Morphogenesis of the branching reef coral Madracis mirabilis

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.     

A mathematical model for pH patterns in the rhizospheres of growth zones

In the classical model by Nye (1981), the main process for the change in pH across the rhizosphere is assumed to be diffusion. The classical model focuses on the non-growing part of the root and assumes that the distribution of ion fluxes along the root is spatially uniform. We consider the rhizosphere of the growth zone and take into account the root growth rate and spatially varying flux along the root surface. We present both analytical (dimensional analysis) and experimental (computational) evidence of the importance of taking into account the root growth rate. We describe a conceptual and mathematical model to analyse the pH field around the root tip over time. The model is used with published data to show that, for typical growth rates in sandy soil, the pH field becomes steady (independent of time) after 6 h. Dimensional analysis reveals that a version of the Péclet number, related to the quotient of root elongation rate and proton diffusivity, can be used to predict the extent of the rhizosphere and the time required for it to become steady. For Péclet numbers much greater than 1 (soils), the root influences soil pH for distances on the millimetre scale. In contrast, for Péclet numbers much less than one (agar, aqueous solution), the root influences substrate pH for radial distances on the scale of centimetres. We also present some evidence that agar-contact techniques to measure the soil pH may not be appropriate for measuring the millimetre-scale gradients in soil pH.     

Simulation and analysis of flow patterns around the scleractinian coral Madracis mirabilis (Duchassaing and Michelotti)

Three-dimensional morphologies of Madracis mirabilis were obtained using X-ray computed tomography scanning techniques. The morphologies were used to simulate the flow patterns around the colony. In the simulations, the thin-branching low-flow morph with a relatively larger branch-spacing was compared with the more compact high-flow morph of M. mirabilis. For both morphologies, the inside-colony flow velocities were computed for Reynolds numbers ranging from 154 to 3840. In the high-flow morph, it was found that in the range of investigated Reynolds numbers a stagnant region develops within the colony, whereas in the low-flow morph the stagnant region disappeared. Experiments done under natural conditions suggest that a morph is adapted to a certain external flow velocity and develops a stagnant region below a particular threshold for the external flow velocity. When the external flow velocity exceeds a certain threshold, which is characteristic for the growth form, the core velocity becomes equal to the external velocity. A potential application of a profile of core velocities for a range of Reynolds numbers for a certain morph is the prediction of the optimal external flow velocity for a certain morph, and this can be used to assess the state of the physical (palaeo-) environment.     

Needle cell elongation and maturation timing derived from pine needle cellulose delta18O

Estimates of the timing of Pinus arizonica Engelm. needle development in 1998 and 1999 were derived from the leaf-cellulose delta18O of weekly growth increments. Significant correlations were noted between time series of local humidity and leaf-cellulose delta18O for needles growing near Tucson, Arizona. Correlations with temperature were also significant, but much lower, suggesting these variations in cellulose delta18O were determined mostly by changes in humidity. The timing of all significant correlations lags the timing of the appearance of the new needle growth, and is interpreted as indicating 16-23 d were required for cell enlargement in 1998 and 13-17 d in 1999. Similarly, properties of the environmental time series, when significantly correlated, are interpreted as indicating the duration of cellulose deposition (7-27 d in 1998, 13-21 d in 1999). Variations in stable-isotope back diffusion (the Péclet effect) and the synthesis of cellulose using stored photosynthate are discussed as explanations for departures from a Craig and Gordon-type model of leaf water delta18O. The Péclet effect, use of stored photosynthate, and variations in the growing-season source-water delta18O, probably confound the development of a high-resolution paleohumidity proxy from subfossil needle cellulose delta18O in this region.     

A Model for Bacterial Colonization of Sinking Aggregates

Sinking aggregates provide important nutrient-rich environments for marine bacteria. Quantifying the rate at which motile bacteria colonize such aggregations is important in understanding the microbial loop in the pelagic food web. In this paper, a simple analytical model is presented to predict the rate at which bacteria undergoing a random walk encounter a sinking aggregate. The model incorporates the flow field generated by the sinking aggregate, the swimming behavior of the bacteria, and the interaction of the flow with the swimming behavior. An expression for the encounter rate is computed in the limit of large Péclet number when the random walk can be approximated by a diffusion process. Comparison with an individual-based numerical simulation is also given.     

The enigma of effective path length for 18O enrichment in leaf water of conifers

The Péclet correction is often used to predict leaf evaporative enrichment and requires an estimate of effective path length (L). Studies to estimate L in conifer needles have produced unexpected patterns based on Péclet theory and leaf anatomy. We exposed seedlings of six conifer species to different vapour pressure deficits (VPD) in controlled climate chambers to produce steady‐state leaf water enrichment (in ¹⁸O). We measured leaf gas exchange, stable oxygen isotopic composition (δ¹⁸O) of input and plant waters as well as leaf anatomical characteristics. Variation in bulk needle water δ¹⁸O was strongly related to VPD. Conifer needles had large amounts of water within the vascular strand that was potentially unenriched (up to 40%). Both standard Craig–Gordon and Péclet models failed to accurately predict conifer leaf water δ¹⁸O without taking into consideration the unenriched water in the vascular strand and variable L. Although L was linearly related to mesophyll thickness, large within‐species variation prevented the development of generalizations that could allow a broader use of the Péclet effect in predictive models. Our results point to the importance of within needle water pools and isolating mechanisms that need further investigation in order to integrate Péclet corrections with ‘two compartment’ leaf water concepts.     

Leaf vein fraction influences the Péclet effect and 18O enrichment in leaf water

The process of evaporation results in the fractionation of water isotopes such that the lighter ¹⁶O isotope preferentially escapes the gas phase leaving the heavier ¹⁸O isotope to accumulate at the sites of evaporation. This applies to transpiration from a leaf with the degree of fractionation dependent on a number of environmental and physiological factors that are well understood. Nevertheless, the ¹⁸O enrichment of bulk leaf water is often less than that predicted for the sites of evaporation. The advection of less enriched water in the transpiration stream has been suggested to limit the back diffusion of enriched evaporative site water (Péclet effect); however, evidence for this effect has been varied. In sampling across a range of species with different vein densities and saturated water contents, we demonstrate the importance of accounting for the relative ‘pool’ sizes of the vascular and mesophyll water for the interpretation of a Péclet effect. Further, we provide strong evidence for a Péclet signal within the xylem that if unaccounted for can lead to confounding of the estimated enrichment within the mesophyll water. This has important implications for understanding variation in the effective path length of the mesophyll and hence potentially the δ¹⁸O of organic matter.     

Pore‐network modeling of biofilm evolution in porous media

The influence of bacterial biomass on hydraulic properties of porous media (bioclogging) has been explored as a viable means for optimizing subsurface bioremediation and microbial enhanced oil recovery. In this study, we present a pore network simulator for modeling biofilm evolution in porous media including hydrodynamics and nutrient transport based on coupling of advection transport with Fickian diffusion and a reaction term to account for nutrient consumption. Biofilm has non‐zero permeability permitting liquid flow and transport through the biofilm itself. To handle simultaneous mass transfer in both liquid and biofilm in a pore element, a dual‐diffusion mass transfer model is introduced. The influence of nutrient limitation on predicted results is explored. Nutrient concentration in the network is affected by diffusion coefficient for nutrient transfer across biofilm (compared to water/water diffusion coefficient) under advection dominated transport, represented by mass transport Péclet number >1. The model correctly predicts a dependence of rate of biomass accumulation on inlet concentration. Poor network connectivity shows a significantly large reduction of permeability, for a small biomass pore volume. Biotechnol. Bioeng. 2011;108: 2413–2423. © 2011 Wiley Periodicals, Inc.     

A Malthusian curb on spatial structure in microorganism populations

That all organisms are born in the company of a parent but die alone is a fundamental biological asymmetry. It has been suggested that this provides a deep-rooted source of spatial pattern formation for microorganisms even at the scale of the population. Such a theory, however, neglects the strong influence in nature of the limited and spatially variable availability of food. The tendency, first recognized by Thomas Malthus in the 18th century, of a population to out-strip its food resources will eventually lead, through local starvation, to the suppression of a heterogeneity growing within a population. Using a generic model it is demonstrated that including local food limitation of breeding strongly dampens spatial structure otherwise resulting from birth and death. The extent of this damping is shown to be a function of the strength of the coupling between organisms and their food and of the total abundance of organic material. Moreover, this work provides an example of a density-dependent process acting to diminish spatial structure rather than to create it and highlights the rich variety of behaviour that is missed by continuum models which fail to represent such local dynamics.     

Models of coral growth: spontaneous branching,compactification and the Laplacian growth assumption

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 and Sloot, J. Theor. Biol 209 (2001) 257). 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. The applicability of the Laplacian growth paradigm to understand coral growth is discussed.     

Importance of fish behaviour in modelling conservation problems: food limitation as an example

Simulation experiments using the inSTREAM individual-based brown trout Salmo trutta population model explored the role of individual adaptive behaviour in food limitation, as an example of how behaviour can affect managers' understanding of conservation problems. The model includes many natural complexities in habitat (spatial and temporal variation in characteristics such as depth and velocity, temperature, hiding and feeding cover, drift-food supply and predation risk), fish physiology (especially, how food intake and growth vary with hydrodynamics, cover, fish size and temperature) and behaviour. When drift-food concentration was increased over a wide range in 7 year simulations, the simulated population was always food limited. In fact, as food supply increased, the population increased at an increasing rate and consumed a higher percentage of the food supply, apparently because higher food concentrations make more stream area energetically profitable for drift feeders. The behaviour most responsible for this response was activity selection: when food was abundant, fish chose to feed less frequently and more nocturnally, thereby reducing predation mortality so more fish survived longer. These results indicate that the traditional concept of food limitation, that food availability stops limiting population size when it exceeds some threshold level, may not be useful and can be misleading. Results also strongly contradict the concept that a salmonid population is not food limited if the total food supply is greater than the population's consumption. Explicit consideration of adaptive behaviour produced a novel but believable understanding of food effects on salmonid populations. Published 2011. This article is a U.S. Government work and is in the public domain in the USA.     

Modelling genetic regulation of growth and form in a branching sponge

We present a mathematical model of the genetic regulation controlling skeletogenesis and the influence of the physical environment on a branching sponge with accretive growth (e.g. Haliclona oculata or Lubomirskia baikalensis). From previous work, it is known that high concentrations of silicate induce spicule formation and upregulate the silicatein gene. The upregulation of this gene activates locally the production of spicules in the sponge and the deposition of the skeleton. Furthermore, it is known that the expression of the gene Iroquois induces the formation of an aquiferous system, consisting of exhalant and inhalant pores. We propose a model of the regulatory network controlling the separation in time and space of the skeletogenesis and the formation of the aquiferous system. The regulatory network is closely linked with environmental influences. In building a skeleton, silicate is absorbed from the environment. In our model, silicate is transported by diffusion through the environment and absorbed at the surface of a geometric model of the sponge, resulting in silicate gradients emerging in the neighbourhood of the sponge. Our model simulations predict sponge morphology and the positioning of the exhalant pores over the surface of the sponge.     

Evolution of cell motility in an individual-based model of tumour growth

Tumour invasion is driven by proliferation and importantly migration into the surrounding tissue. Cancer cell motility is also critical in the formation of metastases and is therefore a fundamental issue in cancer research. In this paper we investigate the emergence of cancer cell motility in an evolving tumour population using an individual-based modelling approach. In this model of tumour growth each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. With this model we have investigated the impact of the micro-environment on the emergence of a motile invasive phenotype. The results show that when a motile phenotype emerges the dynamics of the model are radically changed and we observe faster growing tumours exhibiting diffuse morphologies. Further we observe that the emergence of a motile subclone can occur in a wide range of micro-environmental growth conditions. Iterated simulations showed that in identical growth conditions the evolutionary dynamics either converge to a proliferating or migratory phenotype, which suggests that the introduction of cell motility into the model changes the shape of fitness landscape on which the cancer cell population evolves and that it now contains several local maxima. This could have important implications for cancer treatments which focus on the gene level, as our results show that several distinct genotypes and critically distinct phenotypes can emerge and become dominant in the same micro-environment.     

A size threshold governs Caenorhabditis elegans developmental progression

The growth of organisms from humans to bacteria is affected by environmental conditions. However, mechanisms governing growth and size control are not well understood, particularly in the context of changes in food availability in developing multicellular organisms. Here, we use a novel microfluidic platform to study the impact of diet on the growth and development of the nematode Caenorhabditis elegans. This device allows us to observe individual worms throughout larval development, quantify their growth as well as pinpoint the moulting transitions marking successive developmental stages. Under conditions of low food availability, worms grow very slowly, but do not moult until they have achieved a threshold size. The time spent in larval stages can be extended by over an order of magnitude, in agreement with a simple threshold size model. Thus, a critical worm size appears to trigger developmental progression, and may contribute to prolonged lifespan under dietary restriction.     

Monte Carlo simulation of damascene electroplating: effects of additives

The influence of additives on the filling process of via holes in damascene electroplating is investigated with use of a kinetic Monte Carlo method. The basic system is the solid-by-solid model for crystal growth which includes the vacancy formation during the growth of thin film. Three kinds of additives are included in the model to control the local surface growth rate. Inhibitors and levelers have the effect of preventing the deposition, while accelerators increase the local growth rate. Levelers are modeled to stick to the tips of the surface. We performed a series of simulations by changing the parameters which characterize the additives to see their influence on the filling mechanism. It is shown that void-free filling is possible by the combination of the effects of the additives.     

On the progressive enrichment of the oxygen isotopic composition of water along a leaf

A model has been derived for the enrichment of heavy isotopes of water in leaves, including progressive enrichment along the leaf. In the model, lighter water is preferentially transpired leaving heavier water to diffuse back into the xylem and be carried further along the leaf. For this pattern to be pronounced, the ratio of advection to diffusion (Péclet number) has to be large in the longitudinal direction, and small in the radial direction. The progressive enrichment along the xylem is less than that occurring at the sites of evaporation in the mesophyll, depending on the isolation afforded by the radial Péclet number. There is an upper bound on enrichment, and effects of ground tissue associated with major veins are included. When transpiration rate is spatially nonuniform, averaging of enrichment occurs more naturally with transpiration weighting than with area‐based weighting. This gives zero average enrichment of transpired water, the modified Craig–Gordon equation for average enrichment at the sites of evaporation and the Farquhar and Lloyd (In Stable Isotopes and Plant Carbon‐Water Relations, pp. 47–70. Academic Press, New York, USA, 1993) prediction for mesophyll water. Earlier results on the isotopic composition of evolved oxygen and of retro‐diffused carbon dioxide are preserved if these processes vary in parallel with transpiration rate. Parallel variation should be indicated approximately by uniform carbon isotope discrimination across the leaf.     

Mathematical Modelling of the Phloem: The Importance of Diffusion on Sugar Transport at Osmotic Equilibrium

Plants rely on the conducting vessels of the phloem to transport the products of photosynthesis from the leaves to the roots, or to any other organs, for growth, metabolism, and storage. Transport within the phloem is due to an osmotically-generated pressure gradient and is hence inherently nonlinear. Since convection dominates over diffusion in the main bulk flow, the effects of diffusive transport have generally been neglected by previous authors. However, diffusion is important due to boundary layers that form at the ends of the phloem, and at the leaf-stem and stem-root boundaries. We present a mathematical model of transport which includes the effects of diffusion. We solve the system analytically in the limit of high Münch number which corresponds to osmotic equilibrium and numerically for all parameter values. We find that the bulk solution is dependent on the diffusion-dominated boundary layers. Hence, even for large Péclet number, it is not always correct to neglect diffusion. We consider the cases of passive and active sugar loading and unloading. We show that for active unloading, the solutions diverge with increasing Péclet. For passive unloading, the convergence of the solutions is dependent on the magnitude of loading. Diffusion also permits the modelling of an axial efflux of sugar in the root zone which may be important for the growing root tip and for promoting symbiotic biological interactions in the soil. Therefore, diffusion is an essential mechanism for transport in the phloem and must be included to accurately predict flow.     

DIATOM COLONY FORMATION: A COMPUTATIONAL STUDY PREDICTS A SINGLE MECHANISM CAN PRODUCE BOTH LINKAGE AND SEPARATION VALVES DUE TO AN ENVIRONMENTAL SWITCH1

The morphological plasticity and adaptive behavior exhibited during diatom colony formation in Aulacoseira is explored through computer simulation to study how the interplay of mechanisms such as cytoskeletal‐driven membrane protrusions, silica deposition, and environmental factors may contribute to the generation of two distinct spine morphologies on linkage and separation valves. A multiscale agent‐based computational model was developed, which showed that a single cytoskeleton‐driven, competitive growth mechanism could generate either of the two characteristic phenotypes, given only a single switch in the environment (as might be experienced by a change in light regime). Hypotheses are formulated from the model, and predictions made for potential follow‐up experiments.     

Spatial variation of deuterium enrichment in bulk water of snowgum leaves

Deuterium enrichment of bulk water was measured and modeled in snowgum (Eucalyptus pauciflora Sieber ex Sprengel) leaves grown under contrasting air and soil humidity in arid and wet conditions in a glasshouse. A map of the enrichment was constructed with a resolution of 4 mm by using a newly designed cryodistillation method. There was progressively increasing enrichment in both longitudinal (along the leaf midrib) and transversal (perpendicular to the midrib) directions, most pronounced in the arid-grown leaf. The whole-leaf average of the enrichment was well below the value estimated by the Craig-Gordon model. The discrepancy between model and measurements persisted when the estimates were carried out separately for the leaf base and tip, which differed in temperature and stomatal conductance. The discrepancy was proportional to the transpiration rate, indicating the significance of diffusion-advection interplay (Péclet effect) of deuterium-containing water molecules in small veins close to the evaporating sites in the leaf. Combined Craig-Gordon and desert-river models, with or without the Péclet number, P, were used for predicting the leaf longitudinal enrichment. The predictions without P overestimated the measured values of deltadeuterium. Fixed P value partially improved the coincidence. We suggest that P should vary along the leaf length l to reconcile the modeled data with observations of longitudinal enrichment. Local values of P, P(l), integrating the upstream fraction of water used or the leaf area, substantially improved the model predictions.     

The genetic signature of rapid range expansions: How dispersal,growth and invasion speed impact heterozygosity and allele surfing

As researchers collect spatiotemporal population and genetic data in tandem, models that connect demography and dispersal to genetics are increasingly relevant. The dominant spatiotemporal model of invasion genetics is the stepping-stone model which represents a gradual range expansion in which individuals jump to uncolonized locations one step at a time. However, many range expansions occur quickly as individuals disperse far from currently colonized regions. For these types of expansion, stepping-stone models are inappropriate. To more accurately reflect wider dispersal in many organisms, we created kernel-based models of invasion genetics based on integrodifference equations. Classic theory relating to integrodifference equations suggests that the speed of range expansions is a function of population growth and dispersal. In our simulations, populations that expanded at the same speed but with spread rates driven by dispersal retained more heterozygosity along axes of expansion than range expansions with rates of spread that were driven primarily by population growth. To investigate surfing we introduced mutant alleles in wave fronts of simulated range expansions. In our models based on random mating, surfing alleles remained at relatively low frequencies and surfed less often compared to previous results based on stepping-stone simulations with asexual reproduction.