Estimating and comparing microbial diversity in the presence of sequencing errors

Estimating and comparing microbial diversity are statistically challenging due to limited sampling and possible sequencing errors for low-frequency counts, producing spurious singletons. The inflated singleton count seriously affects statistical analysis and inferences about microbial diversity. Previous statistical approaches to tackle the sequencing errors generally require different parametric assumptions about the sampling model or about the functional form of frequency counts. Different parametric assumptions may lead to drastically different diversity estimates. We focus on nonparametric methods which are universally valid for all parametric assumptions and can be used to compare diversity across communities. We develop here for the first time a nonparametric estimator of the true singleton count to replace the spurious singleton count. Our estimator of the true singleton count is in terms of the frequency counts of doubletons, tripletons and quadrupletons. To quantify microbial diversity, we adopt the measure of Hill numbers (effective number of taxa) under a nonparametric framework. Hill numbers, parameterized by an order q that determines the measures’ emphasis on rare or common species, include taxa richness (q=0), Shannon diversity (q=1), and Simpson diversity (q=2). Based on the estimated singleton count and the original non-singleton frequency counts, two statistical approaches are developed to compare microbial diversity for multiple communities. (1) A non-asymptotic approach based on standardizing sample size or sample completeness via seamless rarefaction and extrapolation sampling curves of Hill numbers. (2) An asymptotic approach based on a continuous diversity (Hill number) profile which depicts the estimated asymptotes of diversities as a function of order q. Replacing the spurious singleton count by our estimated count, we can greatly remove the positive biases associated with diversity estimates due to spurious singletons in the two approaches and make fair comparison across microbial communities, as illustrated in applying our method to analyze sequencing data from viral metagenomes.

Estimating and comparing microbial diversity in the presence of sequencing errors

Estimating and comparing microbial diversity are statistically challenging due to limited sampling and possible sequencing errors for low-frequency counts, producing spurious singletons. The inflated singleton count seriously affects statistical analysis and inferences about microbial diversity. Previous statistical approaches to tackle the sequencing errors generally require different parametric assumptions about the sampling model or about the functional form of frequency counts. Different parametric assumptions may lead to drastically different diversity estimates. We focus on nonparametric methods which are universally valid for all parametric assumptions and can be used to compare diversity across communities. We develop here for the first time a nonparametric estimator of the true singleton count to replace the spurious singleton count. Our estimator of the true singleton count is in terms of the frequency counts of doubletons, tripletons and quadrupletons. To quantify microbial diversity, we adopt the measure of Hill numbers (effective number of taxa) under a nonparametric framework. Hill numbers, parameterized by an order q that determines the measures’ emphasis on rare or common species, include taxa richness (q=0), Shannon diversity (q=1), and Simpson diversity (q=2). Based on the estimated singleton count and the original non-singleton frequency counts, two statistical approaches are developed to compare microbial diversity for multiple communities. (1) A non-asymptotic approach based on standardizing sample size or sample completeness via seamless rarefaction and extrapolation sampling curves of Hill numbers. (2) An asymptotic approach based on a continuous diversity (Hill number) profile which depicts the estimated asymptotes of diversities as a function of order q. Replacing the spurious singleton count by our estimated count, we can greatly remove the positive biases associated with diversity estimates due to spurious singletons in the two approaches and make fair comparison across microbial communities, as illustrated in applying our method to analyze sequencing data from viral metagenomes.

Comment on the papers ‘Creep and recrystallization of large polycrystalline masses’ by Faria and co-authors

In a series of three papers, Faria and co-authors have presented the application of the theory of mixture with continuous diversity to the creep and recrystallization processes of large polycrystalline masses. In this approach, a material point of the continuum is composed of a huge number of grains defined by their crystallographic orientation. The polycrystal is then seen as a continuous mixture of lattice orientations. All the balance equations are expressed to describe the response of the polycrystal and of a group of crystallites sharing the same lattice orientation (i.e. a species). To go further, Faria and co-authors have to make the hypothesis that the strain rate of every species is equal to the strain rate of the polycrystal, and is therefore independent of its lattice orientation. Furthermore, Faria and co-authors insist on the fact that this hypothesis is negligible and has no relation at all to any kind of Taylor-type constraint on the deformation of individual grains, arguing that in their theory of strain rate inhomogeneities on the grain level are smeared out because each species is composed of a very large number of crystals. In this comment, I show that the results obtained with a full-field model suggest that this hypothesis is not insignificant.

Creep and recrystallization of large polycrystalline masses. III. Continuum theory of ice sheets

This work sets forth the first thermodynamically consistent constitutive theory for ice sheets undergoing strain-induced anisotropy, polygonization and recrystallization effects. It is based on the notion of a mixture with continuous diversity, by picturing the ice sheet as a ‘mixture of lattice orientations’. The fabric (texture) is described by an orientation-dependent field of mass density which is sensitive not only to lattice spin, but also to grain boundary migration. No constraint is imposed on stress or strain of individual crystallites, aside from the assumption that basal slip is the dominant deformation mechanism on the grain scale. In spite of the fact that individual ice crystallites are regarded as micropolar media, it is inferred that couples on distinct grains counteract each other, so that the ice sheet behaves on a large scale as an ordinary (non-polar) continuum. Several concepts from materials science are translated to the language of continuum theory, like, for example, lattice distortion energy , grain boundary mobility and Schmid tensor , as well as some fabric (texture) parameters like the so-called degree of orientation and spherical aperture. After choosing suitable expressions for the stored energy and entropy of dislocations, it is shown that the driving pressure for grain boundary migration can be associated to differences in the dislocation potentials ( viz . the Gibbs free energies due to dislocations) of crystallites with distinct c -axis orientations. Finally, the generic representation derived for the Cauchy stress is compared with former generalizations of Glen's flow law, namely the Svendsen–Gödert–Hutter stress law and the Azuma–Goto–Azuma flow law.

Creep and recrystallization of large polycrystalline masses. II. Constitutive theory for crystalline media with transversely isotropic grains

By combining the theory of mixtures with continuous diversity with Liu's method of Lagrange multipliers, a thermodynamically consistent constitutive theory is derived for large polycrystalline masses made up of transversely isotropic crystallites. The media under study are supposed to be incompressible and subjected to strain-induced anisotropy and recrystallization effects. Owing to the fabric (texture) changes caused by lattice rotation and polygonization, the polycrystal and its composing grains are modelled as polar media. Among other results of the theory, the existence of a dislocation potential is inferred, which represents for polycrystals the counterpart to the chemical potential of physical chemistry. Furthermore, exploitation of the dissipation inequality gives rise to the notion of a driving pressure for grain boundary migration . Besides, the vanishing of the Voigt couple stress is analysed together with the existence of internal stresses and couples responsible for the bending/twisting of crystallites by polygonization and heterogeneous strain.

Creep and recrystallization of large polycrystalline masses. I. General continuum theory

This is the first of a series of works on the continuum mechanics and thermodynamics of creep and recrystallization of large polycrystalline masses. The general continuum theory presented here is suited to mono- and multi-mineral rocks. It encompasses several symmetry groups (e.g. orthotropic and transversely isotropic) and diverse crystal classes of triclinic, monoclinic and rhombic systems, among others. The cornerstone of the current approach is the theory of mixtures with continuous diversity, which allows one to regard the polycrystal as a ‘mixture of lattice orientations’. Following this picture, balance equations of mass, linear momentum, lattice spin, energy, dislocations, and entropy are set forth to describe the response of the polycrystal (i.e. the ‘mixture’), as well as of a group of crystallites sharing the same lattice orientation ( viz . a ‘species’). The connection between the balance equations for a species and those for the mixture is established by homogenization rules, formulated for every field of the theory.

On the role of grain growth, recrystallization and polygonization in a continuum theory for anisotropic ice sheets

We outline how to incorporate microscale effects of polycrystalline ice into a continuum description. Actually, analyses of ice cores in Antarctica show that different microstructures generally produce different responses, i.e. a non-uniform distribution of c axes gives rise to anisotropic behaviour. It has been recognized that, to describe certain microstructural processes, like recrystallization or polygonization, we need a parameter able to switch them on (e.g. dislocation density or its associated lattice distortion energy). With this in mind, balance equations for a continuum theory of an anisotropic ice sheet undergoing recrystallization have been recently proposed. In this work, we examine relations for some constitutive quantities, in order to take into account the effects of grain-boundary migration, nucleation and polygonization. We check our assumptions by explicit comparison with the first 1200 m of the Byrd (Antarctica) ice core. Current literature usually gives a relation between normal grain growth and grain boundary migration rate. Here, an equation for normal grain growth which also incorporates the influence of polygonization is suggested. It is based on experimental data from the same core in Antarctica. Polygonization is a microscopic process, but here we present a continuum description of the bending stresses which promote the fragmentation of crystallites in terms of the theory of mixtures with continuous diversity.