Coral growth modeling
Jaap A. Kaandorp, Peter M. A. Sloot, Roeland M. H. Merks, Rolf P.M. Bak, Mark J. A. Vermeij, and Cornelia Maier. 2005.
Morphogenesis of the branching reef coral Madracis mirablis.
Proceedings of the  Royal Society B, 272, 127-134. [ link ]
Understanding external deciding factors in growth and morphology of reef corals is essential to elucidate the role of corals in marine ecosystems, and to explain their susceptibility to pollution and global climate change. Here, we extend on a previously presented model for simulating the growth and form of a branching coral and we compare the simulated morphologies to three-dimensional (3D) images of the coral species Madracis mirabilis. Simulation experiments and isotope analyses of M. mirabilis skeletons indicate that external gradients of dissolved inorganic carbon (DIC) determine the morphogenesis of branching, phototrophic corals. In the simulations we use a first principle model of accretive growth based on local interactions between the polyps. The only species-specific information in the model is the average size of a polyp. From flow tank and simulation studies it is known that a relatively large stagnant and diffusion dominated region develops within a branching colony. We have used this information by assuming in our model that growth is entirely driven by a diffusion-limited process, where DIC supply represents the limiting factor. With such model constraints it is possible to generate morphologies that are virtually indistinguishable from the 3D images of the actual colonies.
Roeland M. H. Merks, Alfons G. Hoekstra, Jaap A. Kaandorp, and Peter M. A. Sloot. 2004.
Polyp Oriented Modelling of Coral Growth. Journal of Theoretical Biology, 228, 559-576. [ link ]
The morphogenesis of colonial stony corals is the result of the collective behaviour of many coral polyps depositing coral skeleton on top of the old skeleton on which they live. Yet, models of coral growth often consider the polyps as a single continuous surface. In the present work, the polyps are modelled individually. Each polyp takes up resources, deposits skeleton, buds off new polyps and dies. In this polyp oriented model, spontaneous branching occurs. We argue that branching is caused by a so called "polyp fanning effect" by which polyps on a convex surface have a competitive advantage relative to polyps on a flat or concave surface. The fanning effect generates a more potent branching mechanism than the Laplacian growth mechanism that we have studied previously (J. Theor. Biol. 224 (2003) 153). We discuss the application of the polyp oriented model to the study of environmentally driven morphological plasticity in stony corals. In a few examples we show how the properties of the individual polyps influence the whole colony morphology. In our model, the spacing of polyps influences the thickness of coral branches and the overall compactness of the colony. Density variations in the coral skeleton may also be important for the whole colony morphology, which we address by studying two variants of the model. Finally, we discuss the importance of small scale resource translocation in the coral colony and its effects on the morphology of the colony.
Roeland M. H. Merks, Alfons G. Hoekstra, Jaap A. Kaandorp, and Peter M. A. Sloot. 2003.
Models of coral growth: Spontaneous branching, compactification and the Laplacian growth assumption. Journal of Theoretical Biology, 224, 153--166.
In stony corals it is often observed that specimens collected from a sheltered growth site have more open and more thinly branched growth forms than specimens of the same species from more exposed growth sites, where stronger water currents are found. This observation was explained using an abiotic computational model inspired by coral growth, in which the growth velocity depended locally on the absorption of a resource dispersed by advection and diffusion (Kaandorp & Sloot (2001), J. Theor. Biol 209, 257-274). In that model a morphological range was found; as the Péclet-number (indicating the relative importance of advective and diffusive nutrient transport) was increased, more compact and spherical growth forms were found. Two unsatisfactory items have remained in this model, which we address in the present paper. First, an explicit curvature rule was responsible for branching. In this work we show that the curvature rule is not needed: the model exhibits spontaneous branching, provided that the resource field is computed with enough precision. Second, previously no explanation was given for the morphological range found in the simulations. Here we show that such an explanation is given by the conditions under which spontaneous branching occurs in our model, in which the compactness of the growth forms depends on the ratio of the rates of growth and nutrient transport. We did not find an effect of flow. This suggests that the computational evidence that hydrodynamics influences the compactness of corals in laminar flows may not be conclusive. We discuss the applicability of the Laplacian growth paradigm to understand coral growth is discussed.
R. M. H. Merks, A. G. Hoekstra, J. A. Kaandorp, and P. M. A. Sloot. 2003.
Diffusion limited growth in laminar flows. International Journal of Modern Physics C, 14 (9), 1171-1182. [ link ]
In the diffusion-limited aggregation (DLA) model, pioneered by Witten and Sander (Phys. Rev. Lett. 47, 1400 (1981)), diffusing particles irreversibly attach to a growing cluster which is initiated with a single solid seed. This process generates clusters with a branched morphology. Advection-diffusion-limited aggregation (ADLA) is a straightforward extension to this model, where the transport of the aggregating particles not only depends on diffusion, but also on a fluid flow. The authors studying two-dimensional and three-dimensional ADLA in laminar flows reported that clusters grow preferentially against the flow direction. The internal structure of the clusters was mostly reported to remain unaffected, except by Kaandorp et al. (Phys. Rev. Lett. 77, 2328 (1996)) who found compact clusters ``as the flow becomes more important''. In the present paper we present three-dimensional simulations of ADLA. We did not find significant effects of low Reynolds-number advection on the cluster structure. The contradicting results by Kaandorp et al. (1996) were recovered only when the relaxation into equilibrium of the advection-diffusion field was too slow, in combination with the synchronous addition of multiple particles.
Ph.D thesis
Roeland M. H. Merks. 2003.
Branching Growth in Stony Corals: a modelling approach. PhD thesis, University of Amsterdam.
Popular write-ups
Roeland Merks. 2005.
Droogzwemmen in het koraalrif. Computersimulaties verklaren koraalgroei (in Dutch).
Nederlands Tijdschrift voor Natuurkunde, 71 (10), 314-317.
Geen meter onder water ben ik geweest tijdens mijn onderzoek naar steenkoralen. Driedimensionaal groeiden ze in tientallen aan elkaar gekoppelde computers in de polders van de Watergraafsmeer, thuisbasis van de sectie Computational Science van de Universiteit van Amsterdam. Wachten op echte, langzaam groeiende koralen had mijn promotie laten duren tot ver na mijn pensioen. En waarom ook wachten? Een computermodel heeft grote voordelen, omdat je eenvoudig kan bepalen welke factoren bijdragen aan de groei.
Conference papers
Roeland Merks, Alfons Hoekstra, Jaap Kaandorp, and Peter Sloot. 2002.
Spontaneous branching in a polyp oriented model of stony coral growth.
In P.M.A. Sloot, C.J. Kenneth Tan, Jack J. Dongarra, and Alfons G. Hoekstra, editors, International Conference on Computational Science (ICCS), volume 2329 of Lecture Notes in Computer Science, pages 88-96, Amsterdam, the Netherlands. Springer-Verlag, Berlin.
Roeland Merks, Alfons Hoekstra, Jaap Kaandorp, and Peter Sloot. 2002.
Branching and compactification in a model of coral growth: a critical reinvestigation of the effect of hydrodynamics.
In V. Capasso, editor, Mathematical Modelling and Computing in Biology and Medicine; 5th ESMTB Conference, pages 539-544, Milano, Italy, 2003. MIRIAM.