FPKM and TPM Units for RNA Seq after Pachter

See Pachter, L. Models for transcript quantification from RNA-seq.

Count-based models

RPKM/FPKM

Simplest version: transcriptome is a set of transcripts with different abundances, and a read is produced by choosing a site in a transcript for the read uniformly at random from among all positions.

• $T$ is the set of transcripts.
• $\ell(t)$ is the length of transcript $t$ for $t\in T$.
• $\rho(t)$ is the relative abundance of transcript $t$ in the transcriptome. (That is, it is the number of mRNA molecules of type $t$, over the total number of such molecules).
• $m$ is the length of the reads
• $F$ is the set of all possible reads and $F_t$ is the set of reads that map to transcript $t$.
• The effective length $\tilde{\ell}(t)$ is $\ell(t)-m+1$, the total number of places that a read of length $m$ could begin.

What’s the chance of choosing a read from a transcript $t$? Let $m_t$ be the number of transcripts of type $t$ and $M=\sum m_{t}$ be the total number of transcripts. The number of reads coming from $m_t$ is $m_t \tilde{\ell}(t)$. The total number of reads from all transcripts is $\sum_{i\in T} m_{i}\tilde{l}(i).$ Therefore the probability of getting a read from $t$ is $\alpha_t = \frac{m_{t}\tilde{\ell}(t)}{\sum_{t} m_{t}\tilde{\ell}(t)}$ and, since $\rho_t=m_{t}/M$, this is the same as $\alpha_t=\frac{\rho_t\tilde{\ell}(t)}{\sum_{t} \rho_{t}\tilde{\ell}(t)}$

It’s worth observing that $\sum \alpha_{t}=1$ and the $\alpha$’s are non-negative.

The $\alpha$ and $\rho$ distributions are related (via the lengths $\tilde{\ell}$) in the opposite direction by the equation $\rho_{t} = \frac{(\alpha_t/\tilde{\ell(t)})}{\sum_{i\in T} \alpha_{i}/\tilde{\ell}(i)}$

The maximum likelihood estimate for the $\alpha_t$ are $X_t/N$ where $N$ is the total number of mapped reads and $X_t$ is the number of reads mapped to $t$.

The maximum likelihood estimate for $\rho_t$, which is the relative abundance of transcript $t$ among the expressed transcripts, is (using the above equations) PROPORTIONAL TO $\hat{\rho}_{t}\sim\frac{X_{t}}{N\tilde{\ell}(t)}.$ (You need to divide by the sum of the terms on the right to get equality).

The number $\frac{X_{t}}{\tilde{\ell}(t)N}(10^{9})$ is the “Reads per Kilobase per millions of mapped reads” because $X_{t}/(N/10^{6})$ is the “Reads per million mapped reads” and then you divide that by $\tilde{\ell}(t)/10^3$ to get “Reads per million mapped reads per kilobase.”

TPM

The ‘transcripts per million’ measure (maximum likelihood) is $10^6\rho_{t}$ so to compute TPM you take the abundance measured in RPKM and divide by the sum of RPKM over all the transcripts.