ratio was used to model a single branch of branching corals with conical branches. Some branching corals have branches that mimic high aspect cones. This model also applies to some massive corals form colonies that can mimic low aspect cones. A constant aspect ratio (α = h_t/r_t) during growth was assumed, where h is branch length/height and r is radius. The calcification rate $\left(\mathrm{G}=\frac{△\mathrm{V}\rho }{{\mathrm{S}}_{\mathrm{t}}}\right)$ and volume $\left(\mathrm{V}=\frac{1}{3}\mathrm{h}\pi {\mathrm{r}}^2=\frac{\pi {\mathrm{h}}^3}{3{\alpha }^2}\right)$ equations were used to solve for extension rate (C_t = h_t+1 − h_t). To calculate surface area productivity or SAP $\left(\frac{\Delta \mathrm{S}}{{\mathrm{S}}_{\mathrm{t}}}=\frac{{\mathrm{S}}_{\mathrm{t}+1}-{\mathrm{S}}_{\mathrm{t}}}{{\mathrm{S}}_{\mathrm{t}}}\right)$, the calcification equations was solved for h _t+1 and substituted into the equation where surface area. The effects of aspect ratio and cone length/height on SAP was calculated.

Results

The derivations of the mathematical relationships for each geometric shape are illustrated in Appendix A. Given a constant calcification rate (G) and density (ρ), the size-dependence of the linear extension rate (C_t) for the four geometric shapes depends on whether the surface area to volume ratio remains constant or changes as the coral colony grows in size (Table 1). For a circular hemispheroid, the size-dependent relationship between calcification rate and (radial) linear extension rate can be expressed $\mathrm{G}=\frac{△\mathrm{V}\rho }{{\mathrm{S}}_{\mathrm{t}}}=\frac{\rho \left(3{\mathrm{C}}_{\mathrm{t}}{\mathrm{r}}_1^2+3{\mathrm{C}}_{\mathrm{t}}^2{\mathrm{r}}_{\mathrm{t}}+{\mathrm{C}}_{\mathrm{t}}^3\right)}{3{\mathrm{r}}_{\mathrm{t}}^2}$, so as the colony reaches a larger size (r_t → ∞), G → ρC_t. Therefore, the linear extension rate increases to an asymptotic maximum value of ${\mathrm{C}}_{\mathrm{t}}\to \frac{\mathrm{G}}{\rho }$ (Fig. 1). For hemispheroids with nonzero eccentricity, the relationships quantitatively depart from the circular hemisphere (equatorial radial extension rate slower for prolate and faster for oblate) but the asymptotic pattern is qualitatively the same.

For a circular flat disc with constant thickness (h), the linear extension rate increases linearly with colony size (radius), ${\mathrm{C}}_{\mathrm{t}}={\mathrm{r}}_{\mathrm{t}}\left(\sqrt{\frac{\mathrm{G}}{\rho \mathrm{h}}+1}-1\right)$ (Fig. 1) and exponentially across time (Kahng et al., 2023).

For a cone with constant aspect ratio, the linear extension rate for height increases asymptotically, ${\mathrm{C}}_{\mathrm{t}}=\sqrt[3]{\frac{3\mathrm{G}\sqrt{{\alpha }^2+1}}{\rho }{\mathrm{h}}_{\mathrm{t}}^2+{\mathrm{h}}_{\mathrm{t}}^3}-{\mathrm{h}}_{\mathrm{t}}$ , where C_t = h_t+1 − h_t. For high aspect cones (e.g.,  α>4), the linear extension rate will not approach the asymptotic maximum value $\left({\mathrm{C}}_{\mathrm{t}}\to \frac{\mathrm{G}\sqrt{{\alpha }^2+1}}{\rho }\right)$until branch length/height become unrealistically long (e.g., h>50 cm) (Fig. 1). The linear extension rate for basal radius also increases asymptotically, ${\mathrm{C}}_{\mathrm{t}}=\sqrt[3]{\frac{3\mathrm{G}\sqrt{{\alpha }^2+1}}{\rho \alpha }{\mathrm{r}}_{\mathrm{t}}^2+{\mathrm{r}}_{\mathrm{t}}^3}-{\mathrm{r}}_{\mathrm{t}}$ , where C_t = r_t+1 − r_t . For low aspect cones (e.g.,  α<0.5), the linear extension rate will approach the asymptotic maximum value $\left({\mathrm{C}}_{\mathrm{t}}\to \frac{\mathrm{G}\sqrt{{\alpha }^2+1}}{\rho \alpha }\right)$ at relatively low values (e.g., h>25 cm) (Fig. 1).

For a branching coral with cylindrical branches of constant diameter, the relationship between calcification rate and total linear extension rate for all branches can be expressed as ${\mathrm{C}}_{\mathrm{t}}=\frac{2\mathrm{G}{\mathrm{h}}_{\mathrm{t}}}{\rho \mathrm{r}}+\frac{\mathrm{G}}{\rho }$ , where h _t is the total lengths of all branches combined. Dividing both sides by the number of branches β _t provides the average branch extension rate $\frac{{\mathrm{C}}_{\mathrm{t}}}{{\beta }_{\mathrm{t}}}=\frac{2\mathrm{G}}{\rho \mathrm{r}}\frac{{\mathrm{h}}_{\mathrm{t}}}{{\beta }_{\mathrm{t}}}+\frac{\mathrm{G}}{\rho {\beta }_{\mathrm{t}}}$ , where $\frac{{\mathrm{C}}_{\mathrm{t}}}{{\beta }_{\mathrm{t}}}$ is average branch extension rate and $\frac{{\mathrm{h}}_{\mathrm{t}}}{{\beta }_{\mathrm{t}}}$ is average branch length. As the colony grows more branches (i.e.,  ρ β _t>>G), the average branch extension rate decreases asymptotically to a value proportional to average branch length, $\frac{{\mathrm{C}}_{\mathrm{t}}}{{\beta }_{\mathrm{t}}}\to \frac{2\mathrm{G}}{\rho \mathrm{r}}\frac{{\mathrm{h}}_{\mathrm{t}}}{{\beta }_{\mathrm{t}}}$ . The proportionality constant decreases within increasing branch diameter.

Given a constant calcification rate (G) and density (ρ), the size-dependence of the surface area productivity (SAP) also depends on whether the surface area to volume ratio remains constant or changes as the coral colony grows in size (Table 1). For hemispheroids with constant eccentricity (e), the equations for SAP are listed below:

• Circular hemisphere $\frac{\Delta \mathrm{S}}{{\mathrm{S}}_{\mathrm{t}}}=\frac{{\left(\frac{3\mathrm{G}{\mathrm{r}}_{\mathrm{t}}^2}{\rho }+{\mathrm{r}}_{\mathrm{t}}^3\right)}^{2/3}}{{\mathrm{r}}_{\mathrm{t}}^2}-1$

• Prolate hemispheroid $\frac{\Delta \mathrm{S}}{{\mathrm{S}}_{\mathrm{t}}}=\frac{{\left(\left({\mathrm{a}}_{\mathrm{t}}^3+\frac{{\mathrm{a}}_1^{2}3\mathrm{G}\sqrt{\left(1-{\mathrm{e}}^2\right)}}{2\rho }\left(1+\frac{{sin}^{-1}\mathrm{e}}{\mathrm{e}\sqrt{\left(1-{\mathrm{e}}^2\right)}}\right)\right)\right)}^{2/3}}{{\mathrm{a}}_1^2}-1$

• Oblate hemispheroid $\frac{\Delta \mathrm{S}}{{\mathrm{S}}_{\mathrm{t}}}={\mathrm{a}}_{\mathrm{t}}^{-2}{\left({\mathrm{a}}_{\mathrm{t}}^3+\frac{3\mathrm{G}{\mathrm{a}}_{\mathrm{t}}^2\left(1+\frac{\left(1-{\mathrm{e}}^2\right)}{2\mathrm{e}}ln\left(\frac{1+\mathrm{e}}{1-\mathrm{e}}\right)\right)}{2\rho \sqrt{1-{\mathrm{e}}^2}}\right)}^{2/3}-1$

Using average G and ρ values for massive Porites (Lough & Barnes, 2000; Lough, 2008; Lough & Cantin, 2014; Lough et al., 2016) were used to calculated SAP as a function of colony size (equatorial radius) and eccentricity (Fig. 2). Flatter hemispheroids (i.e., larger c/a ratio) have higher SAP than taller ones. For all hemispheroids, SAP quickly declines asymptotically with increasing size as a coral colony grows.

For a cone with constant aspect ratio, SAP is a size dependent and decreases asymptotically with increasing size in a manner analogous to hemispheroids, $\frac{\Delta \mathrm{S}}{{\mathrm{S}}_{\mathrm{t}}}={\left(\frac{3\mathrm{G}\alpha \sqrt{1+\frac{1}{{\alpha }^2}}}{\rho {\mathrm{h}}_{\mathrm{t}}}+1\right)}^{2/3}-1$. Using average G and ρ values for branching Acropora (Hughes, 1987; Morgan & Kench, 2012; Pratchett et al., 2015), the SAP for a single conical branch was calculated as a function of branch length and aspect ratio (Fig. 3). Aspect ratio is positively correlated with SAP.

For a branching coral with cylindrical branches, surface area to volume ratio remains constant and all growth parameters are linearly related to total branch length (assuming a constant density). SAP is a size-independent and a function of calcification rate and branch radius/diameter (r), $\frac{\Delta \mathrm{S}}{{\mathrm{S}}_{\mathrm{t}}}=\frac{2\mathrm{G}}{\rho \mathrm{r}}$ . SAP declines with increasing branch thickness. Using average G and ρ values for branching Acropora spp. (Hughes, 1987; Morgan & Kench, 2012; Pratchett et al., 2015), the SAP was calculated as a function of branch thickness (Fig. 4).

For a flat circular disc, SAP is a size independent and is a function of calcification rate and thickness (h), $\frac{\Delta \mathrm{S}}{{\mathrm{S}}_{\mathrm{t}}}=\frac{\mathrm{G}}{\rho \mathrm{h}}$ . Thinner colonies have a higher SAP. Using average G and ρ values for deep-water Leptoseris spp. from 70–111 m (Kahng et al., 2023), the SAP was calculated as a function of average skeletal thickness (Fig. 5).

To isolate impact of geometry alone, SAP was calculated using equivalent values for G and ρ for the contrasting shapes (Fig. 6). Flat, thin plate-like shapes have the highest SAP which is equivalent to planar area productivity (PAP) by virtue of their horizonal orientation. Cylindrical branching corals with small branch diameters also exhibit high SAP, especially in comparison to their massive coral counterparts (hemispheroids and low aspect cones) which have very low SAP as they reach larger size.

Discussion

Supplemental Information