Mar Biol DOI 10.1007/s00227-009-1380-1
ORIGINAL PAPER
Linking local to global properties in branching modular networks: gorgonian coral colonies Nini Johanna Cadena • Camilo Rey • Marcela Herna´ndez-Hoyos • J. Darı´o Sa´nchez • Stanislas Teillaud • Nestor Ardila • Juan A. Sa´nchez
Received: 25 February 2009 / Accepted: 21 December 2009 Springer-Verlag 2010
Abstract Branching growth is present both in plants and animals, either marine or terrestrial. Although cellular and other modular levels of organization in plants and animals have evolved through different molecular and physiological mechanisms, several aspects of their branching modular system and morphology are similar. We studied vessel organization and colony integration, in order to comprehend underlying relationships between different structural components in a gorgonian coral network. The theoretical formalism was validated in the gorgonian coral Eunicea mammosa (Plexauridae, Octocorallia) in Belize. As in vascular plants, these colonial animals create a complex network of connections among modular branches integrated in stem canals downstream toward the base. A new formalism is proposed for describing gorgonian branching. A global property of a colony is for instance the size of its base or its weight whereas a local property is the size of
branch in a particular place of the colony. However, a global property is not the simple addition of local modular properties, as the case of stem canals in the colony base. Theoretically, the process of branching is tightly intertwined with the internal network organization. The colony network centralization is driven by a linear relationship between the total number of branches and the stem canals at the base of the colony. If stem canals play important roles in the transport of nutrients throughout the colony and the biomechanical support from the base up to the tips, we can assume that there is an underlying association between the number of stem canals at the base and the number of for example, terminal branches. These associations may provide new findings that extend our understanding of the functional organization of tree-like networks in octocorals and their vascular systems. The idea that the external components of a tree-like plant network are directly correlated and connected down to the main trunk seems to be analogous in an animal system.
Communicated by T. L. Goulet. N. J. Cadena N. Ardila J. A. Sa´nchez (&) Laboratorio de Biologı´a Molecular Marina—BIOMMAR, Departamento de Ciencias Biolo´gicas-Facultad de Ciencias, Universidad de los Andes, P.O. Box 4976, Bogota´, Colombia e-mail:
[email protected] C. Rey M. Herna´ndez-Hoyos Grupo Imagine, Grupo de Ingenierı´a Biome´dica. Facultad de Ingenierı´a, Universidad de los Andes, Bogota´, Colombia J. D. Sa´nchez Departamento de Matema´ticas, Universidad Nacional del Colombia, Bogota´, Colombia S. Teillaud Facultad de Ciencias Naturales, Programa de Biologı´a Marina, Universidad de Bogota´ Jorge Tadeo Lozano, Bogota´, Colombia
Keywords Tree-like networks Animal branching colonies Gorgonian coral Modular organisms Pipe-model theory Stem canals
Introduction Despite the apparent simplicity of marine modular colonial organisms some of them grow by forming increasingly complex tree-like networks (Kaandorp and Ku¨bler 2001). Growth in branching colonies is usually portrayed as the change in height or occupied surface area by approximating the measurement with bounding shapes disregarding the actual elements of the network (e.g., branches, nodes, connections, modules). There is, in fact, no clear
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explanation of how this multi-branched network grows or how the genetic impact on colony form, in modular colonial invertebrates, intervenes (Lasker and Sa´nchez 2002; Rinkevich 2002). Bifurcation does not provide a realistic explanation of branching and network growth (Sa´nchez et al. 2003). Moreover, observations of growth show a completely different picture with branching as a sub-apical lateral process exhibiting a dynamic development (Sa´nchez et al. 2004). New modular structures generated by branching (i.e., branches and internodes) can evolve semi independently of the modular unit per se or polyps in the case of octocorals (Sa´nchez and Lasker 2003; Sa´nchez 2004; Sa´nchez et al. 2007). Moreover, many other aspects of gorgonian networks have not been theoretically or empirically studied.
Fig. 1 Colony of Eunicea mammosa a Digital photo taken at Carrie Bow Cay, Belize, 12 m depth. b Branch ordering following a left to right scheme, where branches 1 and 2 are successors of branch 3 or branch 4 is successor of 5. Bottom up, branch 3 is a predecessor of branch 1 to branch 5, which coincides with the main stem. c Skeleton of the Colony obtained by image processing. Octocoral image (Muricea muricata) obtained by high-resolution computed tomography. d Axial slice. e Sagittal slice. GA Gorgonian axis; SC Stem canal; PO Polyp.; S Sclerite
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As an analogy with plants, gorgonian octocorals (Octocorallia: Cnidaria) construct tree-like networks that present internally, equally diametric-sized hollow ‘‘pipes’’ or stem canals all along the branches (Bayer 1973). Stem canals are composed by coenemchymal tissue and tiny sclerites. They are present all along the colony, surrounding the gorgonian axis of each branch of the octocoral (Fig. 1). They are connected with the gastrovascular solenias (which arise from the polyps) and their main function is supposed to be the exchange of water and nutrients throughout the colony (Gateno et al. 1998). Although the nature and function of the stem canals are not yet understood, it is clear that the cross-sectional area of the tree-like gorgonian colony increases toward the base so as the number of stem canals from the tips to the holdfast of the
Mar Biol
colony. However, there is no information about the structural and numerical organization of the gorgonian stem canals. Another analogy with most plants (Prusinkiewicz 1998) involves the growth and branching process: these corals also exhibit apical dominance during growth and branching is always sub-apical (Sa´nchez et al. 2004). Branching in modular organisms, such as gorgonians corals, is a sub-apical process (Fig. 1) where all the branches structurally are the same. Nevertheless, in pinnate corals, some branches called ‘‘mother branches,’’ produce new branches known as ‘‘daughter branches’’ at roughly fixed distances or internodes (Lasker and Sa´nchez 2002; Lasker et al. 2003; Sa´nchez et al. 2004). Daughter branches have determinate growth, and their length can be considered constant (Lasker et al. 2003; Sa´nchez et al. 2004). The relationship between mother branches and daughter branches follows power law dependence, where only a few mother with many daughter branches are allowed and most mother branches have fewer daughter branches (Sa´nchez et al. 2004). Such hierarchical branching process is highly dependent on three different factors: a species-specific constant, which is the ratio between the number of total branches and the number of mother branches; the intrinsic rate of branching per mother branch; and a branch-charge capacity per colony (Sa´nchez 2004). The process of branching can be explained in terms of mother branches in pinnate corals (Sa´nchez et al. 2004). In others, the difficulty to determine the hierarchic level of every branch promotes our investigation to find other models to describe the branching and growing process of gorgonian corals. The aim of this work was to study the branching process in the gorgonian coral Eunicea mammosa (Lamouroux) and its association with the stem canals present at the base and at the terminal segment of the branches of the colony. Concerning the stem canals, we were particularly interested in: (1) establishing a correlation between the total number of branches and the number of stem canals at the base of the colony and (2) studying the correspondence between the number of the stem canals and the length of the branches. In addition, we include a mathematical formalism to describe gorgonian corals, which unambiguously defines the concepts involved in our study: (1) the branching that generalizes the concept of mother and daughter branches and (2) the geometric properties of this model and the possible underlying associations between geometric properties of gorgonian corals. The study of the branching network and the stem canals distribution in Eunicea mammosa Lamouroux was then outlined by describing the protocol used to acquire the data and the parameters that have been measured.
A formal description of gorgonian corals Gorgonian corals grow, forming stunningly complex networks. A clear and concise description of the underlying structural relationships within gorgonian corals should start with a brief definition of what a gorgonian coral is in a formal manner. Indeed, the structure of a gorgonian coral can be described by means of an ordering scheme within its branches. Such ordering simplifies the study of their geometrical and biological properties and the interaction between them. Definition 1 (Gorgonian coral) A gorgonian coral G can be regarded as a set of branches G along a binary relation[ that reflects the intuitive concept of ‘‘ancestor of’’. Indeed if a and b are branches of a colony G, and b spurs on a we write a [ b:
ð1Þ
Abusing the notation, we will write a [ G for ‘‘a is a branch of G’’. Notice the following properties of a gorgonian coral G regarding the relation [: • •
There exists a branch a0 of G from which all other branches spur. That is, there exists a primordial branch. Every branch a [ G is comprised by the ordering relation: every branch is either an ancestor or has ancestors.
Given a particular branch a [ G, we define the set of all branches growing directly- or indirectly- as the set of its successors. Definition 2 (Successors of a branch) Given a branch r of a coral G, we define S(a) as the set of the successors of a. SðaÞ ¼ fb 2 Gja [ bg:
ð2Þ
The notion of successors comes to replace the notions of mother and daughter branches: it generalizes them. To see this, we propose the following definition. Definition 3 (Terminal branch) A branch r of a colony G is a terminal branch if and only if it has empty successors, that is, if and only if SðaÞ ¼ ;:
ð3Þ
A non-terminal branch can analogously be defined as a branch with at least one successor. Furthermore, a primordial branch can be defined as a branch whose successors are the entire colony except for itself. Definition 4 (Primordial branch) A branch a^ 2 G: is called the primordial branch or main stem of G if and only if a^ has as successors all other branches in the colony apart from itself, that is:
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Sð^ aÞ ¼ Gnf^ ag:
ð4Þ
In order to describe change among distinct branches (for example, width or height of branches directly connected) within a given gorgonian coral, we need to distinguish all branches that spur directly on a given branch a. Definition 5 (Immediate successor branch) Given branches a, b [ G, we say that b is an immediate successor branch of a if: • •
b [ S(a). b spurs on a. In such case we write a C b for simplicity.
The preceding formal definitions around gorgonian corals aim to simplify the description of relations among the components of gorgonian corals, i.e., their branches. In general, we wish to find architectural rules describing their growth and development and we will describe them using the definitions given in this section. Properties of gorgonian corals We can think of a gorgonian coral as a three-dimensional geometrical object with measurable properties. Volume and surface area or stem canals number are, among others, intrinsic properties of gorgonian corals whose values are supposed to be given by rules that respond to biological or functional constraints, possibly set by environmental factors surrounding the colony (see Sec. ‘‘Discussion’’). The formalism introduced previously can be used to define the concept of property with the purpose of unveiling relations among them that can be explained biologically. Let P be a property of a gorgonian coral G. The measurement of property P on branch a [ G is denoted by P(a). We can understand P as a function on the set of branches G. Our task is to find functions relating distinct properties of G and expressing them in a mathematical language. We must distinguish two levels of expression of properties within gorgonian corals: global and local. By global we designate all properties that pertain to the colony as a whole: The network extension L or surface area A are clear examples, while the width of a branch a, W(a) or its volume V(a) are properties that only concern a. We can think of some global properties as being aggregative in the sense that their values depend on the sum of their value at every branch of the colony. More specifically, if P(a) denotes a local property, the equivalent global property P is defined by X P¼ PðaÞ ð5Þ a2G
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Not all global properties are the aggregate of their local equivalent. For example, the number of stem canals at the base of the primordial branch, U is a global property of the colony that is not obtained by means of simple addition. We wish to find relations among different properties of a gorgonian coral. In order to do so, we define such relations in the following way: Definition 6 (Underlying relation) Let P and Q stand for two properties of a coral colony G (local or global). An underlying relation between P and Q is a function F such that Q ¼ F ðPÞ:
ð6Þ
In the case of a local property, we denote the underlying relation by QðaÞ ¼ F PðaÞ; a 2 G:
ð7Þ
Materials and methods Study of branching network and stem canals in Eunicea mammosa A total of 25 colonies of the candelabrum gorgonian coral Eunicea mammosa were photographed and their stem canals sampled during June 2003 in Carrie Bow Cay fore-reef terrace (Belize, 12 m depth, Fig. 1a). This photographic registry was made with a reference grid (Fig. 1a), in order to count the number of terminal and non-terminal branches, and to measure the network length. The images were transformed for distortion and correction in Adobe-Photoshop (Lasker et al. 2003). Following, the measurements were made with the aid of ImageJ 1.37 (NIH). In addition, a portion of the surrounding tissue cortex at the colony base and at the tip of some haphazardly chosen terminal branches was removed. The widths of these tissue samples were approximately 1 cm and in the case of the terminal branches the samples were taken at variable distance from the tip. The tissues were then observed under an optical microscope, in order to estimate the number of stem canals in chosen segments from each colony. The local properties measured were: individual branch length at every colony, from which network length L was obtained following equation (12); number of stem canals at a certain distances from the tip w. From global properties measured, we estimated the average number of stem canals at a fixed distance from the tip w and the number of stem canals at the base of the colony U (Table 1). These results were then analyzed and statistically tested using SPSS v 16.
Mar Biol Table 1 Summary of network properties (variables and parameters) found in gorgonian corals that were estimated from Eunicea mammosa colonies in Belize Property Definition
Acquisition
N
Total number of branches, it includes all colony tips
Direct counts from digital underwater images
S
Non-terminal branches (mother branches), branches where other successor branches attach
Direct counts from digital underwater images
L
Total network length, which is the sum of the lengths from all branches in the colony
Digital measurements on scale-corrected images using ImageJ (NIH)
U
Number of stem canals at the base of the colony. This is a global Underwater dissection of a ring of tissue (2 cm wide) on the same property of the colony likewise the variables above colonies that digital images were taken. Following direct counts under the scope
w
Number of stem canals close to a branch tip, a local property in a Terminal branch tips collected from the same colonies mentioned spot of the colony. It is supposed to be a nearly fixed speciesabove and dissection under scope and direct count specific parameter, which approached *16 stem canals in Eunicea mammosa
Results Linking local to global properties in a gorgonian coral The studied colonies had an average of 23 branches (min–max: 7–48), 47 stem canals at the base (27–84) and a network length of 210 cm (54.8–703.8). A linear trend between the number of non-terminal branches S and the total number of branches N was found in Eunicea mammosa (Fig. 2a) given by the underlying relation N = F(S), where F(x) = 3.08 x-0.53 (15). Such claim was found with a confidence of r2 = 0.82, (P \ 0.001, F = 106.38, n = 25 see Fig. 2a). Similarly, N was related to the length of the colony L, (L = F(N)); where F(x) = 8.56 x ? 51.88 (16) and a confidence of r2 = 0.71 (P \ 0.001, F = 53.21, n = 24; Fig. 2b). Furthermore, N was related to the number of stem canals at the base of the colony U, (U = F(N)) with F(x) = 1.08 x ? 23.38 (17), where the confidence value is r2 = 0.75 (P \ 0.001, F = 69.46, n = 25; Fig. 2c). Additional findings were that the number of stem canals at branch tips w has a mean value of w = 16.60 (min–max = 14–22; SD = 1,725; Fig. 2d), and the value of w increases proportionally with the distance from the tip downwards to the network (Fig. 2e).
Discussion The gorgonian coral Eunicea mammosa presented a balance around a c ratio value between total and non-terminal branches, which suggests that a growth model similar to Sa´nchez et al. (2004) is applicable in this case. Moreover, the formalism for describing gorgonian corals ordering among branches allowed the inclusion of local versus global properties of the network, which in turn permitted to
address ecological or biomechanical factors as playing roles in determining relations at either level. The relationship between branches and stem canals, which was met in colonies of E. mammosa at Belize indicates a linear relationship through the network, between branching variables, e.g., total number of branches N, and internal colony connections or stem canals U at the base of the colony. If stem canals play important roles in the transport of nutrients throughout the colony and the biomechanical support from the base up to the tips, we can assume that there is an underlying association between the number of stem canals at the base and the number of for example, terminal branches. The new formalism allows us to establish relationships between variables by replacing them in the founded equations. It was not necessary to account for mother and daughter branches given that it is clear that terminal branches are connected down to the main stem through series of predecessors’ branches following the same rules. It is possible for example to express a direct relationship between the number of stem canals in the holdfast, U and the total length of the colony L. Moreover, the extension of the network was also related to the number of total branches N (Fig. 2b) and the number of stem canals at nonterminal branches w increased with the distance from the tips downward the network (Fig. 2c). Indeed, the finding of relationships between local versus global properties within this gorgonian is a step toward understanding the integration and functionality of modular tree-like network in animal colonies. Stem canals extend down from the tips and out the edge of the gorgonian holdfast. The holdfast is the region in the colony with more stem canals. Hypothetically, colony growth starts at the base in response to new resource opportunities at the colony tips. The correlation of stem canals at the base with the number of branches in a colony suggests a coordinated effort in
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Fig. 2 Branching network organization in Eunicea mammosa and stem canals measurement results. a Relationship between number of non-terminal branches S versus total number of branches N per colony. Lines with the trend line in between indicate the 95% confidence intervals (r2 = 0.82, P \ 0.001, F = 106.38, n = 25). b Relationship between total number of branches N versus network length L. Lines with the trend line in between indicate the 95% confidence intervals (r2 = 0.71, P \ 0.001, F = 53.21, n = 24).
c Relationship between total number of branches N versus the number of stem canals in the colony base of E. mammosa. Lines indicate the 95% confidence intervals (r2 = 0.75, P \ 0.001, F = 69.46, n = 25). d Number of stem canals at branch tips w, including composite data from 82 non-terminal branches from 27 colonies. e Relationship between the number of stem canals at branch tips w versus distance to the tip (n = 86)
gorgonian corals for growing and branching without compromising proper attachment to the substrate. Sessile marine benthic organisms are subject to water motion drag forces, which represent a selection pressure shaping efficient attachment mechanisms to the substrate as well as material strength (Wainwright et al. 1976). Drag forces also impose trade-offs between size/area and capacity to attach in sessile organisms (Johnson 2001; Pratt and Johnson 2002), which has also been observed in fresh water aquatic plants (Puijalon et al. 2005). Algae subject to drag forces change shape and size in response to water flow speed (Boller and Carrington 2006). The stalk cross-sectional area in marine algae is positively correlated with the drag forces present in the environment (Kitzes and Denny 2005). Consequently, mechanisms allowing branching sessile organisms to respond and integrate their size and shape with respect to water motion can have a great adaptive value with respect to both survivorship and nutrient uptake (Carrington 1990; Fonseca et al. 2007). Phenotypic plasticity and morphologic integration of different modular traits occurs in octocorals along environmental gradients (Sa´nchez et al. 2007). Both plants and branching sessile animals are subject to similar constraints, and there are surprising parallels toward the functional solutions and phenotypic plasticity to counteract those
limitations in the two groups (Borges 2005). Gorgonian octocorals, also subject to strong water motion and dislodgment, are driven to evolve mechanisms to integrate and balance drag forces acting on the colony. The stem canals seem to have an essential role in the branching process of gorgonian corals. For instance, the axial skeleton of a gorgonian coral has growth rings, which are predictably related with age, modular growth, and biomass (Goffredo and Lasker 2006). Bamboo corals, a group of deep-sea octocorals (family Isididae), also have axial rings apparently due to lunar periodicity, and the stem canals are clearly grooved and visible in the rings (Tracey et al. 2007). The stem canals, which are of equal diameter, surround the gorgonian axis and their numbers are consequently dependent upon the skeletal axis age and thickness. The function of stem canals could be related with intracolonial nutrient transport and morphogenesis (Bayer 1973; Sa´nchez 2004). There is an allometric constraint to resource capture in colonial cnidarians (Kim and Lasker 1998), which could be related with the transport and capacity of the internal network of canals. The key role of stem canals for colony growth can also be appreciated with clipping and regeneration experiments. Clipped lower parts of gorgonian colonies have a growth response related to the original number of branches in the colony (Sa´nchez and
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Lasker 2004). This dependency of an internal network of vessels for support and transport provides a compelling prediction on the analogy of animal and plant branching systems. Tree-like network organization according to the pipemodel theory of plants predicts an association between the numbers of leaves in the crown and the cross-sectional area of the trunk (Shinozaki et al. 1964), an idea formerly sketched by Leonardo D’Vinci in one of his observations on botany (Richter 1939). This model portrays a plant as a series of equally diametric sized pipes that connect every leaf to the trunk. Plant ramification and morphology (Chiba 1991), tree vigor and height (Robichaud and Methven 1991), carbon metabolism (Valentine 1997), and physiological growth and balance (Sieva¨nen 1993; Koskela 2000) among others, have been studied using the pipe-model theory. Among the predictions of the pipe-model theory are (1) a linear relationship between foliage and woody organs (Shinozaki et al. 1964; Chiba 1991) in a plant and (2) an exponential growth of the stem cross-sectional area toward the base. According to these properties, it is interesting to see that these two very different systems may follow some analogies where (1) the foliage of a plant would correspond to the terminal branches of a gorgonian coral and (2) the cross-sectional area at the base of a plant would correspond to the number of stem canals (analogous to xylem and phloem system) that surround the base of the cylindrical gorgonian axis of a colony. This suggests that there are similar constraints and pressures acting on both branching animals and plants and that analogous solutions and adaptations have arisen in parallel. It is compelling that the similar structural patterns in nature such as the tree-like networks, independent of phylogeny, afford very similar ways of organization and function. Given the omnipresence of branching patterns in all systems of nature, e.g., fungi, algae, vascular plants, corals, or even an arterial or pulmonary tree, the existence of common rules for the development of all organic tree-like networks deserves more theoretical and experimental attention. The formalism and the relations presented in this paper provide new findings that may extend our understanding of the structural organization of tree-like networks and their vascular systems. Acknowledgments We want to thank our sponsors, Vicerrectorı´a de Investigaciones (Proyecto interfacultades), Facultad de Ingenierı´a, Facultad de Ciencias and Department of Biological Sciences (Universidad de los Andes, Bogota´, Colombia), COLCIENCIAS (Grant # 1204-09-177774, J.A. Sa´nchez), and Smithsonian Institution. Special thanks to H. R. Lasker (U. Buffalo), whom ideas and discussions were of great influence to develop this study. We are also grateful to the Smithsonian Institution Marine Science Network (K. Ruetzler and M. Lang) for their field support (NMNH-CCRE contribution No. 874), and to M. Orkisz (CREATIS Laboratory, Lyon, France) and J. Adrien (MATEIS Laboratory, Lyon, France) for their contribution in
acquiring the high-resolution computed tomography images of corals. Comments from T. Goulet and three anonymous reviewers greatly improved the manuscript.
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