Asymptotic rates of growth of the extinction probability of a mutant gene

We prove that a result of Haldane (1927) that relates the asymptotic behaviour of the extinction probability of a slightly supercritical Poisson branching process to the mean number of offspring is true for a general Bienaymé-Galton-Watson branching process, provided that the second derivatives of the probability-generating functions converge uniformly to a non-zero limit. We show also by examples that such a result is true more widely than our proof suggests and exhibit some counter-examples.Research supported by NSERC     

On the Genealogy of a Rare Allele

The gene genealogy is derived for a rare allele that is descended from a mutant ancestor that arose at a fixed time in the past. Following Thompson (1976,Amer. J. Human Genet.28, 442–452), the fractional linear branching process is used as a model of the demography of a rare allele. The model does not require the total population size to be constant or the mutant class to be neutral; so long as individuals in the class are selectively equivalent, the class as a whole may have a selective advantage, or disadvantage, relative to other alleles in the population. An exact result is given for the joint probability distribution of the coalescence times among a sample of alleles descended from the mutant. A method is described for rapidly simulating these coalescence times. The relationship between the genealogical structure of a discrete generation branching process and a continuous generation birth–death process is elucidated. The theory may be applied to the problem of estimating the ages of rare nonrecurrent mutations.     

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.     

On the survival probability of a slightly advantageous mutant gene with a general distribution of progeny size—A branching process model

A branching process method is employed to study the survival probability of a slightly advantageous mutant gene with a general distribution of progeny size in a large population. A counter-example to a classic proposition is given. A somewhat weaker result is proved.Supported in part by NIH Grant 5R01 GM10452-18     

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.     

On the survival probability of a slightly advantageous mutant gene in a multitype population: A multidimensional branching process model

The estimated survival probability of a slightly supercritical Galton-Watson process is generalized to a multitype branching process. The result is used to estimate the probability of initial success of a mutant gene whose effect on the individual carrier depends on the carrier's sex, class, etc. The probability of initial success is also estimated in a case where the effect of the mutation is manifested in terms of the distribution of types within one's progeny, e.g. in a case of a change in the sex ratio.     

An application of a general branching process in the study of the genetics of aging.

A general branching process model is developed to analyse familial dependence in longevity data. A general formula for the survival function of a randomly chosen sibling of an individual of a specified age is derived. The branching process model takes into account that siblings' ages may be censored. This is applied to a data set consisting of lifelengths of siblings of centenarians. Age distributions used in the branching process model are estimated from US Census data from the relevant period. It is demonstrated that there is a marked difference in the survival function according to the formula assuming no familial effect and the empirical survival function estimated from the data; thus, indicating a strong familial component.     

Hypothesis: A New Role for the Renin-Angiotensin System in Ureteric Bud Branching

Branching morphogenesis in the developing mammalian kidney involves growth and branching of the ureteric bud (UB), leading to formation of its daughter collecting ducts, calyces, pelvis and ureters. Even subtle defects in the efficiency and/or accuracy of this process have profound effects on the ultimate development of the kidney and result in congenital abnormalities of the kidney and urinary tract. This review summarizes current knowledge regarding a number of genes known to regulate UB development and emphasizes an emerging role for the renin-angiotensin system (RAS) in renal branching morphogenesis. Mutations in the genes encoding components of the RAS in mice cause renal papillary hypoplasia, hydronephrosis, and urinary concentrating defect. These findings imply that UB-derived epithelia are targets for angiotensin (ANG) II actions during metanephric kidney development. Here, it is proposed that papillary hypoplasia in RAS-deficient mice is secondary to an intrinsic defect in the development of the renal medulla. This hypothesis is based on the following observations: (a) UB and surrounding stroma express angiotensinogen (AGT) and ANG II AT₁ receptors in vivo; (b) ANG II stimulates UB cell process extension, branching and cord formation in collagen gel cultures in vitro; and (c) AT₁ blockade inhibits ANG II-induced UB cell branching. It is further postulated that ANG II is a novel stroma-derived factor involved in stroma/UB cross-talk which regulates UB branching morphogenesis.Key Words: kidney development, branching morphogenesis, renin-angiotensin, stromal mesenchyme, ureteric bud     

A logistic branching process for population genetics

A logistic (regulated population size) branching process population genetic model is presented. It is a modification of both the Wright-Fisher and (unconstrained) branching process models, and shares several properties including the coalescent time and shape, and structure of the coalescent process with those models. An important feature of the model is that population size fluctuation and regulation are intrinsic to the model rather than externally imposed. A consequence of this model is that the fluctuation in population size enhances the prospects for fixation of a beneficial mutation with constant relative viability, which is contrary to a result for the Wright-Fisher model with fluctuating population size. Explanation of this result follows from distinguishing between expected and realized viabilities, in addition to the contrast between absolute and relative viabilities.     

Geographical distribution of a neutral allele considered as a branching process

The limiting spacial correlations are derived for a population of neutral alleles migrating among K colonies. The allelic population is modeled as a subcritical branching process and the limiting correlations are obtained conditional on nonextinction of the population.     

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.     

Hypothesis: A New Role for the Renin-Angiotensin System in Ureteric Bud Branching

Branching morphogenesis in the developing mammalian kidney involves growth and branching of the ureteric bud (UB), leading to formation of its daughter collecting ducts, calyces, pelvis and ureters. Even subtle defects in the efficiency and/or accuracy of this process have profound effects on the ultimate development of the kidney and result in congenital abnormalities of the kidney and urinary tract. This review summarizes current knowledge regarding a number of genes known to regulate UB development and emphasizes an emerging role for the renin-angiotensin system (RAS) in renal branching morphogenesis. Mutations in the genes encoding components of the RAS in mice cause renal papillary hypoplasia, hydronephrosis, and urinary concentrating defect. These findings imply that UB-derived epithelia are targets for angiotensin (ANG) II actions during metanephric kidney development. Here, it is proposed that papillary hypoplasia in RAS-deficient mice is secondary to an intrinsic defect in the development of the renal medulla. This hypothesis is based on the following observations: a) UB and surrounding stroma express angiotensinogen (AGT) and ANG II AT1 receptors in vivo; b) ANG II stimulates UB cell process extension, branching and cord formation in collagen gel cultures in vitro; and c) AT1 blockade inhibits ANG II-induced UB cell branching. It is further postulated that ANG II is a novel stroma-derived factor involved in stroma/UB cross-talk which regulates UB branching morphogenesis.     

On a periodic-like behavior of a delayed density-dependent branching process

Most of natural populations seem to be regulated in their sizes in complex ways. Particularly, the sizes of some populations change in time or generation roughly periodically. There are many theoretical studies on such population dynamics. This paper develops stochastic population models for a periodic-like population dynamics. To see the nature of such mechanism, we consider simple models of a delayed density-dependent branching process, and present by numerical simulations how such a branching process shows periodic population changes. The effects of randomly changing stationary environments on the population dynamics are also considered.     

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.     

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.     

A Reconsideration of Galton's Problem (Using a Two-Sex Population)

The main purposes of this paper are to promote and expound the bisexual Galton–Watson branching process as a relevant model for the consideration of Francis Galton's problem regarding the extinction of surnames of “men of note.” A scheme for adapting the bisexual process to consider Galton's problem is introduced. A necessary and sufficient condition for the certain extinction of a male-induced property in a two-sex species is presented. An approach for calculating the extinction of a male-generated characteristic in the two-sex species is proposed. That approach is then used to find the probability of the extinction of surnames in a bisexual population for Alfred Lotka's data based on an United States Census. Finally, these results are then compared with the classic extinction probabilities (from Lotka) associated with the traditional Galton–Watson branching process using asexual reproduction.     

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.     

Strigolactones,a novel class of plant hormone controlling shoot branching

For several decades, auxin and cytokinin were the only two hormones known to be involved in the control of shoot branching through apical dominance, a process where the shoot apex producing auxin inhibits the outgrowth of axillary buds located below. Grafting studies with high branching mutants and cloning of the mutated genes demonstrated the existence of a novel long distance carotenoid derived signal which acted as a branching inhibitor. Recently, this branching inhibitor has been shown to belong to the strigolactones, a group of small molecules already known to be produced by roots, exuded in the rhizosphere and as having a role in both parasitic and symbiotic interactions.     

Ecological specialization of mixotrophic plankton in a mixed water column

In recent years, the population dynamics of plankton in light- or nutrient-limited environments have been studied extensively. Their evolutionary dynamics, however, have received much less attention. Here, we used a modeling approach to study the evolutionary behavior of a population of plankton living in a mixed water column. Initially, the organisms are mixotrophic and thus have both autotrophic and heterotrophic abilities. Through evolution of their trophic preferences, however, they can specialize into separate autotrophs and heterotrophs. It was found that the light intensity gradient enables evolutionary branching and thus may result in the ecological specialization of the mixotrophs. By affecting the gradient, other environmental properties also acquire influence on this evolutionary process. Intermediate mixing intensities, large mixing depths, and high nutrient densities were found to facilitate evolutionary branching and thus specialization. Later results may explain why mixotrophs are often more dominant in oligotrophic systems while specialist strategies are associated with eutrophic systems.