Fractal geometry of mosaic pattern demonstrates liver regeneration is a self-similar process.

Partial hepatectomy causes compensatory, nonneoplastic growth and regeneration in mammalian liver. Compensatory liver growth can be used to examine aspects of patterns of cell division in regenerating tissue. Chimeric animals provide markers of cell lineage which are independent of growth and can be used to follow cell division patterns. Previous experimental evidence suggests that compensatory liver growth is uniform, without focal centers of proliferation. In this study we have extended that observation to include genes important in regeneration and cell cycle control in order to establish that nascent growth centers are not present in regenerating liver. There is a uniform spatial distribution of expression of these genes which is not related to mosaic pattern in the chimeras. While these genes may help regulate hepatocyte proliferation they do not appear to regulate patch pattern in the chimeras. With this information confirming uniform growth it was possible to use fractal analysis to test various hypothesized patterns of regenerative growth in the liver. The results of this analysis indicate that mosaic pattern does not change substantially during the regenerative process. Patch area and perimeter (the area occupied by or perimeter around cells of like lineage) increase during compensatory liver growth in chimeric rats without alteration of the geometric complexity of patch boundaries (boundaries around cells of like lineage). These tissue findings are consistent with previously reported computer models of growth in which repetitive application of simple decisions assuming uniform growth created complex mosaic patterns. They support the notion that an iterating (repeating), self-similar (a pattern in which parts are representative of, but not identical to the whole) cell division program is sufficient for the regeneration of liver tissue following partial hepatectomy. Iterating, self-similar cell division programs are important because they suggest a way in which complex patterns (or morphogenesis) can be efficiently created from a small amount of stored information.     

Predicting root biomass from branching patterns of Douglas-fir root systems

There are many examples of branching networks in nature, such as tree crowns, river systems, arteries and lungs. These networks have often been described as being self-similar, or following scale-invariant branching rules, and this property has been used to derive several scaling laws. In this paper we model root systems of Douglas-fir ( Pseudotsuga menziesii var. glauca (Beissn.) Franco) as branching networks following several simple branching rules. Our objective is to establish a relationship between trunk diameter and root biomass. We explore the effect of the self-similar branching assumption on this relationship. Using data collected from a mature stand in British Columbia, we find that branching asymmetry and the rate of root taper change with root size, thereby violating the assumption of self-similarity. However, the data are in general agreement with Leonardo da Vinci's area-preserving branching hypothesis. We use the field data to parameterize two models, one assuming self-similar branching and a second incorporating the measured size dependencies of branching parameters. The two models differ by only a small amount (≈8%) in their predictions. For both models, the predicted relationship between trunk diameter and root biomass is in good concordance with previously published empirical data. We conclude that the assumption of self-similar branching, although violated by the data, nevertheless provides a useful tool for predicting the allometric relationship between trunk diameter and root biomass. Finally, we use our models to show that the geometric properties of individual bifurcations fundamentally change the root biomass cost of different root topologies.     

Fractal morphometry of cell complexity.

Irregularity and self-similarity under scale changes are the main attributes of the morphological complexity of both normal and abnormal cells and tissues. In other words, the shape of a self-similar object does not change when the scale of measurement changes, because each part of it looks similar to the original object. However, the size and geometrical parameters of an irregular object do differ when it is examined at increasing resolution, which reveals more details. Significant progress has been made over the past three decades in understanding how irregular shapes and structures in the physical and biological sciences can be analysed. Dominant influences have been the discovery of a new practical geometry of Nature, now known as fractal geometry, and the continuous improvements in computation capabilities. Unlike conventional Euclidean geometry, which was developed to describe regular and ideal geometrical shapes which are practically unknown in nature, fractal geometry can be used to measure the fractal dimension, contour length, surface area and other dimension parameters of almost all irregular and complex biological tissues. We have used selected examples to illustrate the application of the fractal principle to measuring irregular and complex membrane ultrastructures of cells at specific functional and pathological stage.     

Branch mode selection during early lung development

Many organs of higher organisms, such as the vascular system, lung, kidney, pancreas, liver and glands, are heavily branched structures. The branching process during lung development has been studied in great detail and is remarkably stereotyped. The branched tree is generated by the sequential, non-random use of three geometrically simple modes of branching (domain branching, planar and orthogonal bifurcation). While many regulatory components and local interactions have been defined an integrated understanding of the regulatory network that controls the branching process is lacking. We have developed a deterministic, spatio-temporal differential-equation based model of the core signaling network that governs lung branching morphogenesis. The model focuses on the two key signaling factors that have been identified in experiments, fibroblast growth factor (FGF10) and sonic hedgehog (SHH) as well as the SHH receptor patched (Ptc). We show that the reported biochemical interactions give rise to a Schnakenberg-type Turing patterning mechanisms that allows us to reproduce experimental observations in wildtype and mutant mice. The kinetic parameters as well as the domain shape are based on experimental data where available. The developed model is robust to small absolute and large relative changes in the parameter values. At the same time there is a strong regulatory potential in that the switching between branching modes can be achieved by targeted changes in the parameter values. We note that the sequence of different branching events may also be the result of different growth speeds: fast growth triggers lateral branching while slow growth favours bifurcations in our model. We conclude that the FGF10-SHH-Ptc1 module is sufficient to generate pattern that correspond to the observed branching modes.     

Branching pattern of pulmonary arterial tree in anesthetized dogs

The geometry of the pulmonary arterial tree of six adult dogs was measured by a high-speed, volume-scanning, X-ray tomographic technique. After the dogs were anesthetized a catheter was advanced to the right ventricular outflow tract and 2 mL/kg Renovist contrast agent injected rapidly. During the subsequent pulmonary arterial phase of the angiogram the dogs were scanned. Three-dimensional geometry of the pulmonary arterial tree was measured in terms of vessel segment cross-sectional area, branching angles and interbranch segment lengths along axial pathways. The effect of lung inflation and phase of the cardiac cycle on geometry was shown to be most marked on vessel cross-sectional area. The geometric branching patterns in all dogs were similar. The observed, in-vivo branching pattern behaved somewhat like the branching pattern predicted from optimized models proposed by Murray, Zamir, and Uylings.     

Investigation of the structure of human coronary vasculature

Some results of a morphometric study of the parameters of coronary arteries are presented. The parameters that characterize the structure of the arterial vasculature as an optimal branching system have been calculated. Statistically reliable correlations between the diameter of the bigger of two daughter vessels in a bifurcation with the diameter of the parent vessel as well as between the diameter of the smaller daughter vessel and the asymmetry coefficient have been obtained. Differences in the structural parameters of the two types of coronary arteries that provide blood delivery and distribution have been revealed. The relationships between the lengths and diameters of the arteries of different subsystems have been obtained. It is shown that asymmetrical branching is characteristic of the coronary vasculature, and self-similar asymmetric tree-like systems may be used for its modeling.     

The complex structure of hunter-gatherer social networks

In nature, many different types of complex system form hierarchical, self-similar or fractal-like structures that have evolved to maximize internal efficiency. In this paper, we ask whether hunter-gatherer societies show similar structural properties. We use fractal network theory to analyse the statistical structure of 1189 social groups in 339 hunter-gatherer societies from a published compilation of ethnographies. We show that population structure is indeed self-similar or fractal-like with the number of individuals or groups belonging to each successively higher level of organization exhibiting a constant ratio close to 4. Further, despite the wide ecological, cultural and historical diversity of hunter-gatherer societies, this remarkable self-similarity holds both within and across cultures and continents. We show that the branching ratio is related to density-dependent reproduction in complex environments and hypothesize that the general pattern of hierarchical organization reflects the self-similar properties of the networks and the underlying cohesive and disruptive forces that govern the flow of material resources, genes and non-genetic information within and between social groups. Our results offer insight into the energetics of human sociality and suggest that human social networks self-organize in response to similar optimization principles found behind the formation of many complex systems in nature.     

On the significance of fiber branching in the human myocardium

Myocardial tissue exhibits a high degree of organization in that the cardiac muscle fibers are both systematically aligned and highly branched. In this study, the influence and significance of fiber branching is analyzed mathematically. In order to allow for analytic solutions, a regular geometry and simplified constitutive relations are considered. It is found that branching is necessary to stabilize the ventricular wall.     

Developmental morphology of the shoot in Weddellina squamulosa and implications for shoot evolution in the Podostemaceae

BACKGROUND AND AIMS: In angiosperms, the shoot apical meristem produces a shoot system composed of stems, leaves and axillary buds. Podostemoideae, one of three subfamilies of the river-weed family Podostemaceae, have a unique 'shoot' that lacks a shoot apical meristem and is composed only of leaves. Tristichoideae have been interpreted to have a shoot apical meristem, although its branching pattern is uncertain. The shoot developmental pattern in Weddellinoideae has not been investigated with a focus on the meristem. Weddellinoideae are in a phylogenetically key position to reveal the process of shoot evolution in Podostemaceae. METHODS: The shoot development of Weddellina squamulosa, the sole species of Weddellinoideae, was investigated using scanning electron microscopy and semi-thin serial sections. KEY RESULTS: The shoot of W. squamulosa has a tunica-corpus-organized apical meristem. It is determinate and successively initiates a new branch extra-axillarily at the base of an immediately older branch, resulting in a sympodial, approximately plane branching pattern. Large scaly leaves initiate acropetally on the flanks of the apical meristem, as is usual in angiosperms, whereas small scaly leaves scattered on the stem initiate basipetally in association with the elongation of internodes. CONCLUSIONS: Weddellinoideae, like Tristichoideae, have a shoot apical meristem, leading to the hypothesis that the meristem was lost in Podostemoideae. The patterns of leaf formation in Podostemoideae and shoot branching in Weddellinoideae are similar in that these organs arise at the bases of older organs. This similarity leads to another hypothesis that the 'branch' in Weddellinoideae (and possibly Tristichoideae) and the 'leaf' in Podostemoideae are comparable, and that the shoot apical meristem disappeared in the early evolution of Podostemaceae.     

Disto-proximal regional determination and intercalary regeneration in planarians, revealed by retinoic acid induced disruption of regeneration.

The mechanisms that define the body pattern during development and regeneration are the object of major concern in developmental biology. To understand the process and sequence of antero-posterior pattern formation of planarian body regions during regeneration, regenerating organisms were treated with exogenous retinoic acid, which affects development and regeneration in other systems, and the sequence of regional determination has been monitored by a specific molecular marker for the central region, which includes the pharynx. The sequence of gross regional specification have never been analysed in planarians using molecular regional markers or by direct disruption of the regeneration process. Exogenous retinoic acid administration on regenerating planarians disrupts anterior, but not posterior regeneration. The period of maximum sensitivity to exogenous retinoic acid is one day after amputation, during which time the determination of the head has been reported to occur. The data obtained allow us to suggest that gross regional specification during planarian regeneration is disto-proximal, from the regenerative blastema to the old stump, and thus takes place by intercalation of the central region between the anterior and posterior ones.     

The regulation of intramuscular nerve branching during normal development and following activity blockade

In vertebrates, approximately 50% of the lumbosacral motoneurons die during a short period of development that coincides with synaptogenesis in the limb. Although it has been postulated that these motoneurons die because they fail to obtain adequate trophic support from the muscles, it is not clear how this factor is supplied. The mechanism by which activity blockade prevents motoneurons cell death is also unknown. In order to begin to understand the nature of these proposed trophic interactions, we have examined the temporal sequence of axonal invasion and ramification within two muscles of the chick hindlimb, the predominantly slow iliofibularis and the fast posterior iliotibialis, during the cell death period. We found striking differences in intramuscular nerve ingrowth and branching between fast and slow muscle. We also observed differences in the molecular composition of fast and slow myotubes that may contribute to the nerve pattern differences. In addition, we observed a progressive increase in the degree of intramuscular nerve fasciculation as well as a precise temporal sequence of nerve branching. The earliest detectable response to chronic curarization was a dramatic decrease in the degree of intramuscular nerve fasciculation. Activity blockade also greatly enhanced nerve branching within the muscles from the time that nerve branches normally formed, and, additionally, interfered with the normal cessation of axon growth. Our results support the idea that nerve endings are the sites of trophic uptake. Furthermore, although our results do not allow us to exclude other activity-dependent influences on motoneuron survival, they suggest the following testable hypotheses: (1) the normal regulation of motoneuron survival may result from the precise control of intramuscular nerve branching, (2) activity blockade may increase motoneuron survival by enhancing intramuscular nerve branching, and (3) anything which affects this complex process of nerve branching may also alter motoneuron survival.     

Development of the trigeminal motor neurons in parrots: Implications for the role of nervous tissue in the evolution of jaw muscle morphology

Vertebrates have succeeded to inhabit almost every ecological niche due in large part to the anatomical diversification of their jaw complex. As a component of the feeding apparatus, jaw muscles carry a vital role for determining the mode of feeding. Early patterning of the jaw muscles has been attributed to cranial neural crest‐derived mesenchyme, however, much remains to be understood about the role of nonneural crest tissues in the evolution and diversification of jaw muscle morphology. In this study, we describe the development of trigeminal motor neurons in a parrot species with the uniquely shaped jaw muscles and compare its developmental pattern to that in the quail with the standard jaw muscles to uncover potential roles of nervous tissue in the evolution of vertebrate jaw muscles. In parrot embryogenesis, the motor axon bundles are detectable within the muscular tissue only after the basic shape of the muscular tissue has been established. This supports the view that nervous tissue does not primarily determine the spatial pattern of jaw muscles. In contrast, the trigeminal motor nucleus, which is composed of somata of neurons that innervate major jaw muscles, of parrot is more developed compared to quail, even in embryonic stage where no remarkable interspecific difference in both jaw muscle morphology and motor nerve branching pattern is recognized. Our data suggest that although nervous tissue may not have a large influence on initial patterning of jaw muscles, it may play an important role in subsequent growth and maintenance of muscular tissue and alterations in cranial nervous tissue development may underlie diversification of jaw muscle morphology. J. Morphol. 275:191–205, 2014. © 2013 Wiley Periodicals, Inc.     

Hoxa11 and Hoxd11 regulate branching morphogenesis of the ureteric bud in the developing kidney

Hoxa11 and Hoxd11 are functionally redundant during kidney development. Mice with homozygous null mutation of either gene have normal kidneys, but double mutants have rudimentary, or in extreme cases, absent kidneys. We have examined the mechanism for renal growth failure in this mouse model and find defects in ureteric bud branching morphogenesis. The ureteric buds are either unbranched or have an atypical pattern characterized by lack of terminal branches in the midventral renal cortex. The mutant embryos show that Hoxa11 and Hoxd11 control development of a dorsoventral renal axis. By immunohistochemical analysis, Hoxa11 expression is restricted to the early metanephric mesenchyme, which induces ureteric bud formation and branching. It is not found in the ureteric bud. This suggests that the branching defect had been caused by failure of mesenchyme to epithelium signaling. In situ hybridizations with Wnt7b, a marker of the metanephric kidney, show that the branching defect was not simply the result of homeotic transformation of metanephros to mesonephros. Absent Bf2 and Gdnf expression in the midventral mesenchyme, findings that could by themselves account for branching defects, shows that Hoxa11 and Hoxd11 are necessary for normal gene expression in the ventral mesenchyme. Attenuation of normal gene expression along with the absence of a detectable proliferative or apoptotic change in the mutants show that one function of Hoxa11 and Hoxd11 in the developing renal mesenchyme is to regulate differentiation necessary for mesenchymal-epithelial reciprocal inductive interactions.     

Modeling of intraorgan arterial vasculatures. II. Propagation of pressure waves

The dependences of the wave conductivity of self-similar dichotomous branching models of intraorgan arterial vasculatures on the model parameters have been calculated. It was found that, with different sets of parameters, it is possible to model the sucking effect connected with negative wave reflection at the arterial branching as well as the resonant properties of arterial beds. It was shown that the selection of an adequate model for a given intraorgan vasculature should be based on the agreement between the biophysical characteristics of the model and the vasculature that reflect the propagation and reflection of pulse waves.     

Morphometric analysis of dendritic tree of cerebellar Purkinje-cells

Basic neural processes of sensorimotor adaptation can be observed by cellular-level studies on cerebellar cortex, both by electrophysiological and morphological means. In earlier studies we demonstrated double (sometimes triple or quadruple) rhythmic prespike activity patterns in dendritic microelectrode records taken from cerebellar Pc. By their active and passive interactions, the actual input pattern will turn into an arrhythmic output spiking, realizing a nonlinear and phase-sensitive integration. This curious complex spike-generating process can give rise to novel cerebellar functional models. The acting membrane dynamics rely not just upon the ionic current machinery but also upon the specific micromorphology of the dendritic tree, especially that of the branching areas. Although, statistical analyses on Pc dendrites generally followed the graph-theory, elaborated abstract parameters for complexity and hierarchy, and denied realistic geometry. Thus a novel specific morphometric study should be carried out, first applied to the main branching sites.     

Distinct roles for adhesion molecules during innervation of embryonic chick muscle

In vitro studies have suggested that the cell adhesion molecules NCAM and G4/L1 contribute to a variety of events during neural development. We have directly tested the role played by these molecules in the process of initial nerve ingrowth and ramification in the embryonic chick iliofibularis muscle by in ovo injections of specific adhesion-blocking antibodies and analysis of the resultant nerve branching pattern in muscle whole mounts. Antibodies against both molecules produced axonal defasciculation, which resulted in an enhanced transverse projection to the fast region of the muscle. In the case of anti-G4/L1, we also observed a large increase in the number of side branches that form from nerve trunks in the slow region and an enhancement of nerve branching in the fast region. Conversely, anti-NCAM produced a striking decrease in both the number and length of side branches in the slow region, and a reduction in nerve branching in the fast region. A similar reduction of nerve branching was obtained following injection of an endosialidase, which removes sialic acid from NCAM, and which was observed to enhance fiber-fiber apposition, presumably by increasing cell adhesion. Based on their biochemical properties in vitro and their in vivo distribution, both NCAM and G4/L1 are in a position to contribute to axon-axon adhesive interactions, whereas NCAM would be expected to also promote axon-myotube interactions. Our observations in fact indicate that these two adhesion molecules play different but complementary roles during muscle innervation and, specifically, that axon-axon fasciculation is influenced by both NCAM and G4/L1 in an anatomically distinct manner to regulate the overall pattern of nerve branching and that NCAM-mediated axon-myotube interactions are necessary for the attainment of the normal stereotyped pattern of nerve branching in both fast and slow regions of this muscle.     

Anatomically based finite element models of the human pulmonary arterial and venous trees including supernumerary vessels.

Studies of the origin of pulmonary blood flow heterogeneity have highlighted the significant role of vessel branching structure on flow distribution. To enable more detailed investigation of structure-function relationships in the pulmonary circulation, an anatomically based finite element model of the arterial and venous networks has been developed to more accurately reflect the geometry found in vivo. Geometric models of the arterial and venous tree structures are created using a combination of multidetector row X-ray computed tomography imaging to define around 2,500 vessels from each tree, a volume-filling branching algorithm to generate the remaining accompanying conducting vessels, and an empirically based algorithm to generate the supernumerary vessel geometry. The explicit generation of supernumerary vessels is a unique feature of the computational model. Analysis of branching properties and geometric parameters demonstrates close correlation between the model geometry and anatomical measures of human pulmonary blood vessels. A total of 12 Strahler orders for the arterial system and 10 Strahler orders for the venous system are generated, down to the equivalent level of the terminal bronchioles in the bronchial tree. A simple Poiseuille flow solution, assuming rigid vessels, is obtained within the arterial geometry of the left lung, demonstrating a large amount of heterogeneity in the flow distribution, especially with inclusion of supernumerary vessels. This model has been constructed to accurately represent available morphometric data derived from the complex asymmetric branching structure of the human pulmonary vasculature in a form that will be suitable for application in functional simulations.     

Form matters: morphological aspects of lateral root development

Background

The crucial role of roots in plant nutrition, and consequently in plant productivity, is a strong motivation to study the growth and functioning of various aspects of the root system. Numerous studies on lateral roots, as a major determinant of the root system architecture, mostly focus on the physiological and molecular bases of developmental processes. Unfortunately, little attention is paid either to the morphological changes accompanying the formation of a lateral root or to morphological defects occurring in lateral root primordia. The latter are observed in some mutants and occasionally in wild-type plants, but may also result from application of external factors.

Scope and Conclusions

In this review various morphological aspects of lateral branching in roots are analysed. Morphological events occurring during the formation of a typical lateral root are described. This process involves dramatic changes in the geometry of the developing organ that at early stages are associated with oblique cell divisions, leading to breaking of the symmetry of the cell pattern. Several types of defects in the morphology of primordia are indicated and described. Computer simulations show that some of these defects may result from an unstable field of growth rates. Significant changes in both primary and lateral root morphology may also be a consequence of various mutations, some of which are auxin-related. Examples reported in the literature are considered. Finally, lateral root formation is discussed in terms of mechanics. In this approach the primordium is considered as a physical object undergoing deformation and is characterized by specific mechanical properties.