The authors have declared that no competing interests exist.
Conceived and designed the experiments: NC JAK AG. Performed the experiments: NC. Analyzed the data: NC JAK TM AG. Contributed reagents/materials/analysis tools: NC JAK CC TM AG. Wrote the paper: NC JAK TM AG.
The growth of scleractinian corals is strongly influenced by the effect of water motion. Corals are known to have a high level of phenotypic variation and exhibit a diverse range of growth forms, which often contain a high level of geometric complexity. Due to their complex shape, simulation models represent an important option to complement experimental studies of growth and flow. In this work, we analyzed the impact of flow on coral's morphology by an accretive growth model coupled with advectiondiffusion equations. We performed simulations under noflow and unidirectional flow setup with the
A longstanding question in marine biology and coral biology is the morphological plasticity of corals, sponges and other marine sessile organisms and the influence of water movement. Usually branching species tend to develop symmetrical colonies where branches are being formed in all directions. There is a long standing discussion if this process in which colonies develop symmetrical colonies is controlled by genes or by the environment. In this work, we address this question for the scleractinian coral
Some corals are known to have a high degree of morphological plasticity along different environmental conditions
There is evidence to suggest that different morphologies emerge in response to changes in hydrodynamic energy
Water motion also influences the coral's physiological mechanisms during photosynthesis and respiration
In two studies by Nakamura et al.
(A) A controlled coral exposed to ambient current with the average nearbottom velocity of 5 cm s^{−1}
Former simulation models were used to address the influence of flow on the morphology of scleractinian corals. The first approach used a threedimensional aggregation model (represented on a cubic lattice) coupled with a hydrodynamic model
In three more recent papers
In order to study the emergence of symmetrical and asymmetrical growth forms under different hydrodynamic conditions, we used a previously published computational growth model
The experimental data used in this work originated from samples of
The simulations in this study are based on the accretive growth model
(A) A spherical object represents an initial growth state of the simulation (first growth step) (B) A simulation phase involves solving the NavierStokes equations (i) and the advectiondiffusion equation (ii). (C) Accretion phase translocates absorbed nutrients from previous simulation phase to a new growth layer hence, after a few consecutive growth steps, spontaneous branching occurs.
(A–C) Accretive growth steps; vertex v_{i} represents a simulated corallite. The new layer is constructed along the direction of normal vector n_{i} of the vertex v_{i}. A, B and C are three consecutive growth steps where triangles are inserted once the surface of the object increases.
The simulation domain consists of two compartments: a rectangular channel (60 cm length × 60 cm width × 40 cm height), where fluid is supposed to enter and exit the domain from left to right. The simulation is initialized with a triangulated spherical object with a diameter of 6 cm representing the (initial) simulated coral. The discretization of the simulation domain is done using the Galerkin finite element method in COMSOL Multiphysics
The arrow indicates the direction of the flow.
The impact of the flow is described using the nondimensional Reynolds number, defined as
After the solutions of NVS equations had been found, simulated nutrient entered the simulation domain from all sides and were absorbed by the simulated coral. Subsequently, the amount of absorbed nutrients were determined by solving the advectiondiffusion equation,
These absorbed quantities were translocated to the neighboring vertices by means of surface diffusion,
After calculating surface diffusion, the translocated concentrations of nutrients at each vertex were used to determine the thickness of a new growth layer.
For a vertex
To compare the influence of hydrodynamics to the morphology obtained from the simulations and CTscanned corals, we used the
Figure/Label  Velocity (m s^{−1})  Dynamic Viscosity (Pa s)  Fluid Density (kg m^{−3})  Diffusion Coefficient (m^{2} s^{−1})  Surface Diffusion (m^{2} s^{−1})  dc (m) 


Surface area (m^{2})  Volume (m^{3})  Surface Volume Ratio (m^{−1})  Sm_ mag_{mean} (m) 
6A/SIM_NO_FLOW  0  N/A  N/A  1  3.00e^{−4}  2.42e^{−3}  0  0  4.95e^{−2}  1.19e^{−4}  416  3.27e^{−3} 
6B/SIM_FLOW_D1  0.05  5.00e^{−2}  1.00e^{3}  1.00e^{−1}  3.00e^{−4}  2.26e^{−3}  1.13e^{−3}  2.26  4.59e^{−2}  1.12e^{−4}  410  3.98e^{−3} 
6C/SIM_FLOW_D2  0.05  5.00e^{−2}  1.00e^{3}  1.00e^{−2}  3. 00e^{−4}  2.10e^{−3}  1.05e^{−2}  2.10  3.64e^{−2}  8.98e^{−5}  405  11.25e^{−3} 
6D/SIM_FLOW_D3  0.05  5.00e^{−2}  1.00e^{3}  1.00e^{−3}  3. 00e^{−4}  1.94e^{−3}  9.7e^{−2}  1.9  3.22e^{−2}  7.74e^{−5}  416  28.77e^{−3} 
6E/SIM_FLOW_D4  0.05  5.00e^{−2}  1.00e^{3}  1.00e^{−4}  3.00e^{−4}  2.26e^{−3}  1.13  2.26  2.30e^{−2}  6.34e^{−5}  363  33.45e^{−3} 
6F/SIM_FLOW_D5  0.05  5.00e^{−2}  1.00e^{3}  1.00e^{−5}  3.00e^{−4}  N/A  ∼11  ∼2  3.45e^{−2}  1.03e^{−4}  335  N/A 
1A/CT_456  0.05  1.00e^{−3}  1.00e^{3}  1.00e^{−3}  N/A  2.71e^{−3}  1.36e^{−1}  135.5  1.32e^{−1}  1.53e^{−4}  863  2.4e^{−3} 
1B/TS_002  0.01  1.00e^{−3}  1.00e^{3}  1.00e^{−3}  N/A  1.63e^{−3}  1.63e^{−2}  16.3  4.97e−^{2}  8.46e^{−5}  587  1.93e^{−3} 
1C/TS_001  0.15  1.00e^{−3}  1.00e^{3}  1.00e^{−3}  N/A  1.25e^{−3}  1.88e^{−1}  187.5  8.64e^{−2}  1.28e^{−4}  675  3.96e^{−3} 
1D/TS_003  0.15  1.00e^{−3}  1.00e^{3}  1.00e^{−3}  N/A  1.51e^{−3}  2.27e^{−1}  226.5  4.13e^{−2}  7.15e^{−5}  578  7.36e^{−3} 
1E/CT_455  0.15  1.00e^{−3}  1.00e^{3}  1.00e^{−3}  N/A  1.92e^{−3}  2.88e^{−1}  288  1.03e^{−1}  1.38e^{−4}  746  20.53e^{−3} 
While for simulated objects exact
Due to the complexity of coral's geometry, the morphological analysis required an alternative approach from traditional landmarkbased morphometrics
An important preprocessing step of the morphometric analysis is the construction of a morphological skeleton of a 3D object, shown in
(A) A skeleton graph with the increased level of occlusion of the volumetric data in the background, (B–C) Visualization of spheres used for calculating morphometric traits  diameter of a sphere at the terminal branch is defined as terminal branch thickness –
To quantify the symmetry of branch formation in corals and simulated objects, we introduce two extra morphometric variables  the symmetry angles
Furthermore, consider a reference point
The distribution of the symmetry angles
Under diffusionlimited conditions (noflow) in the simulations, branches emerge in all directions leading to a relatively symmetrical shape (
(A) Simulated coral in a noflow condition. (B–F) Simulated corals from various flow simulations (B)
In the subsequent simulations we increased the impact of flow over diffusion by gradually lowering the diffusion coefficient (
Similarly to the the coral from the flume tank experiment (
Considering the flow patterns around the simulated objects, we observe an asymmetrical branching trend with a high degree of compactification with the increasing
(A) SIM_FLOW_D1,
To quantify the degree of compactification of the corals and the simulated forms under the influence of different flow conditions, we computed the surface/volume ratio of each form. In general, the surface/volume ratio of
The surface/volume ratios calculated from the
The degree of symmetry was analyzed by examining the distribution of symmetry angles (
Red lines indicate projected branches vector on the substratum plane (visualized from the bottom up perspective). For the simulation with flow, flow is directed from right to left. The morphometric traits measured here are as follow: symmetry angles
We verify the same trend in the CTscanned corals, for which the sums of the symmetry magnitudes are small for a highly symmetric growth form, but there is a distinction in symmetry angles between the controlled coral and the
Red lines indicate projected branches vector on the substratum plane (visualized from the bottom up perspective). For the
The simulation approach provides an indication of how the simulated corals can change their degree of symmetry by the increasing
Using a simulation approach, we studied the impact of hydrodynamics on the growth of the scleractinian coral
Error bars indicate 95% confidence interval. (C–D) shows regression plot of skewness and kurtosis against
Error bars indicate 95% confidence interval. (CD) surface/volume ratio of simulated and CTscanned corals versus
Our simulation model predicts a decreasing surface/volume ratio when
The surface/volume ratio provides a significant implication of how corals occupy a certain volume without taking into consideration the temporal scale of their growth. After a period of simulation time, objects from advectiondominated simulations occupy less volume and become more compact, reducing their surface/volume ratio. However, if spatial scale of the growth is used to evaluate the surface/volume ratio, at any rate, flowinduced object will exhibit a higher ratio. For example, considering the interim object (growth step 98) of the diffusiondominated simulation (
Although our simulations provide a reasonable approximation of a coral growth process and various growth forms emerge in response to the varying
In our model, we also address the relevant importance of the growth function (
The methods presented in this paper for modeling accretive growth and the impact of hydrodynamics, in combination with a method for the quantitative analysis of threedimensional complex shape can be applied to a large class of marine sessile organisms (e.g. scleractinian corals, hydro corals, sponges, rhodoliths). Morphological plasticity is a major issue in different fields of marine and coral biology (e.g. ecology, taxonomy, paleontology) with applications in environmental studies (e.g. coral bleaching and ocean acidification
To date, our coupled accretive growth model is the first example of a computational model of growth form that can be used to generate objects with a high resemblance to biological growth forms under different hydrodynamic conditions. We can compare and quantify our simulated objects and the real corals using three dimensional morphometrics. Our study also shows that the formation of symmetrical branching forms
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We would like to thank Paula RamosSilva (Section Computational Science, University of Amsterdam) for her contribution in proofreading the previous and final versions of this manuscript. We are grateful to Y. Mass, I. Berenstein, T. Idan, O. Ben Shaprut, M. Ohavia for assistance with the coral growth experiment in the reef and to the Israel Science Foundation for supporting that study. We also wish to thank the anonymous reviewers for their valuable comments and suggestions on the earlier versions of this manuscript.