Characters related to skeletal light scattering and micro-architecture of the skeleton are described below; the remaining characters are described here. Skeletal density (n), the maximum mass of skeletal material per cubic cm (g·cm⁻³), is affected by skeletal extension and thickening of nanograins (Bucher et al., 1998) and species-specific values were either obtained from the literature or directly measured from museum specimens of Marcelino et al. (2013) using standard methods (mass divided by volume determined by displacement of water; Hughes, 1987). Species-specific skeletal growth rate values, characterized by the maximum linear increase in skeleton deposits measured over a year (mm·yr⁻¹), were obtained from the literature and related to n and D_f by a coral growth model (see equations 9–13). Species-specific corallite arrangement (corallite size measured as corallite max diameter, corallite complexity, and percent of surface area composed of coenosteum) were also obtained. Corallite complexity, was parameterized by counting the number of septa within each cycle, and weighing septal counts within cycles by their extension from the corallite wall toward the columella (sum of the number of septa per cycle · extension weight) within 10 corallites for each species from the museum specimens of Marcelino et al. (2013). Primary septal cycles (i.e., those that extend to the columella) were weighted by a factor of three, intermediate cycles were weighted by a factor of two, and cycles that extend minimally from the corallite wall were unweighted. The percent of surface area composed of coenosteum (% SA_coenosteum) was parameterized through direct measurements of museum specimens of Marcelino et al. (2013) captured in size-standardized digital photos and subtracting the area occupied by corallites from the total area of skeletal surface in photographs using ImageJ (version 1.47; NIH). Colony growth morphology was estimated by calculating cumulative fractality for each of 10 growth forms characterizing the 88 coral species, which is the product of apparent surface fractality of the colony and mass-fractal dimension D_f. Apparent surface fractality was measured by visually scoring the structural complexity (i.e., rugosity, or the 3D surface area occupying a given volume) of individual specimens representing each growth form such that encrusting forms were scored the lowest given their low surface area per volume ratio and branching forms (especially closed branches) were scored the highest (modified from Young et al., 2017).

Bulk skeletal scattering (bulk-μs′, characterized by the reduced light scattering coefficient μs′ measured in mm⁻¹), is the inverse distance a photon travels before it is randomized in direction, and results from light interaction with voids and skeletal structures at different length scales throughout the entire skeleton (Enríquez et al., 2005; Stambler and Dubinsky, 2005; Terán et al., 2010). Micro-skeletal light scattering (micro-μs′ or μs,m′) is the reduced light scattering coefficient within the top ~200 μm of the skeleton and is not influenced by voids (Marcelino et al., 2013; Swain et al., 2016b). Micro-μs′ was measured using low-coherence enhanced backscattering spectroscopy (LEBS) on tissue-free coral skeletons. The LEBS instrument and principles were described in detail by Marcelino et al. (2013) and references therein, but briefly, LEBS can measure microscopic light-scattering through broadband partial spatial coherence of illumination (Lsc) which selectively isolates the scattering spectrum from photons that propagate paths such that the distance between light entry and light exit points is comparable to Lsc.

Measurements of μs,m′ of corallites and coenosteum were taken for a subset of 14 species (previously characterized for skeletal μs,m′ at the colony level by Marcelino et al., 2013), with corallite diameters and distances between corallites (i.e., coenosteum) larger than the instrument spot size (~1 mm in diameter). Their ratio was calculated as, (1)μS,m_Ratio′= μS,m, corallite′μS,m, coenosteum′ Values for μs,m_corallite′ and μs,m_corallite′ are reported as the means of ten replicate measurements. To evaluate the potential for coenosteum to transport light into adjacent corallites, we calculated the effective light transport distance of coenosteum (ETD_coenosteum), which measures how much light can be transported by coenosteum given its intrinsic μs,m′ (and therefore transport mean free path length, l′S,m= 1/μs,m′) and inter-corallite distance (D_{inter−corallite}) such that: (2)ETDcoenosteum= Dinter−corallitels,m′ where l′s,and D_{inter−corallite} are measured in millimeters. A high ETD_coenosteum value indicates low potential for coenosteum to transport light into adjacent corallites. D_{inter−corallite} is the mean distance between ten haphazardly chosen adjacent corallites captured in size-standardized digital photos of the museum specimens of Marcelino et al. (2013), extracted using ImageJ (version 1.47; NIH).

The effect of light transport in the coenosteum on corallite irradiance was modeled based on the diffusion approximation to the light transport equation (Zonios et al., 1999). Corallites were approximated as semi-infinite axially-symmetrical cylindrical regions of turbid media, surrounded by semi-infinite regions of coenosteum, with the scattering properties measured by LEBS. In this approximation, irradiances relative to the incident intensity that would be observed at the surface layer of the corallite (R₁) and the coenosteum (R₂) regions due to a uniform illumination of the coral surface are, respectively, (3)R1(r)=∫0rc∫02πR(r2+r′2−2rr′cosϕ;μ′s1,μa1)r′dϕdr′ where r is the radial distance from the center of the corallite, r_c is the radius of the corallite, μ′s1 and μ_a1 are the reduced scattering and the absorption coefficients of the corallite, respectively, and R(x;μ′s,μa) is the irradiance observed at a distance x from a point source at a surface of a semi-infinite slab of turbid media with the reduced scattering and absorption coefficients μ′s and μ_a, respectively. The point-source reflectance is derived from the diffusion approximation as follows from Zonios et al. (1999): (4)R(λ,r)=z04πμs′μs′+μa⌊(μ+1r1)exp(−μr1)r12+(1+43A)(μ+1r2)exp(−μr2)r22⌋ with μ=⌊3μa(μa+μs′⌋1/2; z0=1μs′+μa; r1=(z02+r2)1/2; r2=⌊z02(1+43A)2+r2⌋1/2

Some light illuminating the corallite at any given point gets diffusely scattered within the corallite and returns to its surface after which it leaves the coral. A portion of the light, however, may reach the coenosteum region and reach the surface of the coenosteum after being multiply scattered in that region. This irradiance is modeled as the result of the diffusion of light reaching the outer radius of the corallite: (5)R1→2(r)=R1(rc)∫02πR(r2+r′2−2rr′cosϕ;μ′s2,μa2)dϕ∫02πR(2rc1−cosϕ;μ′s2,μa2)dϕ where μ′s2 and μ_a2 are the reduced scattering and the absorption coefficients of the coenosteum.

Similarly, the surface irradiance relative to the illumination intensity in the coenosteum region due to the illumination of the coenosteum is (6)R2(r)=∫rcrc+L/2∫02πR(r2+r′2−2rr′cosϕ;μ′s2,μa2)r′dϕdr′ where L is the average intercorallite distance, and the irradiance of light that is transported from the coenosteum and reaches the corallite surface is (7)R2→1(r)=R2(rc)∫02πR(r2+r′2−2rr′cosϕ;μ′s1,μa1)dϕ∫02πR(2rc1−cosϕ;μ′s1,μa1)dϕ

Taken together, the irradiance in each of the two regions, corallite and coenosteum, has two components: the “endogenous” light due to the illumination of the specific region and the light transported from the other region: (8)R(r)={R1(r)+R2→1(r), r<rcR2(r)+R1→2(r), r≥rc

Skeletal growth can be expressed as a function of (1) the size of the elementary unit block (nanograin), which has been observed between 30 and 100 nm in a few species using atomic force microscopy (Cuif and Dauphin, 2005a) and 40–50 nm in Stylophora pistillata using scanning helium ion microscopy (Von Euw et al., 2017), and its density (assumed to be 2.73 g/cm³ or the density of calcium carbonate), (2) the size of larger clusters as biomineralization occurs over different length scales through a mass-fractal equation of crystal growth (Uwaha and Saito, 1990; Peitgen et al., 1992; Miyashita et al., 2005), and (3) the distance between the cellular release of ions (e.g., calcium, carbonate and protons) and organic matrix molecules by calicoblastic cells and their incorporation into the mineralization site (incorporation diffusion length, l) assuming a diffusion-limited process (Furukawa et al., 2001; Allemand et al., 2004). (9)M=Mmin(Rr)Df (10)n=MV=MminrDfR3−Df=(rR)3−Dfn0 (11)nn0=(rR)3−Df

where, M represents the mass of a biomineralized fractal cluster, M_min represents the mass of the elementary unit block (i.e., nanograin), D_f is the mass-fractal dimension of the coral skeleton, r represents radius of the nanograin, R represents the radius of a biomineralized cluster (possibly the “mesocrystals”; domains in “calcification centers” of Porites sp skeletons consisting of nanograins organized crystallographically as one large crystal over several micrometers; Benzerara et al., 2011), n represents the average skeletal density, V is the volume of the fractal cluster, and n0=Mminr3 represents the density of the nanograin. Incorporation diffusion length l, (Furukawa et al., 2001) can be expressed as: (12)l=2Diff/G where, Diff is the diffusion coefficient of the ions and other elements involved in the mineralization process and G is the growth rate (interface velocity or the rate of linear increase of the skeletal surface at the deposition interface). Since the radius of a cluster R should be half the diffusion length (Uwaha and Saito, 1990; Furukawa et al., 2001; Miyashita et al., 2005), then: (13)R=12l=>R=DiffG =>G∝1R ∝ (nn0)13−Df  

Since D_f < 3 (Furukawa et al., 2001), it follows that denser skeletons (i.e., higher n) tend to grow faster than less dense skeletons and the relationship between G and n is a power-law with an exponent depending on D_f.

In order to test these model predictions, we used previously measured mass-fractal dimension D_f values for the 88 targeted species. The data showed that the majority of coral skeletons had D_f < 3 (Marcelino et al., 2013; see Text S2). This implies that the density auto-correlation function typically follows an inverse power-law relationship, between the minimal length scale r and the maximal length scale R, thus supporting the notion that coral skeletons are mass fractal-like structures. We tested the experimental relationship between growth rate G (characterized by maximum linear extension rates of species-specific skeletons), skeletal density n, and D_f, of each skeleton and compared it to the relationship obtained by the model (equation 13). Using available datasets and literature we compiled species-specific (1) maximum linear extension rates of skeleton (mm·yr⁻¹) as a proxy for growth rates and (2) maximum skeletal density values (g CaCO₃·cm⁻³) (Table S1).

Coral species characters were mapped onto a molecular phylogeny for assessment of homoplasy, phylogenetically-corrected regression, and phylogenetically-corrected principal components analysis. Patterns among species traits may be determined by a combination of ecological and evolutionary processes, and the ecological process can only by accurately assessed once the patterns of shared evolutionary history are accounted for (Rezende et al., 2007; Ivens et al., 2016). The patterns of shared evolutionary history violate an assumption of standard statistical analyses that individual data points are independent, and must be corrected for phylogeny (Revell, 2010). The comprehensive coral phylogeny of Huang (2012) was trimmed to include only the targeted species, the 88 species tree and a 14 species sub-set tree, using the Phylotools R package (Revell, 2012) to preserve proper branch lengths. These trimmed phylogenetic trees were used to define the evolutionary relationships and distances between species for phylogenetic correction of the raw data.

Characters were mapped onto the trimmed Huang (2012) phylogenies using Mesquite 2.75 (Maddison and Maddison, 2011) and visualized with Evolview (Zhang et al., 2012). Homoplasy of categorical data were assessed through the Retention Index (RI) in Mesquite, which is the fraction of apparent synapomorphy (character present in ancestor and shared exclusively by its evolutionary descendants) retained after mapping to the phylogeny, where highly homoplasious characters have low RI values (Farris, 1989). Relationships between continuous variables were assessed with Phylogenetic Independent Contrasts (PIC) analysis within the Phenotypic Diversity Analysis Programs (PDAP) of Mesquite, which corrects an ordinary least squares linear regression for non-independence due to evolutionary relationships among species (Midford and Garland, 2010). We report the results of phylogenetic independent contrasts through the Pearson Product-Moment Correlation Coefficient, or PCC, as well as its p-value. Skeletal features that were not found to be associated with bleaching response through PIC analyses were further assessed to determine if combinations of these features could be correlated with bleaching. Phylogenetically-corrected linear combinations of skeletal features were identified through phylogenetic principal component analysis (phylo-PCA) (Revell, 2009) using the Phytools R package (Revell, 2012). Phylogenetically corrected logistic regressions (Ives and Garland, 2010) of dichotomous variables (branching/massive colony morphology) were performed in Phyloglm v2.4 (Phylogenetic Generalized Linear Model; Ho and Ané, 2014) in R. Because of incomplete taxon sampling for n and G, and the order of magnitude greater parameters for solitary species (which made them extreme outliers in our analyses), the taxa included in the phylo-PCA were reduced to 59 species (Table S1). Phylo-PCAs were based on variable correlation rather than covariance due to their vastly different distributions. Principal components are reported in non-corrected space, therefore assessment of correlation with bleaching response was performed with PIC as above.

Bleaching responses of the 88 targeted coral species are highly diverse (bleaching response index, taxon-BRI, ranged from 2.75 to 72.85%, Swain et al., 2016a) and appear to have multiple independent evolutionary origins (i.e., bleaching response is homoplasious across the phylogeny resulting in a low RI = 0.33, Figure 1), providing an opportunity to detect correlations with skeletal characters that may otherwise closely reflect evolutionary history. Homoplasious characters provide a better opportunity to reveal the underlying relationships between character states, after correction for phylogenetic non-independence (Revell, 2010) among species.

The pattern of increased taxon-BRI associated with decreased μs,m′ (Phylogenetic independent contrast tests, PCC = −0.24, p = 0.024, n = 88; Figure 1; Table S2), is consistent with earlier work of our lab (Marcelino et al., 2013) and supports the optical feedback-loop hypothesis (Enríquez et al., 2005; Marcelino et al., 2013; Swain et al., 2016b; Wangpraseurt et al., 2017). This pattern is robust to decreased sample size (88, rather than 95 taxa), increased precision of the phylogeny used for correction of non-independence (using the updated and more comprehensive phylogeny of Huang, 2012), and increased precision of the species-specific estimation of bleaching response (using the updated and more comprehensive bleaching response values of Swain et al., 2016a). Association with bleaching response is specific to the light-scattering properties within the top ~200 μm of the coral skeleton (μs,m′) and was not detected in the light scattering properties within the overall skeleton (bulk-μs′, PCC = 0.06, p = 0.708, n = 42), mirroring our earlier findings using 10 experimentally bleached coral species (Swain et al., 2016b).

While we cannot currently explain these patterns, we hypothesize that they may be due, in part, to significant structural and chemical differences between the top ~200 μm layer and the rest of the skeleton. Separation between the most recently deposited biomineral and older parts of the skeleton is made by exothecal dissepiments; thin horizontal plates of biomineralized calcium carbonate laid as the colony extends upwards which varies among species. Dissepiment spacing was measured at 60–460 μm in Porites lobata, 180–720 μm in Goniastrea favulus, 200–1,000 μm in Pocillopora damicornis, (overall mean of ~400 μm; Nothdurft and Webb, 2007), 550–620 μm in Orbicella annularis and Montastrea faveolata, 800–850 μm in Montastrea cavernosa, and 680–720 μm in Montastrea franski (overall mean of ~600 μm; Dávalos-Dehullu et al., 2008). Micro-μs′ may result from light scattering by the most recently biomineralized calcium carbonate and organic matrix located above the latest dissepiment (e.g., the >60 nm voids in Goniastrea stelligera skeleton; Frankowiak et al., 2016), while bulk-μs′ would result from light scattering by new and old skeleton where older voids could potentially be filled with water or gas due to degradation of organic material. However, how the structural and chemical composition of the top skeletal layer is associated with bleaching susceptibility will require further investigation.

Differential photosynthetic efficiencies between polyp (above corallite) and coenosarc (above coenosteum) tissues have been attributed to high variability in within-colony light environments (Kühl et al., 1995; Ralph et al., 2002; Ulstrup et al., 2006, 2007). We reasoned that this high variability could be explained, in part, by differences in skeletal light scattering and transport between corallites and coenosteum. To that effect, we first measured light scattering properties and morphometrics of corallites and coenosteum in 14 coral species and then modeled the irradiance that reaches the corallite from coenosteum-transported light in three coral species with differing efficacy of light transport.

We measured μs,m′ of corallites (μs,m_corallite′) and coenosteum (μs,m_coenosteum′), calculated their ratio (μ_{S,m_Ratio}′ Equation 1) and determined the effective light transport distances of coenosteum (ETD_coenosteum) for 14 coral species (Figure 2). The ETD_coenosteum(equation 2), corresponds to the probability that light that enters the coenosteum could be diffusely transported into a corallite and is dependent on the distance traveled by a photon before scatter and randomization (transport mean free path in the top ~200 μm of the skeleton, lS′m, where l′S,m= 1/μs,m′), and on the mean inter-corallite distance (D_{inter−corallite}) within a colony. A high ETD_coenosteum indicates low probability that light entering coenosteum will be transported to neighboring corallites; similarly to within-tissue transport, the coenosteum could effectively transport light into corallites with inter-corallite distances up to ~20 × l_S′_{m_coenosteum}, (Ishimaru, 1999). In corals with (μs,m_corallite′/μs,m_coenosteum′) > 1, and D_{inter−corallite} < 20 × lS′m, the coenosteum could transport light to corallites which could then work as “light-trapping devices” where photons could be scattered multiple times by corallite septa and walls until absorbed by Symbiodinium. The 14 corals examined had (mean ± standard deviation) l_S′_{m_coenosteum} = 0.24 ± 0.12 mm; D_{inter−corallite} = 2.3 ± 1.8 mm; ETD_coenosteum = 13.48 ± 12.97 and μ_{S,m_Ratio}′ 1.11 ± 0.33, indicating that coenosteum could effectively transport light into corallites and be trapped by multiple scattering. Accordingly, corals with low-scattering coenosteum and high scattering corallites that are spaced within effective transport distances would be more likely to increase light availability to their Symbiodinium and precipitate a more rapid bleaching response under increased thermal stress, as predicted by the optical feedback-loop hypothesis (Enríquez et al., 2005; Marcelino et al., 2013; Swain et al., 2016b; Wangpraseurt et al., 2017). Indeed, higher μ_{S,m_Ratio}′ (i.e., μs,m_corallite′ > μs,m_coenosteum′, PCC = 0.69, p = 0.006, n = 14) and lower ETD_coenosteum (i.e., more effective transport, PCC = −0.603, p = 0. 022, n = 14) are significantly associated with increased bleaching response among the coral species examined here. This pattern is consistent with the within-tissue micro-sensor measurements of Wangpraseurt et al. (2017), who demonstrated much larger increases in scalar irradiance enhancement within corallites than at either the coenosarc surface or oral surface of the polyps between bleached and non-bleached corals.

Using the light scattering properties and morphometrics of corallites and coenosteum, we then modeled the irradiance at the corallite surface due to light transport within the coenosteum, assuming that no other light hits the corallite and absorption is negligible (equations 4–8). The distribution of coenosteum-originated irradiance at the corallite surface (R_{2 → 1}(r)) was modeled for three representative coral species with increasingly higher taxon-BRI and correspondingly progressively smaller ETD_coenosteum (Montastraea cavernosa, Seriatopora caliendrum, and Montipora informis, Figures 2, 3A–C). This can be quantified as the fraction of light that “escapes” the coenosteum and leaks into the corallite: T2→1=〈R2→1〉(〈R2→1〉+〈R2〉), where 〈R2→1〉=1πrc2∫0rc∫02πR2→1(r)rdϕdr is the average irradiance at the corallite surface due to the light transported from the coenosteum and 〈R2〉= 1π((rc+l/2)2− rc2)∫rcrc+l/2∫02πR2(r)rdϕdr is the average irradiance at the coenosteum. In case of M. cavernosa (high-ETD_coenosteum), only a small fraction of the coenosteum illumination reaches corallites (6%, Table S3). This fraction is much higher for the two species with more efficient light transport in the coenosteum (lower ETD_coenosteum); in S. caliendrum, and M. informis, nearly half of the light irradiating from the coenosteum gets transported into corallites (43 and 35%, respectively, Table S3). Figures 3D,E shows corresponding profiles of R_{2 → 1} as a function of the radial distance from the center of the corallites. In M. cavernosa, the species with the least efficient ETD_coenosteum (low bleaching susceptibility, high-ETD_coenosteum), the small fraction of light that diffuses into the corallite stays in its periphery, hardly ever reaching the center (〈R_{2 → 1}〉 = 0.04, Figure 3D). Conversely, corallites of the more efficient light transporters, S. caliendrum and M. informis (higher bleaching susceptibility and lower ETD_coenosteum) receive more light from the coenosteum, which also propagates further into the center of the corallites (〈R_{2 → 1}〉 = 0.07 and 〈R_{2 → 1}〉 = 0.2, respectively, Figures 3E,F). Thus, S. caliendrum and M. informis corallites work as “light-trapping devices,” diffusing light all the way through its center. This light-trapping effect of corallites is even more pronounced in M. informis which shows the highest bleaching susceptibility in our dataset (taxon-BRI = 73%, Figure 3F). Concomitantly, the fraction of the average corallite irradiance due light transport from the coenosteum, F2→1=〈R2→1〉(〈R2→1〉+〈R2〉), where 〈R1〉=1πrc2∫0rc∫02πR1(r)rdϕdr, progressively increases from M. cavernosa to S. caliendrum and M. informis. While coenosteum-transported irradiance plays only a small role in the total irradiation of the corallites of M. cavernosa (5%, Table S3), the fraction is considerably higher for S. caliendrum (46%, Table S3), and M. informis corallites (64%, Table S3).

Both observational and simulated results suggest that higher μ_{S,m_Ratio}′ and lower ETD_coenosteum correlate with increased bleaching susceptibility in corals. The variable light environment between corallites and coenosarc is likely also a function of other effects, such as the spatially heterogeneous distribution of host pigments in the tissue; green fluorescent protein-like (GFP) pigments in the host are more abundant in polyps then in coenosarc (Salih et al., 2000; Kaniewska et al., 2011; Lyndby et al., 2016; Ong et al., 2018). Additionally, morphological plasticity in corals has been implicated in regulation of within-colony light levels to optimize light capture and photosynthesis (Kaniewska et al., 2008, 2014) and temperature microenvironments within the colony have been found to result from variations in colony morphology, irradiance, and flow patterns within and around the colony (Jimenez et al., 2011; Ong et al., 2017). Future work could look at the combined effect of tissue and skeleton in modulating light environment to the algae, in particular at the level of the polyp/corallite and coenosarc/coenosteum.

Rate of deposition of calcium carbonate material (calcification rate) is the result of the linear extension of bio-aragonite deposited at the “centers of calcification” over time (linear growth or extension rate; mm yr⁻¹) and increased density from infilling by “thickening deposits” (bulk density or density of the deposited material and the volume of voids in the overall architecture; gCaCO₃ cm⁻³) (reviewed in Stolarski, 2003; Cuif and Dauphin, 2005b; Lough and Cooper, 2011). The fractal coral growth model derived here relates linear extension rate of skeleton (G) and skeletal density (n) through a power-law (which has the property of invariance across a range of length-scales, thus representing fractal growth) with an exponent depending on D_f (Equations 3–7). Since D_f < 3 (Marcelino et al., 2013), the model showed that denser skeletons (i.e., higher n) tend to grow faster. This trend was confirmed by observational data with measured G and n for each coral species showing a positive correlation; branching and corymbose corals showed the highest skeletal density (1.94–2.33 gCaCO₃ cm⁻³, n = 26) and linear extension rates (48.1–86.2 mm yr⁻¹) while massive, encrusting and columnar corals had the lowest skeletal density and linear extension rates (1.61–1.99 gCaCO₃ cm⁻³ and 12.38–20.44 mm yr⁻¹, n = 29; Figure 4A, r = 0.35, p = 0.02). These results agree with previously reported differences in growth strategies (i.e., higher investment in linear extension and/or skeletal density) which are due, in part, to the expected mechanical properties of different growth forms. Branching and digitate forms typically have higher extension rates and denser skeletons, while massive and encrusting forms show lower extension rates and lower skeletal density (Hughes, 1987; Jimenez and Cortes, 1993; Marshall, 2000; Morgan and Kench, 2012). However, rapidly growing branch tips of Acropora species are much less dense than their colony base due to secondary infilling of older parts of the colony to provide strength (Hughes, 1987; Bucher et al., 1998).

On average, dependence G(n) followed a power-law form G ∝ n^α with exponent α ~ 3.8 (i.e., power law best fit to the experimental data). From Equation 13, α=1(3-Df), this exponent corresponds to the average value of D_f ~ 2.74, which approximates the fractal dimension of diffusion-limited aggregation in three-dimensions (Peitgen et al., 1992). The model also predicts that G and D_f are inversely related: corals that exhibit faster linear growth rates are expected to form skeletons with lower fractal dimensions. This trend was also confirmed in our observational data with independently measured species-specific G and D_faveraged for each one of the 10 growth forms, which showed a significant negative correlation (in spite of the large variations in D_f for some growth forms, Figure 4B, r = −0.30, p = 0.025, n = 59). Furthermore, the fractal growth model was able to predict growth rates of coral taxa. We compared measured species-specific linear extension rates with linear extension rates predicted by the model (G in Equation 13) based on reported maximal skeletal density for each coral species and measured D_f for each coral skeleton. Although skeletal density is affected by processes that are not directly related to the linear fractal growth (e.g., infilling; Stolarski, 2003; Cuif and Dauphin, 2005b; Nothdurft and Webb, 2007), model-predicted G correlated well with measured growth rates (Figure 4C, r = 0.39, p = 0.003, n = 59). Additionally, the model prediction that skeletal density is inversely related to D_f was also validated (r = 0.35, p = 0.009, n = 59).

Predictions of the skeletal growth model, that corals with faster linear extension rates will have denser skeletons and will form skeletons with lower mass-fractal skeletal dimension, are in agreement with experimental data. D_f is one of the parameters that influence light scattering: according to light scattering theory (Rogers et al., 2014), μs,m′ is expected to be proportional to D_f. This positive relationship between D_f and μs,m′was confirmed by our data (PCC = 0.224, p = 0.036, n = 88). Furthermore, the model shows inverse dependence between growth rate of the skeleton (i.e., linear extension rate) and the size of the biomineralized cluster R (Equation 13), which light theory describes as positively correlated with μs,m′ (Rogers et al., 2009). Consequently, and although growth rate is influenced by several factors (reviewed in Dávalos-Dehullu et al., 2008; Lough and Cooper, 2011), the model predicts an inverse relationship between growth rate and μs,m′, which is confirmed by our data (PCC = −0.292, p = 0.017). Since an inverse dependence between μs,m′ and bleaching response has been shown through experimental and correlational evidence (Marcelino et al., 2013; Swain et al., 2016b), corals that grow faster are expected to have denser skeletons, lower skeletal D_f, lower μs,m′, and are likely to show higher bleaching susceptibility. However, growth rate was not found to be significantly associated with bleaching susceptibility with phylogenetically-corrected regression analysis (Table S2) although taxon-BRI and measured G show an increasing linear relationship that plateaus (Figure S1).

Several structures demonstrated to either directly modulate the internal light microenvironment, or to change in response to variation in external light fields at different length scales, were evaluated for possible association with bleaching susceptibility. Since μs,m′ is inversely correlated with bleaching susceptibility (Marcelino et al., 2013; Swain et al., 2016b), we reasoned that skeletal structures that partially determine μs,m′ could also conceivably be associated with bleaching susceptibility.

Micro-μs′, similarly to light-scattering in tissue, is determined by interaction with several skeletal structures: (1) average size of nanograin (30–100 nm, Cuif and Dauphin, 2005a; Von Euw et al., 2017), and (2) extent of the fractal range in the biomineralized cluster R (which relate positively to μs,m′; Rogers et al., 2009 and are currently unknown for individual species), (3) the shape of the optical refractive index correlation function C(r) given by mass-fractal dimension D_f (which inversely correlates with μs,m′; Rogers et al., 2009) and was previously determined for all 88 targeted species (Marcelino et al., 2013), (4) the variance of the of the optical refractive index correlation function, C(r) (unknown for each species), and (5) macroscopic structures with voids (which will lower bulk- or micro-μs′; Enríquez et al., 2005; Stambler and Dubinsky, 2005; Rogers et al., 2009, 2014; Terán et al., 2010; Marcelino et al., 2013; and have been measured for some species).

Although D_f showed a significant association with μs,m′ as expected from light theory, when D_f was tested for possible association with bleaching response, phylogenetically-corrected regression analysis revealed a non-significant negative relationship, indicating that although the mass-fractal organization of the skeleton is implicated in light scattering it is not correlated with bleaching susceptibility via the μs,m′ pathway.

Macroscopic structures at higher length scales were also evaluated for possible association with bleaching susceptibility: colonial module dimensions (corallite size) and organization (corallite complexity, percent colony surface area composed of coenosteum—% SA_coenosteum). While it is clear that these features are related to each other in specific patterns across the coral phylogeny, since larger diameter corallites are also more complex (PCC = 0.809, p < 0.001, n = 88) and are surrounded by less coenosteum (PCC = −0.359, p < 0.001, n = 88; Figure 1; Table S2), none of these structures was significantly associated with bleaching response (Figure 1; Table S2). Finally, colony morphology (10 different growth forms listed in Table S1) was examined for correlation with bleaching susceptibility, since massive and encrusting colonies are historically reported to bleach and die at lower rates than branching and digitate colonies (Marshall and Baird, 2000; McClanahan, 2004; van Woesik et al., 2011).

We quantified each growth form by calculating cumulative fractality, or the product of the apparent surface fractality of each growth form, and D_f measured for each species. No significant correlation between taxon-BRI and cumulative fractality was detected (PCC = −0.08, p = 0.47, n = 81). We further performed phylo-log regression analysis on coral species that could be classified as massive (coded as 0, n = 44) or branching (coded as 1, n = 27) and could not identify a significant association with bleaching response (Z = −0.58, p = 0.56). Additionally, we used growth form-specific ratios of surface area to volume described by Madin et al. (2016) (see Text S2) to quantify growth forms and also found no significant correlation with bleaching response. Recently it has been shown that the perceived relationship between growth forms and bleaching susceptibility is partially the result of their phylogentic relationships (McCowan et al., 2012; Swain et al., 2018); for example, while in the Faviidae family branching species bleached less than massive, the opposite was observed in the Acroporidae and Poritidae families (McCowan et al., 2012). Furthermore, due to the largely unknown effect of phenotypic plasticity within species (observed along gradients of light and nutrient flow) and species boundaries on colony morphology (reviewed in Veron, 2013), it has been challenging to accurately quantify and examine possible associations between species-specific growth forms and their bleaching response.

We reasoned that there may not be a clear relationship between bleaching response and individual features, but in combination with other skeletal characters known to modulate the light microenvironment, we may reveal their relationship with bleaching response. To that effect, we applied phylogenetically-corrected principal components analysis (phylo-PCA; Figure 5) to the seven characters that were not associated with taxon-BRI, to identify groups of characters that could be linearly combined in multi-dimensional space into principal components that explain the data variance: PC1 explained 41.2% and PC2 explained 22.3% of the data variance (PC3 explained only 15% of the data and was not considered further). Correlations between the two main principal components and the seven skeletal variables show that D_f, n, corallite complexity, and % SA_coenosteum were important components of PC1 while growth rate, corallite size and cumulative fractality were important components of PC2. Neither PC1 (PCC = 0.07, p = 0.6, n = 59) nor PC2 (PCC = 0.14, p = 0.31, n = 59) were significantly related to bleaching response.

These results show that variation in the skeletal structures targeted in this study, which have either been directly associated with modulating the light microenvironment surrounding Symbiodinium or shown to respond to variation in external light, are not related to bleaching response of these species. The lack of significant relationships should not be taken as an indication that these features do not actually modulate the internal light microenvironment of in hospite Symbiodinium, just that variation in these features and their abilities to modulate light microenvironments were not able to explain the observed variation in bleaching response across species. Variation in bleaching response among coral species has been attributed to diverse factors both intrinsic to the holobiont (e.g., thermotolerance of Symbiodinium and the interactions between coral, Symbiodinium, and other microbes; Baird et al., 2009; Leggat et al., 2011; Cunning and Baker, 2013; Krediet et al., 2013) and extrinsic from the environment (e.g., site-specific environmental conditions and frequency of thermal anomalies; McClanahan and Maina, 2003; Guest et al., 2012; Pratchett et al., 2013). Furthermore, the diversity of skeletal macro features observed among coral species are the result of simultaneous optimization of light capture and other vital functions including respiration and metabolite exchange between tissues and the environment (mass transfer; van Woesik et al., 2012), particle capture (Sebens et al., 1997), reproduction (Soong and Lang, 1992), and structural stability (Baldock et al., 2014). These forms are constrained by physical limitations imposed by the materials (Jimenez and Cortes, 1993; Boller et al., 2002) and three dimensional geometry of assembling modules (polyps) into a colony (Barbeitos, 2012), the physiological integration of those units (Coates and Oliver, 1973; Coates and Jackson, 1987; Soong and Lang, 1992 and Swain et al., 2018), and the evolutionary history of the species (Barbeitos et al., 2010; Budd et al., 2012). Perhaps the multitude of competing selection pressures directed at skeletal features may reveal correlations with bleaching susceptibility via pathways that are not light-scattering related, such as life-history strategies which have their own sets of specific morphological features (Darling et al., 2012, 2013). Corals with stress-tolerant strategies typically have massive morphologies, large corallites or meandroid polyp organization, grow slowly and have high fecundity, while corals with competitive strategies have large branching and plating morphologies, with smaller corallites, grow quickly, are more sensitive to storm breakages and bleach and die at greater rates during thermal stress events (Darling et al., 2012). Additionally, different morphologies are known to differ in their mass transfer abilities and within-tissue temperatures. Corals with higher surface area to volume ratios have faster gas and metabolite exchange across tissues (van Woesik et al., 2012), although ciliary movement (across the entire colony surface) can actively enhance mass transfer rates by 400% relative to passive molecular diffusion (Shapiro et al., 2014). Massive corals, due to their thick tissues and thicker thermal boundary layers, have higher within-tissue temperatures than branching corals with a thin veneer of tissue overlaying the skeleton and higher surface area to volume ratios (Jimenez et al., 2008, 2011). Furthermore, several non-photosynthetic fluorescent pigments present in host cells, such as GFP-like proteins, UV-absorbing compounds and other pigments, have been implicated in modulating the light microenvironment of Symbiodinium (Schlichter et al., 1986; Salih et al., 2000; Smith et al., 2013; Lyndby et al., 2016). In the case of GFP-like proteins, their role in modulating light microenvironment of Symbiodinium changes with pigment density. When present at low densities in the tissue, GFP-pigments scatter downwelling light, thus increasing light absorption and heating in the tissue, but at high pigment densities strong scattering reduces absorption and heating (Lyndby et al., 2016). The presence of GFP-pigments in the tissues may partially account for differential coral bleaching susceptibility and should be carefully investigated.