Properties of phylogenetic trees generated by Yule-type speciation models

We investigate some discrete structural properties of evolutionary trees generated under simple null models of speciation, such as the Yule model. These models have been used as priors in Bayesian approaches to phylogenetic analysis, and also to test hypotheses concerning the speciation process. In this paper we describe new results for three properties of trees generated under such models. Firstly, for a rooted tree generated by the Yule model we describe the probability distribution on the depth (number of edges from the root) of the most recent common ancestor of a random subset of k species. Next we show that, for trees generated under the Yule model, the approximate position of the root can be estimated from the associated unrooted tree, even for trees with a large number of leaves. Finally, we analyse a biologically motivated extension of the Yule model and describe its distribution on tree shapes when speciation occurs in rapid bursts.     

Partitioning of root and shoot competition and the stability of savannas

A classic problem in coexistence theory is how grasses and trees coexist in savannas. A popular deterministic model of savannas, the rooting niche separation model, is based on an assumption that is not empirically supported in many savannas. Alternative models that do not rely on the rooting niche assumption invoke intricate stochastic mechanisms that limit their attractiveness as general models of savannas. In this article we develop an alternative deterministic model of grass-tree interactions and use it to analyze the conditions under which grass-tree coexistence is possible. The novel feature of this model is that it partitions aboveground and belowground competition and simulates the fact that fire and herbivory remove only aboveground biomass. The model predicts that stable coexistence of grasses and trees is possible, even when grasses and trees do not have separate rooting niches. We show that when aboveground competition is intense, grasses can be excluded by trees; under such conditions, fire can prevent grasses from exclusion and induce a stable savanna state. The model provides a general framework for exploring the interactive effects of competition, herbivory, and fire on savanna systems.     

Revisiting the Two-Layer Hypothesis: Coexistence of Alternative Functional Rooting Strategies in Savannas

The two-layer hypothesis of tree-grass coexistence posits that trees and grasses differ in rooting depth, with grasses exploiting soil moisture in shallow layers while trees have exclusive access to deep water. The lack of clear differences in maximum rooting depth between these two functional groups, however, has caused this model to fall out of favor. The alternative model, the demographic bottleneck hypothesis, suggests that trees and grasses occupy overlapping rooting niches, and that stochastic events such as fires and droughts result in episodic tree mortality at various life stages, thus preventing trees from otherwise displacing grasses, at least in mesic savannas. Two potential problems with this view are: 1) we lack data on functional rooting profiles in trees and grasses, and these profiles are not necessarily reflected by differences in maximum or physical rooting depth, and 2) subtle, difficult-to-detect differences in rooting profiles between the two functional groups may be sufficient to result in coexistence in many situations. To tackle this question, I coupled a plant uptake model with a soil moisture dynamics model to explore the environmental conditions under which functional rooting profiles with equal rooting depth but different depth distributions (i.e., shapes) can coexist when competing for water. I show that, as long as rainfall inputs are stochastic, coexistence based on rooting differences is viable under a wide range of conditions, even when these differences are subtle. The results also indicate that coexistence mechanisms based on rooting niche differentiation are more viable under some climatic and edaphic conditions than others. This suggests that the two-layer model is both viable and stochastic in nature, and that a full understanding of tree-grass coexistence and dynamics may require incorporating fine-scale rooting differences between these functional groups and realistic stochastic climate drivers into future models.     

Theoretical relationship between the optimal models of the vascular tree

Two models of optimal branching structure of the vascular tree are compared. Murray’s minimum work model derived from minimum energy loss due to flow and volume in the duct system is proved to be included as a mathematical group in the authors’ model defined by the minimum volume under determinant pressure, flow and position at the terminals. The problem about heterotypical trees which are identical at the terminal conditions but different in the topological order of branch combinations are discussed, applying the results of analyses on the equivalent duct of uniform terminal pressure trees. It is proved that the minimum work tree has the least energy loss compared with its heterotypical minimum volume trees and is a better model of branching structure of the vascular tree.     

Spontaneous Patchiness in a Host-Parasitoid Integrodifference Model

Observations of a host-parasitoid interaction in which victims are significantly less motile than their exploiters suggest the possibility of stable spatial pattern in a fairly homogeneous environment. Findings of pattern formation in continuous-time models are not fully able to account for this behavior. Those findings often rely on questionable biological conditions, and more fundamentally, the continuous nature of time in such models does not reflect the reality of the observed interaction. In this paper, we introduce a discrete-time spatial model of the interaction. The final state of our model is often a striking spatial pattern, similar to those observed. We analyze the model, describe its transient behavior, and find the conditions under which these spatial patterns occur, as well as an estimate of maximum possible patch size under those conditions. We also discuss the existence of such conditions in the natural system.     

The Algebra of the General Markov Model on Phylogenetic Trees and Networks

It is known that the Kimura 3ST model of sequence evolution on phylogenetic trees can be extended quite naturally to arbitrary split systems. However, this extension relies heavily on mathematical peculiarities of the associated Hadamard transformation, and providing an analogous augmentation of the general Markov model has thus far been elusive. In this paper, we rectify this shortcoming by showing how to extend the general Markov model on trees to include incompatible edges; and even further to more general network models. This is achieved by exploring the algebra of the generators of the continuous-time Markov chain together with the “splitting” operator that generates the branching process on phylogenetic trees. For simplicity, we proceed by discussing the two state case and then show that our results are easily extended to more states with little complication. Intriguingly, upon restriction of the two state general Markov model to the parameter space of the binary symmetric model, our extension is indistinguishable from the Hadamard approach only on trees; as soon as any incompatible splits are introduced the two approaches give rise to differing probability distributions with disparate structure. Through exploration of a simple example, we give an argument that our extension to more general networks has desirable properties that the previous approaches do not share. In particular, our construction allows for convergent evolution of previously divergent lineages; a property that is of significant interest for biological applications.     

The growth,activity and distribution of the fruit tree root system

Summary The paper reviews information, much of it obtained from studies using the East Malling root observation laboratories, on the growth and development of the fruit tree root system. The production of new white root varies from year-to-year, generally being highest in the early years. As trees age, woody roots constitute an increasing fraction of total root length although the contribution made by new root growth to the total root length of established trees is also affected by soil management, being higher for trees under grass than under herbicide. Soil management also affects the balance of short (lateral) to long (extension) roots; under grass there are more lateral roots.Calculation of the rate of water uptake per unit root length needed at various times in the year to meet transpirational demand, suggests that woody roots, which recent experimental work has shown to be capable of absorbing water, must be responsible for much of total water supply.Measurements of VA mycorrhizal infection in field-grown trees indicated, for part of the season, higher per cent infection in trees grown under irrigated grass than under herbicide management. It is suggested that this, which is associated with raised leaf phosphorus levels, may be due at least partly to higher numbers of lateral roots, the root type which becomes infected. The growth and functioning of the root system under field conditions depend upon the production and integration of a range of root types.     

Growth–mortality relationships as indicators of life-history strategies: a comparison of nine tree species in unmanaged European forests

Forest succession depends strongly on the life history strategies of individual trees. An important strategic element is the ability to survive unfavourable environmental conditions that result in strongly reduced tree growth. In this study, we investigated whether the relationship between growth and mortality differs among tree species and site conditions. We analysed 10 329 trees of nine tree species (Picea abies , Taxus baccata , Fagus sylvatica , Tilia cordata , Carpinus betulus , Fraxinus excelsior , Quercus robur , Betula spp. and Alnus glutinosa ) from unmanaged forests of Europe: the continental Białowieża forest (Poland) and several oceanically influenced Swiss forest reserves. For each species, we calculated a set of flexible logistic regression models with the explanatory variables growth (as measured by relative basal area increment), tree size and site. We selected the species-specific model with the highest goodness-of-fit and calculated its discriminatory power (area under the receiver operating characteristic curve, AUC) and calibration measures. Most models achieved at least a good discriminatory power (AUC>0.7) and the AUC ranged from 0.62 to 0.87; calibration curves did not indicate any overfitting. Almost all growth–mortality relationships differed among species and sites, i.e. there is no universal growth–mortality relationship. Some species such as F. excelsior showed reduced survival probabilities for both unfavourable and very good growth conditions. We conclude that the growth–mortality relationships presented here can contribute to the life-history classification of trees and that they should also help to improve projections of forest succession models.     

Tree seasonality in a warming climate

Climate warming has increased researchers' interest in plant phenology and its modelling. Although the main focus is on projections of accelerated springtime phenological events, also a further extension of the growing season by delayed growth cessation is often projected. However, ecophysiological studies indicate that, for boreal and temperate trees, such generalisations are precluded owing to differential climatic conditions and inter- and intraspecific genetic differences. The annual cycle of these trees is an integrated system, where one phase affects subsequent phases, resulting in delayed impacts, which are only partially addressed in current ecophysiological models. Here, we outline an updated integrated conceptual model of the annual cycle by identifying ecophysiological phenomena that are particularly significant under climate warming.     

Approaches for missing covariate data in logistic regression with MNAR sensitivity analyses

Data with missing covariate values but fully observed binary outcomes are an important subset of the missing data challenge. Common approaches are complete case analysis (CCA) and multiple imputation (MI). While CCA relies on missing completely at random (MCAR), MI usually relies on a missing at random (MAR) assumption to produce unbiased results. For MI involving logistic regression models, it is also important to consider several missing not at random (MNAR) conditions under which CCA is asymptotically unbiased and, as we show, MI is also valid in some cases. We use a data application and simulation study to compare the performance of several machine learning and parametric MI methods under a fully conditional specification framework (MI-FCS). Our simulation includes five scenarios involving MCAR, MAR, and MNAR under predictable and nonpredictable conditions, where “predictable” indicates missingness is not associated with the outcome. We build on previous results in the literature to show MI and CCA can both produce unbiased results under more conditions than some analysts may realize. When both approaches were valid, we found that MI-FCS was at least as good as CCA in terms of estimated bias and coverage, and was superior when missingness involved a categorical covariate. We also demonstrate how MNAR sensitivity analysis can build confidence that unbiased results were obtained, including under MNAR-predictable, when CCA and MI are both valid. Since the missingness mechanism cannot be identified from observed data, investigators should compare results from MI and CCA when both are plausibly valid, followed by MNAR sensitivity analysis.     

Lifespan Extension in a Semelparous Chordate Occurs via Developmental Growth Arrest Just Prior to Meiotic Entry

It is proposed that the ageing process is linked to signaling from the germline such that the rate of ageing can be adjusted to the state of the reproductive system, allowing these two processes to co-evolve. Mechanistic insight into this link has been primarily derived from iteroparous reproductive models, the nematode C. elegans, and the arthropod Drosophila. Here, we examined to what extent these mechanisms are evolutionarily conserved in a semelparous chordate, Oikopleura dioica, where we identify a developmental growth arrest (GA) in response to crowded, diet-restricted conditions, which can extend its lifespan at least three-fold. Under nutritional stress, the iteroparative models sacrifice germ cells that have entered meiosis, while maintaining a reduced pool of active germline stem cells (GSCs). In contrast, O. dioica only entered GA prior to meiotic entry. Stress conditions encountered after this point led to maturation in a normal time frame but with reduced reproductive output. During GA, TOR signaling was inhibited, whereas MAPK, ERK1/2 and p38 pathways were activated, and under such conditions, activation of these pathways was shown to be critical for survival. Direct inhibition of TOR signaling alone was sufficient to prevent meiotic entry and germline differentiation. This inhibition activated the p38 pathway, but did not activate the ERK1/2 pathway. Thus, the link between reproductive status and lifespan extension in response to nutrient-limited conditions is interpreted in a significantly different manner in these iteroparative versus semelparous models. In the latter case, meiotic entry is a definitive signal that lifespan extension can no longer occur, whereas in the former, meiotic entry is not a unique chronological event, and can be largely erased during lifespan extension in response to nutrient stress, and reactivated from a pool of maintained GSCs when conditions improve.     

Mathematical Models of Localized Muscle Fatigue: Sensitivity Analysis and Assessment of Two Occupationally-Relevant Models

Muscle fatigue models (MFM) have broad potential application if they can accurately predict muscle capacity and/or endurance time during the execution of diverse tasks. As an initial step toward facilitating improved MFMs, we assessed the sensitivity of selected existing models to their inherent parameters, specifically that model the fatigue and recovery processes, and the accuracy of model predictions. These evaluations were completed for both prolonged and intermittent isometric contractions, and were based on model predictions of endurance times. Based on a recent review of the literature, four MFMs were initially chosen, from which a preliminary assessment led to two of these being considered for more comprehensive evaluation. Both models had a higher sensitivity to their fatigue parameter. Predictions of both models were also more sensitive to the alteration of their parameters in conditions involving lower to moderate levels of effort, though such conditions may be of most practical, contemporary interest or relevance. Although both models yielded accurate predictions of endurance times during prolonged contractions, their predictive ability was inferior for more complex (intermittent) conditions. When optimizing model parameters for different loading conditions, the recovery parameter showed considerably larger variability, which might be related to the inability of these MFMs in simulating the recovery process under different loading conditions. It is argued that such models may benefit in future work from improving their representation of recovery process, particularly how this process differs across loading conditions.     

Habitat modeling with single- and multi-target trees and ensembles

Habitat modeling studies the influence of abiotic factors on the abundance of a given taxonomic group of organisms. In this work, we investigate the effect of environmental conditions on communities of organisms in three different ecosystems. Namely, we consider the diatom community in Lake Prespa, Macedonia, the Collembola community in the soils of Denmark and 14 organisms living in Slovenian rivers. The data for these case studies consist of physical and chemical properties of the environment as well as the relative abundances or presence of the organisms under investigation.The multi-species data are analyzed by constructing habitat models for each species separately (single-target decision trees) or by constructing a single habitat model for all the species (multi-target predictive clustering trees). Typically, habitat models are constructed for each species individually and thus do not exploit the interactions between/among species. While approaches for building a single habitat model of a group of organisms exist, they typically construct models that are not readily interpretable and, thus, are seldom used by the research community. In this work, we explore in detail the construction of interpretable models of both types. Furthermore, we construct ensembles of decision trees and ensembles of predictive clustering trees to increase the predictive performance of the models.The key outcomes of the interpretation and discussion of the obtained models for each case study are as follows. First, we show that multi-target predictive clustering trees are a very useful method for the analysis of multi-species data and that they are more efficient and produce more concise models than single-target decision trees. The obtained multi-target habitat models are readily interpretable and identify the environmental conditions that influence the composition and structure of a given community of organisms. Second, we conclude that the temperature and magnesium are the most important factors influencing the complete diatom community in Lake Prespa, while the nitrates and the temperature influence more the most abundant species. Third, the biological oxygen demand is the most influential factor for the abundance of river dwelling species, while the river community structure is mostly influenced by the NO₂ concentration. Finally, the structure of the community of soil microarthropods is mostly influenced by the soil type and the crop history.     

Heterogeneous perfusion is a consequence of uniform shear stress in optimized arterial tree models

Using optimized computer models of arterial trees we demonstrate that flow heterogeneity is a necessary consequence of a uniform shear stress distribution. Model trees are generated and optimized under different modes of boundary conditions. In one mode flow is delivered to the tissue as homogeneously as possible. Although this primary goal can be achieved, resulting shear stresses between blood and the vessel walls show very large spread. In a second mode, models are optimized under the condition of uniform shear stress in all segments which in turn renders flow distribution heterogeneous. Both homogeneous perfusion and uniform shear stress are desirable goals in real arterial trees but each of these goals can only be approached at the expense of the other. While the present paper refers only to optimized models, we assume that this dual relation between the heterogeneities in flow and shear stress may represent a more general principle of vascular systems.     

Empirical phylogenies and species abundance distributions are consistent with preequilibrium dynamics of neutral community models with gene flow

Community characteristics reflect past ecological and evolutionary dynamics. Here, we investigate whether it is possible to obtain realistically shaped modeled communities–that is with phylogenetic trees and species abundance distributions shaped similarly to typical empirical bird and mammal communities–from neutral community models. To test the effect of gene flow, we contrasted two spatially explicit individual‐based neutral models: one with protracted speciation, delayed by gene flow, and one with point mutation speciation, unaffected by gene flow. The former produced more realistic communities (shape of phylogenetic tree and species‐abundance distribution), consistent with gene flow being a key process in macro‐evolutionary dynamics. Earlier models struggled to capture the empirically observed branching tempo in phylogenetic trees, as measured by the gamma statistic. We show that the low gamma values typical of empirical trees can be obtained in models with protracted speciation, in preequilibrium communities developing from an initially abundant and widespread species. This was even more so in communities sampled incompletely, particularly if the unknown species are the youngest. Overall, our results demonstrate that the characteristics of empirical communities that we have studied can, to a large extent, be explained through a purely neutral model under preequilibrium conditions.     

A mixed branch length model of heterotachy improves phylogenetic accuracy

Evolutionary relationships are typically inferred from molecular sequence data using a statistical model of the evolutionary process. When the model accurately reflects the underlying process, probabilistic phylogenetic methods recover the correct relationships with high accuracy. There is ample evidence, however, that models commonly used today do not adequately reflect real-world evolutionary dynamics. Virtually all contemporary models assume that relatively fast-evolving sites are fast across the entire tree, whereas slower sites always evolve at relatively slower rates. Many molecular sequences, however, exhibit site-specific changes in evolutionary rates, called "heterotachy." Here we examine the accuracy of 2 phylogenetic methods for incorporating heterotachy, the mixed branch length model--which incorporates site-specific rate changes by summing likelihoods over multiple sets of branch lengths on the same tree--and the covarion model, which uses a hidden Markov process to allow sites to switch between variable and invariable as they evolve. Under a variety of simple heterogeneous simulation conditions, the mixed model was dramatically more accurate than homotachous models, which were subject to topological biases as well as biases in branch length estimates. When data were simulated with strong versions of the types of heterotachy observed in real molecular sequences, the mixed branch length model was more accurate than homotachous techniques. Analyses of empirical data sets confirmed that the mixed branch length model can improve phylogenetic accuracy under conditions that cause homotachous models to fail. In contrast, the covarion model did not improve phylogenetic accuracy compared with homotachous models and was sometimes substantially less accurate. We conclude that a mixed branch length approach, although not the solution to all phylogenetic errors, is a valuable strategy for improving the accuracy of inferred trees.     

Root location in random trees: A polarity property of all sampling consistent phylogenetic models except one

Neutral macroevolutionary models, such as the Yule model, give rise to a probability distribution on the set of discrete rooted binary trees over a given leaf set. Such models can provide a signal as to the approximate location of the root when only the unrooted phylogenetic tree is known, and this signal becomes relatively more significant as the number of leaves grows. In this short note, we show that among models that treat all taxa equally, and are sampling consistent (i.e. the distribution on trees is not affected by taxa yet to be included), all such models, except one (the so-called PDA model), convey some information as to the location of the ancestral root in an unrooted tree.     

Incorporating variability in simulations of seasonally forced phenology using integral projection models

Phenology models are becoming increasingly important tools to accurately predict how climate change will impact the life histories of organisms. We propose a class of integral projection phenology models derived from stochastic individual‐based models of insect development and demography. Our derivation, which is based on the rate summation concept, produces integral projection models that capture the effect of phenotypic rate variability on insect phenology, but which are typically more computationally frugal than equivalent individual‐based phenology models. We demonstrate our approach using a temperature‐dependent model of the demography of the mountain pine beetle (Dendroctonus ponderosae Hopkins), an insect that kills mature pine trees. This work illustrates how a wide range of stochastic phenology models can be reformulated as integral projection models. Due to their computational efficiency, these integral projection models are suitable for deployment in large‐scale simulations, such as studies of altered pest distributions under climate change.     

A basic limitation on inferring phylogenies by pairwise sequence comparisons

Distance-based approaches in phylogenetics such as Neighbor-Joining are a fast and popular approach for building trees. These methods take pairs of sequences, and from them construct a value that, in expectation, is additive under a stochastic model of site substitution. Most models assume a distribution of rates across sites, often based on a gamma distribution. Provided the (shape) parameter of this distribution is known, the method can correctly reconstruct the tree. However, if the shape parameter is not known then we show that topologically different trees, with different shape parameters and associated positive branch lengths, can lead to exactly matching distributions on pairwise site patterns between all pairs of taxa. Thus, one could not distinguish between the two trees using pairs of sequences without some prior knowledge of the shape parameter. More surprisingly, this can happen for any choice of distinct shape parameters on the two trees, and thus the result is not peculiar to a particular or contrived selection of the shape parameters. On a positive note, we point out known conditions where identifiability can be restored (namely, when the branch lengths are clocklike, or if methods such as maximum likelihood are used).     

Local Diameter Fully Constrains Dendritic Size in Basal but not Apical Trees of CA1 Pyramidal Neurons

Computational modeling of dendritic morphology is a powerful tool for quantitatively describing complex geometrical relationships, uncovering principles of dendritic development, and synthesizing virtual neurons to systematically investigate cellular biophysics and network dynamics. A feature common to many morphological models is a dependence of the branching probability on local diameter. Previous models of this type have been able to recreate a wide variety of dendritic morphologies. However, these diameter-dependent models have so far failed to properly constrain branching when applied to hippocampal CA1 pyramidal cells, leading to explosive growth. Here we present a simple modification of this basic approach, in which all parameter sampling, not just bifurcation probability, depends on branch diameter. This added constraint prevents explosive growth in both apical and basal trees of simulated CA1 neurons, yielding arborizations with average numbers and patterns of bifurcations extremely close to those observed in real cells. However, simulated apical trees are much more varied in size than the corresponding real dendrites. We show that, in this model, the excessive variability of simulated trees is a direct consequence of the natural variability of diameter changes at and between bifurcations observed in apical, but not basal, dendrites. Conversely, some aspects of branch distribution were better matched by virtual apical trees than by virtual basal trees. Dendritic morphometrics related to spatial position, such as path distance from the soma or branch order, may be necessary to fully constrain CA1 apical tree size and basal branching pattern.