The survival probability of a mutant in a multidimensional population

We prove a general result about the asymptotic behaviour of the survival probability of a slightly supercritical multitype Bienaymé-Galton-Watson branching process. This is the complete analogue of a result which Ewens (1968) obtained for a Poisson branching process.Research supported by NSERC     

Adaptive branching in source-sink habitats

Evolution and ecological diversification in a heterogeneous environment is driven by an often complex interplay between local adaptation and dispersal between different habitat types. Heterogeneous environments also easily generate source-sink dynamics of populations coupled by dispersal. It follows that local adaptation and possible adaptive radiation almost by necessity involves adaptation to a (pseudo-)sink habitat, which is considered unlikely. We here study a model of ‘parapatric branching’ with this special focus on the spatial ecology of the process. We find that evolutionary branching can display a sequence of alternating adaptations to the source or the sink. In some circumstances a true sink can become a pseudo-sink through adaptation to the corresponding source habitat. The evolutionary endpoint is a spatially structured community consisting of two source populations with one corresponding sink or pseudo-sink each. Our results shed new light on the interpretation of extant source-sink systems and the process of parapatric branching.     

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.     

The effective founder effect in a spatially expanding population

The gradual loss of diversity and the establishment of clines in allele frequencies associated with range expansions are patterns observed in many species, including humans. These patterns can result from a series of founder events occurring as populations colonize previously unoccupied areas. We develop a model of an expanding population and, using a branching process approximation, show that spatial gradients reflect different amounts of genetic drift experienced by different subpopulations. We then use this model to measure the net average strength of the founder effect, and we demonstrate that the predictions from the branching process model fit simulation results well. We further show that estimates of the effective founder size are robust to potential confounding factors such as migration between subpopulations. We apply our method to data from Arabidopsis thaliana. We find that the average founder effect is approximately three times larger in the Americas than in Europe, possibly indicating that a more recent, rapid expansion occurred.     

Diameters, generations, and orders of branches in the bronchial tree

Studies of bronchial tree data by West et al. (J. Appl. Physiol. 60: 1089-1097, 1986) have shown that plots of mean diameter against generation, using log-log scales, can be represented by a power function with harmonic modulations. Other studies have shown that the mean diameter of the airways is exponentially related to order of branching. This paper demonstrates that both observations are compatible with a fractal model of branching, and because airway branching is fractal, this may explain why both are also true of the bronchial tree. Furthermore, the exponential relationship of mean diameter with generation in the larger airways, demonstrated by Weibel, is shown to result from the exponential relation of diameter with order in the fractal model.     

Characterisation of a new regulator of BDNF signalling, Sprouty3, involved in axonal morphogenesis in vivo

During development, many organs, including the kidney, lung and mammary gland, need to branch in a regulated manner to be functional. Multicellular branching involves changes in cell shape, proliferation and migration. Axonal branching, however, is a unicellular process that is mediated by changes in cell shape alone and as such appears very different to multicellular branching. Sprouty (Spry) family members are well-characterised negative regulators of Receptor tyrosine kinase (RTK) signalling. Knockout of Spry1, 2 and 4 in mouse result in branching defects in different organs, indicating an important role of RTK signalling in controlling branching pattern. We report here that Spry3, a previously uncharacterised member of the Spry family plays a role in axonal branching. We found that spry3 is expressed specifically in the trigeminal nerve and in spinal motor and sensory neurons in a Brain-derived neurotrophin factor (BDNF)-dependent manner. Knockdown of Spry3 expression causes an excess of axonal branching in spinal cord motoneurons in vivo. Furthermore, Spry3 inhibits the ability of BDNF to induce filopodia in Xenopus spinal cord neurons. Biochemically, we show that Spry3 represses calcium release downstream of BDNF signalling. Altogether, we have found that Spry3 plays an important role in the regulation of axonal branching of motoneurons in vivo, raising the possibility of unexpected conservation in the involvement of intracellular regulators of RTK signalling in multicellular and unicellular branching.     

Tests of a Cellular Model for Constant Branch Distribution in the Filamentous Fungus Neurospora crassa

The growth of mycelial fungi is characterized by the highly polarized extension of hyphal tips and the formation of subapical branches, which themselves extend as new tips. In Neurospora crassa, tip growth and branching are crucial elements for this saprophyte in the colonization and utilization of organic substrates. Much research has focused on the mechanism of tip extension, but a cellular model that fully explains the known phenomenology of branching by N. crassa has not been proposed. We described and tested a model in which the formation of a lateral branch in N. crassa was determined by the accumulation of tip-growth vesicles caused by the excess of the rate of supply over the rate of deposition at the apex. If both rates are proportional to metabolic rate, then the model explains the known lack of dependence of branch interval on growth rate. We tested the model by manipulating the tip extension rate, first by shifting temperature in both the wild type and hyperbranching (colonial) mutants and also by observing the behavior of both tipless colonies and colonyless tips. We found that temperature shifts in either direction result in temporary changes in branching. We found that colonyless tips also pass through a temporary transition phase of branching. The tipless colonies produced a cluster of new tips near the point of damage. We also found that branching in colonial mutants is dependent on growth rate. The results of these tests are consistent with a model of branching in which branch initiation is controlled by the dynamics of tip growth while being independent of the actual rate of this growth.     

Establishment success and extinction risk in autocorrelated environments

We consider establishment success (and extinction risk of small populations) in fluctuating environments, by means of an inhomogeneous branching process model. In this model it is assumed that individuals reproduce asexually during discrete reproduction periods. Within each period individuals reproduce independently and have random numbers of offspring. Expected numbers of offspring vary over reproduction periods due to random environmental changes. Previous simulation results indicated that there is a positive autocorrelation between the establishment probabilities of invaders in successive reproduction periods when environmental states are independently distributed. This result was never formally proved. In this paper we prove that this is indeed true, regardless of the form of the distribution of environmental states or the offspring distribution (under a monotonicity condition, which holds for biologically realistic models). Furthermore, we prove that it is also true for positively autocorrelated environmental states. We show by a counterexample that in environments with a strong negative autocorrelation establishment probabilities can be negatively autocorrelated. This was further examined through simulations. Our results imply that in independent, positively autocorrelated and weakly negatively autocorrelated environments the probability of success of invasion in different independently varying sites is the highest, followed by sequential invasion. For environments with a strong negative autocorrelation, sequential invasion has the highest probability of success. Effects of autocorrelation were further examined with simulations. From the results it appears that the expected length of 'runs of bad luck' is the most crucial factor for establishment success.     

Potential role of RGD-binding integrins in mammalian lung branching morphogenesis.

Cell-matrix interactions are generally considered critical for normal lung development. This is particularly likely to be true during the glandular stage, when the primitive airways are formed through a process termed branching morphogenesis. Integrins, transmembrane receptors that bind to extracellular matrices, are likely to mediate important interactions between embryonic cells and their matrices during branching morphogenesis. In this report, we examine the role of integrin receptors in this process. Immunohistochemical studies revealed that the integrins VLA 3, VLA 5 and integrin receptors to vitronectin are expressed in the epithelium and/or mesenchyme during the glandular stage of murine lung development. To correlate expression with function, an in vitro model of murine lung branching morphogenesis was utilized to examine branching in the presence of inhibitors of ligand binding to integrin receptors. One such reagent, a hexapeptide containing the RGD (Arg-Gly-Asp) sequence, diminished branching and resulted in an abnormal morphology, whereas a control peptide RGESP (Arg-Gly-Glu-Ser-Pro) had no effect. These findings suggest a critical role for cell-matrix interactions mediated via integrin receptors in early stages of mammalian lung development.     

Melatonin Inhibits Embryonic Salivary Gland Branching Morphogenesis by Regulating Both Epithelial Cell Adhesion and Morphology

Many organs, including salivary glands, lung, and kidney, are formed by epithelial branching during embryonic development. Branching morphogenesis occurs via either local outgrowths or the formation of clefts that subdivide epithelia into buds. This process is promoted by various factors, but the mechanism of branching morphogenesis is not fully understood. Here we have defined melatonin as a potential negative regulator or “brake” of branching morphogenesis, shown that the levels of it and its receptors decline when branching morphogenesis begins, and identified the process that it regulates. Melatonin has various physiological functions, including circadian rhythm regulation, free-radical scavenging, and gonadal development. Furthermore, melatonin is present in saliva and may have an important physiological role in the oral cavity. In this study, we found that the melatonin receptor is highly expressed on the acinar epithelium of the embryonic submandibular gland. We also found that exogenous melatonin reduces salivary gland size and inhibits branching morphogenesis. We suggest that this inhibition does not depend on changes in either proliferation or apoptosis, but rather relates to changes in epithelial cell adhesion and morphology. In summary, we have demonstrated a novel function of melatonin in organ formation during embryonic development.     

Evolutionary relationships of eukaryotic kingdoms

The evolutionary relationships of four eukaryotic kingdoms—Animalia, Plantae, Fungi, and Protista—remain unclear. In particular, statistical support for the closeness of animals to fungi rather than to plants is lacking, and a preferred branching order of these and other eukaryotic lineages is still controversial even though molecular sequences from diverse eukaryotic taxa have been analyzed. We report a statistical analysis of 214 sequences of nuclear small-subunit ribosomal RNA (srRNA) gene undertaken to clarify these evolutionary relationships. We have considered the variability of substitution rates and the nonindependence of nucleotide substitution across sites in the srRNA gene in testing alternative hypotheses regarding the branching patterns of eukaryote phylogeny. We find that the rates of evolution among sites in the srRNA sequences vary substantially and are approximately gamma distributed with size and shape parameter equal to 0.76. Our results suggest that (1) the animals and true fungi are indeed closer to each other than to any other crown group in the eukaryote tree, (2) red algae are the closest relatives of animals, true fungi, and green plants, and (3) the heterokonts and alveolates probably evolved prior to the divergence of red algae and animal-fungus-green-plant lineages. Furthermore, our analyses indicate that the branching order of the eukaryotic lineages that diverged prior to the evolution of alveolates may be generally difficult to resolve with the srRNA sequence data.     

The risk of competitive exclusion during evolutionary branching: Effects of resource variability, correlation and autocorrelation

Evolutionary branching has been suggested as a mechanism to explain ecological speciation processes. Recent studies indicate however that demographic stochasticity and environmental fluctuations may prevent branching through stochastic competitive exclusion. Here we extend previous theory in several ways; we use a more mechanistic ecological model, we incorporate environmental fluctuations in a more realistic way and we include environmental autocorrelation in the analysis. We present a single, comprehensible analytical result which summarizes most effects of environmental fluctuations on evolutionary branching driven by resource competition. Corroborating earlier findings, we show that branching may be delayed or impeded if the underlying resources have uncorrelated or negatively correlated responses to environmental fluctuations. There is also a strong impeding effect of positive environmental autocorrelation, which can be related to results from recent experiments on adaptive radiation in bacterial microcosms. In addition, we find that environmental fluctuations can lead to cycles of repeated branching and extinction.     

Mechanisms of Side Branching and Tip Splitting in a Model of Branching Morphogenesis

Recent experimental work in lung morphogenesis has described an elegant pattern of branching phenomena. Two primary forms of branching have been identified: side branching and tip splitting. In our previous study of lung branching morphogenesis, we used a 4 variable partial differential equation (PDE), due to Meinhardt, as our mathematical model to describe the reaction and diffusion of morphogens creating those branched patterns. By altering key parameters in the model, we were able to reproduce all the branching styles and the switch between branching modes. Here, we attempt to explain the branching phenomena described above, as growing out of two fundamental instabilities, one in the longitudinal (growth) direction and the other in the transverse direction. We begin by decoupling the original branching process into two semi-independent sub-processes, 1) a classic activator/inhibitor system along the growing stalk, and 2) the spatial growth of the stalk. We then reduced the full branching model into an activator/inhibitor model that embeds growth of the stalk as a controllable parameter, to explore the mechanisms that determine different branching patterns. We found that, in this model, 1) side branching results from a pattern-formation instability of the activator/inhibitor subsystem in the longitudinal direction. This instability is far from equilibrium, requiring a large inhomogeneity in the initial conditions. It successively creates periodic activator peaks along the growing stalk, each of which later on migrates out and forms a side branch; 2) tip splitting is due to a Turing-style instability along the transversal direction, that creates the spatial splitting of the activator peak into 2 simultaneously-formed peaks at the growing tip, the occurrence of which requires the widening of the growing stalk. Tip splitting is abolished when transversal stalk widening is prevented; 3) when both instabilities are satisfied, tip bifurcation occurs together with side branching.     

Conversion of fibrinogen to fibrin. The correlation between clotting kinetics and morphology of fibrin polymerized in different solvent media

The morphology of fibrin strongly depends on solvent medium, as shown by clotting experiments carried out in the presence of different salts. The clots were characterized by electron microscopy and spectrophotometric methods; the kinetics of gelation were determined. In the presence of electrolytes which strongly delay clotting, the strands are thin and few branching points are observed; opposite morphological changes are induced by salts which act as accelerating agents. On the basis of this correlation, and of previous data on the structure of fibrin, a kinetic model of the self-assembly process is outlined. It accounts well for the observed solvent effects on the morphology of the network. An important result emerging from our experiments is that the fibers undergo branching prior to gelation. Branching points arise from the defective growth of the fibers; a simple explanation of the occurrence of branching may be obtained by our self-assembly model.     

A branching model for the spread of infectious animal diseases in varying environments

This paper is concerned with a stochastic model, describing outbreaks of infectious diseases that have potentially great animal or human health consequences, and which can result in such severe economic losses that immediate sets of measures need to be taken to curb the spread. During an outbreak of such a disease, the environment that the infectious agent experiences is therefore changing due to the subsequent control measures taken. In our model, we introduce a general branching process in a changing (but not random) environment. With this branching process, we estimate the probability of extinction and the expected number of infected individuals for different control measures. We also use this branching process to calculate the generating function of the number of infected individuals at any given moment. The model and methods are designed using important infections of farmed animals, such as classical swine fever, foot-and-mouth disease and avian influenza as motivating examples, but have a wider application, for example to emerging human infections that lead to strict quarantine of cases and suspected cases (e.g. SARS) and contact and movement restrictions.     

Branching in Amyloid Fibril Growth

Using the peptide hormone glucagon and Aβ(1-40) as model systems, we have sought to elucidate the mechanisms by which fibrils grow and multiply. We here present real-time observations of growing fibrils at a single-fibril level. Growing from preformed seeds, glucagon fibrils were able to generate new fibril ends by continuously branching into new fibrils. To our knowledge, this is the first time amyloid fibril branching has been observed in real-time. Glucagon fibrils formed by branching always grew in the forward direction of the parent fibril with a preferred angle of 35-40°. Furthermore, branching never occurred at the tip of the parent fibril. In contrast, in a previous study by some of us, Aβ(1-40) fibrils grew exclusively by elongation of preformed seeds. Fibrillation kinetics in bulk solution were characterized by light scattering. A growth process with branching, or other processes that generate new ends from existing fibrils, should theoretically give rise to different fibrillation kinetics than growth without such a process. We show that the effect of adding seeds should be particularly different in the two cases. Our light-scattering data on glucagon and Aβ(1-40) confirm this theoretical prediction, demonstrating the central role of fibril-dependent nucleation in amyloid fibril growth     

A branching process, its application in biology: influence of demographic parameters on the social structure in mammal groups

Branching processes are widely used in biology. This theoretical tool is used in cell dynamics, epidemics and population dynamics. In population dynamics, branching processes are mainly used to access extinction probabilities of populations, groups or families, with the Galton-Watson branching process. Many mammal species live in socially-structured groups, and the smallest units of these groups are lineages (or families) of kin-related individuals. In many primate species, these lineages are matrilines, as females remain in their natal groups most of the time, whereas males generally disperse. Lineage parameters, such as numbers of matrilines, size of each matriline and average degree of relatedness, could strongly influence the genetic composition of groups. Evidence indicates that division along matrilines could induce substantial differentiation among fission groups. Here, we develop a novel mathematical model based on the branching process theory describing demographic dynamics of groups. The main result of this model is an explicit analytical expression of the joint distribution of numbers of lineages and sizes of socially-structured groups. We investigated the influence of parameters such as natality and mortality on the outcome of the process, including extinction probability. Finally, we discuss this theoretical result with respect to biological significance.     

The interplay of matrix metalloproteinases, morphogens and growth factors is necessary for branching of mammary epithelial cells.

The mammary gland develops its adult form by a process referred to as branching morphogenesis. Many factors have been reported to affect this process. We have used cultured primary mammary epithelial organoids and mammary epithelial cell lines in three-dimensional collagen gels to elucidate which growth factors, matrix metalloproteinases (MMPs) and mammary morphogens interact in branching morphogenesis. Branching stimulated by stromal fibroblasts, epidermal growth factor, fibroblast growth factor 7, fibroblast growth factor 2 and hepatocyte growth factor was strongly reduced by inhibitors of MMPs, indicating the requirement of MMPs for three-dimensional growth involved in morphogenesis. Recombinant stromelysin 1/MMP3 alone was sufficient to drive branching in the absence of growth factors in the organoids. Plasmin also stimulated branching; however, plasmin-dependent branching was abolished by both inhibitors of plasmin and MMPs, suggesting that plasmin activates MMPs. To differentiate between signals for proliferation and morphogenesis, we used a cloned mammary epithelial cell line that lacks epimorphin, an essential mammary morphogen. Both epimorphin and MMPs were required for morphogenesis, but neither was required for epithelial cell proliferation. These results provide direct evidence for a crucial role of MMPs in branching in mammary epithelium and suggest that, in addition to epimorphin, MMP activity is a minimum requirement for branching morphogenesis in the mammary gland.     

Dynamic modeling of branching morphogenesis of ureteric bud in early kidney development

In the early kidney development, a simple epithelial tube called ureteric bud is derived from the intermediate mesoderm and undergoes a complex process of growth and terminal bifid branching. The branching of the ureteric bud is achieved by different cellular behaviors including cell proliferation and chemotaxis. In this paper, we examine how the branching morphology depends on different physical or chemical factors by constructing a cell-based model to describe the simple tube branching in the early kidney development. We conclude that a proper balance between growth speed of epithelial sheet due to cell proliferation and cell mobility due to chemotaxis is necessary to realize the development of normal Y-shaped pattern. When cell proliferation is fast compared to chemotaxis, kinked pattern is formed, and when cell proliferation is slow, bloated pattern is formed. These are consistent with experimental observations in different morphological anomalies of mutants. We show that the different branching patterns are accurately predicted by growth-chemotaxis ratio.     

Branching patterns in phylogenies cannot distinguish diversity‐dependent diversification from time‐dependent diversification

One of the primary goals of macroevolutionary biology has been to explain general trends in long‐term diversity patterns, including whether such patterns correspond to an upscaling of processes occurring at lower scales. Reconstructed phylogenies often show decelerated lineage accumulation over time. This pattern has often been interpreted as the result of diversity‐dependent (DD) diversification, where the accumulation of species causes diversification to decrease through niche filling. However, other processes can also produce such a slowdown, including time dependence without diversity dependence. To test whether phylogenetic branching patterns can be used to distinguish these two mechanisms, we formulated a time‐dependent, but diversity‐independent model that matches the expected diversity through time of a DD model. We simulated phylogenies under each model and studied how well likelihood methods could recover the true diversification mode. Standard model selection criteria always recovered diversity dependence, even when it was not present. We correct for this bias by using a bootstrap method and find that neither model is decisively supported. This implies that the branching pattern of reconstructed trees contains insufficient information to detect the presence or absence of diversity dependence. We advocate that tests encompassing additional data, for example, traits or range distributions, are needed to evaluate how diversity drives macroevolutionary trends.