# Composite material in the sea urchin Cidaris rugosa: ordered and disordered
micron-scale bicontinuous geometries111Manuscript accepted for publication in
Journal of the Royal Society Interface, https://doi.org/10.1098/rsif.2023.0597

Anna-Lee Jessop1222Author for correspondence: Annie Jessop
(Annie.Jessop@murdoch.edu.au), Allan J Millsteed1, Jacob JK Kirkensgaard3,4,
Jeremy Shaw2,
Peta L Clode2,6, Gerd E Schröder-Turk1,53331School of Mathematics, Statistics,
Chemistry and Physics, Murdoch University, Australia, 2Centre for Microscopy,
Characterisation, and Analysis, University of Western Australia, Australia,
3Niels Bohr Institute, University of Copenhagen, Denmark, 4Department of Food
Science, University of Copenhagen, Denmark, 5Research School of Physics, The
Australian National University, Australia, 6 School of Biological Sciences,
University of Western Australia, Australia

## Abstract

The sponge-like biomineralised calcite materials found in echinoderm skeletons
are of interest in terms of both structure formation and biological function.
Despite their crystalline atomic structure, they exhibit curved interfaces
that have been related to known triply-periodic minimal surfaces. Here, we
investigate the endoskeleton of the sea urchin Cidaris rugosa that has long
been known to form a microstructure related to the Primitive surface. Using
X-ray tomography, we find that the endoskeleton is organised as a composite
material consisting of domains of bicontinuous microstructures with different
structural properties. We describe, for the first time, the co-occurrence of
ordered Primitive and Diamond structures and of a disordered structure within
a single skeletal plate. We show that these structures can be distinguished by
structural properties including solid volume fraction, trabeculae width, and
to a lesser extent, interface area and mean curvature. In doing so, we present
a robust method that extracts interface areas and curvature integrals from
voxelized datasets using the Steiner polynomial for parallel body volumes. We
discuss these very large scale bicontinuous structures in the context of their
function, formation, and evolution.

##  1 Introduction

Nature’s ability to construct specialized composite materials through highly
controlled biomineralization processes has inspired many decades of research
(reviewed in [41, 8]). Echinoderms, such as sea urchins and star fish, have
been a focus of much of this research due to their ability to form skeletons
with optimized physical properties [46, 47, 27] and the ease of imaging their
optically clear embryos in real time (reviewed in [43]). They form a
hierarchical multi-element endoskeleton consisting of stroma and stereom: the
stereom consists primarily of magnesium-calcite, small amounts of stable
amorphous calcium carbonate, water, and intracrystalline organic molecules;
the stroma consists of sclerocytes, collagen fibrils, and other extracellular
matrix components [10, 34]. The stereom is formed through the
biomineralization of calcium carbonate into magnesium-rich calcite that
diffracts X-rays like a single crystal [33, 26, 7, 46].

Figure 1: Reconstructed micro-CT data of the sea urchin C. rugosa. (A) Three-
dimensional visualization of the whole urchin skeleton depicting the
approximate location of the sectioned interambulacral plate shown in B. The
oral surface of the skeleton is pointing upwards and the aboral surface is
pointing downwards. Scale bar = 1 cm. (B) Three-dimensional visualization of
the sectioned interambulacral plate depicting the location of the slice shown
in C. Scale bar = 1 mm. (C) A single slice through the interambulacral plate
shown in C approximately oriented with the tubercle in the positive
Z-direction. The outlined section depicts a close up of this slice shown in D.
Scale bar = 1 mm. (D) Close up of outlined section in C, where the red
highlight shows the region of the interambulacral plate that was scanned with
a resolution of $1.69\,\mu{m}$ and the blue highlight shows the region of the
plate scanned with a resolution of 732 nm. Scale bar = $100\,\mu{m}$.

Stereom types include dense solid layers of calcite and diverse three-
dimensional mesh-works of trabeculae that exhibit complex sponge-like
morphologies with curved surfaces [34]. Many of these sponge-like stereom
types can be thought of as bicontinuous geometries that partition space into
two separate domains each of which forms a single connected network: one
filled with stereom and the other with stroma. They can be disordered
structures such as the ’laminar’ or ’labyrinthic’ stereom or they can exhibit
a high degree of order such as in the ’rectilinear’ stereom [34]. In a few
extraordinary cases, these ordered bicontinuous calcite structures have been
shown to relate to known triply-periodic minimal surfaces (TPMS), albeit with
solid volume fractions ¡50% [26, 7, 46, 11]. These structures exhibit
periodicity in three spatial directions and partition space into two
interconnected volumes, generating a porous light-weight structure with
optimized properties such as high-energy absorption capabilities [46].
Remarkably, in all cases, the length scales (manifest in the lattice
parameters) of these TPMS-related calcite structures are enormous
(¿$10\mu{m}$; given they are thought to be templated by soft matter [16]),
being more than an order of magnitude larger than related structures found
across most kingdoms of life (reviewed in [14]). For example similar
structures are found in insect cuticles and scales [23, 31, 44, 45], in
intracellular membranes [2, 20, 19], and in the keratin nanostructures of bird
feathers [30]. Ordered bicontinuous structures related to TPMS geometries are
hence common throughout the natural world, as they also are in physical
chemistry (see reviews in [21, 15, 22]); however, the structures found in
these echinoderms represent the only examples at such large length scales.

One of these examples was first described many decades ago in the sea urchin
Cidaris rugosa [26]. Using scanning electron microscopy, Nissen (1969)
described a stereom type within the interambulacral plates of C. rugosa that
exhibited a ”regular network of trabeculae” and appeared to resemble one of
the simplest TPMS, the so-called Primitive surface (’Plumber’s nightmare’).
Since then, authors have designated the ’rectilinear’ stereom type, found in
several species of Cidaroids [34], to the Primitive surface [10, 11, 12],
however a three-dimensional quantitative descriptive of this stereom type is
lacking. Recently, a related ordered bicontinuous form, the so-called Diamond
surface, was discovered in the ossicles of the starfish Protoreaster nodosus
[46] and in the fossilized skeleton of a 385 million year old crinoid
Haplocrinites boitardi [11], sparking a renewed interest into the calcite
microstructures of echinoderms including their potential in biomimetics [27,
28, 47, 40].

In the present study we re-examine the skeletal microstructures of C. rugosa
using X-ray tomography and X-ray scattering, providing a three-dimensional
analysis of three of these microstructures and a preliminary analysis on their
crystallinity.

##  2 Methods

###  2.1 Sample preparation

A single specimen of C. rugosa was donated by Prof. Charles Messing (Nova
Southeastern University). This specimen had been collected during a manned
submersible dive in the North Atlantic Ocean (26°07.1849’ N 79°50.0250’W) at a
depth of 302 m in June 2006 and was identified by Dr David Pawson (Smithsonian
Institute).

The spines of the whole urchin were removed and the whole urchin skeleton was
glued to an aluminium mount and imaged. Subsequent to this, a single
interambulacral plate was sectioned from the whole urchin using a 20 mm
circular saw blade and glued to a pin for imaging.

###  2.2 Image acquisition and processing

Urchin samples were scanned with a Versa 520 XRM (Zeiss, Pleasanton, CA, USA)
running Scout and Scan software (v15.0.17350.39816) at the Centre for
Microscopy, Characterisation, and Analysis at the University of Western
Australia. The whole urchin was scanned with a voltage of 65 kV at a
magnification of 0.4X using the LE3 source filter delivering an isotropic
voxel size of 22.8 $\mu$m. The sectioned interambulacral plate was scanned
with a voltage of 80 kV at a magnification of 4X using the LE2 source filter
delivering an isotropic voxel size of 3.72 $\mu$m. Two higher magnification
scans were also conducted on an inner section of the interambulacral plate
using a voltage of 80 kV, a magnfication of 4X, and the LE2 filter, delivering
isotropic voxel sizes of 1.69 $\mu$m and 732 nm. Raw projection data were
reconstructed using XRM Reconstructor software (v15.0.17350.39816) following a
standard centre shift and beam hardening correction.

Three-dimensional micro-CT data were visualized and processed in Avizo 8.1
(Thermo Fisher Scientific, MA, USA), Dragonfly (v2022.1.0.1249, Object
Research Systems, Montreal, Canada), and MATLAB 2021B (MathWorks®). The 16-bit
grey scale intensity values of the inner interambulacral plate (isotropic
voxel sizes of $1.69\,\mu{m}$ and 732 nm) had a clear bimodal distribution for
values above zero. The grey scale data were converted to binary data by
thresholding at the lowest frequency of the intensity distribution between the
two modes. The 16-bit grey scale values of the whole interambulacral plate
(isotropic voxel sizes of 3.72 $\mu$m) were first filtered using the three-
dimensional ’Delineate’ filter in Avizo 8.1 and then binarized using the same
process described above. The binarized data set with isotropic voxel sizes of
$1.69\,\mu{m}$ was used in all subsequent analyses.

###  2.3 Structure identification

Subvolumes of $94^{3}$ voxels corresponding to $(160\,\mu{m})^{3}$ resembling
the $Fd\overline{3}m$ single Diamond structure and the $Pm\overline{3}m$
single Primitive structure were identified by visual inspection. To ensure the
D-like and P-like stereom were in fact resemblances of these structures, nodal
approximations with solid volume fractions and lattice parameters closely
matching the structures were simulated and fit using manual registration in
Dragonfly. Nodal approximations of the Primitive surface were modeled as level
sets of the equation $\cos(2\pi x/a)+\cos(2\pi y/a)+\cos(2\pi z/a)=f$. Nodal
approximations of the Diamond surface were modelled as level sets of
$\phi_{D}(x,y,z)=f$ where

$\displaystyle\phi_{D}(x,y,z)$ $\displaystyle=$
$\displaystyle\sin\left(\frac{2\pi x}{a}\right)\sin\left(\frac{2\pi
y}{a}\right)\sin\left(\frac{2\pi z}{a}\right)+\sin\left(\frac{2\pi
x}{a}\right)\cos\left(\frac{2\pi y}{a}\right)\cos\left(\frac{2\pi
z}{a}\right)$ $\displaystyle+$ $\displaystyle\cos\left(\frac{2\pi
x}{a}\right)\sin\left(\frac{2\pi y}{a}\right)\cos\left(\frac{2\pi
z}{a}\right)+\cos\left(\frac{2\pi x}{a}\right)\cos\left(\frac{2\pi
y}{a}\right)\sin\left(\frac{2\pi z}{a}\right)$

The constant $f$ is adapted to give the desired volume fraction and $a$ is the
lattice parameter. The common Miller index notation [hkl] is used to describe
crystallographic directions; for example, the [100] direction are the
coordinate axes (x,y,z) in eq. 2.3, and [111] the body diagonal (the notation
is as in reference [13]).

The solid volume fraction was calculated for each subvolume as described
below. The lattice parameter was approximated using the mean pore size
(calculated as described below) of the subvolume relative to the mean pore
size of a simulated nodal approximation with the same solid volume fraction
and a lattice parameter equal to one.

In total, eight D-like stereom subvolumes were identified. The $[111]$
direction of the Diamond structure (as shown in Figure 3B) was identified in
each of subvolumes and the angle it forms with the spine direction (the
positive Z-direction of our tomography data set) analysed.

We also analysed eight subvolumes with P-like stereom and eight subvolumes
with disordered stereoms. For these, no orientation analysis was carried out.

###  2.4 Distribution of solid volume fraction and pore sizes

Our analyses are based on a segmentation of the tomography grey scale data
into a binary data set consisting of voxels that have values of either one
(part of the solid phase) or zero (part of the void phase). Mathematically,
this is the discrete (’voxelised’) analogue of the continuous indicator
function $\chi(\mathbf{r})$ which equals one and zero for points $\mathbf{r}$
in the solid and void phase, respectively. We define the body representing the
solid phase as $K=\\{\mathbf{r}\,|\,\chi(\mathbf{r})=1\\}$ and the void phase
as $\overline{K}=\\{\mathbf{r}\,|\,\chi(\mathbf{r})=0\\}$

The solid volume fraction $\phi$ is the volume of the stereom (that is, the
solid phase) divided by the overall sample size. A localised version of the
solid volume fraction can be calculated for points across tomography data sets
when dividing the sample into subvolumes and then calculating the same ratio
for the binary data for each subvolume (for our main data set, we use
subvolumes of $50^{3}$ voxels corresponding to $(84.5\,\mu{m})^{3}$).

Pore size and trabeculae width distributions are defined based on the so-
called ’covering radius transform’ (CRT, [39, 24]) which in turn is derived
from the Euclidean distance transform (EDM) which, for each voxel of a given
phase, indicates the distance to the nearest voxel of the other phase.
Mathematically, the Euclidean distance map is the function
$D_{EDM}(\mathbf{r})$ defined for any point $\mathbf{r}\in\overline{K}$ in one
phase as the distance to the nearest point of the other (which necessarily
sits on the interface between $K$ and $\overline{K}$:

$D_{EDM}(\mathbf{r})=\min_{(\mathbf{r^{\prime}}\in
K)}\left(|\mathbf{r}-\mathbf{r^{\prime}}|\right).$ (2)

The covering radius transform [39, 24] associates with each point $\mathbf{r}$
of the phase $K$, the value $D_{CRT}(\mathbf{r})$ of the largest EDM sphere
that covers $\mathbf{r}$; an EDM sphere is a sphere centered at a point
$\mathbf{r}^{\prime}\in K$ with radius $D_{EDM}(\mathbf{r}^{\prime})$.

The distribution of the values of $D_{CRT}$ can be used as a meaningful pore
size (providing the radii of the pores) or trabeculae width (providing the
half-width of trabeculae) distributions of the body $K$.

###  2.5 Surface area and mean curvature

We have developed an algorithm, described in the Supplementary Materials, to
estimate the surface area, the average mean curvature and the topological
Euler characteristic. For a binary 3D data set, the algorithm computes the
Euclidean distance function of a void phase. By counting the voxels with EDM
$\leq l$, it determines the volume $V_{p}(l)$ of the these sets for values of
$l$ that are less than the so-called ’reach’. These sets are dilations (or
parallel bodies) of the solid phase. The so-called Steiner formula prescribes
that $V_{p}(l)$ is a second-order polynomial in $l$

$V_{p}(l)=(Al)\left(1+\langle H\rangle l+\frac{\langle
G\rangle}{3}l^{2}\right)=Al+A\langle H\rangle l^{2}+\frac{2\pi\chi}{3}l^{3}$
(3)

where $A$ is the surface area of the solid-void interface $S$, $\langle
H\rangle=\frac{1}{A}\int_{S}H(p)dA$ the average mean curvature of $S$ and
$\langle G\rangle=\frac{1}{A}\int_{S}G(p)dA=\frac{2\pi\chi}{A}$ the average
Gauss curvature of $S$ with the Euler characteristic $\chi$. The values of
$A$, $\langle H\rangle$ and $\langle G\rangle$ can therefore be obtained by a
fit to $V_{p}(l)$. For minimal surface model structures, we find that the fit
produces accurate results for relatively low voxel discretisations; accuracy
is further increased by combining data for $V_{p}(l)$ both the solid and the
void phase. The algorithm works well for homogeneous structures where $l$ is
relatively large.

###  2.6 Small- and Wide-angle X-ray Scattering (SAXS/WAXS)

SAXS/WAXS measurements were performed using a Nano-inXider instrument from
Xenocs SAS (Grenoble, France). The instrument is equipped with a Rigaku
(Rigaku-Denki, Co., Tokyo, Japan) 40 W micro-focused Cu source producing
X-rays with a wavelength of $\lambda$ = 1.54 Å. Scattered radiation is
detected by two Pilatus detectors from Dectris (Baden, Switzerland) covering
both SAXS and WAXS simultaneously. The two-dimensional scattering data were
radially and azimuthally averaged using standard reduction software (XSACT,
Xenocs). The scattering patterns for the radially averaged intensity were
recorded as a function of the scattering vector q =
4$\pi$*sin$\theta$/$\lambda$ where 2$\theta$ is the scattering angle. Samples
were mounted and measured in vacuum. Measuring times were 900 s. Samples were
prepared by gentle drilling either from the external/internal side of the
plate to create a bevelled region where, by small lateral adjustments of the
X-ray beam, the sample thickness could be varied; measurements were ultimately
conducted on samples approximately 0.5 mm thick which enabled appropriate
flux.

##  3 Results

The endoskeleton of the sea urchin C. rugosa has an oblate spheroid shape
(major axis diameter = 22.2 mm) with pentaradial symmetry, dividing the
skeleton into five interambulacral zones (comprising the interambulacral
plates) and five ambulacral zones (comprising the ambulacral plates). Each
interambulacral zone consists of between 10 and 11 interambulacral plates of
varying sizes with the smallest occurring near the oral and aboral surfaces
and the largest occurring through the centre of the skeleton. All
interambulacral plates are covered in protrusions called tubercles (which
house the spines of the sea urchin via a ball-and-socket joint [36]). The
interambulacral plate examined here is located centrally on the skeleton and
is approximately 3.7 mm in width (Figure 1A, B). It comprises diverse types of
microstructures that are easily visualized at different resolutions both in
cross-section and volume rendering ($3.72\,\mu{m}$, $1.69\,\mu{m}$, 732 nm;
Figure 1C, D). An imperforate stereom is found around the external surface of
the tubercle and the remaining microstructures are porous. The porous
microstructures sampled here can be grouped into either ordered or disordered
bicontinuous structures depending on the regularity of their trabeculae
networks.

###  3.1 Ordered stereom types resemble Diamond and Primitive bicontinuous
geometries

We identified two ordered stereom types that both have a regular network of
trabeculae but can be distinguished by their topological and geometric
differences. These are based on companion surfaces of Schwarz Diamond minimal
surface and Schwarz Primitive minimal surface. In line with Nissen’s
finding[26], we identified a stereom type that closely resembles the Primitive
minimal surface geometry and is located in a layer of stereom that lies
between the tubercle and approximately 1 mm below the tubercle (Figure 2).
This P-like stereom is a slightly distorted (likely monoclinic or triclinic)
version of the Primitive geometry with cubic $Pm\overline{3}m$ symmetry. Three
translational repeat vectors can be identified in each sample with the
$[100]$, $[010]$, and $[001]$ directions of the bicontinuous Primitive
geometry. The average angle between these directions is 77°$\pm 4$°. Figure 2B
displays a sample of the P-like stereom (in grey) such that the three
orthogonal directions of the cube surrounding it roughly align with the
$[100]$, $[010]$, and $[001]$ directions of the Primitive geometry. Adjacent
to the P-like stereom is a nodal approximation of the Primitive geometry (in
yellow) which, in the example in Figure 2B, has a lattice parameter equal to
30 microns and a solid volume fraction of 0.38. The lattice parameters of the
P-like stereom samples, as estimated from the $\langle D_{CRT}\rangle$ value
(Table 1), varied between 23 $\mu{m}$ and 25 $\mu{m}$. These estimates agree,
up to approximately $\pm 1\,\mu{m}$, with estimates for the lattice parameter
by manual registration of the nodal surface model.

Figure 2: A representative subvolume of the stereom that closely resembles a
single Primitive surface. (A) Cross-section through the interambulacral plate
showing the location of the representative subvolume. Scale bar = 1 mm. (B)
The subvolume of the sea urchin stereom (in grey; isotropic voxel size of 732
nm) and a simulated nodal approximation of the single Primitive surface with a
solid volume fraction $\phi=0.38$ (in yellow) and lattice parameter $a=30$
microns. Scale bar = $100\,\mu{m}$.(C) Perpendicular slices through the sea
urchin subvolume depicted in (B). Scale bar = $50\,\mu{m}$

We also identified a stereom type that closely resembles the $Fd\overline{3}m$
Diamond geometry (Figure 3). This D-like stereom is located in a layer of
stereom that lies in the centre of the plate below the P-like stereom.
Remarkably, even large subvolumes containing more than 64 unit cells exhibit
an almost perfect resemblance of the Diamond geometry (Figure 3B, C). Figure
3B displays a sample of the D-like stereom (in grey) such that the three
orthogonal directions of the cube surrounding it align with the $[111]$,
$[11\overline{2}]$, and $[1\overline{1}0]$ directions of the Diamond geometry.
Adjacent to the D-like stereom in Figure 3B is a simulated nodal approximation
of the Diamond geometry where the lattice parameter is equal to 39 microns and
the solid volume fraction is 0.3. Figure 3C shows the close correspondence
between cross-sections through each direction of the D-like stereom (in blue)
and the simulated nodal approximation of the Diamond geometry (in grey). The
lattice parameters, as estimated from the $\langle D_{CRT}\rangle$ value
(Table 1), of the D-like stereom samples varied between 33 $\mu{m}$ and 39
$\mu{m}$. These estimates agree, up to approximately $\pm 1\,\mu{m}$, with
estimates for the lattice parameter by manual registration of the nodal
surface model. The structural properties of both the P-like and D-like stereom
samples are discussed below and presented in Table 1.

Figure 3: A representative subvolume of the stereom that closely resembles a
single Diamond surface. (A) Cross-section through the interambulacral plate
showing the location of the representative subvolume. Scale bar = 1 mm. (B)
The subvolume of the sea urchin stereom (in grey; isotropic voxel size of 732
nm) and a simulated nodal approximation of the a single Diamond surface with a
solid volume fraction $\phi=0.3$ (in yellow) and lattice parameter
$a=39\,\mu{m}$ . Scale bar = $50\,\mu{m}$. The rectangular domain is oriented
such that its orthogonal axes are the $[11\overline{2}]$, $[1\overline{1}0]$
and $[111]$ crystallographic directions of the Diamond geometry (in Miller
index notation). (C) Slices through this subvolume showing the urchin and
simulated data in the $[111]$, $[11\overline{2}]$, and $[1\overline{1}0]$
planes. Scale bar = $50\,\mu{m}$.

Eight subvolumes of the D-like stereom were sampled across the interambulacral
plate such that representative samples were taken across an approximately
radial distribution centred on the tubercle (Figure 4A, B). The $[111]$
direction was easily identified in cross section and was shown to be almost
uniformly oriented across all eight samples. All samples were oriented within
6 degrees from one another (Figure 4C). In relation to the spine (the positive
Z-direction), the mean $[111]$ direction was oriented 16 $\pm$ 2 degrees
approximately towards the positive X-direction (Figure 4C).

Figure 4: Orientation of the $[111]$ direction of the sea urchin D-like
stereom relative to the direction of the spine (the positive Z-direction of
the tomography dataset). (A, B) Volume render of the intermabulacral plate
viewed along the Z-axis (A) and Y-axis (B) showing locations of eight sampled
D-like stereom. (C) Orientation of the $[111]$ direction of each sample (black
points) relative to the spine (grey point) represented by vectors projected
onto a sphere. The sphere is presented in (C) as a perspective along the
Z-axis. The mean orientation of the $[111]$ direction relative to the spine
(the positive Z-direction) is 16$\pm$2 degrees.

###  3.2 Structural properties of ordered and disordered stereom types

In addition to the P-like and D-like ordered stereom types, a Disordered
stereom type was identified and a representative sample is shown in Figure 5.
The Disordered stereom type has an irregular network of trabeculae without
pore alignment even in small subvolumes. It is found on the internal surface
of the interambulacral plate and can easily be distinguished from other
stereom types by its solid volume fraction ($0.48\pm 0.03$; Figure 6A) and
trabeculae widths ($11.0\pm 0.4\,\mu$m; Figure 6C), that are approximately 30%
and 50% larger than the D-like stereom located above it, respectively. This
difference creates a sharp interface between Disordered and D-like stereom
types as shown in Figure 6A and Figure 6C where no apparent gradual shift
between the two occurs.

Figure 5: A representative subvolume of the Disordered stereom. (A) Cross-
section through the interambulacral plate showing the location of the
representative subvolume. Scale bar = 1 mm. (B) The subvolume of the sea
urchin stereom (isotropic voxel size of $1.69\,\mu{m}$). Scale bar =
$100\,\mu{m}$. (C) Perpendicular slices through the subvolume depicted in (B).
Scale bar = $50\,\mu{m}$. Figure 6: The distribution of solid volume fraction,
pore sizes, and trabeculae widths across the interambulacral plate. (A) The
distribution of solid volume fraction showing distinct differences across the
plate. Scale bar = 1 mm. Squares P, D, and Di show locations of subsampled
stereom where P is the P-like, D is the D-like, and Di is the Disordered
stereom. (B) and (C) show the maximum pore sizes and trabeculae widths,
respectively. (D), (E), and (F) show the distribution of solid volume
fraction, pore sizes, and trabeculae widths across a subsample of each stereom
type.

Table 1 summarizes the differences in structural properties between each of
the identified stereom types. The solid volume fraction of the D-like stereom
was relatively lower ($0.31\pm 0.04$), compared to the P-like ($0.38\pm 0.03$)
and Disordered ($0.48\pm 0.03$) stereom types. Figure 6D displays the
distribution of solid volume fraction across each of the representative
samples and shows the clear distinction between stereom types.

The three stereom types have clear differences in the typical trabeculae
width, that is, in the material thickness of the solid phase (Figure 6F). The
D-like stereom has thinner solid elements ($\langle D_{CRT}^{t}\rangle=5.2\pm
0.7$), compared to the P-like ($7.1\pm 0.7$) and compared to the Disordered
stereom ($11\pm 0.4$).

By contrast, the typical pore sizes within the three stereom types are much
more uniform. The pore sizes are $\langle D_{CRT}^{p}\rangle=9.4\pm 0.6\mu{m}$
for the D-like stereom, $10.3\pm 0.9\mu{m}$ for the P-like stereom, and
$11.8\pm 0.6\mu{m}$ for the Disordered stereom. This is reflected also in the
distributions of pore sizes for the representative samples shown in Figure 6E.
We also show in the Supplementary Materials Figure S1, that this pore space
percolates throughout the entire plate, connecting the external environment to
the inside of the sea urchin.

Compared to simulated nodal approximations (Supplementary Table S1), the ratio
$\langle D_{CRT}^{p}\rangle/\langle D_{CRT}^{t}\rangle$ of pore sizes to
trabeculae widths were larger in the sea urchin D-like and P-like stereom
types. The ratio for the D-like stereom was between 6.8% and 19.5% larger than
the simulated Diamond structure. The ratio for the P-like stereom was between
9.6% and 12.7% larger than the simulated Primitive structure.

The interface areas of the trabeculae pore interfaces and their average mean
curvatures were calculated using the Steiner method and are included in Table
1. There are small but significant differences between the stereom types that
are consistent with the structural assignment of Diamond and Primitive
geometries: The interface areas $A/V$ per volume are larger for the Diamond
stereom (on average across the eight sampled domains, $A/V=0.10$) than for the
P-like stereom (on average, $A/V=0.09$). The average mean curvatures are
larger in absolute value for the D-like stereoms (on average $0.030$) than for
the P-like stereom (on average $0.015$). These average values are consistent
with the values for the exact Diamond and Primitive geometries in Table S1, at
the corresponding volume fractions and lattice parameters.

Compared to the ordered P-like and D-like stereom geometries, the disordered
stereom has lower average mean curvature (on average across the eight sample
domains $0.003$). That is, it is closer to a minimal surface, in line with the
more balanced distribution of trabeculae (solid) material and pore space
($\phi\approx 0.48$).

Table 1:  Key morphological properties of representative samples from the interambulacral plate of C. rugosa. Three samples from three types of stereom were examined including a D-like stereom, a P-like stereom, and a Disordered stereom. We provide estimates of the solid volume fraction $\phi$, the lattice parameter $a$, the maximal value of the Euclidean distance map $m_{EDM}$, the average $\langle D_{CRT}\rangle$ and the standard deviation of the maximal covering radius transform $\Delta D_{CRT}$, the surface area $A$ per volume, and the mean curvature $\langle H\rangle$ of the solid phase (representing the trabeculae) and pore (or void) phase of each sample. The superscripts ’t’ and ’p’ indicate ’trabeculae’ and ’pore’, respectively. The volume of each sample is $0.004mm^{3}$. |  |  | solid phase (trabeculae) | void phase (pore) |  |
---|---|---|---|---|---|---
Surface | $\phi$ | $a$ | $m_{EDM}^{t}$ | $\langle D_{CRT}^{t}\rangle$ | $\Delta D_{CRT}^{t}$ | $m_{EDM}^{p}$ | $\langle D_{CRT}^{p}\rangle$ | $\Delta D_{CRT}^{p}$ | $A/V$ | $\langle H\rangle$
| [1] |  [$\mu{m}$] |  [$\mu{m}$] |  [$\mu{m}$] |  [$\mu{m}$] |  [$\mu{m}$] |  [$\mu{m}$] |  [$\mu{m}$] |  [$\frac{1}{\mu{m}}$] |  [$\frac{1}{\mu{m}}$]
D-like stereom 1 | 0.30 | 39 | 8.45 | 5.43 | 1.07 | 12.65 | 10.07 | 1.27 | 0.09 | 0.0209
D-like stereom 2 | 0.28 | 38 | 7.17 | 4.79 | 0.84 | 14.04 | 10.07 | 1.38 | 0.10 | 0.0376
D-like stereom 3 | 0.29 | 37 | 7.37 | 4.80 | 0.84 | 12.98 | 9.66 | 1.27 | 0.10 | 0.0484
D-like stereom 4 | 0.40 | 38 | 9.71 | 6.48 | 1.11 | 12.42 | 8.82 | 1.21 | 0.10 | 0.0158
D-like stereom 5 | 0.31 | 39 | 8.45 | 5.46 | 1.00 | 13.62 | 10.00 | 1.46 | 0.09 | 0.0252
D-like stereom 6 | 0.28 | 35 | 6.97 | 4.52 | 0.81 | 12.65 | 9.04 | 1.35 | 0.11 | 0.0509
D-like stereom 7 | 0.28 | 33 | 6.76 | 4.36 | 0.79 | 11.95 | 8.69 | 1.25 | 0.11 | 0.0277
D-like stereom 8 | 0.36 | 37 | 8.45 | 6.03 | 1.01 | 13.94 | 8.99 | 1.43 | 0.11 | 0.0090
P-like stereom 1 | 0.37 | 23 | 9.71 | 7.25 | 1.22 | 13.52 | 10.29 | 1.44 | 0.09 | 0.0199
P-like stereom 2 | 0.38 | 25 | 10.82 | 7.70 | 1.40 | 15.67 | 10.98 | 1.78 | 0.08 | 0.0200
P-like stereom 3 | 0.45 | 24 | 11.08 | 8.44 | 1.37 | 12.76 | 10.17 | 1.50 | 0.09 | 0.0056
P-like stereom 4 | 0.35 | 29 | 10.14 | 6.98 | 1.28 | 16.30 | 11.63 | 2.08 | 0.08 | 0.0170
P-like stereom 5 | 0.35 | 27 | 9.85 | 6.82 | 1.22 | 14.44 | 10.76 | 1.55 | 0.09 | 0.0219
P-like stereom 6 | 0.36 | 25 | 9.26 | 6.68 | 1.21 | 13.62 | 10.35 | 1.51 | 0.09 | 0.0243
P-like stereom 7 | 0.40 | 28 | 10.01 | 6.31 | 1.17 | 13.62 | 8.80 | 1.57 | 0.10 | 0.0062
P-like stereom 8 | 0.41 | 27 | 10.42 | 6.91 | 1.32 | 14.54 | 9.31 | 1.72 | 0.11 | 0.0058
Disord. stereom 1 | 0.45 | - | 14.64 | 10.64 | 1.85 | 17.96 | 12.53 | 2.45 | 0.07 | 0.0062
Disord. stereom 2 | 0.44 | - | 14.64 | 10.41 | 1.77 | 17.48 | 12.42 | 2.24 | 0.07 | 0.0049
Disord. stereom 3 | 0.47 | - | 14.64 | 10.80 | 1.96 | 16.39 | 12.27 | 2.28 | 0.07 | 0.0080
Disord. stereom 4 | 0.51 | - | 15.21 | 11.43 | 1.84 | 15.48 | 11.10 | 2.19 | 0.07 | 0.0003
Disord. stereom 5 | 0.47 | - | 14.93 | 10.91 | 2.00 | 18.89 | 11.17 | 2.36 | 0.07 | 0.0024
Disord. stereom 6 | 0.48 | - | 15.30 | 11.19 | 1.96 | 14.83 | 11.04 | 1.91 | 0.07 | 0.0013
Disord. stereom 7 | 0.49 | - | 15.11 | 10.94 | 1.87 | 16.81 | 11.75 | 2.37 | 0.07 | 0.0005
Disord. stereom 8 | 0.51 | - | 16.29 | 11.47 | 1.94 | 16.98 | 11.80 | 2.31 | 0.07 | 0.0039

###  3.3 X-ray Scattering of the Disordered and Primitive stereom

The structural X-ray tomography analyses at the micron-scale are complemented
with some preliminary SAXS and WAXS analyses of the Ångstrom and nanometer
scale structure within the constituent material. We analysed two samples
representing different regions of the sea urchin stereom. One of these regions
is on the internal surface of the plate where we expect the porous structure
to be disordered; this sample is labelled ”Internal (Disordered)”. The other
region is closer to the external surface in a position where we expect the
microstructure to be the Primitive surface; this sample is labelled ”External
(P-like)”. Figure 7A and B show the SAXS and WAXS results, respectively.

Figure 7: Azimuthally averaged intensities of SAXS (A) and WAXS (B) data from
internal and external regions of the stereom that are characterised by micron-
scale ”Internal (Disordered)” and ”External (P-like)” surface structures,
respectively. The data labelled ”Calcite” is reference data for calcite [32].
(C,D) 2D detector images of and WAXS for the ”External (P-like)” and ”Internal
(Disordered)” samples.

For both samples, the 2D scattering intensity exhibits a pattern that is not
consistent with a single crystal structure (discrete spots), but rather with a
polycrystalline structure of randomly oriented crystallites (continuous
concentric circles $I(\mathbf{q})=I(|\mathbf{q}|)$); we attribute the remnant
granularity to the polycrystalline nature with relatively large
crystallites.444Due to the experimental setup the 2D detector only register
approximately a fifth of the full pattern, thus any larger crystallites in the
‘right’ orientation will contribute disproportionately if registered - we
believe this to be the reason for the different intensities between the two
data sets, for example the [2,0,2] reflection at 43.1 degrees.

From the WAXS data in Figure 7B we see that both samples have the same Bragg-
peaks, consistent with the crystal structure of calcite [32].

However, the sample from the internal region where the stereom has a
disordered micron-scale microstructure has a clear underlying amorphous
contribution for 2$\theta$ angles over approximately 20 degrees that must stem
from a material with an amorphous structure at the molecular (nanometer)
scale. It is tempting to speculate that the amorphous scattering could be a
signal that stems from an amorphous calcium carbonate phase present in the
native sea urchin. Given that sample preparation in regard to the ordered and
the disordered regions was the same, it is unlikely but not impossible that
the amorphous signal stems from sample preparation artefacts (e.g.
mechanochemical effects such as grinding).

We attribute the difference in nano-domain size deduced from the SAXS data to
this molecular scale difference. A fit with a model combining a low-q power
law with a Unified scattering model term [4] allows the determination of a
structural measure in the form of a radius of gyration which can be
interpreted as an average domain size in the calcite. We find that the average
domain size is larger in the Disordered stereom compared to the ordered P-like
stereom. The radii of gyration are 1.57 nm and 1.32 nm for the Internal sample
(Disordered stereom) and External sample (P-like stereom), respectively.

##  4 Discussion and conclusions

By combining high-resolution three-dimensional data with structural modelling
and mapping, we have here, for the first time, identified the co-occurence of
micron-scale Diamond and Primitive TPMS-related structures along with a
disordered structure in the endoskeleton of a sea urchin (Figure 2, 3, 5).
Structural properties including solid volume fraction and trabeculae widths
differ to the extent that contrasting domain boundaries are clearly visible
(Figure 6). Additionally, small differences in interface area and mean
curvature were also found between the three bicontinuous structures (Table 1).
The pore sizes across the plate remain relatively consistent and we show that
the void space of the plate is a single connected component, connecting the
external environment with the inside of the sea urchin (Figure S1).

Our structural analyses have established both differences and similarities
between the three identified bicontinuous geometries (D-like, P-like and
Disordered). A standout feature is that the pore sizes in all three structures
are fairly uniform, whereas both the solid volume fraction and the trabeculae
widths vary considerably.

In terms of the network topology of the disordered structure, it remains to be
seen whether and how it relates to the ordered structures, but observations in
smaller scale biological photonic crystals have shown that the amorphous
networks in weevil scales are related to their ordered counterparts [6, 3].

While our X-ray scattering analyses are preliminary, they allow us to consider
structural models for the relationship between the Disordered and P-like
stereom. There is no difference between the calcite crystallite sizes, as
extracted from WAXS, but there is a difference in the radius of gyration, as
extracted from SAXS. A structural hypothesis, could thus be that the
Disordered stereom is a mixture of calcite crystallites dispersed in an
amorphous matrix leading to a larger effective domain size of combined
crystalline/amorphous material. If the increased nano-domains are interpreted
as a ‘swelling’ of the crystal by amorphous calcium carbonate, this may
explain the observed larger trabeculae widths and solid volume fractions of
the Disordered stereom; the increase in dimensions of the crystallites (as
determined by SAXS) is approximately 19% (1.57 nm/1.32 nm) whereas the
increase in trabeculae width (as determined by tomography) is approximately
35% ($10.5\,\mu{m}/7.7\,\mu{m})$. Further investigations, through synchrotron
analysis where greater flux enables a spatially more fine-grained analysis
including on thicker samples, could test this hypothesis.

The unique distribution of ordered (Diamond and Primitive) and disordered
bicontinuous structures in C. rugosa offers an opportunity to examine the
possible functions of different stereom types, aside from their obvious role
in mechanical support [46]. In living urchins, the spines are attached to the
tubercles via a ball-and-socket joint that is connected to the plate by
collagenous fibres that penetrate the long galleries of the surface stereom
(that Smith refers to as ’rectilinear’ stereom) [34]. As Smith (1980)
highlighted in his comprehensive study, the type of investing soft tissue is
clearly related to the morphology of the stereom and perhaps even influences
its growth. Smith (1980) reported that the ’rectilinear’ stereom found in
cidaroids was associated with a diverse set of soft tissue including ectoderm,
muscle fibres, and collagenous fibres, whereas the ’labyrinthic’ stereom
(Disordered stereom in our study) was associated with an endothelial layer.
Our results now show that this ’rectilinear’ stereom, while made up of ordered
bicontinuous structures, does include different types of ordered structures
with different structural properties. It would therefore be very informative
to the functional interpretation of these structures to examine the specific
types of investing soft tissue that are associated with the D-like and P-like
stereom types, respectively.

While the Primitive surface related structure reported here was observed many
decades ago [26], the occurrence of the Diamond surface related structure is
novel and represents only the third occurrence of this structure at a micron-
sized length scale [46, 11]. Despite there being very few examples of ordered
bicontinuous structures at such large length scales, the recent discoveries of
Diamond surface related structures in the knobby starfish and a fossilized
crinoid highlight that this structure is found across distantly related clades
[46, 11]. Gorzelak et al. (2023) suggest that this may indicate the
independent evolution of this structure between echinoderm clades or
alternatively may indicate a more widespread genomically encoded trait that
emerges during increased predation/mechanical stress periods. Indeed, one of
the unique properties of this Diamond structure is enhanced damage tolerance
[46], however whether this property evolved for this purpose or is merely a
by-product of the formation process is yet to be seen.

The biogenesis of the ordered bicontinuous structures remains a key question,
despite substantive progress in understanding the biomineralisation
mechanisms. We share the view expressed by Hyde & Meldrum [16] that the
”ultimate question of what determines the ultrastructure remains open” and
that we can ”not explain how those structures can form with huge lattice
parameters ($>10\,\mu{m}$)”.

One aspect concerns the formation of smooth interfaces in a material usually
associated with sharp crystalline features (i.e., geogenic calicte). It is now
accepted that rather than being true single crystals, the biogenic calcite of
echinoderm stereom is better described as ’mesocrystalline’ comprising small
($0.1-0.3\,\mu{m}$) domains of co-oriented crystalline calcite and it is
thought that this property allows the crystal lattice to form curved surfaces
[16, 33]. It is believed that the mesocrystalline structure is achieved as a
consequence of the crystallization pathway whereby calcite forms via transient
amorphous precursor phases involving nanoparticles of amorphous calcium
carbonate (ACC) [16].

Another fascinating aspect of the formation of calcite structures in sea
urchins and other marine organisms is the high degree of alignment of the
c-axis of calcite with both the macroscopic features of the sea urchin (such
as the direction of the spines [17]) and with the orientation of the micron-
scale ordered porous structures (such as the reported alignment of calcite
c-axis with the [111] direction of the micron-scale Diamond structure in a
starfish [46]). We have here reported that the orientation of the [111]
direction of the D-like stereom is uniform across the interambulacral plate,
at a constant angle of about 15o to the spine direction. We have not been able
to identify any other obvious morphological feature (such as the skeleton
surface normal) that this [111] direction aligns with. Whether the calcite
c-axis aligns with the [111] direction of our D-like stereom, such as is the
case in the starfish, is yet to be seen.

The even more intriguing question is, in our view, how the formation of
highly-ordered Diamond-like or Primitive-like structures takes place at the
very large length scales of microns. These length scales are so large that the
entropic or enthalpic mechanisms responsible for the formation of related
structures in lipid (bilayer) phases – at much smaller scales – are irrelevant
and cannot drive this structure formation. Yet, closely related structures
form. Skeletogenesis in sea urchins has been a well studied topic, however
almost all studies have focused on skeletogenesis in larval sea urchins not in
the adult forms where we find these highly-ordered large-scale structures.
Nonetheless, skeletogenesis in sea urchin larvae can certainly inform some
aspects of how these structures form. For example, it is well understood that
the larval skeleton forms within a syncytium, a thin cytoplasmic cable (3
$\mu$m in diameter) formed by many connected skeletogenic cells [5, 42, 43,
29]. The ultrastructural morphology of the larval skeleton is therefore
dictated by the organic matrix that forms the syncytium. Over the last two
decades it has become apparent that certain signalling molecules, namely
vascular endothelial growth factor, play an integral role in skeletogenic cell
patterning and gene expression, syncytium formation, and therefore in
controlling the gross geometry of the larval echinoderm skeleton [9, 18, 1,
35, 25, 37]. Many of the same molecular and cellular processes directing
skeletogenesis in larval echinoderms also play a role in juvenile and adult
skeletogenesis [38], therefore it is likely that the very same signalling
molecules are influencing the development of the adult ultrastructural forms.

It is clear that the formation process of the bicontinuous geometries in the
sea urchin (and other echinoderms) is substantially different from the
formation of bicontinuous lipid membrane forms – the length scale is hugely
different and it does not involve a nanoscale intracellular membrane sculpted
to a bicontinuous form (the role of the shape of the plasma membrane of the
syncytium remains an open question). It seems likely that the formation
mechanism will involve some elements that can be explained by generic
mathematical pattern formation processes, some elements that will relate to
calcium carbonate biomineralisation, some elements that will be specific to
the biochemistry, and some elements that relate to the evolutionary benefits
of the microstructural forms. It will be exciting to see how tomography
studies, such as are conducted here, of mature and developing sea urchins will
help to elucidate both the mathematical, physical, chemical, and biological
elements of this process.

## Acknowledgements

All data in the article are available on Dryad
(https://doi.org/10.5061/dryad.wdbrv15vf).

We gratefully acknowledge funding by the Australian Research Council (ARC)
through the Discovery Project DP200102593. A.J.M is supported by an Australian
Government Research Training Program (RTP) Scholarship.

We thank Dr Charles Messing (Nova Southeastern University of Florida) for a
donation of the C. rugosa specimen. We thank Rasmus Klarskov Nordal E Petersen
for help with X-ray sample preparation. We are grateful to Prof Bodo Wilts and
the three anonymous reviewers for insightful comments and helpful suggestions
on the mansucript. The authors acknowledge the facilities of Microscopy
Australia at the Centre for Microscopy, Characterisation and Analysis at The
University of Western Australia.

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