Additive set functions as linear functionals

We discuss the connection of additive set functions to dual function spaces by means of integrals. Isometrical isomorphisms are presented between: a) bounded contents and the dual of uniformly approximable functions, b) charges on Baire algebras and the duals of bounded, continuous functions over A-topological spaces, c) Hewitt measures on Baire -algebras and the duals of continuous functions over A-topological spaces.Wir behandeln die Verbindung additiver Mengenfunktionen zu dualen Funktionenräumen im Sinne des Integrals. Isometrische Isomorphismen werden vorgestellt zwischen: a) Beschränkten Inhalten und den Dualräumen gleichmäßig approximierbarer Funktionen, b) Ladungen (charges) auf Baire Algebren und den Dualräumen beschränkter, stetiger Funktionen A-topologischer Räume, c) Hewitt Maßen auf Baire σ-Algebren und den Dualräumen stetiger Funktionen Atopologischer Räume

Bacterial diversification through geological time

Numerous studies have estimated plant and animal diversification dynamics; however, no comparable rigorous estimates exist for bacteria—the most ancient and widespread form of life on Earth. Here, we analyse phylogenies comprising up to 448,112 bacterial lineages to reconstruct global bacterial diversification dynamics. To handle such large phylogenies, we developed methods based on the statistical properties of infinitely large trees. We further analysed sequencing data from 60 environmental studies to determine the fraction of extant bacterial diversity missing from the phylogenies—a crucial parameter for estimating speciation and extinction rates. We estimate that there are about 1.4–1.9 million extant bacterial lineages when lineages are defined by 99% similarity in the 16S ribosomal RNA gene, and that bacterial diversity has been continuously increasing over the past 1 billion years (Gyr). Recent bacterial extinction rates are estimated at 0.03–0.05 per lineage per million years (lineage^(–1) Myr^(–1)), and are only slightly below estimated recent bacterial speciation rates. Most bacterial lineages ever to have inhabited this planet are estimated to be extinct. Our findings disprove the notion that bacteria are unlikely to go extinct, and provide a valuable perspective on the evolutionary history of a domain of life with a sparse and cryptic fossil record

Bacterial diversification through geological time

Numerous studies have estimated plant and animal diversification dynamics; however, no comparable rigorous estimates exist for bacteria—the most ancient and widespread form of life on Earth. Here, we analyse phylogenies comprising up to 448,112 bacterial lineages to reconstruct global bacterial diversification dynamics. To handle such large phylogenies, we developed methods based on the statistical properties of infinitely large trees. We further analysed sequencing data from 60 environmental studies to determine the fraction of extant bacterial diversity missing from the phylogenies—a crucial parameter for estimating speciation and extinction rates. We estimate that there are about 1.4–1.9 million extant bacterial lineages when lineages are defined by 99% similarity in the 16S ribosomal RNA gene, and that bacterial diversity has been continuously increasing over the past 1 billion years (Gyr). Recent bacterial extinction rates are estimated at 0.03–0.05 per lineage per million years (lineage^(–1) Myr^(–1)), and are only slightly below estimated recent bacterial speciation rates. Most bacterial lineages ever to have inhabited this planet are estimated to be extinct. Our findings disprove the notion that bacteria are unlikely to go extinct, and provide a valuable perspective on the evolutionary history of a domain of life with a sparse and cryptic fossil record

Moving forward in circles: challenges and opportunities in modelling population cycles

Population cycling is a widespread phenomenon, observed across a multitude of taxa in both laboratory and natural conditions. Historically, the theory associated with population cycles was tightly linked to pairwise consumer–resource interactions and studied via deterministic models, but current empirical and theoretical research reveals a much richer basis for ecological cycles. Stochasticity and seasonality can modulate or create cyclic behaviour in non-intuitive ways, the high-dimensionality in ecological systems can profoundly influence cycling, and so can demographic structure and eco-evolutionary dynamics. An inclusive theory for population cycles, ranging from ecosystem-level to demographic modelling, grounded in observational or experimental data, is therefore necessary to better understand observed cyclical patterns. In turn, by gaining better insight into the drivers of population cycles, we can begin to understand the causes of cycle gain and loss, how biodiversity interacts with population cycling, and how to effectively manage wildly fluctuating populations, all of which are growing domains of ecological research

The ecology of microbial metabolic pathways

Microbial metabolic activity drives biogeochemical cycling in virtually every ecosystem. Yet, microbial ecology and its role in ecosystem biochemistry remain poorly understood, partly because the enormous diversity found in microbial communities hinders their modeling. Despite this diversity, the bulk of global biogeochemical fluxes is driven by a few metabolic pathways encoded by a small set of genes, which through time have spread across microbial clades that can replace each other within metabolic niches. Hence, the question arises whether the dynamics of these pathways can be modeled regardless of the hosting organisms, for example based on environmental conditions. Such a pathway-centric paradigm would greatly simplify the modeling of microbial processes at ecosystem scales. Here I investigate the applicability of a pathway-centric paradigm for microbial ecology. By examining microbial communities in replicate "miniature" aquatic environments, I show that similar ecosystems can exhibit similar metabolic functional community structure, despite highly variable taxonomic composition within individual functional groups. Further, using data from a recent ocean survey I show that environmental conditions strongly explain the distribution of microbial metabolic functional groups across the world's oceans, but only poorly explain the taxonomic composition within individual functional groups. Using statistical tools and mathematical models I conclude that biotic interactions, such as competition and predation, likely underlie much of the taxonomic variation within functional groups observed in the aforementioned studies. The above findings strongly support a pathway-centric paradigm, in which the distribution and activity of microbial metabolic pathways is strongly determined by energetic and stoichiometric constraints, whereas additional mechanisms shape the taxonomic composition within metabolic guilds. These findings motivated me to explore concrete pathway-centric mathematical models for specific ecosystems. Notably, I constructed a biogeochemical model for Saanich Inlet, a seasonally anoxic fjord with biogeochemistry analogous to oxygen minimum zones. The model describes the dynamics of individual microbial metabolic pathways involved in carbon, nitrogen and sulfur cycling, and largely explains geochemical depth profiles as well as DNA, mRNA and protein sequence data. This work yields insight into ocean biogeochemistry and demonstrates the potential of pathway-centric models for microbial ecology.Science, Faculty ofGraduat

Supplemental Code 1

R code demonstrating the use of the flow algorithm for fitting dSSE models to large phylogenetic trees. This example fits a BiSSE model to a dated angiosperm tree and woodiness trait data

Data from: A general and efficient algorithm for the likelihood of diversification and discrete-trait evolutionary models

As the size of phylogenetic trees and comparative data continue to grow and more complex models are developed to investigate the processes that gave rise to them, macroevolutionary analyses are becoming increasingly limited by computational requirements. Here we introduce a novel algorithm, based on the "flow" of the differential equations that describe likelihoods along tree edges in backward time, to reduce redundancy in calculations and efficiently compute the likelihood of various macroevolutionary models. Our algorithm applies to several diversification models, including birth-death models and models that account for state- or time-dependent rates, as well as many commonly used models of discrete-trait evolution, and provides an alternative way to describe macroevolutionary model likelihoods. As a demonstration of our algorithm's utility, we implemented it for a popular class of state-dependent diversification models - BiSSE, MuSSE, and their extensions to hidden-states. Our implementation is available through the R package castor. We show that, for these models, our algorithm is one or more orders of magnitude faster than existing implementations when applied to large phylogenies. Our algorithm thus enables the fitting of state-dependent diversification models to modern massive phylogenies with millions of tips, and may lead to potentially similar computational improvements for many other macroevolutionary models

Trait biases in microbial reference genomes

17 pagesCommon culturing techniques and priorities bias our discovery towards specific traits that may not be representative of microbial diversity in nature. So far, these biases have not been systematically examined. To address this gap, here we use 116,884 publicly available metagenome-assembled genomes (MAGs, completeness ≥80%) from 203 surveys worldwide as a culture-independent sample of bacterial and archaeal diversity, and compare these MAGs to the popular RefSeq genome database, which heavily relies on cultures. We compare the distribution of 12,454 KEGG gene orthologs (used as trait proxies) in the MAGs and RefSeq genomes, while controlling for environment type (ocean, soil, lake, bioreactor, human, and other animals). Using statistical modeling, we then determine the conditional probabilities that a species is represented in RefSeq depending on its genetic repertoire. We find that the majority of examined genes are significantly biased for or against in RefSeq. Our systematic estimates of gene prevalences across bacteria and archaea in nature and gene-specific biases in reference genomes constitutes a resource for addressing these issues in the future

Supplement 4. C++ source code for the numerical simulations of the OUSS model and the population models. This code was also used to construct the CDF grid for the FAP correction.

<h2>File List</h2><div> <p><a href="Supplement4/Makefile">Makefile</a> (MD5: 1571a268f7b017373cf02034235d025d)</p> <p><a href="Supplement4/Readme.rtf">Readme.rtf</a> (MD5: 558b9c308189cdadd7f4c18b746eb18f)</p> <p><a href="Supplement4/source/Auxiliary.cpp">Auxiliary.cpp</a> (MD5: 02eeb96d83fe95532f8ccdda6ced3b9e)</p> <p><a href="Supplement4/source/Config.h">Config.h</a> (MD5: )d511149a4f32261c77b7d8be553ce491</p> <p><a href="Supplement4/source/FitOU.cpp">FitOU.cpp</a> (MD5: )eacb4f61818e9a5717eadf12a5759e3d</p> <p><a href="Supplement4/source/FitOUSS.cpp">FitOUSS.cpp</a> (MD5: )87d50d0ff06f874b68cca35a4ae16bc9</p> <p><a href="Supplement4/source/GompertzGrowthModel.h">GompertzGrowthModel.h</a> (MD5: 245e19cf1649a4ac5423926558a657f2)</p> <p><a href="Supplement4/source/Grid2DInterpolator.cpp">Grid2DInterpolator.cpp</a> (MD5: 4bf646da95188cea56e26d1f81caf589)</p> <p><a href="Supplement4/source/Grid2DInterpolator.h">Grid2DInterpolator.h</a> (MD5: 2cc0f61b0825331b6ba524205e75dfba)</p> <p><a href="Supplement4/source/Grid3DInterpolator.cpp">Grid3DInterpolator.cpp</a> (MD5:48e47b221aae950337526dd17d61e46e )</p> <p><a href="Supplement4/source/Grid3DInterpolator.h">Grid3DInterpolator.h</a> (MD5: ce9d9f631a393b98fe5b2b3ef6d0e24b)</p> <p><a href="Supplement4/source/GridMultilinearInterpolator.cpp">GridMultilinearInterpolator.cpp</a> (MD5: )8d6e62da84f6f1655e737b75249321de</p> <p><a href="Supplement4/source/GridMultilinearInterpolator.h">GridMultilinearInterpolator.h</a> (MD5: c9a326cf6b4ef6accf63cd6cd7b5983d)</p> <p><a href="Supplement4/source/InternalDefs.h">InternalDefs.h</a> (MD5: )e562ebdcb0ce950769b16f7e65f3b1d1</p> <p><a href="Supplement4/source/LogisticGrowthModel.h">LogisticGrowthModel.h</a> (MD5: )dff1b2ca16acb293e01efc2e941f0045</p> <p><a href="Supplement4/source/LombScargleSpectrum.h">LombScargleSpectrum.h</a> (MD5: 9eca3f459dbd96bcd7087184cac5a0fd)</p> <p><a href="Supplement4/source/main.cpp">main.cpp</a> (MD5: 1d24789f3c9589f753e282272597db76)</p> <p><a href="Supplement4/source/OrnsteinUhlenbeckModel.h">OrnsteinUhlenbeckModel.h</a> (MD5: e6bb199e430a68766491e68d5f5b5503)</p> <p><a href="Supplement4/source/PeakSignificance.cpp">PeakSignificance.cpp</a> (MD5: 00904b009f3ea273ce558ac6b9ebde04)</p> <p><a href="Supplement4/source/Points.h">Points.h</a> (MD5: )3ce2099bbbdcc4840aa4bf45d1f1b873</p> <p><a href="Supplement4/source/StochasticRungeKutta.h">StochasticRungeKutta.h</a> (MD5: )78e274a281beb35ed8617309a61de347</p> <p><a href="Supplement4/source/VectorArithmetics.h">VectorArithmetics.h</a> (MD5: f06222ddadecbb6ec5d0d3a7388b5f06)</p> <p><a href="Supplement4/source/STPlot/Makefile">STPlot/Makefile</a> (MD5: )</p> <p><a href="Supplement4/source/STPlot/sources/STColor.cpp">STColor.cpp</a> (MD5: f06222ddadecbb6ec5d0d3a7388b5f06)</p> <p><a href="Supplement4/source/STPlot/sources/STColor.h">STColor.h</a> (MD5: d2fbe6d7067fe383c9c67d3fb59289fa)</p> <p><a href="Supplement4/source/STPlot/sources/STPipe.cpp">STPipe.cpp</a> (MD5: da5e65bb44f7edcb6917e0b236bdcd5b)</p> <p><a href="Supplement4/source/STPlot/sources/STPipe.h">STPipe.h</a> (MD5: f0f692f8c8e08400ae9cace9fde76874)</p> <p><a href="Supplement4/source/STPlot/sources/STPlot.cpp">STPlot.cpp</a> (MD5: 2ce77e64526b46048926e2a293e5a79b)</p> <p><a href="Supplement4/source/STPlot/sources/STPlot.h">STPlot.h</a> (MD5: f783a60895433f031f9f5d388340126d)</p> <p><a href="Supplement4/source/STPlot/sources/STPlotDefaults.h">STPlotDefaults.h</a> (MD5: bc6bad182a2ad1ba233b288d4f72c8c7)</p> </div><h2>Description</h2><div> <p>C++ source code for the numerical simulations of the OUSS model and the population models. This code was also used to construct the CDF grid for the FAP correction. You will need a C++ compiler (e.g., GCC) to compile the code, as well as the ALGLIB library (free C++ version). The latter is needed for the L-BFGS optimization algorithm.</p> <p>The Makefile contains all necessary commands for compilation using GCC. Use "make" in your terminal/command line to execute it. More details are given in the Readme.rtf file as well as in the source files.</p> </div