Normalizing need not be the norm: count-based math for analyzing single-cell data

Abstract

Counting transcripts of mRNA are a key method of observation in modern biology. With advances in counting transcripts in single cells (single-cell RNA sequencing or scRNA-seq), these data are routinely used to identify cells by their transcriptional profile, and to identify genes with differential cellular expression. Because the total number of transcripts counted per cell can vary for technical reasons, the first step of many commonly used scRNA-seq workflows is to normalize by sequencing depth, transforming counts into proportional abundances. The primary objective of this step is to reshape the data such that cells with similar biological proportions of transcripts end up with similar transformed measurements. But there is growing concern that normalization and other transformations result in unintended distortions that hinder both analyses and the interpretation of results. This has led to an intense focus on optimizing methods for normalization and transformation of scRNA-seq data. Here, we take an alternative approach, by avoiding normalization and transformation altogether. We abandon the use of distances to compare cells, and instead use a restricted algebra, motivated by measurement theory and abstract algebra, that preserves the count nature of the data. We demonstrate that this restricted algebra is sufficient to draw meaningful and practical comparisons of gene expression through the use of the dot product and other elementary operations. This approach sidesteps many of the problems with common transformations, and has the added benefit of being simpler and more intuitive. We implement our approach in the package countland, available in python and R.

Appendix 1. Properties of the count-space over zero and the natural numbers

This appendix provides additional background and information on mathematical concepts relevant to the analysis of transcript count matrices.

The natural numbers, $\mathbb {N}_0$

The natural numbers are all positive integers. Depending on the definition, the natural numbers may also include zero (i.e. all non-negative integers), a convention we follow here so that $\mathbb {N}_0=\{0,1,2,3,\ldots \}$. We can classify sets of numbers by the operations under which they are closed, meaning operations on elements in the set result in elements that are also in the set. One familiar set is a field, which is closed under addition and multiplication, along with their inverses, subtraction and division, and includes rational numbers $\mathbb {Q}$ and real numbers $\mathbb {R}$. The natural numbers form a semiring, because $\mathbb {N}_0$ is closed for multiplication and addition, but not subtraction or division.

$\mathbb {N}_0$ contains the additive identity element, 0, that can be added to any element to return the same element (e.g. $x+0=x$). $\mathbb {N}_0$ also contains the multiplicative identity element, 1, that can be multiplied with any element to return the same element (e.g. $x1=x$).

Inverses are elements that return the identity elements under specified operations. For example, the negative numbers are inverses under addition, because $x+(-x)$ returns the additive identity element, 0. However, because $\mathbb {N}_0$ does not contain negative numbers, it doesn’t have additive inverses. Similarly, reciprocal values are inverses under multiplication, because $x(1/x)$ returns the multiplicative identity element, 1. $\mathbb {N}_0$ likewise does not contain reciprocals, so therefore doesn’t have multiplicative inverses.

Subtraction can be defined as the addition of an additive inverse (a negative number), and division can be defined as multiplication with a multiplicative inverse (a reciprocal). Because these two inverses are not contained in $\mathbb {N}_0$, there is no subtraction or division. This is equivalent to the observation that $\mathbb {N}_0$ is not closed for subtraction or division.

Count-space over $\mathbb {N}_0$

The count-space over $\mathbb {N}_0$ contains the vectors $\textbf{V}$ such that

$$\begin{aligned} \textbf{V} = {(a_1,a_2,a_3,...a_n): a_1,a_2,a_3,...a_n \in \mathbb {N}_0 } \end{aligned}$$

The count-space over $\mathbb {N}_0$ is a semimodule that is restricted to the integer-lattice that is located over the upper right quadrant of a coordinate system, inclusive of the origin and axes. Certain operations are possible in this restricted space, while others are not. For example, we can apply the operation of the dot product because this operation requires only multiplication and addition of vector elements. However, unlike a vector space over a field, in the count-space over $\mathbb {N}_0$ there are no angles between vectors. Calculating angles from dot product requires division by vector length, and $\mathbb {N}_0$ is not closed for division.

Furthermore, in count-space, vector length is not a Euclidean measure of distance as there is no equivalent measure of distance in a space without subtraction or square roots. Instead of Euclidean distance, we can use the number of integer steps in a positive direction as a measure of length, which is equivalent to the magnitude of the Manhattan distance between the vector terminus and the origin, though calculating the Manhattan distance generally requires subtraction.

Vector rotation is not possible in count-space as it would require rotation matrices with new basis vectors that include negative elements. Vector reflections are possible, because we are free to permute our count matrix, as are shears of vectors. Some vector projections are possible, but not all. For example, if we project vector $\textbf{b}$ onto vector $\textbf{a}$, the result will be a multiple $q$ of $\textbf{a}$, $q\textbf{a}$. The value of that multiple will be equal to $q=(\textbf{a}^T\textbf{b})/(\textbf{a}^T\textbf{a})$, which requires division to calculate unless $\textbf{a}^T\textbf{a} = 1$. Over $\mathbb {N}_0$, that only happens when there is a single entry that is 1, i.e. when vector $\textbf{a}$ is one of the original basis vectors. Therefore we can project vectors onto basis vectors, but not onto arbitrary vectors (e.g. we cannot project one cell vector onto another). Projecting onto basis vectors is the equivalent of multiplying some values by 0 while retaining others.

Without rotation and vector projection, it is clear that certain complex operations like principal component analysis that rely on these are not possible in this vector space.

High-dimensional, low-magnitude count-space over $\mathbb {N}_0$

While the above pertains to the count-space over $\mathbb {N}_0$ in general, there are interesting properties of the very high dimensional, low magnitude spaces that describe scRNA-seq count data.

Most gene expression datasets contain measurements for many thousands of genes, meaning this count-space has many thousands of dimensions. Furthermore, due to the sparse nature of these count matrices, it is difficult if not impossible to find a reasonable lower-dimensional approximation. In other words, because many features contain only a few, non-overlapping observations, there is no way to reduce the rank of this matrix without discarding features.

Because most measures of gene expression are low-magnitude integers, most cell vectors terminate only a few steps from the origin in any given direction. This does not mean that vectors are close to the origin overall. Vector length is non-Euclidean; it is calculated as the sum of steps back to the origin (Manhattan distance), not the distance along a diagonal (Euclidean distance).

Because the vast majority of values in the count matrix are zero, cell vectors are perpendicular to each other in many directions. The may result in the outcome that cell-cell similarity has more to do with the number and distribution of non-zero observations than expression magnitude. This has been demonstrated by the fact that binary transformations of scRNA-seq data to zero/non-zero contain enough information to recapitulate major patterns in the data (Qiu 2020).