Coralline algae are important habitat formers found on all rocky shores.
While the impact of future ocean acidification on the physiological
performance of the species has been well studied, little research has
focused on potential changes in structural integrity in response to climate
change. A previous study using 2-D Finite Element Analysis (FEA) suggested
increased vulnerability to fracture (by wave action or boring) in algae
grown under high CO

Since the pre-industrial era CO

Rhodoliths (Fig. 1), non-geniculate free living (live and dead) coralline
red algae (Foster, 2001), are an extremely diverse group of benthic
calcifying organisms found from the polar to the tropical regions and low
intertidal zones to 150m deep (Foster, 2001). They are major contributors to
the global inorganic carbon budget in shallow water ecosystems (Mackenzie et
al., 2004). The largest rhodolith bed found on the Abrolhos shelf
contributes roughly 5 % to the global calcium carbonate budget
(Amado-Filho et al., 2012). Additionally these benthic ecosystems support a
high level of biodiversity by forming structurally and functionally complex
habitats (Nelson, 2009) for many organisms including polychaetes,
crustaceans and mollusks (Foster, 2001), as well as being important nursery
grounds to commercial species including scallops (Grall and Hall-Spencer,
2003). Coralline algae biodiversity and habitat complexity are directly
correlated; species richness and abundance of, in particular, arthropods,
annelids and cnidarians, are known to increase as rhodolith size and
branching increases (Foster et al., 2013). However coralline algae are
ecologically fragile due to their slow growth rate of

A

Furthermore, as they form high Mg-calcite skeletons, the most soluble
polymorph of calcium carbonate (CaCO

Coralline algae have many pressures to withstand in their natural
environments, including wave action and bioerosion, while maintaining a
structurally and functionally complex habitat. Under elevated CO

Finite Element Analysis (FEA) is a technique that reconstructs the stress, strain and deformation in structures (Zienkiewicz et al., 2005). Originally developed for mathematical and engineering applications, it has recently become an established technique in zoology and palaeontology to understand morphology, function and evolution of hard tissue structures (Rayfield, 2007).

FEA works by transforming a continuous structure into a discrete number of
elements which are connected to each other via nodes. The combination of
elements and the interconnecting nodes form the mesh. Appropriate material
properties (Young's modulus and Poisson's ratio) are assigned to the
elements to mimic the elasticity of the structure. Adequate boundary
conditions (magnitude and direction of loading and constraints) are applied
and then nodal displacements are calculated in response to the applied
boundary conditions and material properties of the model. The nodal
displacement is used to calculate the strain and subsequently stress (using
the Young's modulus, see Eq. 1) and hence mechanical performance of
complex structures can be inferred. (For mathematical equations see
Mathematics of FEA; Rayfield, 2007; Supplement).

These simple 2-D models represented the mechanical performance of a
cross-section of uniform thickness (1

The effect of elevated CO

We then used our improved understanding of the performance of the 3-D models
to re-assess the impact of current and future ppm CO

Four 3-D FE models with different coralline algal features were created based on the measurements and properties of Ragazzola et al. (2012) models. All 3-D geometric models were created and analysed in the Finite Element software package, Abaqus/CAE, v.6.10, (Simula, USA, Dassault Systémes, Simula, Providence, RI, USA), following the protocol established by Ragazzola et al. (2012).

Firstly, the 2-D models from Ragazzola et al. (2012) were expanded, using the
extrude command in Abaqus, to create a 3-D model with the same area, all
length and width dimensions (78.92

The four computer derived models;

Although the cellular width and height in the Corridor model represented the
dimensions recorded by Ragazzola et al. (2012) from SEM images, the cells
are not represented as discrete entities and instead are represented as
hollow calcite “tubes”. As algal cells grow apically, they form a lattice of
individual cells, hence a second model, “Compartment model” (Fig. 3b) was
created in which cell walls were added in the z direction to create discrete
cellular spaces. All dimensions apart from the depth of the individual cells
and divisions between cells in the

Computed tomography (CT) scans of

Dimensions used in the 3-D corridor model. Units

A final model, “the OA model” (Fig. 3d), was created by adjusting the cell
size and spacing to represent the dimensions of the future 589

Orthoslice projections in the different planes of a rhodolith
thallus:

In keeping with Ragazzola et al. (2012), a load pressure of 20 000 Pa was
applied to the top left corner, 40

Convergence tests were performed for each mesh type in order to determine the minimum mesh size required. The mesh size was decreased until the average von Mises value no longer changed relative to mesh size. Hypothetically, all refined meshes should converge to similar results yet our converged von Mises stress value was an order of magnitude different between the hexagonal and tetrahedral mesh. This was due to the shape of the tetrahedral elements and the way tetrahedral elements interlock together, making a tetrahedral model stiffer than a hexagonal model. Whereas Dumont et al. (2005) found that comparing a converged 4-node linear and a stiffer 10-node quadrilateral tetrahedral mesh of the same model gave different mean stress values, but within 10 %. This shows that even when comparing different forms of the same tetrahedral element, variation in stress is still apparent. Hence when comparing different element types (hexagonal and tetrahedral), we find an even greater difference in variation. Therefore in order to compare the 2-D to 3-D geometric models, the corridor model was meshed with 4-node linear hexagonal elements. As tetrahedral elements were better at capturing the complex geometry of the biological model and to account for variation in results depending on element type, all models were then meshed with 4-node linear tetrahedral elements in order to be compared to the biologically realistic model and to each other.

Loads and Boundary constraints. All models had the same loads and
constraints applied. Loads, representing wave erosion, were applied to the
top left hand corner along a strip 40

In keeping with Ragazzola et al. (2012), all models were assumed to be composed of a linearly elastic, isotropic, homogeneous material with a Young's Modulus of 36 GPa and a Poisson ratio of 0.31 (properties of calcite; Tanur et al., 2010). Even though these material properties do not accurately represent the heterogeneities in the specimen and their material properties this approach gives a comparative insight into how different geometries affect the overall strength of a structure.

Initially, to analyse how sensitive the models were to changing material
properties, a set of 2-D and 3-D corridor models with different Young's
modulus (maximum and minimum Young's modulus values of two different
bivalves –

Mesh type, number of elements, average von Mises stress, 95th percentile of von Mises stress and total strain energy for the different models.

In order to compare the impact of predators on the different 3-D geometric models, stress and strain results were calculated. Stress, generated by the applied load (force) on a given area, is represented by the von Mises stress, a function of each of the principle stresses that represents tensile or compressive stress (Rayfield, 2007). Average stresses were calculated by dividing the von Mises stress by the element size to account for differing number of elements between models. Total strain energy refers to the energy stored in a system as a load is applied, which is a useful variable to record in mineralised structures as the more strain in a system leads to a larger amount of potential energy available for fracture (Gordon, 1978). The 95th percentile of von Mises stress was additionally used as a comparison between the corridor, compartment and biological models as this metric highlighted the extremes of the von Mises Stress distribution – an important parameter to highlight fracture potential.

Stress and strain energy are linearly dependent on surface area and volume
respectively (Dumont et al., 2009), hence the applied pressure was rescaled
for the biological model to rule out the effect of increased Mg-calcite
volume on modelling stress and strain. To account for the increase in volume
of calcite between the 2-D and the 3-D model, as strain energy is dependent on
volume, the strain energy was calculated for the 3-D model using Eq. (3),
outlined by Dumont et al. (2009);

The von Mises stress was displayed graphically on the model, with warm colours (red/orange) indicating areas of high stress and cool colours (blue) indicating areas of low stress. Stress distribution throughout the model was very similar to surface stress distribution. The surfaces of the model were more sensitive to the loads and constraints, due to immediate contact with the boundary conditions. The minimum and maximum von Mises stress values were found on the surfaces of the models, being more influenced by the position of the boundary conditions and complexities in the geometry. Hence, the minimum and maximum values did not provide any additional information on the overall structural integrity of the model than that provided by the surface contour plots. Average von Mises stress values, total strain energies and 95th percentile of von Mises stress can be found in Table 1.

The biological and the compartment model were exposed to different loading scenarios in Abaqus. This included the original load setup explained earlier in Sect. 2.1.4 (Fig. 7a); the compressive loads, where the load was applied to the top of the cube opposite the constraint (Fig. 7b); and shear loads, where the load was applied on the face adjacent to the bottom constraint (Fig. 7c).

As this part of the study moved on from the initial research of Ragazzola et al. (2012), it was decided to use loads defined experimentally based on
real-wave velocities. Starko et al. (2015) used wave velocities of up to 3.5 m s

The von Mises stress patterns on the biological model

A similar stress distribution was observed in the 2-D and 3-D geometric model
(Corridor model; Fig. 8a–b), with areas of high stress occurring along the
intra-cell walls and adjacent to the constrained surface. The average von
Mises stress of the two models was very similar (Table 1), indicating that
the simple 2-D model (Fig. 8a) was an accurate representation of the stress
in a simple 3-D geometric model (Fig. 8b). As expected, the total strain
energy in the 3-D model was over 500

Once the complexity of the 3-D model was increased to better represent the
natural structural complexity, differences between the simple 3-D model
(Corridor model) and the more complex models became evident (Fig. 9a–c). In
the compartment model of current CO

The results of the comparison between the 3-D geometric models (the Corridor and Compartment model) and the realistic model of similar dimensions generated from CT scans (the Biological model) showed that a similar average stress, strain energy (Table 1) and stress distribution (Fig. 9b–c) was observed between the compartment model and the biological model. Comparison of the internal morphology between the compartment model and the biological model also showed similarities. Both models had regularly distributed cavities. However unlike the compartment model the biological model cavities were spheroidal and, due to the natural variation within these specimens, the arrangement of cavities was not as regimented as in the compartment model (Fig. 10). Both the biological model and the compartment model had the same percentage volumes of calcite and cavities whereas the corridor model had a lower percentage volume of calcite (Table 2).

Comparison of all 3-D models.

The inside spheroidal cavities of the biological model.

As the compartment model was similar in performance to the biological model,
we used this model to assess the impact of ocean acidification (Fig. 2c–d).
The change in wall thickness and cell size in the 589

Percentage volumes of calcite and cavities in the biological model, the corridor and the compartment model.

However, using more environmentally significant forces in the shear and
compressive comparison tests, we can see that the stresses and strains
exerted by these organisms were not as large as those taken from Ragazzola
et al. (2012). Accounting for the change in units, the differences between
the von Mises stress results (Pa) are on the order of magnitude of 10

In the biological model, under the original load setup, stress dissipated throughout the model from the corner where the load was applied to the constrained corner (Fig. 7a). While under the compressive load setup, the stress had a top to bottom distribution (from the loaded surface to the constrained surface) with a slight increase in stress surrounding the cavities in the model (Fig. 7b) and under the shear load setup, two thin bands of higher stress perpendicular to each other were observed (Fig. 7c). The average von Mises stress, 95th percentile of von Mises stress and total strain energy were slightly larger under the shear load setup compared to the compressive load setup (Table 3). All three values were larger than the compressive or shear model in the original load set up (Table 3).

In the compartment model, under the original load setup the stress dissipated throughout the model from the corner where the load was applied to the constrained corner (Fig. 7d). While under the compressive load setup, the area of higher stress was restricted to the top of the model where the load was applied (Fig. 7e) and under the shear load setup, the area of high stress spread from the right hand side near the constrained corner (Fig. 7f). The average von Mises stress, 95th percentile of von Mises stress and the total strain energy were largest under sole shear loads and smallest in the compressive load model, with values for the original set up falling in between (Table 3).

Average von Mises stress, 95th percentile of von Mises stress and total strain energy for the different load types exerted on the biological, compartment and corridor models. Total strain energy for the biological model has been corrected for calcite volume (Eq. 2). The compartment model under the shear loading type is highlighted in bold to reiterate that the loading setup is different to the biological model under a shear loading type.

Note the shear load in the compartment model was applied differently to the
arrangement for the biological model. As the compartment model was not able
to run under a sole shear load, like the biological model, a small
constraint on the opposite face (1

The sensitivity test of the 2-D model and the 3-D corridor model highlighted that increasing the Young's modulus by 120 % did not result in any change in stress, whereas the total strain energy decreased with increasing Young's modulus (Table 4).

Average von Mises stress and total strain energy for the comparison of the different material properties in the 2-D and 3-D corridor models.

Ragazzola et al. (2012) used 2-D FE models to predict if changes to coralline
algae cellular morphology induced by higher CO

Coralline algae grow apically with lateral cell fusion being very common (Irvine and Chamberlain, 1994). This creates a vast network of individual cells able to oppose stress in all directions (Gordon, 1978). The importance of these structures is highlighted by the geometric model with compartments (the Compartment model) being the most stable of the geometric structures assessed and also most comparable – in terms of percentage volume of calcite, stress distribution and magnitudes of average stress and total strain energy – to the biological model (Tables 1 and 2). This highlights the importance of geometry changes, which our method accurately captured, to the distribution and magnitude of stress. This occurrence was also observed by Romeed et al. (2006) who previously found that changes in the geometry between their 2-D and 3-D models of a restored premolar tooth also affected their displacement and profile stresses.

Creating the FE model of the biologically realistic structure (the
Biological model) from CT data was a time consuming process compared to the
user effort required to generate the compartmentalised geometric model (the
Compartment model). As these different models produced very similar measures
of average stress (2.75E

Hence this model was used to assess the impact of ocean acidification
changes to the cell growth on structural integrity. When the cellular size
of the 3-D model was adjusted to reflect 589

Although our geometric and biological models show congruence, they are still
simplifications of the heterogeneities in the algal skeleton. A factor which
was not considered in this study is the potential effect of changing
material properties due to global change on the algal skeleton. Material
properties are affected by the concentration of Mg in the skeleton (Ma et
al., 2008), which is dependent on temperature (Kamenos et al., 2008) and
potentially pH (Ragazzola et al., 2013; Ries, 2011). Mg incorporated into
the calcite lattice increases the lattice distortion, which causes an
increase in the sliding resistance and deformation resistance to crystals
(Wang et al., 1997). Ma et al. (2008) found that due to a much larger
difference in mol % of MgCO

Sea surface temperatures in the North Atlantic are predicted to rise
2.5

Proteins are also known to affect material properties as the incorporation
of organic macromolecules reduces the brittleness and enables plastic
deformation (Berman et al., 1988; Ma et al., 2008; Wang et al., 1997; Weiner
et al., 2000). The presence of chitin and collagen within the skeleton of

However, using these more biologically accurate models, we have further supported previous results that state future climate change will lead to a loss in the structural integrity of coralline algae. We have shown that by increasing the complexity of a simple 2-D geometric model to a 3-D geometric model we can obtain informative data on the effect of ocean acidification on the structural integrity of the coralline algal skeleton, without the need for complex real biological models derived from CT scanning that take ample computer time to construct and analyse. As responses to climate change are species-specific, we are therefore able to create models tailor made to individual species and analyse how they react to future climate change. We have also shown the susceptibility these models have to shear loads rather than compressive loads.

As the oceans are becoming more acidic, with concurrent calcification
pressure, it is vital to understand the potential effect of ocean
acidification on the skeletons of these habitat-forming organisms to infer
whether they are able to maintain habitats in the future. As coralline algae
are major habitat formers, with the diversity and abundance of species
dependent on their structural complexity, weakening of the skeleton under
high CO

It is important to note that recent long-term studies have shown calcifying organisms acclimating to ocean acidification for example cold-water corals sustaining growth rates (Form and Riebesell, 2012) or coralline algae decreasing growth rates to maintain cell wall thickness (Ragazzola et al., 2013). The consequence of this sustained growth on the material properties and structural integrity has not been assessed and poses an open question with regards to their ability to provide habitats in the future.

Leanne Melbourne and Julia Griffin carried out the experiments under the guidance of Daniela Schmidt and Emily Rayfield. Leanne Melbourne prepared the manuscript with contributions from all co-authors.

The authors would like to thank NERC studentship award [NE/L501554/1] and the Natural History Museum, London for LAM and a Royal Society URF for DNS for providing funding, Federica Ragazzola for allowing access to FE models and Jen Bright and Phil Anderson for general help with modelling aspects. The tomographic scans from Ragazzola et al. (2012) were taken on the TOMCAT beamline at the Swiss Light Source, Paul Scherrer Institut, Villigen, Switzerland. European Commission under the 7th Framework Programme: Research Infrastructures. We also would like to thank our reviewers, C. Evenhuis and an anonymous reviewer, whose valuable comments provided improvements to our paper. Edited by: J. Middelburg