Why Sequencing Counts Have a Shared Total
A sequencing table may look like a collection of separate measurements: one count for each gene, taxon, or other feature in each sample. But within a sample, those counts share a finite sequencing library. If more reads are assigned to one feature, fewer reads remain available to be assigned to others, given the total number counted. This creates a constraint that matters for interpreting both counts and proportions.
The constraint does not mean every sample has the same number of reads. Library sizes can differ substantially across samples. It means that, for a given sample and a defined set of counted features, the feature counts add to that sample’s library size. The total is a count of sequenced observations, not automatically a measure of the total amount of biological material in the original specimen.
The choice of features matters. If the table includes only reads assigned to a selected set of genes, \(N_j\) is the total across those genes—not necessarily the total number of reads generated by the instrument. Reads that are unassigned or excluded are outside this particular sum. Changing the included features can therefore change the denominator and the resulting proportions.
Counts, Proportions, and What They Measure
A sequencing count records how many reads or assigned fragments were observed for a feature. It depends on the amount of material from that feature that entered the assay, but also on sequencing depth and technical processes such as capture, amplification, and mapping. A count is therefore not, by itself, a direct measurement of the feature’s absolute number of molecules per cell, per unit of tissue, or per sample.
Dividing a feature count by the library size answers a different question: what fraction of the counted sequencing observations in this sample were assigned to this feature? That is a relative quantity. It describes a feature’s share of the measured composition, not its absolute abundance. The distinction is especially important when samples have different library sizes or when the overall composition changes.
Worked Example: Equal Proportions, Different Library Sizes
Consider two hypothetical samples with counts for two genes. Sample 1 has 800 reads for gene A and 200 for gene B; sample 2 has 1,600 for gene A and 400 for gene B.
For sample 1, the library size is \(N_1=800+200=1{,}000\). The proportions are \(800/1{,}000=0.800\) for A and \(200/1{,}000=0.200\) for B. For sample 2, \(N_2=1{,}600+400=2{,}000\), giving \(1{,}600/2{,}000=0.800\) for A and \(400/2{,}000=0.200\) for B.
The second sample has twice as many observed reads for each gene, but also twice the library size. The proportions are identical. These counts alone show that more reads were collected and assigned in sample 2; they do not establish that either gene had twice the absolute molecular abundance. A change in sequencing depth is one explanation consistent with the counts.
A useful idealized model makes the constraint explicit. Suppose a sample’s \(N_j\) reads are independent draws from a fixed composition over the \(D\) features, and let \(p_{ij}\) be the probability that a draw is assigned to feature \(i\). The probabilities satisfy \(\sum_{i=1}^{D}p_{ij}=1\). Conditional on \(N_j\), the count vector follows a multinomial distribution:
This model is a simplifying description, not a claim that every sequencing pipeline generates perfectly independent, equally representative reads. It assumes a fixed total, a defined set of categories, and a constant assignment probability for each feature within the sample. Technical biases or biological heterogeneity can make the model incomplete. Even so, it clarifies the basic consequence of a shared total: the features are not unconstrained measurements.
Why a Relative Share Can Change Without an Absolute Change
A proportion depends on its denominator. If the amount assigned to one feature stays constant while other features increase, its share of the total declines. Conversely, a feature’s share can rise because other features decline, even if the feature itself does not increase. Relative change and absolute change are therefore different claims.
Worked Example: A Falling Share Does Not Prove a Falling Amount
Suppose, purely hypothetically, that sample 1 contains 60 units of feature A and 40 units of feature B, for a total of 100 units across these two features. Their shares are \(60/100=0.600\) and \(40/100=0.400\). In sample 2, imagine that A remains at 60 units while B rises to 140 units. The total is now 200 units, so A’s share is \(60/200=0.300\), while B’s share is \(140/200=0.700\).
A’s share fell from 0.600 to 0.300, but its absolute amount remained 60 units in this hypothetical construction. The change came from the increase in B. Sequencing proportions alone would show A becoming a smaller fraction of the measured total; they would not show that A’s absolute amount fell.
The same observed proportions can also arise at different overall scales. For example, amounts of 60 and 140 have shares 0.300 and 0.700, just as amounts of 120 and 280 do: \(120/(120+280)=0.300\) and \(280/(120+280)=0.700\). A composition identifies relative shares, not the overall scale of the amounts that produced them. In real sequencing data, counts are reads rather than direct measurements of these hypothetical units, which adds further reasons not to equate a count with an absolute amount.
This is not a flaw in proportions: they are useful when the scientific target is explicitly relative, such as a feature’s share of the observed sequencing composition. The limitation arises when a relative measurement is interpreted as though it were an absolute measurement. To make an absolute claim, the study needs relevant information beyond the sequencing proportions, such as a suitable external calibration or measurement of a defined biological quantity.
The Shared-Total Constraint Induces Dependence
Under the multinomial model, two different feature counts tend to vary in opposite directions conditional on the library size. This is a mathematical consequence of allocating a fixed number of draws among categories. It does not imply that the underlying biological quantities must oppose one another.
A short derivation uses indicator variables. For each read \(r\), let \(I_{ir}=1\) if it is assigned to feature \(i\), and 0 otherwise. Then \(Y_{ij}=\sum_{r=1}^{N_j} I_{ir}\). For two distinct features, a read cannot be assigned to both, so \(I_{ir}I_{kr}=0\); therefore \(\operatorname{Cov}(I_{ir},I_{kr})=0-p_{ij}p_{kj}=-p_{ij}p_{kj}\). Assignments for different reads are independent under the model, so their cross-read covariances are zero. Summing the \(N_j\) same-read covariances gives \(-N_jp_{ij}p_{kj}\), as stated.
Worked Example: Negative Covariance Under a Fixed Library Size
Assume a hypothetical library has \(N=100\) reads and two features with assignment probabilities \(p_1=0.400\) and \(p_2=0.600\). The probabilities sum to one, so these features exhaust the categories in this example.
The multinomial covariance is
The negative value follows from the fixed total: a read assigned to feature 1 is not available to be assigned to feature 2. This model-based covariance is not a claim that two biological abundances are negatively associated. Nor does it guarantee that every real data set will show a negative empirical association, because library sizes can vary and biological and technical processes can add other sources of variation.
Checking the Constraint and Choosing an Interpretation
A first diagnostic is to calculate the total count for each sample over the stated feature set. Inspecting these totals reveals whether library sizes vary and whether samples with large totals may dominate comparisons made on raw counts. Also check that proportions were calculated using a clearly stated denominator and that they sum to one, subject to rounding. These checks clarify what the table represents; they do not establish that counts are unbiased measures of biological abundance.
Interpretation should follow the measurement target. Raw counts retain information about the number of reads observed, but their magnitudes depend on library size. Within-sample proportions put samples on a common total, but describe relative composition. External calibration can provide information about an absolute scale when the calibration procedure is appropriate and its assumptions are defensible. No one representation answers every scientific question.
| Representation | What it describes | Main interpretive caution |
|---|---|---|
| Raw sequencing counts | Observed reads or assigned fragments for each feature | Counts are affected by library size and are not direct absolute quantities. |
| Within-sample proportions | Each feature’s share of the specified counted total | A share can change because other features change; proportions do not give the overall scale. |
| Externally calibrated measurements | An estimate tied to a defined absolute quantity, if calibration is valid | Interpretation depends on the calibration, its reference, and the measurement assumptions. |
Diagnostics can also flag mismatches between the intended interpretation and the data. If a conclusion concerns molecules per cell but the only evidence is a change in read proportion, the measurement target and conclusion do not align. If totals differ greatly, raw-count comparisons need particular caution. If the analyzed feature set changes between samples or processing steps, the denominator may no longer represent the same composition. These are reasons to investigate the data and design, not proof that any one sample is erroneous.
Common Pitfall
Wrong reasoning: “The count for gene A doubled between samples, so the amount of gene A doubled.” This ignores that the total number of reads may also have doubled, or that the composition of the other features may have changed.
Correction: First state what the count records and compare it with the relevant library size. A count is an observed sequencing tally; its within-sample proportion is a share of the counted total. Neither quantity alone identifies absolute molecular abundance. An absolute claim requires suitable calibration or other information that establishes an absolute scale.
The core lesson is that sequencing counts are linked by the total available library. They can be valuable measurements, but their interpretation must respect that shared total. Relative abundance is about shares within a defined composition; absolute abundance is about amounts on a scale that those shares alone cannot recover.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- A sample has counts of 300, 500, and 200 across three features. What is its library size, and what is the proportion for the second feature?
- Two samples have counts of 400 and 100 for features A and B, and 800 and 200 for A and B. What do the counts and proportions show, and what do they not establish?
- Under a multinomial model, why is the covariance between two distinct feature counts nonpositive?
- Feature C’s proportion falls between two samples. Does this alone establish that its absolute amount fell? Explain.
- Name one useful comparison with proportions and one limitation of that comparison.
Solutions. (1) The library size is \(300+500+200=1{,}000\); the second feature’s proportion is \(500/1{,}000=0.500\). (2) The library sizes are 500 and 1,000. A’s proportions are \(400/500=0.800\) and \(800/1{,}000=0.800\); B’s are 0.200 in both samples. The observed counts doubled along with the library size, but this does not establish that either feature’s absolute amount doubled. (3) With a fixed total, a single draw cannot be assigned to both features; the multinomial covariance is \(-Np_ip_k\), which is nonpositive. (4) No. Its share could fall because other features increased, even if C’s absolute amount stayed constant. (5) Proportions allow comparison of feature shares across samples with different library sizes; they do not identify the overall absolute scale or, by themselves, absolute changes in a feature.