Comprehensive and accurate comparisons of transcriptomic distributions of cells from samples taken from two different biological states, such as healthy versus diseased individuals, are an emerging challenge in single-cell RNA sequencing (scRNA-seq) analysis. Current methods for detecting differentially abundant (DA) subpopulations between samples rely heavily on initial clustering of all cells in both samples. Often, this clustering step is inadequate since the DA subpopulations may not align with a clear cluster structure, and important differences between the two biological states can be missed. Here, we introduce DA-seq, a targeted approach for identifying DA subpopulations not restricted to clusters. DA-seq is a multiscale method that quantifies a local DA measure for each cell, which is computed from its k nearest neighboring cells across a range of k values. Based on this measure, DA-seq delineates contiguous significant DA subpopulations in the transcriptomic space. We apply DA-seq to several scRNA-seq datasets and highlight its improved ability to detect differences between distinct phenotypes in severe versus mildly ill COVID-19 patients, melanomas subjected to immune checkpoint therapy comparing responders to nonresponders, embryonic development at two time points, and young versus aging brain tissue. DA-seq enabled us to detect differences between these phenotypes. Importantly, we find that DA-seq not only recovers the DA cell types as discovered in the original studies but also reveals additional DA subpopulations that were not described before. Analysis of these subpopulations yields biological insights that would otherwise be undetected using conventional computational approaches.
Nucleation of various ordered phases in block copolymers is studied by examining the free-energy landscape within the self-consistent field theory. The minimum energy path (MEP) connecting two ordered phases is computed using a recently developed string method. The shape, size, and free-energy barrier of critical nuclei are obtained from the MEP, providing information about the emergence of a stable ordered phase from a metastable phase. In particular, structural evolution of embryonic gyroid nucleus is predicted to follow two possible MEPs, revealing an interesting transition pathway with an intermediate perforated layered structure.
We consider n-by-n matrices whose (i, j)-th entry is f (X T i X j ), where X 1 , . . . , Xn are i.i.d. standard Gaussian random vectors in R p , and f is a real-valued function. The eigenvalue distribution of these random kernel matrices is studied at the "large p, large n" regime. It is shown that, when p, n → ∞ and p/n = γ which is a constant, and f is properly scaled so that V ar(f (X T i X j )) is O(p −1 ), the spectral density converges weakly to a limiting density on R. The limiting density is dictated by a cubic equation involving its Stieltjes transform. While for smooth kernel functions the limiting spectral density has been previously shown to be the Marcenko-Pastur distribution, our analysis is applicable to non-smooth kernel functions, resulting in a new family of limiting densities.
This paper studies the limiting behavior of Tyler's M-estimator for the scatter matrix, in the regime that the number of samples n and their dimension p both go to infinity, and p/n converges to a constant y with 0 < y < 1. We prove that when the data samples x1, . . . , xn are identically and independently generated from the Gaussian distribution N (0, I), the operator norm of the difference between a properly scaled Tyler's Mestimator and n i=1 xix ⊤ i /n tends to zero. As a result, the spectral distribution of Tyler's M-estimator converges weakly to the Marčenko-Pastur distribution.
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