Polyp oriented modelling of coral growth

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.     

Modeling of Branching Patterns in Plants

A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations. Recently, a number of models connected with the genetic network or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factors for pattern formation and phyllotaxis. However, all these models do not take into consideration the whole plant morphogenesis, concentrating on the events in the shoot or root apex. On the other hand, other approaches for modeling phyllotaxis, where the whole plant is considered, usually are mostly phenomenological, and due to it, do not describe the details of plant growth and branching mechanism. In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism where the main physiological factors of plant growth and development are taken into consideration. We model a growing plant as a system of intervals, which we will consider as branches. We assume that the number and location of the branches are not given a priori, but appear and grow according to certain rules, elucidated by the application of mathematical modeling. Four variables are included in our model: concentrations of the plant hormones auxin and cytokinin, proliferation and growth factor, and nutrients—we observe a wide variety of plant forms and study more specifically the involvement of each variable in the branching process. Analysis of the numerical simulations shows that the process of pattern formation in plants depends on the interaction of all these variables. While concentrations of auxin and cytokinin determine the appearance of a new bud, its growth is determined by the concentrations of nutrients and proliferation factors. Possible mechanisms of apical domination in the frame of our model are discussed.     

Effects of competition and herbivory on interactions between a hard coral and a brown alga

Despite widespread acceptance of the negative effects of macroalgae on corals, very few studies have experimentally tested the competitive nature of the interaction, and most have ignored the potential effects of corals on algae. We report the effects of herbivory and competition on the growth of the branching scleractinian coral Porites cylindrica Dana and the creeping foliose brown alga Lobophora variegata (Lamouroux) Womersley, on an inshore fringing reef of the central Great Barrier Reef. L. variegata overgrows branches of P. cylindrica from the base up, forming a distinct boundary between the alga and the coral tissue. The experiment used exclusion cages to test for effects of herbivores, and removal of algae and coral tissue, at their interaction boundary, to test for inhibition of the competitors by each other. Comparisons of coral branches with the algae present or removed showed that the presence and overgrowth of the alga caused significant coral tissue mortality. Comparisons of branches with coral tissue unmanipulated or damaged showed that the coral inhibited the overgrowth by L. variegata, but that the algae were markedly superior competitors. Importantly, reduced herbivory resulted in faster algal growth and consequent overgrowth and mortality of coral tissue, demonstrating the critical importance of herbivory to the outcome of the competitive interaction.     

Generation model of positional values as cell operation during the development of multicellular organisms

Many conventional models have used the positional information hypothesis to explain each elementary process of morphogenesis during the development of multicellular organisms. Their models assume that the steady concentration patterns of morphogens formed in an extracellular environment have an important property of positional information, so-called “robustness”. However, recent experiments reported that a steady morphogen pattern, the concentration gradient of the Bicoid protein, during early Drosophila embryonic development is not robust for embryo-to-embryo variability. These reports encourage a reconsideration of a long-standing problem in systematic cell differentiation: what is the entity of positional information for cells? And, what is the origin of the robust boundary of gene expression? To address these problems at a cellular level, in this article we pay attention to the re-generative phenomena that show another important property of positional information, “size invariance”. In view of regenerative phenomena, we propose a new mathematical model to describe the generation mechanism of a spatial pattern of positional values. In this model, the positional values are defined as the values into which differentiable cells transform a spatial pattern providing positional information. The model is mathematically described as an associative algebra composed of various terms, each of which is the multiplication of some fundamental operators under the assumption that the operators are derived from the remarkable properties of cell differentiation on an amputation surface in regenerative phenomena. We apply this model to the concentration pattern of the Bicoid protein during the anterior-posterior axis formation in Drosophila, and consider the conditions needed to establish the robust boundary of the expression of the hunchback gene.     

Rewiring coral: Anthropogenic nutrients shift diverse coral–symbiont nutrient and carbon interactions toward symbiotic algal dominance

Improving coral reef conservation requires heightened understanding of the mechanisms by which coral cope with changing environmental conditions to maintain optimal health. We used a long‐term (10 month) in situ experiment with two phylogenetically diverse scleractinians (Acropora palmata and Porites porites) to test how coral–symbiotic algal interactions changed under real‐world conditions that were a priori expected to be beneficial (fish‐mediated nutrients) and to be harmful, but non‐lethal, for coral (fish + anthropogenic nutrients). Analyzing nine response variables of nutrient stoichiometry and stable isotopes per coral fragment, we found that nutrients from fish positively affected coral growth, and moderate doses of anthropogenic nutrients had no additional effects. While growing, coral maintained homeostasis in their nutrient pools, showing tolerance to the different nutrient regimes. Nonetheless, structural equation models revealed more nuanced relationships, showing that anthropogenic nutrients reduced the diversity of coral–symbiotic algal interactions and caused nutrient and carbon flow to be dominated by the symbiont. Our findings show that nutrient and carbon pathways are fundamentally “rewired” under anthropogenic nutrient regimes in ways that could increase corals’ susceptibility to further stressors. We hypothesize that our experiment captured coral in a previously unrecognized transition state between mutualism and antagonism. These findings highlight a notable parallel between how anthropogenic nutrients promote symbiont dominance with the holobiont, and how they promote macroalgal dominance at the coral reef scale. Our findings suggest more realistic experimental conditions, including studies across gradients of anthropogenic nutrient enrichment as well as the incorporation of varied nutrient and energy pathways, may facilitate conservation efforts to mitigate coral loss.     

Thermal stress markers in <Emphasis Type="Italic">Colpophyllia</Emphasis><Emphasis Type="Italic">natans</Emphasis> provide an archive of site-specific bleaching events

Tropical coral reef monitoring relies heavily on in situ diver observations. However, in many reef regions resources are not available to regularly monitor reefs. This lack of historical baseline data makes it difficult to determine how different reefs respond to environmental stressors and what the implications are for management. To test whether coral cores could be used to identify bleaching events retrospectively, three sites in Tobago with pre-existing reef data including water quality and bleaching observations were identified. Colpophyllia natans cores were examined for growth anomalies which occurred during periods of thermal stress. If present, anomalies were compared to in situ, real-time bleaching observations and water quality data. Interestingly, sites with better water quality during the 2005 thermal anomaly were less prone to bleaching. We suggest that by reducing terrestrial run-off (e.g., sediment and nutrients), and therefore improving marine water quality, reef managers could enhance near-shore coral reef resilience during high-temperature events.     

Organic Matter Degradation Drives Benthic Cyanobacterial Mat Abundance on Caribbean Coral Reefs

Benthic cyanobacterial mats (BCMs) are impacting coral reefs worldwide. However, the factors and mechanisms driving their proliferation are unclear. We conducted a multi-year survey around the Caribbean island of Curaçao, which revealed highest BCM abundance on sheltered reefs close to urbanised areas. Reefs with high BCM abundance were also characterised by high benthic cover of macroalgae and low cover of corals. Nutrient concentrations in the water-column were consistently low, but markedly increased just above substrata (both sandy and hard) covered with BCMs. This was true for sites with both high and low BCM coverage, suggesting that BCM growth is stimulated by a localised, substrate-linked release of nutrients from the microbial degradation of organic matter. This hypothesis was supported by a higher organic content in sediments on reefs with high BCM coverage, and by an in situ experiment which showed that BCMs grew within days on sediments enriched with organic matter (Spirulina). We propose that nutrient runoff from urbanised areas stimulates phototrophic blooms and enhances organic matter concentrations on the reef. This organic matter is transported by currents and settles on the seabed at sites with low hydrodynamics. Subsequently, nutrients released from the organic matter degradation fuel the growth of BCMs. Improved management of nutrients generated on land should lower organic loading of sediments and other benthos (e.g. turf and macroalgae) to reduce BCM proliferation on coral reefs.     

A comparison between coral colonies of the genus Madracis and simulated forms

In addition to experimental studies, computational models provide valuable information about colony development in scleractinian corals. Using our simulation model, we show how environmental factors such as nutrient distribution and light availability affect growth patterns of coral colonies. To compare the simulated coral growth forms with those of real coral colonies, we quantitatively compared our modelling results with coral colonies of the morphologically variable Caribbean coral genus Madracis. Madracis species encompass a relatively large morphological variation in colony morphology and hence represent a suitable genus to compare, for the first time, simulated and real coral growth forms in three dimensions using a quantitative approach. This quantitative analysis of three-dimensional growth forms is based on a number of morphometric parameters (such as branch thickness, branch spacing, etc.). Our results show that simulated coral morphologies share several morphological features with real coral colonies (M. mirabilis, M. decactis, M. formosa and M. carmabi). A significant correlation was found between branch thickness and branch spacing for both real and simulated growth forms. Our present model is able to partly capture the morphological variation in closely related and morphologically variable coral species of the genus Madracis.     

Effects of nutrient enhancement on the fecundity of a coral reef macroalga

On coral reefs, there is concern that increased nutrient supply (e.g. due to eutrophication) causes increased algal growth and hence increased algal abundance, in turn causing colonisation and invasions of coral populations, resulting in reef degradation, or a coral-algal phase shift. For example, species of Sargassum, a highly seasonal, large, brown seaweed, are suggested to be colonising corals on inshore coral reefs of the Great Barrier Reef, as a result of anthropogenic increases in terrestrial runoff of sediments and nutrients. However, implicit in this argument is the assumption that nutrient-related increases in growth will lead to increased fecundity (and/or propagule success), since without such changes, increased abundance can only occur by vegetative means. Whilst plausible, there is no experimental evidence for this assumption in coral reef algae.We here present an initial study in which experimental increases in nutrient supply apparently did not lead to increased fecundity of Sargassum siliquosum; rather, density and biomass of receptacles were reduced in nutrient-enhanced algae. There was little effect of nutrient treatments on the proportional allocation of biomass to reproductive and vegetative structures: nutrient enhancement led to similar decreases in biomass of reproductive and vegetative tissue. Tissue nutrient levels indicated effective enhancement of nitrogen supply, although phosphorus levels were not different across nutrient treatments.The reduced fecundity with increased nutrients may reflect either a genuine inhibition by higher nutrient levels, as found in previous studies, or accelerated maturation, causing increased tissue losses due to more advanced seasonal senescence. Either way, this exploratory experiment provides initial evidence that nutrient effects on tropical coral reef macroalgae may be complex, and does not support the assumption that increased nutrient supply will result in a numeric increase in populations of Sargassum spp. Our results should not be taken as an unequivocal demonstration that nutrients inhibit fecundity overall, but illustrate the need to distinguish between effects on different life-history processes (e.g. growth and reproduction). For increased growth of pre-existing individuals to contribute to algal invasions and phase shifts, that growth must result in either increased fecundity, or increased propagule success.     

Four-dimensional imaging with virtual reality to quantitatively explore jigsaw puzzle-like morphogenesis of Arabidopsis cotyledon pavement cells

In most dicotyledonous plants, leaf pavement cells exhibit complex jigsaw puzzle-like cell morphogenesis during leaf expansion. Although detailed molecular biological information and mathematical modeling of this jigsaw puzzle-like cell morphogenesis are now available, a full understanding of this process remains elusive. Recent reports have highlighted the importance of three-dimensional (3D) structures (i.e., anticlinal and periclinal cell wall) in understanding the mechanical models that describe this morphogenetic process. We believe that it is important to acquire 3D shapes of pavement cells over time, i.e., acquire and analyze four-dimensional (4D) information when studying the relationship between mechanical modeling and simulations and the actual cell shape. In this report, we have developed a framework to capture and analyze 4D morphological information of Arabidopsis thaliana cotyledon pavement cells by using both direct water immersion observations and computational image analyses, including segmentation, surface modeling, virtual reality and morphometry. The 4D cell models allowed us to perform time-lapse 3D morphometrical analysis, providing detailed quantitative information about changes in cell growth rate and shape, with cellular complexity observed to increase during cell growth. The framework should enable analysis of various phenotypes (e.g., mutants) in greater detail, especially in the 3D deformation of the cotyledon surface, and evaluation of theoretical models that describe pavement cell morphogenesis using computational simulations. Additionally, our accurate and high-throughput acquisition of growing cell structures should be suitable for use in generating in silico model cell structures.     

Transplantation and cultivation of fragments of coral colonies of various scleractinian species on a reef in Vietnam

Six coral species of the genus Acropora and two species of the genus Porites were studied during experiments on cultivation of reef-building scleractinian corals. The research has established species-specific factors and others affecting regeneration of fragments and growth of new colonies in these coral species. The accretion of donor fragments and new branches averaged from 40 to 160 mm per year, depending on the coral species, colony size, and season of transplantation. An average monthly accretion of medium and larger transplants and growth of new branches were 1.2–1.3 times higher at spring cultivation than at autumn transplanting. When transplanted, coral fragments of medium and larger sizes survived well and showed higher growth rates in all species studied. These transplants developed the highest number of new branches, and their buds and formed the largest colonies. Prolongation of the cultivation time from 1 to 1.5 years caused a 1.2–1.4 fold accretion of transplants.     

Bitten down to size: Fish predation determines growth form of the Caribbean coral reef sponge Mycale laevis

Interactions between organisms add complexity to ecosystem function, particularly on coral reefs. The Caribbean orange icing sponge Mycale laevis is semi-cryptic, often growing under coral colonies or between coral branches. This association is reportedly a mutualism, with the sponge deterring boring sponges from invading the coral skeleton and the coral providing an expanding surface for sponge growth. But is there an alternative explanation for the proximity of sponge and coral? We examined the importance of fish predation on the growth of the sponge. While the semi-cryptic growth form of M. laevis predominates on reefs off the Florida Keys and the Bahamas Islands, M. laevis grows with a non-cryptic, erect morphology off Bocas del Toro, Panama. Surveys revealed that sponge-eating fishes were rare or absent at Bocas del Toro compared to sites in the Florida Keys. Because past studies were inconsistent about the palatability of M. laevis to fish predators, we conducted feeding experiments with sponges from all three sites. Crude organic extracts of M. laevis from all three sites were palatable to generalist fish predators in aquarium assays, and field feeding assays and caging experiments conducted in the Florida Keys confirmed that spongivorous fishes readily ate exposed fragments of M. laevis. Our results suggest that M. laevis is restricted to its semi-cryptic growth form by spongivorous predators, with corals providing a physical refuge from predation. This alternative explanation supports the broader hypothesis that Caribbean reef sponges can be categorized on the basis of chemical defense into defended, palatable, and preferred species, the last of which are restricted to refugia.     

Quantification of porosity in Acropora pulchra (Brook 1891) using X-ray micro-computed tomography techniques

Skeletal density and porosity characteristics are key parameters for investigations into scleractinian coral growth and for assessing the effects of a range of anthropogenic influences on coral reefs. Typically, skeletal density is measured by using planar X-rays of thin slabs cut from mound-shaped colonies or, for branching forms, by using methods based on Archimedean principles. This paper describes a novel non-destructive technique based on micro-computed tomography (micro-CT) to measure porosity of branching coral skeleton. This approach incorporates methods for segmenting internal and external portions of branch and for distinguishing between skeleton and air, whilst accounting for the effects of beam hardening. Measurements were obtained from colonies of branching Acropora pulchra collected across a reef-flat transect at King Reef, central nearshore Great Barrier Reef. The results show significant variation in porosity within and among branches sampled from individual colonies, but not within a reef-flat transect. Micro-CT techniques yield comparable results to traditional methods based on Archimedean principles, but offer advantages in their suitability for a wider range of coral specimens because of the non-destructive nature of the technique and in their more rigorous control of model parameters that can bias results.     

Propagation of the threatened staghorn coral Acropora cervicornis: methods to minimize the impacts of fragment collection and maximize production

Coral reef restoration methods such as coral gardening are becoming increasingly considered as viable options to mitigate reef degradation and enhance recovery of depleted coral populations. In this study, we describe several aspects of the coral gardening approach that demonstrate this methodology is an effective way of propagating the threatened Caribbean staghorn coral Acropora cervicornis: (1) the growth of colonies within the nursery exceeded the growth rates of wild staghorn colonies in the same region; (2) the collection of branch tips did not result in any further mortality to the donor colonies beyond the coral removed for transplantation; (3) decreases in linear extension of the donor branches were only temporary and donor branches grew faster than control branches after an initial recovery period of approximately 3–6 weeks; (4) fragmentation did not affect the growth rates of non-donor branches within the same colony; (5) small branch tips experienced initial mortality due to handling and transportation but surviving tips grew well over time; and (6) when the growth of the branch tips is added to the regrowth of the fragmented donor branches, the new coral produced was 1.4–1.8 times more than new growth in undisturbed colonies. Based on these results, the collection of small (2.5–3.5 cm) branch tips was an effective propagation method for this branching coral species resulting in increased biomass accumulation and limited damage to parental stocks.     

Seascape-scale trophic links for fish on inshore coral reefs

It is increasingly accepted that coastal habitats such as inshore coral reefs do not function in isolation but rather as part of a larger habitat network. In the Caribbean, trophic subsidies from habitats adjacent to coral reefs support the diet of reef fishes, but it is not known whether similar trophic links occur on reefs in the Indo-Pacific. Here, we test whether reef fishes in inshore coral, mangrove, and seagrass habitats are supported by trophic links. We used carbon stable isotopes and mathematical mixing models to determine the minimum proportion of resources from mangrove or seagrass habitats in the diet of five fish species from coral reefs at varying distances (0–2,200 m) from these habitats in Moreton Bay, Queensland, eastern Australia. Of the fish species that are more abundant on reefs near to mangroves, Lutjanus russelli and Acanthopagrus australis showed no minimum use of diet sources from mangrove habitat. Siganus fuscescens utilized a minimum of 25–44 % mangrove sources and this contribution increased with the proximity of reefs to mangroves (R ² = 0.91). Seagrass or reef flat sources contributed a minimum of 14–78 % to the diet of Diagramma labiosum, a species found in higher abundance on reefs near seagrass beds, but variation in diet among reefs was unrelated to seascape structure. Seagrass or reef flat sources also contributed a minimum of 8–55 % to a fish species found only on reefs (Pseudolabrus guentheri), indicating that detrital subsidies from these habitats may subsidize fish diet on reefs. These results suggest that carbon sources from multiple habitats contribute to the functioning of inshore coral reef ecosystems and that trophic connectivity between reefs and mangroves may enhance production of a functionally important herbivore.     

Ediacaran fronds

Leaf-shaped Ediacaran fronds such as Charniodiscus, Charnia, Rangea and Swartpuntia are among the most widespread and readily recognized elements of the soft-bodied Ediacara Biota (late Neoproterozoic). Recent papers have proposed a variety of possible lifestyles for Ediacaran fronds, but functional morphological analysis supports the traditional view that Ediacaran fronds gathered nutrients from the water column through filter feeding or direct nutrient absorption, resulting in the onset of Phanerozoic-style ecological competition complete with denizens occupying distinct tiers in the water column. Within the standard Ediacaran frond morphoshape, four distinct branching architectures have been isolated: Arborea-type branching, characterized by pea-pod like primary branches which house several secondary branches within the protective sheath; Charnia-type branching, resulting in a zigzagging central axis due to the overlapping of alternating sigmoidal primary branches which overly an inferred internal stalk; Rangea-type branching, consisting of petaloids composed of several overlapping primary branches which are self-similar (fractal) over three orders of branching and attached to a straight central stalk; and Swartpuntia-type branching, consisting of petaloids with unornamented tube-like primary branches. Traditional Ediacaran taxonomy has emphasized unity of fronds as a high-level taxon, but the frond morphology most likely represents convergent evolution resulting from competition for nutrients in the water column. It is recommended that any new, higher-order classification schemes highlight the differences in branching architecture as a way of recognizing clades within the Ediacara biota.     

Colimitation of a freshwater herbivore by sterols and polyunsaturated fatty acids

Empirical data providing evidence for a colimitation of an herbivore by two or more essential nutrients are scarce, particularly in regard to biochemical resources. Here, a graphical model is presented, which describes the growth of an herbivore in a system with two potentially limiting resources. To verify this model, life-history experiments were conducted with the herbivore Daphnia magna feeding on the picocyanobacterium Synechococcus elongatus, which was supplemented with increasing amounts of cholesterol either in the presence or the absence of saturating amounts of eicosapentaenoic acid (EPA). For comparison, D. magna was raised on diets containing different proportions of S. elongatus and the cholesterol- and EPA-rich eukaryotic alga Nannochloropsis limnetica. Somatic and population growth of D. magna on a sterol- and EPA-deficient diet was initially constrained by the absence of sterols. With increased sterol availability, a colimitation by EPA became apparent and when the sterol requirements were met, the growth-limiting factor was shifted from a limitation by sterols to a limitation by EPA. These data imply that herbivores are frequently limited by two or more essential nutrients simultaneously. Hence, the concept of colimitation has to be incorporated into models assessing nutrient-limited growth kinetics of herbivores to accurately predict demographic changes and population dynamics.     

Facilitation in Caribbean coral reefs: high densities of staghorn coral foster greater coral condition and reef fish composition

Recovery of the threatened staghorn coral (Acropora cervicornis) is posited to play a key role in Caribbean reef resilience. At four Caribbean locations (including one restored and three extant populations), we quantified characteristics of contemporary staghorn coral across increasing conspecific densities, and investigated a hypothesis of facilitation between staghorn coral and reef fishes. High staghorn densities in the Dry Tortugas exhibited significantly less partial mortality, higher branch growth, and supported greater fish abundances compared to lower densities within the same population. In contrast, partial mortality, branch growth, and fish community composition did not vary with staghorn density at the three other study locations where staghorn densities were lower overall. This suggests that density-dependent effects between the coral and fish community may only manifest at high staghorn densities. We then evaluated one facilitative mechanism for such density-dependence, whereby abundant fishes sheltering in dense staghorn aggregations deliver nutrients back to the coral, fueling faster coral growth, thereby creating more fish habitat. Indeed, dense staghorn aggregations within the Dry Tortugas exhibited significantly higher growth rates, tissue nitrogen, and zooxanthellae densities than sparse aggregations. Similarly, higher tissue nitrogen was induced in a macroalgae bioassay outplanted into the same dense and sparse aggregations, confirming greater bioavailability of nutrients at high staghorn densities. Our findings inform staghorn restoration efforts, suggesting that the most effective targets may be higher coral densities than previously thought. These coral-dense aggregations may reap the benefits of positive facilitation between the staghorn and fish community, favoring the growth and survivorship of this threatened species.     

Nutrient availability affects the response of the calcifying chlorophyte Halimeda opuntia (L.) J.V. Lamouroux to low pH

Atmospheric carbon dioxide emissions cause a decrease in the pH and aragonite saturation state of surface ocean water. As a result, calcifying organisms are expected to suffer under future ocean conditions, but their physiological responses may depend on their nutrient status. Because many coral reefs experience high inorganic nutrient loads or seasonal changes in nutrient availability, reef organisms in localized areas will have to cope with elevated carbon dioxide and changes in inorganic nutrients. Halimeda opuntia is a dominant calcifying primary producer on coral reefs that contributes to coral reef accretion. Therefore, we investigated the carbon and nutrient balance of H. opuntia exposed to elevated carbon dioxide and inorganic nutrients. We measured tissue nitrogen, phosphorus and carbon content as well as the activity of enzymes involved in inorganic carbon uptake and nitrogen assimilation (external carbonic anhydrase and nitrate reductase, respectively). Inorganic carbon content was lower in algae exposed to high CO₂, but calcification rates were not significantly affected by CO₂ or inorganic nutrients. Organic carbon was positively correlated to external carbonic anhydrase activity, while inorganic carbon showed the opposite correlation. Carbon dioxide had a significant effect on tissue nitrogen and organic carbon content, while inorganic nutrients affected tissue phosphorus and N:P ratios. Nitrate reductase activity was highest in algae grown under elevated CO₂ and inorganic nutrient conditions and lowest when phosphate was limiting. In general, we found that enzymatic responses were strongly influenced by nutrient availability, indicating its important role in dictating the local responses of the calcifying primary producer H. opuntia to ocean acidification.     

Formulation and numerical simulations of a continuum model of avascular tumor growth

In this paper we present a continuum mathematical model for a multicellular spheroid that mimics the micro-environment within avascular tumor growth. The model consists of a coupled system of non-linear convection-diffusion-reaction equations. This system is solved using a previously developed conservative Galerkin characteristics method. In the model considered, there are three cell types: the proliferative cells, the quiescent non-dividing cells which stay in the G₀ phase of the cell cycle and the necrotic cells. The model includes viable cell diffusion, diffusion of cellular material and the removal of necrotic cells. We assume that the nutrients diffuse passively and are consumed by the proliferative and quiescent tumor cells depending on the availability of resources (oxygen, glucose, etc.). The numerical simulations are performed using different sets of parameters, including biologically realistic ones, to explore the effects of each of these model parameters on reaching the steady state. The present results, taken together with those reported earlier, indicate that the removal of necrotic cells and the diffusion of cellular material have significant effects on the steady state, reflecting growth saturation, the number of viable cells, and the spheroid size.