Cellular Deconvolution

Cellular deconvolution is an important analytical step for experiments that rely on samples that could comprise several different cell types (e.g. bulk RNA-seq, spatial genomics, etc.). Though these experiments are more cost effective compared to single cell RNA-seq (scRNA-seq), the gene expression data that result are an average over all of the cells in the sample, and the expression profiles of cells of interest that are scarce may be masked by those are that more abundant. At its core, cellular deconvolution is the problem of teasing apart the gene expression patterns of each cell type from bulk data.

In this post, I’m going to reviewing the problem of cellular deconvolution and some of the work done in that area. I’ll use the case of bulk RNA-seq for my discussions, but keep in mind that the problem is much more general than this particular application.

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Background

Deconvolution methods can be divided into two main types: reference-based (also called partial deconvolution methods) and reference-free (also called complete deconvolution methods).

Reference-Based Methods

A majority of the more popular/recent cellular deconvolution methods are reference-based, which makes sense because of the rise of single cell sequencing. In reference-based methods, single cell data (the reference dataset) are required as inputs, and are used to help define the expression signature of each cell type or group.

This often requires defining a set of (sometimes mutually exclusive) marker genes for each cell type, which can be done using prior knowledge like a literature review. However, if we know the cell types for each cell in our reference dataset, we can use it to identify the marker genes with some sort of statistical comparison. For example, the average expression of gene $i$ in cell type $j$ can be compared to the average of gene $i$ across all other cells via a $t$-test or likelihood ratio test. If the expression is statistically significantly different, then gene $i$ can be used as a marker for cell type $j$.

Beyond picking marker genes, the reference dataset is used to compute the expression signatures by averaging the gene expression levels within each cell type. If the cell types are unknown in the reference dataset, then an unsupervised clustering step can be done first to group by cell type.

Reference-Free Methods

In contrast to reference-based procedures, reference-free methods do not require any additional data. From the bulk data itself, marker genes are identified and the expression signatures are estimated, usually with unsupervised machine learning techniques.

One way to select the marker genes is to project the bulk gene expression data into an alternative (often lower-dimensional) space and choose the genes that are close enough to the corners of the simplex in which all of the data fall (LinSeed¹). Another way is to treat the samples as documents and use existing topic modelling algorithms (CellDistinguisher²).

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Regression Framework

A common way that cellular deconvolution methods approach the problem is to frame the average gene expression obtained via sequencing as a linear combination of the gene expression of each cell type.

We’ll assume to measure the expression levels of $n$ marker genes from $m$ tissue samples containing cells of $k$ types. The measured gene expression for $n$ genes will be collected in the matrix \(\mathbf{B} \in \mathbb{R}^{n \times m}_{\geq 0}\) where $\mathbf{B}_{i,l}$ denotes the average gene expression of gene $i$ over all cells in sample $l$. We’ll let \(\mathbf{S} \in \mathbb{R}^{n \times k}_{\geq 0}\) denote the signature matrix, where \(\mathbf{S}_{i,j}\) denotes the gene expression of gene $i$ in cell type $j$. We’ll also have the matrix \(\mathbf{P} \in \mathbb{R}^{k \times m}_{\geq 0}\), where \(\mathbf{P}_{j, l}\) denotes the proportion of cell type $j$ in sample $l$. We enforce that \(\sum_{j = 1}^{k} \mathbf{p}_{j,l} = 1\) for all $l$.

The linear combination framework aims to estimate $\mathbf{P}$ under the assumption that the following model holds:

\[\mathbf{B} = \mathbf{S} \mathbf{P} \label{eq:lin-comb-framework}\]

We can then find the cell type proportions as the solution to:

\[\underset{\mathbf{P} \in \mathbb{R}^{k,m}_{\geq 0} \text{: } \sum_{j = 1}^k \mathbf{P}_{j,\cdot} = 1}{\arg \min} \left\{ \mathcal{L}(\mathbf{B}, \mathbf{S}\mathbf{P}) \right\} \label{eq:reference-based}\]

where $\mathbf{L}$ is some loss or distance function (e.g. constrained least squares, weight constrained least squares, etc.³).

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Probabilistic Framework

An alternative to the linear combination framework is to fit a Bayesian or hierarchical model to the data.

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Dimension Reduction Framework

Dimension reduction frameworks are often reference-free and generally rely upon (non-negative) matrix factorization of $\mathbf{B}$ or independent component analysis. At a very high level, a lower dimensional space may have dimensions corresponding to cell types (compared to the original space that had genes as its dimensions). If the subspace is learned, then this could lead to the discovery of new cell types or cell states.

As is characteristic of this kind of method, the cellular deconvolution via dimensionality reduction is usually very computationally efficient, and the results are often easily interpretable (especially in the case of independent component analysis). However, there is no guarantee that the dimensions of the subspace correspond to cell types/states, and these methods may perform poorly in cases where the data are very noisy or the cell types in the sample are highly similar.⁴

Linseed

Linseed (linear subspace identification for gene expression deconvolution) is a dimension reduction framework that is entirely reference-free. It hinges upon the idea of mutual linearity, which is a property of genes that are markers for the same cell type.

Mutual Linearity

Suppose we have $n$ samples of some cell type $a$ that has three true marker genes, $x$, $y$, and $z$. Let $x_j$, $y_j$, and $z_j$ denote the expression levels of these genes. Mutual linearity assumes that there exist corresponding proportionality constants, $k_{x,y}$ and $k_{x, z}$, such that the following holds:

\[\mathbf{G} = \begin{bmatrix} x_1 & x_2 & \dots & x_n \\ y_1 & y_2 & \dots & y_n \\ z_1 & z_2 & \dots & z_n \end{bmatrix} = \begin{bmatrix} x_1 & x_2 & \dots & x_n \\ k_{x,y} x_1 & k_{x,y} x_2 & \dots & k_{x,y} x_n \\ k_{x,z} x_1 & k_{x,z} x_2 & \dots & k_{x,z} x_n \end{bmatrix}\]

Row Normalization

The first step of Linseed is to normalize by dividing each entry by the sum of its row:

\[\tilde{\mathbf{G}} = \begin{bmatrix} \frac{x_1}{\sum_{j = 1}^n x_j} & \frac{x_2}{\sum_{j = 1}^n x_j} & \dots & \frac{x_n}{\sum_{j = 1}^n x_j} \\ \frac{y_1}{\sum_{j = 1}^n y_j} & \frac{y_2}{\sum_{j = 1}^n y_j} & \dots & \frac{y_n}{\sum_{j = 1}^n y_j} \\ \frac{z_1}{\sum_{j = 1}^n z_j} & \frac{z_2}{\sum_{j = 1}^n z_j} & \dots & \frac{z_n}{\sum_{j = 1}^n z_j} \\ \end{bmatrix} = \begin{bmatrix} \frac{x_1}{\sum_{j = 1}^n x_j} & \frac{x_2}{\sum_{j = 1}^n x_j} & \dots & \frac{x_n}{\sum_{j = 1}^n x_j} \\ \frac{k_{x,y} x_1}{\sum_{j = 1}^n k_{x,y} x_j } & \frac{k_{x,y}x_2}{\sum_{j = 1}^n k_{x,y} x_j } & \dots & \frac{k_{x,y}x_n}{\sum_{j = 1}^n k_{x,y} x_j } \\ \frac{k_{x,z}x_1}{\sum_{j = 1}^n k_{x,z} x_j} & \frac{k_{x,z}x_2}{\sum_{j = 1}^n k_{x,z} x_j} & \dots & \frac{k_{x,z}x_n}{\sum_{j = 1}^n k_{x,z} x_j} \end{bmatrix} = \begin{bmatrix} \frac{x_1}{\sum_{j = 1}^n x_j} & \frac{x_2}{\sum_{j = 1}^n x_j} & \dots & \frac{x_n}{\sum_{j = 1}^n x_j} \\ \frac{x_1}{\sum_{j = 1}^n x_j} & \frac{x_2}{\sum_{j = 1}^n x_j} & \dots & \frac{x_n}{\sum_{j = 1}^n x_j} \\ \frac{x_1}{\sum_{j = 1}^n x_j} & \frac{x_2}{\sum_{j = 1}^n x_j} & \dots & \frac{x_n}{\sum_{j = 1}^n x_j} \end{bmatrix}\]

This implies that all genes have a proportionality constant of $\tilde{k} = 1$ when we consider $\tilde{\mathbf{G}}$.

Simplex

Let $\mathbf{G} \in \mathbb{R}^{g \times m}_{\geq 0}$ denote the expression matrix for $g$ genes across $m$ samples. Let $\mathbf{H} \in \mathbb{R}^{c \times m}$ denote the mixture proportions of $c$ cell types in the $m$ samples.

Transposing both of these matrices yields $\mathbf{G}^\top$ and $\mathbf{H}$^\top, which can be thought of as collections of $g$ and $c$ vectors, respectively, in an $m$-dimensional space defined by the samples. These matrices are then column-normalized (which is equivalent to row-normalizing the original matrices and then transposing) to get $\tilde{\mathbf{G}}^\top$ and $\tilde{\mathbf{H}}^\top$.

Since they are normalized, the columns of $\tilde{\mathbf{G}}^\top$ and $\tilde{\mathbf{H}}^\top$ sum to $1$. The column and therefore correspond to points lying on a $c-1$-dimensional simplex in $m$-dimensional space. The corners of this simplex correspond

A vector like this would correspond to a “pure” sample (only containing a single cell type) if it were a column in $\tilde{\mathbf{H}}^\top$ and would correspond to a true marker gene if it were in $\tilde{\mathbf{G}}^\top$ ()

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Future Work

Though cellular deconvolution has developed rapidly recently, there are still open problems.³

Mutually Exclusive Marker Genes

Most methods assume that the defining genes for each cell types are not expressed in other cell types (mutually exclusive), which does not always hold. For example, subtypes of a given cell type can have overlapping markers due to their shared lineage. Additionally, if the average expression levels in the bulk sample are very similar to a particular cell type, then that cell type’s proportion may be overestimated.

Fixed Number Of Cell Types

All methods assume that the number of cell types is known. Thus, estimates may be biased when a cell type is present in the sample but not represented in $\mathbf{S}$.

Confounding

Most methods also fail to account for confounding effects during estimation. For example, laboratory protocols may introduce bias (e.g. sequencing errors, batch effects). In addition, cells are neither independent nor static. In the case of the immune system, the expression levels of certain cells is affected by the presence and signaling of other cells. Moreover, cells undergo state changes and differentiation that are often defined by changes in gene expression.

Evaluation

Cellular deconvolution methods are quite difficult to validate due to the lack of a clear ground truth. However, this is not a problem unique to cellular deconvolution, and there are plenty proposed solutions. One example is taking two samples: one for bulk RNA-seq and one for scRNA-seq.

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References