# Comparative Identification and Assessment of Single Nucleotide Variants Through Shifts in Base Call Frequencies

## Table of Contents

## 1 Motivation

The advances in high-throughput nucleotide sequencing have lead to new
possibilities of identifying genomic alterations, which is of particular interest
for cancer research. While the detection of genomic variants is often conducted
by comparing against a population-based reference sequence (such as the human GRCh
37), in the investigation of tumors a comparison between matched samples is
predominantly pursued. Considering a test sample \(T\) and a control sample \(C\),
the aim is to identify single nucloetide variants (SNVs) which show significant
changes between the two. Typically, this is applied for comparing
a tumor sample against a matched normal/constitutive sample of the same patient,
focusing on alterations that may have occurred in the transition to the
cancerous tissue, and therefore are often termed *somatic* variants. However, a
broader range of questions can be addressed by extending the focus to include
genomic changes of temporally or spatially separated samples to each other, to
investigate tumor evolution and subclonal shifts [11].

Extending the scope of comparative variant analyses in a cancer related setting
may require an accompanying change in definitions. In the classical tumor
versus normal setting, a *somatic* event is characterized by both the presence
of the variant in the tumor and the absence in the matched normal
[16]. This definition is not satisfactory for
comparing different tumor stages against each other. If we imagine two stages
of a tumor haboring multiple subclones, variants may be present in both stages
with changing abundances. Here, the classical definition of a *somatic* variant
would result in missing the event. In the following, we will therefore focus on
estimating the shift in variant allele frequencies between a test and a control
sample. The classical somatic definition requiring the variant to be absent in the
control constitutes a special case of this.

Numerous approaches focus on more narrowly defined somatic comparisons,
motivated by scientific questions in the field of cancer genomics
[15] [16]. Most existing methods
do not allow an explicit statement about the evidence for the absence of a
putative variant in a sample, and a clear distinction is needed whether a *negative* call
results from a true negative site or the lack of statistical power to detect it.
As an example, imagine a variant calling approach using a hypothesis test with the
null hypothesis that the variant frequencies at a locus are identical for both
samples. While a significant p-value provides evidence for the presence of a
variant, inverting this argument does not work: A non-significant p-value can
both be indicative of the absence or insufficient power. Furthermore, most
methods return only a point estimate for the variant frequency in the tumor,
without stating the confidence of the estimate. However, this information would
be critical to reliably differentiate between variants arising from competing
subclonal populations.

With the `Rariant`

package, we pursue a generalized approach for identifying and
quantifying SNV alterations from high-throughput sequencing data in comparative
settings. By focusing on shifts of the non-consensus base call frequencies,
events like loss of heterozygosity and clonal expansions in addition to somatic
variants can be detected. In the following, we will 1) outline the methodological
framework, 2) provide an example workflow on how to obtain a call set starting
from short-read alignments, and 3) illustrate potential benefits of the methods on
a biological cancer data set. This methodology can be used for a high
performance identification of variant sites as well as to quantitatively assess
the presence or absence in comparisons between matched samples.

## 2 Methodology

### 2.1 Comparative shift of non-consensus base call frequencies

Starting with a set of aligned reads we observe the sequencing depth \(N^{S}_{i}\) and the number of non-consensus base calls \(K^{S}_{i}\) for a sample \(S\) at position \(i\). Non-consensus is here defined as base calls that differ from the consensus sequence, which can be either the reference sequence or, in a comparative setting, the most abundant base call in the control sample. The non-consensus rate \(p^{S}_{i}\), which we assume to be binomially distributed, can then be estimated as

$$\hat{p}^{S}_{i} = \frac{K^{S}_{i}}{N^{S}_{i}}$$

for sites with \(N^{S}_{i} > 0\). Since \(K^{S}_{i} \leq N^{S}_{i}\), the rates are bound to the range \([0,1]\).

The true non-consensus rate

$$p^{S}_{i} = v^{S}_{i} + e^{S}_{i}$$

comprises the presence of a putative variant with a frequency \(v^{S}_{i}\) and a technical error rate \(e^{S}_{i}\). In order to detect and describe the change in the variant frequency, we focus on the shift \(d_{i}\) in non-consensus rates as the difference of the rates between the test and control samples, which we estimate as

$$\hat{d}_{i} = \hat{p}^{T}_{i} - \hat{p}^{C}_{i}.$$

If we assume that the true site-specific technical error rates are identical between the two matched samples [10], the difference of the rates yields an unbiased estimate for the change in the variant frequency. Thus, positions not haboring biological alterations will result in \(\hat{d}_{i} \approx 0\).

### 2.2 Confidence intervals

Distinguishing biological variants from noise requires knowledge about the variance of the point estimate \(\hat{d_{i}}\). By constructing a confidence interval (CI) for \(d_{i}\) with confidence level \(\beta\) [14], we assess the certainty of the estimated shift in non-consensus frequencies. The probability of the true value being outside the confidence interval is less than \(\alpha = 1 - \beta\). This is in concordance with the type I or \(\alpha\) error definition in statistical testing.

Under the assumption that the non-consensus counts \(K^{S}_{i}\) in our samples follow binomial distributions with parameters \(p^{S}_{i}\) and \(N^{S}_{i}\), several methods have been established for estimating confidence intervals for the difference of two rate parameters [12] [7]. The performance of an approach is generally described in terms of its coverage probability indicating the probability of a confidence interval to cover the true value (see 4.2). Coverage probabilities greater and less than the confidence level \(\beta\) describe conservative and liberal behaviors, respectively. Due to the conservative coverage probabilities and high computational effort of exact confidence interval estimates, approximate methods are generally preferred [1] [7].

The Agresti-Caffo (AC) confidence interval [2]

$$\tilde{p}^{T} - \tilde{p}^{C} \pm z \sqrt{ \frac{\tilde{p}^{T} (1 - \tilde{p}^{T})} {\tilde{N}^{T}} + \frac{\tilde{p}^{C}(1 - \tilde{p}^{C})} {\tilde{N}^{C}} }$$

with

$$\tilde{p}^{X} = \frac{K^{X}+\zeta}{N^{X}+2\zeta},$$

$$\tilde{N}^{X} = N^{X} + 2\zeta,$$

$$\zeta = \frac{1}{4} z^2,$$

and \(z = z_{(1-\beta)/2}\) as the upper \((1-\beta)/2\) percentile of the standard normal distribution), is an approximation of the score test-based confidence interval. Several publications emphasize the usefulness and advantages of the AC method over related approaches [7] [3] [5].

#### 2.2.1 Decision making with confidence intervals

While the estimate for the shift in the non-consenus frequency \(\hat{d}\) indicates the change in abundance and direction of a variant, the corresponding confidence interval gives us information about the precision and power of the estimate. Generally, wide confidence intervals will be present at sites with little statistical power, as due to low sequencing depths.

For the case that we compare a tumor to a matched normal sample, we show a set of hypothetical cases that can be distinguished by regarding the point estimate and its confidence interval:

- Presence of a somatic, heterozygous variant
- Presence of a somatic, subclonal variant
- Presence of loss of heterozygosity
- Absence of a somatic variant
- Presence or absence of a variant cannot be distinguished due to the low certainty of the estimate
- No power due to insufficient sequencing depth

### 2.3 Distinguishing event classes

Focusing on the comparative shift of non-consensus frequencies allows us to detect and
distinguish different types of events. Since `Rariant`

does not make explicit
assumptions about the abundance of a potential variant in the control sample, we
are further able to find clonal shifts, for example between different tumor
samples, or losses of heterozygocity. Generally, gains and losses of variant
alleles are characterized by positive and negative values of \(d\), respectively.
For a differentiated interpretation of the results, we classify a variant into
one of four classes:

- somatic
- A somatic variant that does not occur in the control sample
- hetero/LOH
- A shift away from heterozygous SNP in the control sample
- undecided
- Both of the 'somatic' or 'hetero' are possible
- powerless
- A distinction between the two classes cannot be made due to a lack of power

The classification is based on two binomial tests for each position:

- Somatic variants where the variant allele is not present in the control sample, rejecting a binomial test with the alternative hypothesis \(H_{1}: p^{C} > 0\).
- Sites with a loss of heterozygosity with a shift away from a heterozygous variant in the control sample, rejecting a binomial test with the alternative hypothesis \(H_{1}: p^{C} \neq \frac{1}{2}\).

### 2.4 Identifying variant sites in large datasets

The method that we have described before is suited for detecting variant positions efficiently in large sequencing datasets, including whole-exome and whole-genome sequencing. For this purpose, we test for a shift in non-consensus frequencies between two samples at each genomic position individually:

- Form the base counts table for four bases A, C, G, T from the aligned reads. In order to reduce the number of false counts, we can optionally exclude reads with low base calling quality and clip the head of each read.
- Determine the consensus sequence: In a comparative setting, we will use the most abundant base call.
- Calculate the sequencing depth \(N^{S}_{i}\), mismatch counts \(K^{S}_{i}\), and derived statistics for both samples, based on the consensus sequence (see 2.1).
- Find potential variant sites with a Fisher's Exact Test, comparing the number of mismatching and total bases between the samples: \({K^{T}_{i}, N^{T}_{i}, K^{C}_{i}, N^{C}_{i}}\). The p-values are corrected for multiple testing according to the Benjamini-Hochberg procedure. Only positions rejecting the null hypothesis at a significance level \(\alpha\) are further on considered as potential variants.
- Calculate Agresti-Caffo confidence intervals with confidence level \(\beta\), in order to evaluate presence or absence of the variant (see 2.2).
- Classify variant sites into the groups: somatic, LOH, undecided, and powerless (see 2.3).

## 3 Workflow

### 3.1 Multiple Sample Simulation Study

We want to further demostrate the usage and abilities of the `Rariant`

package
on a real-life data set. Due to legal and privacy issues, most human cancer
sequencing data is not publicly accessible and therefore cannot serve as an
example data set here. Alternatively, we conduct an analysis to mimic the
characteristics of current cancer sequencing studies.

For the purpose of the analysis, we compare three samples from the 1000 Genomes
project [6], serving as a
control/normal (`control`

) and two related test/tumor samples (`test`

and
`test2`

). Further, we simulate a clonal mixture (`mix`

) of the two test samples
by combining their reads.

library(Rariant) library(GenomicRanges) library(ggbio)

tp53_region = GRanges("chr17", IRanges(7565097, 7590856))

control_bam = system.file("extdata", "platinum", "control.bam", package = "Rariant", mustWork = TRUE) test1_bam = system.file("extdata", "platinum", "test.bam", package = "Rariant", mustWork = TRUE) test2_bam = system.file("extdata", "platinum", "test2.bam", package = "Rariant", mustWork = TRUE) mix_bam = system.file("extdata", "platinum", "mix.bam", package = "Rariant", mustWork = TRUE)

v_test1 = rariant(test1_bam, control_bam, tp53_region, select = FALSE)

Warning: 'rbind_all' is deprecated. Use 'bind_rows()' instead. See help("Deprecated")

v_test2 = rariant(test2_bam, control_bam, tp53_region, select = FALSE)

Warning: 'rbind_all' is deprecated. Use 'bind_rows()' instead. See help("Deprecated")

v_mix = rariant(mix_bam, control_bam, tp53_region, select = FALSE)

Warning: 'rbind_all' is deprecated. Use 'bind_rows()' instead. See help("Deprecated")

v_all = GenomicRangesList(T1 = v_test1, T2 = v_test2, M = v_mix)

Warning: 'GenomicRangesList' is deprecated. Use 'GRangesList(..., compress=FALSE)' instead. See help("Deprecated")

v_all = endoapply(v_all, updateCalls)

To better understand the evidence for the presence or absence of particular variants across samples, we plot the confidence intervals, colored according to the predicted event type, and abundance shifts for all sites of interest, colored according to the sign of the shift.

t_ci = tracks(lapply(v_all, plotConfidenceIntervals, color = "verdict")) + verdictColorScale() print(t_ci)

Warning: Removed 6 rows containing missing values (geom_pointrange).

t_ci = tracks(lapply(v_all, plotConfidenceIntervals, color = "eventType")) print(t_ci)

Warning: Removed 6 rows containing missing values (geom_pointrange).