From Copy Number to Downstream Measurements
A copy-number alteration changes the number of DNA copies in a genomic region. A gene in that region may show a corresponding change in RNA abundance and, in turn, protein abundance—but these links are not guaranteed. Transcriptional regulation, protein turnover, assay noise, tumor composition and other factors can weaken or obscure the pattern. The aim here is to trace a selected alteration through matched measurements and describe what the data support at each layer.
The analysis is gene- and sample-specific. For each selected gene, the copy-number value, RNA measurement and protein measurement must refer to the same tumor sample and the same gene product. Copy number is often represented by a log2 ratio relative to a reference, while RNA and protein abundance are represented on normalized log2 scales. A log2 ratio of 1 corresponds to a twofold ratio; a difference of 1 between two group means on a log2 abundance scale corresponds to a twofold ratio of their geometric means.
Before analysis, record how each variable was formed. A segmented copy-number log2 ratio is not necessarily an absolute count of gene copies, especially when tumors differ in ploidy or purity. RNA-sequencing counts need appropriate normalization and a suitable transformation or model; proteomic intensities also require assay-appropriate normalization. For the numerical examples below, the supplied RNA and protein values are already normalized log2 abundances, so arithmetic differences have a direct log2 interpretation.
A Focused Analysis and Its Assumptions
First define the selected alteration and the samples to compare. An altered-versus-neutral comparison can be useful when the alteration category was specified using a defensible copy-number rule. A continuous analysis can retain more information by relating the copy-number log2 ratio to RNA abundance. Avoid choosing genes, thresholds or directions because they produce striking downstream results and then treating the same data as an independent confirmation.
For a selected gene, a simple adjusted association model for RNA is
Here \(\mathbf{Z}_i\) contains pre-specified covariates, such as tumor purity, assay batch or a relevant clinical factor; \(\beta_{1g}\) is the expected RNA difference associated with a one-unit increase in the copy-number log2 ratio, conditional on those covariates. A corresponding protein model can relate \(P_{ig}\) to \(C_{ig}\). To describe the total copy-number–protein association, do not automatically adjust that model for RNA: RNA may lie on the pathway being traced, and conditioning on it changes the question.
The usual linear-model interpretation requires an adequately specified mean relationship, independent errors across the analyzed units, and residual variability suitable for the chosen inference. For small samples, outliers and nonlinearity can dominate. If several tumors come from one patient, treating them as independent is not justified without accounting for that dependence. Matching RNA and protein within a tumor makes the measurements aligned; it does not make different tumors independent if the sampling structure says otherwise.
When comparing altered and neutral tumors, the estimated mean difference is the altered-group mean minus the neutral-group mean. Its interpretation depends on comparable groups or appropriate adjustment. Differences in purity, batch, tumor subtype or other features can produce apparent molecular differences even when copy number is not the explanation. With only a few selected loci, emphasize effect sizes, uncertainty and the selection process; do not present a selected result as genome-wide discovery.
Worked Example: A Selected Amplification With RNA and Protein Increases
Suppose a hypothetical study has four neutral and four altered tumors for one selected gene. The table gives matched copy-number log2 ratios, normalized RNA log2 abundance and normalized protein log2 abundance. The category was defined before examining RNA or protein. Assume for this descriptive comparison that all eight samples passed the same assay-quality rules.
| Group | Copy-number log2 ratios | RNA log2 abundance | Protein log2 abundance |
|---|---|---|---|
| Neutral | -0.1, 0.0, 0.1, 0.0 | -0.2, 0.1, 0.0, 0.1 | -0.1, 0.0, 0.1, 0.0 |
| Altered | 0.8, 1.0, 1.2, 1.4 | 0.8, 1.0, 1.1, 1.3 | 0.6, 0.8, 0.9, 1.0 |
The neutral RNA mean is \((-0.2+0.1+0.0+0.1)/4=0.00\). The altered RNA mean is \((0.8+1.0+1.1+1.3)/4=1.05\). Thus the altered-minus-neutral mean difference is \(1.05-0.00=1.05\) log2 units. On the abundance scale this corresponds to a geometric-mean ratio of \(2^{1.05}=2.07\). The group means are consistent with higher RNA abundance in the altered tumors.
For protein, the neutral mean is \((-0.1+0.0+0.1+0.0)/4=0.00\); the altered mean is \((0.6+0.8+0.9+1.0)/4=0.825\). The difference is \(0.825\) log2 units, corresponding to \(2^{0.825}=1.77\) on the abundance ratio scale. Both downstream group differences are positive. This is a concordant descriptive pattern, not proof of a causal chain: the comparison has only eight tumors, and potential group differences such as purity or subtype have not been adjusted for.
Interpreting Continuous Dosage Relationships
A binary alteration category discards variation within groups. When the measured copy-number ratio is informative and the association is plausibly summarized by a straight line, a regression slope provides a direct dosage-response summary. In simple linear regression with an intercept, the least-squares slope is the cross-product of centered copy number and RNA divided by the sum of squared centered copy number. This follows by minimizing the residual sum of squares with respect to the slope.
To see why, write the fitted value as \(\widehat{R}_{ig}=\widehat{\beta}_{0g}+\widehat{\beta}_{1g}C_{ig}\). Minimizing \(\sum_i(R_{ig}-\widehat{R}_{ig})^2\) first gives \(\widehat{\beta}_{0g}=\bar{R}_g-\widehat{\beta}_{1g}\bar{C}_g\). Substituting this intercept into the residual sum of squares and setting its derivative with respect to \(\widehat{\beta}_{1g}\) to zero yields
which gives the stated slope when the copy-number values are not all identical. This is an estimate of a linear association in the observed samples, not an intervention effect.
Worked Example: Estimating a Copy-Number–RNA Slope
For a hypothetical gene, five matched tumors have copy-number log2 ratios \(C=(-0.4,-0.2,0,0.2,0.4)\) and normalized RNA log2 abundances \(R=(-0.5,-0.1,0.2,0.1,0.6)\). These measurements are paired by tumor, and suppose there are no repeated tumors from the same patient. The sample means are \(\bar{C}=0\) and \(\bar{R}=0.06\).
The numerator in the slope formula is \((-0.4)(-0.56)+(-0.2)(-0.16)+(0)(0.14)+(0.2)(0.04)+(0.4)(0.54)=0.480\). The denominator is \((-0.4)^2+(-0.2)^2+0^2+0.2^2+0.4^2=0.400\). Therefore,
The fitted line is \(\widehat{R}=0.060+1.20C\). Within this observed range, a one-unit increase in the copy-number log2 ratio is associated with an estimated 1.20-unit increase in RNA log2 abundance. A one-unit predictor change is large relative to these five observations, so it is also helpful to express the association over a smaller change: an increase of \(0.2\) in \(C\) corresponds to an estimated \(1.20(0.2)=0.24\) log2-unit increase in RNA, or an abundance ratio of \(2^{0.24}=1.18\). The estimate is based on five points; it should not be extrapolated or treated as precise evidence without uncertainty assessment and diagnostics.
When a Downstream Layer Shows No Clear Change
A copy-number change may be accompanied by an RNA shift but not a detectable protein shift. That result can arise because RNA is not translated proportionally, protein abundance is regulated or buffered, the protein assay is less precise, or the sample is too small to resolve a modest effect. “No clear downstream evidence” is therefore a statement about the observed estimate and its uncertainty, not a declaration that the true effect is exactly zero.
For a group mean difference, a confidence interval gives a useful scale for this distinction. A narrow interval centered near zero can rule out some effects of practical importance under the model assumptions; a wide interval that includes zero and appreciable effects is inconclusive. The interval is not the probability that the fixed parameter lies inside it. Its repeated-sampling interpretation is that the confidence procedure would cover the parameter at its stated rate under its assumptions.
Worked Example: Amplification Without Clear Protein Evidence
Consider a different hypothetical selected locus with four neutral and four altered tumors. Their normalized protein log2 abundances are neutral \((-0.1,0,0.1,0)\) and altered \((-0.1,0.1,0,0)\). Assume independent tumors, approximately normal within-group errors and equal group variances for this small-sample pooled interval. The two group means are both 0, so the altered-minus-neutral estimate is \(0.000\) log2 units.
Each group has a sum of squared deviations of \(0.020\), so its sample variance is \(s^2=0.020/(4-1)=0.00667\). The pooled variance is also \(0.00667\), and the standard error of the difference is
With \(df=4+4-2=6\), the 95% two-sided \(t\) critical value is approximately \(2.447\). The interval is \(0.000\pm2.447(0.0577)=[-0.141,0.141]\) log2 units. On the ratio scale its endpoints are \(2^{-0.141}=0.907\) and \(2^{0.141}=1.103\). Under the stated assumptions, these data do not show a protein mean increase; the interval is also compatible with modest decreases or increases. The correct conclusion is that this small example provides no clear evidence of a protein shift, not that protein abundance is unaffected.
Diagnostics, Alternatives and Limits
Check the analysis before interpreting a cross-layer pattern. Confirm sample identifiers and gene mappings, inspect missingness, and verify that RNA and protein values are normalized on comparable, documented scales. Plot each layer against copy number and label samples with unusual values. In regression, inspect residuals versus fitted values for curvature or changing spread, and inspect residual distributions for influential outliers. With small samples these checks have limited power; a plot can reveal a problem but cannot certify assumptions.
Also check whether apparent dosage associations track tumor purity, ploidy, batch or subtype. Broad chromosome-arm events can affect many genes together, while a focal event may be more specific; neither pattern alone proves a gene-specific mechanism. A single extreme sample can create an apparent slope. Sensitivity analyses that report the result with and without a clearly influential observation can show fragility, but observations should not be deleted simply because they weaken the conclusion.
The main alternatives answer different questions. Altered-versus-neutral summaries are easy to explain but depend on a threshold and lose within-group information. Continuous regression uses the measured dosage gradient but relies on a suitable functional form and can be sensitive to leverage. A paired tumor-normal contrast can remove stable person-level differences when matched normal tissue is available, but it does not remove tumor-specific confounding and requires the matching structure to be modeled correctly. These approaches are not interchangeable; state the estimand and design that motivate the choice.
For a short list of pre-specified genes, report estimates and uncertainty gene by gene and disclose how the list was selected. If many genes or alteration definitions are screened, account for multiplicity when making formal claims. Finally, protein measurements may be missing for particular genes, peptides may not uniquely identify a protein, and assay scales may not be directly comparable. A missing or low-quality protein result is not evidence of no protein effect.
Define the copy-number feature, alteration rule, gene mapping and matched set before examining downstream patterns.
Confirm normalization, sample alignment, missingness, purity or batch concerns, and any repeated-measure structure.
Report copy-number–RNA and copy-number–protein estimates, with units, uncertainty and plots; describe RNA–protein correspondence separately.
Call the pattern concordant only when measured directions and evidence align, and distinguish inconclusive results from evidence of negligible effects.
Check Your Understanding
Use the questions to check whether the comparison, scale and conclusion are all clear. Brief solutions follow.
- On a normalized log2 RNA scale, the altered-minus-neutral mean difference is \(0.5\). What abundance ratio does this represent?
- Why should a copy-number–protein model estimating the total association not automatically include RNA as a covariate?
- A protein difference estimate is near zero, but its interval is wide and includes meaningful increases and decreases. What is the appropriate conclusion?
- Name two diagnostics or design checks that could reveal a misleading copy-number–RNA association.
Solutions. (1) \(2^{0.5}=1.41\), so the altered group has an estimated geometric-mean RNA abundance about 1.41 times that of the neutral group. (2) RNA may be an intermediate in the pathway from copy number to protein; conditioning on it changes the estimand from a total association and may obscure part of the pattern. (3) The result is inconclusive: it does not establish no effect, because the interval remains compatible with effects in both directions. (4) For example, plot RNA against copy number and inspect for curvature or an influential point; also check purity, batch, sample matching and repeated tumors from a patient.