Estimating the frequency of multiplets in single-cell RNA sequencing from cell-mixing experiments

Derivation of multiplet frequency from observed numbers of pure and mixed-cell droplets

Consider the case in which cells of two types (e.g., human and mouse) are distributed into individual barcoded droplets, although the same logic applies if the cells are distributed into barcoded wells or combinations of wells. Assume the sequencing data have been analyzed so that each non-empty droplet can be classified as containing at least one cell of type 1, at least one cell of type 2, or cells of both types. I will refer to the number of droplets in each of these three groupings as N₁, N₂, and N_1,2, respectively. For instance, the 10× cellranger pipeline (version 2.1.1) returns these numbers as the “Estimated Number of Cell Partitions.”

The only assumption of the derivation is that the number of cells per droplet is Poisson distributed. Let μ₁ be the average number of cells of type 1 per droplet, and μ₂ be the average number of cells of type 2 per droplet. The average number of cells of any type per droplet is then μ₁ + μ₂. Therefore, the probability that a droplet contains at least one cell of any type is (1) $$\begin{array}{ll}\mathrm{Pr}(c\ge 1)\hfill & =1-\mathrm{Pr}(c=0)\hfill \\ \hfill & =1-e^{-{\text{μ}}_1-{\text{μ}}_2}.\hfill \end{array}$$

Likewise, the probability that a droplet contains multiple cells of any type (e.g., a multiplet) is (2) $$\begin{array}{ll}\mathrm{Pr}(c\ge 2)\hfill & =1-\mathrm{Pr}(c=0)-Pr(c=1)\hfill \\ \hfill & =1-e^{-{\text{μ}}_{\text{1}}-{\text{μ}}_{\text{2}}}-({\text{μ}}_{\text{1}}{\text{+μ}}_{\text{2}})e^{-{\text{μ}}_{\text{1}}+{\text{μ}}_{\text{2}}}.\hfill \end{array}$$

The multiplet frequency M is simply the probability that a droplet with at least one cell actually contains multiple cells, which is (3) $$\begin{array}{ll}M\hfill & =\frac{\mathrm{Pr}(c\ge 2)}{\mathrm{Pr}(c\ge 1)}\hfill \\ \hfill & =1-\frac{({\text{μ}}_{\text{1}}+{\text{μ}}_{\text{2}})e^{-{\text{μ}}_{\text{1}}+{\text{μ}}_{\text{2}}}}{1-e^{-{\text{μ}}_1-{\text{μ}}_{\text{2}}}}.\hfill \end{array}$$

However, evaluating this expression for M requires the values of μ₁ and μ₂.

We can write down equations for μ₁ and μ₂ by again using the fact that the number of cells per droplet is Poisson distributed. Specifically, if N is the total number of droplets (empty and non-empty), then the expected number of droplets that have at least one cell of type 1 is $N\times \mathrm{Pr}\left(c_1\ge 1\right)=N\left(1-e^{-{\text{μ}}_1}\right)$. The observed number of droplets with at least one cell of type 1 is N₁, so setting the observed number equal to the expected number gives us an equation for μ₁, (4) $$N_1=N\left(1-e^{-{\text{μ}}_1}\right).$$

This equation is easily solved for μ₁ to yield (5) $${\text{μ}}_1=-\ln \left(\frac{N-N_1}{N}\right),$$and likewise for μ₂, (6) $${\text{μ}}_2=-\ln \left(\frac{N-N_2}{N}\right).$$

Equations (5) and (6) give us a way to determine the values (μ₁ and μ₂) needed to calculate the multiplet frequency (Eq. (3)) in terms of the experimental observables N₁ and N₂. Unfortunately, these two equations also require knowledge of the total (empty and non-empty) number of droplets N, which is not directly observable from the sequencing data.

However, we can take advantage of another relationship to calculate N. The fraction of all (empty and non-empty) droplets that contain cells of both types is $\frac{N_{\mathrm{1,2}}}{N}$, and this fraction is simply the product of the probability that a droplet contains at least one cell of type 1 with the probability that a droplet contains at least one cell of type 2, which in mathematical terms can be stated as $\mathrm{Pr}\left(c_1\ge 1\wedge c_2\ge 1\right)=\mathrm{Pr}\left(c_1\ge 1\right)\times \mathrm{Pr}\left(c_2\ge 1\right)$. Therefore, (7) $$\frac{N_{\mathrm{1,2}}}{N}=\frac{N_1}{N}\times \frac{N_2}{N}\mathrm{.}$$

This equation can be solved to give (8) $$N=\frac{N_1{N}_2}{N_{\mathrm{1,2}}}\mathrm{,}$$ which can be completely evaluated in terms of the experimental observables. Equations (5), (6), and (8) can be used to calculate μ₁ and μ₂ in terms of the experimental observables, and those results used to calculate the multiplet frequency via Eq. (3). This provides an analytic solution for the multiplet frequency in terms of the three experimental observables.

Implementation and example calculations

A simple function to perform the calculations described in the previous subsection is implemented in Python in the Jupyter notebook found at https://github.com/jbloomlab/multiplet_freq/blob/master/calcmultiplet.ipynb, and in R in the Jupyter notebook found at https://github.com/jbloomlab/multiplet_freq/blob/master/calcmultiplet_R.ipynb (see also Files S1–S4). To illustrate the calculations, I used this function to calculate the multiplet frequency for hypothetical data.

First, consider hypothetical data in which the two types of cells are mixed in equal proportions. Prior papers have approximated the multiplet frequency from such experiments as simply twice the fraction of non-empty droplets that contain cells of both types (Klein et al., 2015; Macosko et al., 2015; Zheng et al., 2017; Cao et al., 2017), which is $\frac{N_{\mathrm{1,2}}}{N_1+N_2-N_{\mathrm{1,2}}}$ in the notation defined in the previous subsection. Table 1 shows that the exact equation derived in the previous subsection gives very similar results to this approximate method as long as the multiplet frequency is low. When the multiplet frequency becomes high, the approximate method starts to overestimate the true multiplet frequency, since it fails to account for the fact that some multiplets will contain more than two cells.

Next, consider hypothetical data in which the two types of cells are mixed in unequal proportions. Table 2 shows the multiplet frequencies for several such experiments. An interesting aspect of the results is that at high multiplet frequencies and very unequal cell proportions, the multiplet frequency is substantially lower than the fraction of droplets containing the rarer cell type that contain a mix of both cell types. The reason is that multiplets (particularly higher-order ones) become more and more likely to contain at least one cell of the rarer type relative to droplets that contain only one cell. For instance, in the final experiment in Table 2, two-thirds of the droplets containing mouse cells have a mix of both cell types, yet less than half the non-empty droplets are multiplets (the multiplet frequency is 0.459). This somewhat non-intuitive result illustrates the importance of using the correct mathematical relationship to calculate the multiplet frequency when cell types are mixed unequally.