A quaternion model for single cell transcriptomics

We present an approach for modeling single cell RNA-sequencing (scRNA-seq) and spatial transcriptomics (ST) data using quaternions. Quaternions are four dimensional hypercomplex numbers that, along with real numbers, complex numbers and octonions, represent one of the four normed division algebras. Quaternions have been primarily employed to represent three-dimensional rotations in computer graphics with most biomedical applications focused on problems involving the structure and orientation of biomolecules, e.g., protein folding, chromatin conformation, etc. In this paper, we detail an approach for mapping the cells/locations in a scRNA-seq/ST data set to quaternions. According to this model, the quaternion associated with each cell/location represents a vector in [R]3 with vector length capturing sequencing depth and vector direction capturing the relative expression profile. Assuming that biologically interesting features of an scRNA-seq/ST data set are preserved within a rank three reconstruction of the unnormalized counts, this representation has several benefits for data analysis. First, it supports a novel approach for scRNA-seq/ST data visualization that captures cell state uncertainty. Second, the model implies that transformations between cell states can be viewed as three-dimensional rotations, which have a corresponding representation as rotation quaternions. The fact that these rotation quaternions can be interpreted as cells enables a novel approach for characterizing cell state transitions with specific relevance to the analysis of pseudo-temporal ordering trajectories. Most importantly, a quaternion representation supports the genome-wide spectral analysis of scRNA-seq/ST data relative to a single variable, e.g., pseudo-time, or two variables, e.g., spatial coordinates, using a one or two-dimensional hypercomplex discrete Fourier transform. An R package supporting this model and the hypercomplex Fourier analysis of ST data along with several example vignettes is available at https://hrfrost.host.dartmouth.edu/QSC.