Analyzing High Density Peptide Array Data using HERON

1 HERON

The goal of HERON (Hierarchical Epitope pROtein biNding) is to analyze peptide binding array data measured from nimblegen or any high density peptide binding array data.

1.1 Installation

You can install the released version of HERON from bioconductor with:

if (!require("BiocManager", quietly = TRUE))
    install.packages("BiocManager")

# The following initializes usage of Bioc devel
BiocManager::install(version='devel')

BiocManager::install("HERON")

And the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("Ong-Research/HERON")
options(warn=2)
library(HERON)
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1.2 Example

These are examples which shows you how to interact with the code. We will be using a subset of the COVID-19 peptide binding array dataset from the https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8245122/ publication.

1.2.1 Pre-process data

First, we start with a sequence data.frame where rows are amino acid sequences of the peptide binding probe and the columns are samples.

data(heffron2021_wuhan)

knitr::kable(assay(heffron2021_wuhan)[1:5,1:5])
Sample_9 Sample_1 Sample_39 Sample_23 Sample_56
AAAYYVGYLQPRTFLL 0.3746667 0.5423724 0.7329991 0.1225761 0.9727648
AADLDDFSKQLQQSMS 2.8056786 3.1205293 9.2330091 4.2324981 9.8410852
AAEASKKPRQKRTATK 0.2704410 0.7198765 2.6648164 1.7079940 4.5664978
AAEIRASANLAATKMS 0.7120529 1.3820778 1.5463912 3.2038894 7.3400386
AAIVLQLPQGTTLPKG 2.9876932 0.1259101 4.6971758 5.6496847 2.2246734

The data is pre-processed by first quantile normalizing on sequence-level data.

seq_ds_qn <- quantileNormalize(heffron2021_wuhan)
knitr::kable(head(assay(seq_ds_qn)[,1:5]))
Sample_9 Sample_1 Sample_39 Sample_23 Sample_56
AAAYYVGYLQPRTFLL 0.3569421 0.8802587 0.5174103 0.1539593 0.8632584
AADLDDFSKQLQQSMS 2.4975483 4.4552679 6.6610043 4.8175685 7.2030291
AAEASKKPRQKRTATK 0.2706685 1.1883782 2.0845928 1.6878615 3.4542907
AAEIRASANLAATKMS 0.6506443 2.1632309 1.1600102 3.6584394 5.0898633
AAIVLQLPQGTTLPKG 2.8027363 0.2307280 3.2037424 6.1906585 1.8298191
AALALLLLDRLNQLES 0.5432972 0.4990815 2.2830699 0.0681958 2.9236376

1.2.2 Calculate probe-level p-values

The quantile normalized data is then used with the colData and probe_meta data frames to calculate probe-level p-values. The colData data frame describes the experimental setup of the samples. for the colData data frame, the important columns are ptid, visit, and SampleName. The SampleName is the column name of the sequence table, ptid is the name of the sample, and visit is either pre or post. The ptid can the same value for one pre and post sample, which would indicate a paired design. A condition column can also be provided which can indicate multiple conditions for the post samples.

knitr::kable(head(colData(heffron2021_wuhan)))
SampleName ptid visit condition
Sample_9 Sample_9 Sample_9 pre Control
Sample_1 Sample_1 Sample_1 pre Control
Sample_39 Sample_39 Sample_39 post COVID
Sample_23 Sample_23 Sample_23 post COVID
Sample_56 Sample_56 Sample_56 post COVID
Sample_53 Sample_53 Sample_53 post COVID

The probe_meta data frame describes the mapping the amino acid sequence (PROBE_SEQUENCE) of the probes to the probe identifier (PROBE_ID). The PROBE_ID contains the name of the protein and the position within the protein of the first acid of the probe sequence separated by a semicolon.

probe_meta <- metadata(heffron2021_wuhan)$probe_meta
knitr::kable(head(probe_meta[,c("PROBE_SEQUENCE", "PROBE_ID")]))
PROBE_SEQUENCE PROBE_ID
76322 MYSFVSEETGTLIVNS NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;1
76323 YSFVSEETGTLIVNSV NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;2
76324 SFVSEETGTLIVNSVL NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;3
76325 FVSEETGTLIVNSVLL NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;4
76326 VSEETGTLIVNSVLLF NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;5
76327 SEETGTLIVNSVLLFL NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;6

The calcCombPValues call will first calculate the probe level p-values using the colData, adjust the probe p-values on a column-by-column basis frame.

The combined p-values can be calculated on the sequence data set, then converted to a probe data set through convertSequenceDSToProbeDS, or calculated on the probe data set. If the data was smoothed prior using consecutive probes, then the p-values should calculated on the smoothed probe data set.

seq_pval_res <- calcCombPValues(seq_ds_qn)
probe_pval_res <- convertSequenceDSToProbeDS(seq_pval_res)
knitr::kable(head(assays(probe_pval_res)$padj))
Sample_39 Sample_23 Sample_56 Sample_53 Sample_43 Sample_22 Sample_51 Sample_49 Sample_25 Sample_33 Sample_30 Sample_55 Sample_59 Sample_27 Sample_36 Sample_54 Sample_31 Sample_45 Sample_26 Sample_58 Sample_60 Sample_32 Sample_24 Sample_42 Sample_29 Sample_34 Sample_35 Sample_44 Sample_38 Sample_21 Sample_40 Sample_57 Sample_47 Sample_46 Sample_41 Sample_28 Sample_48 Sample_52 Sample_50 Sample_37
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;1 1 1.0000000 1.0000000 0.7009991 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.9804672 1 1 1 1.0000000 1 1 1 1 0.0109035 1 1 1 1 1 1 1
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;2 1 1.0000000 1.0000000 1.0000000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.0000000 1 1 1 1.0000000 1 1 1 1 0.0444799 1 1 1 1 1 1 1
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;3 1 0.1693163 0.7731021 1.0000000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.0000000 1 1 1 0.9938559 1 1 1 1 0.1018061 1 1 1 1 1 1 1
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;4 1 1.0000000 1.0000000 1.0000000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.0000000 1 1 1 1.0000000 1 1 1 1 0.6824978 1 1 1 1 1 1 1
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;5 1 1.0000000 1.0000000 1.0000000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.0000000 1 1 1 1.0000000 1 1 1 1 1.0000000 1 1 1 1 1 1 1
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;6 1 1.0000000 1.0000000 1.0000000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.0000000 1 1 1 1.0000000 1 1 1 1 1.0000000 1 1 1 1 1 1 1

1.2.3 Obtain probe-level calls

Once the p-values have been calculated, the makeProbeCalls will make the calls using the adjusted p-values calculated. makeProbeCalls also includes a filter to remove one-hit probes, which are probes that were called only in one sample and do not have a consecutive probe called in a single sample. The return value is HERONProbeDataSet with the calls included as an additional assay.

probe_calls_res <- makeProbeCalls(probe_pval_res)
knitr::kable(head(assay(probe_calls_res, "calls")[,1:5]))
Sample_39 Sample_23 Sample_56 Sample_53 Sample_43
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;1 FALSE FALSE FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;2 FALSE FALSE FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;3 FALSE FALSE FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;4 FALSE FALSE FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;5 FALSE FALSE FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;6 FALSE FALSE FALSE FALSE FALSE

K of N, Fraction, and Percent of calls made per sequence, probe, epitope, or protein can be determined using the getKofN function.

probe_calls_k_of_n <- getKofN(probe_calls_res)

1.2.4 Find Epitope Segments using the unique method

After calculating probe p-values and calls, the HERON can find epitope segments, i.e. blocks of consecutive probes that have been called within a sample or cluster of samples. The unique method finds all consecutive probes for each sample, then returns the unique set of all epitopic regions.


epi_segments_uniq_res <- findEpitopeSegments(
    probe_calls_res,
    segment_method = "unique"
)

knitr::kable(head(epi_segments_uniq_res))
x
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_140_141
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_172_173
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_247_247
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_542_546
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_553_560
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_568_577

The format of the epitope segments are SEQID_Begin_End, where the SEQID is the protein name, Begin is the starting position of the first probe within the consecutive block, and the End is the starting position of the last probe within the consecutive block.

1.2.5 Calculate Epitope-level p-values

With the epitope segments and the probe p-values, HERON can calculate a significance value for the segments using a meta p-value method. Here, we are calculating epitope p-values using Wilkinson’s max meta p-value method.


epi_padj_uniq <- calcEpitopePValues(
    probe_calls_res,
    epitope_ids = epi_segments_uniq_res,
    metap_method = "wmax1"
)

You can add sequence annotations to the row data of the HERONEpitopeDataSet object.

epi_padj_uniq <- addSequenceAnnotations(epi_padj_uniq)

1.2.6 Obtain Epitope-level calls

The makeEpitopeCalls method will work on the epitope adjusted p-values to make epitope-level sample. getKofN can also get the K of N results.


epi_calls_uniq <- makeEpitopeCalls(epi_padj_uniq, one_hit_filter = TRUE)
epi_calls_k_of_n_uniq <- getKofN(epi_calls_uniq)
knitr::kable(head(epi_calls_k_of_n_uniq))
K F P
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_140_141 4 0.100 10.0
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_172_173 9 0.225 22.5
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_247_247 11 0.275 27.5
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_542_546 12 0.300 30.0
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_553_560 34 0.850 85.0
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface_568_577 37 0.925 92.5

1.2.7 Calculate Protein-level p-values

Calculate protein p-values using Tippett’s (Wilkinson’s Min) method.

prot_padj_uniq <- calcProteinPValues(
    epi_padj_uniq,
    metap_method = "tippetts"
)

1.2.8 Obtain Protein-level calls

prot_calls_uniq <- makeProteinCalls(prot_padj_uniq)
prot_calls_k_of_n_uniq <- getKofN(prot_calls_uniq)
knitr::kable(head(prot_calls_k_of_n_uniq))
K F P
NC_045512.2;YP_009724390.1;Wu1-SARS2_surface 40 1.0 100
NC_045512.2;YP_009724393.1;Wu1-SARS2_membrane 40 1.0 100
NC_045512.2;YP_009724397.2;Wu1-SARS2_nucleoca 40 1.0 100
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope 12 0.3 30

1.3 Other

The are other functions to utilize depending upon what you would like to do. For example, there are different methods for finding the epitope segments that involve clustering across samples. There are two main methods, one using hierarchical clustering and another using the skater method from the spdep package. In addition to the two methods, how the distance matrix is calculated can be modified. The following subsections demonstrate how to find epitope segment blocks using hclust or skater and using a binary hamming distance or a numeric euclidean distance for making calls. After the segments are found, you can then use the calcEpitopePValuesMat and makeEpitopeCalls functions as before to find the epitope p-values, protein p-values, and the respective calls.

1.3.1 Find Epitope Segments using the hclust method

1.3.1.1 binary calls with hamming distance

epi_segments_hclust_res <- findEpitopeSegments(
    probe_calls_res,
    segment_method = "hclust",
    segment_score_type = "binary",
    segment_dist_method = "hamming",
    segment_cutoff = "silhouette"
)

1.3.1.2 z-scores with euclidean distance

epi_segments_hclust_res2 <- findEpitopeSegments(
    probe_calls_res,
    segment_method = "hclust",
    segment_score_type = "zscore",
    segment_dist_method = "euclidean",
    segment_cutoff = "silhouette"
)

1.3.2 Find Epitope Segments using the skater method

1.3.2.1 binary calls with hamming distance

epi_segments_skater_res <- findEpitopeSegments(
    probe_calls_res,
    segment_method = "skater",
    segment_score_type = "binary",
    segment_dist_method = "hamming",
    segment_cutoff = "silhouette"
)

1.3.2.2 z-scores with euclidean distance

epi_segments_skater_res <- findEpitopeSegments(
    probe_calls_res,
    segment_method = "skater",
    segment_score_type = "zscore",
    segment_dist_method = "euclidean",
    segment_cutoff = "silhouette"
)

1.3.3 Other meta p-value methods

HERON also allows a choice for a number of meta p-value methods for combining p-values for epitopes (calcEpitopePValuesPDS) and proteins (calcProteinPValuesEDS). The methods supported by HERON are : min_bonf, min, max, fischer/sumlog, hmp/harmonicmeanp, wilkinsons_min1/tippets, wilkinsons_min2/wmin2, wilkinsons_min3, wilkinsons_min4, wilkinsons_min5, wilkinsons_max1/wmax1, wilkinsons_max2/wmax2, and cct.

When choosing a p-value method, keep in mind that the epitope p-value should be one that requires most of the probe p-values to be small (e.g. wmax1) and the protein should be one that requires at least one of the epitope p-values to be small (e.g. wmax1). Other p-value methods such as the cct and the hmp have been shown to be more accurate with p-value that have dependencies.

1.3.4 Making z-score cutoff calls

Some investigators would like to make z-score global level calls rather than using adjusted p-values. HERON is flexible to allow for such a setup. For example, the user wanted to make probe-level calls using a z-score cutoff > 2.


seq_pval_res_z <- calcCombPValues(
    obj = seq_ds_qn, 
    use = "z", 
    p_adjust_method = "none"
)


p_cutoff <- pnorm(2, lower.tail = FALSE)

probe_pval_res_z <- convertSequenceDSToProbeDS(seq_pval_res_z, probe_meta)

probe_calls_z2 <- makeProbeCalls(probe_pval_res_z, padj_cutoff = p_cutoff)
probe_k_of_n_z2 <- getKofN(probe_calls_z2)

knitr::kable(head(assay(probe_calls_z2,"calls")[,1:5]))
Sample_39 Sample_23 Sample_56 Sample_53 Sample_43
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;1 FALSE FALSE FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;2 FALSE FALSE FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;3 FALSE FALSE FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;4 FALSE FALSE FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;5 FALSE FALSE FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;6 FALSE FALSE FALSE FALSE FALSE
knitr::kable(head(probe_k_of_n_z2[probe_k_of_n_z2$K > 0,]))
K F P
NC_045512.2;YP_009724393.1;Wu1-SARS2_membrane;4 1 0.025 2.5
NC_045512.2;YP_009724393.1;Wu1-SARS2_membrane;5 3 0.075 7.5
NC_045512.2;YP_009724393.1;Wu1-SARS2_membrane;6 1 0.025 2.5
NC_045512.2;YP_009724393.1;Wu1-SARS2_membrane;7 4 0.100 10.0
NC_045512.2;YP_009724393.1;Wu1-SARS2_membrane;8 10 0.250 25.0
NC_045512.2;YP_009724393.1;Wu1-SARS2_membrane;156 2 0.050 5.0

The calls can also be used to find epitopes segments, p-values, and calls. Also, can be used to make protein level calls.

epi_segments_uniq_z2_res <- findEpitopeSegments(
    probe_calls_z2,
    segment_method = "unique"
)

If we want to find regions where the z > 2 for all of the consecutive probes, using the max meta p-value method will ensure that.


epi_pval_uniq_z2 <- calcEpitopePValues(
    probe_pds = probe_pval_res_z,
    epitope_ids = epi_segments_uniq_z2_res,
    metap_method = "max",
    p_adjust_method = "none"
)

Again, makeEpitopeCalls will be used in order to find significant regions.

epi_calls_uniq_z2 <- makeEpitopeCalls(
    epi_ds = epi_pval_uniq_z2, 
    padj_cutoff = p_cutoff
)

Other segmentation methods can be used with the called regions through the binary score clustering methods.

epi_segments_skater_z2_res <- findEpitopeSegments(
    probe_calls_z2,
    segment_method = "skater",
    segment_score_type = "binary",
    segment_dist_method = "hamming",
    segment_cutoff = "silhouette")

1.3.5 Smoothing across probes

The pepStat paper (https://pubmed.ncbi.nlm.nih.gov/23770318/) mentions that smoothing can sometimes help reduce the high-variability of the peptide array data. HERON has implemented a running average smoothing algorithm across protein probes that can handle missing values. The function smoothProbeMat will take as input a probe matrix and will return a matrix that has been smoothed. The smoothed matrix can then be processed through the workflow using the calcProbePValuesProbeMat call instead of the calcProbePValuesSeqMat call since now the probe signal is no longer a direct copy from the sequence.


probe_ds_qn <- convertSequenceDSToProbeDS(seq_ds_qn, probe_meta )

smooth_ds <- smoothProbeDS(probe_ds_qn)

After you smoothed the data using the probes, the probe p-value will be calculated on the probe-level rather than the sequence-level.

probe_sm_pval <- calcCombPValues(smooth_ds)

The probe calls can then be made as before.

probe_sm_calls <- makeProbeCalls(probe_sm_pval)
probe_sm_k_of_n <- getKofN(probe_sm_calls)
knitr::kable(assay(probe_sm_calls,"calls")[1:3,1:3])
Sample_39 Sample_23 Sample_56
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;1 FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;2 FALSE FALSE FALSE
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;3 FALSE FALSE FALSE
knitr::kable(head(probe_sm_k_of_n[probe_sm_k_of_n$K > 0,]))
K F P
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;1 1 0.025 2.5
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;2 1 0.025 2.5
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;57 1 0.025 2.5
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;58 1 0.025 2.5
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;59 1 0.025 2.5
NC_045512.2;YP_009724392.1;Wu1-SARS2_envelope;60 2 0.050 5.0

1.3.6 Calculate paired t-tests

Paired t-tests can be utilized and combined with the z-tests as well. Here is an example where we pair the five COVID- samples to the COVID+ samples and run calcProbePValuesSeqMat with the t_paired parameter set.

data(heffron2021_wuhan)
probe_meta <- attr(heffron2021_wuhan, "probe_meta")
colData_paired <- colData(heffron2021_wuhan)

## Make some samples paired by setting the sample ids
pre_idx <- which(colData_paired$visit == "pre")
colData_post <- colData_paired[colData_paired$visit == "post",]
new_ids <- colData_post$SampleName[seq_len(5)]
colData_paired$ptid[pre_idx[seq_len(5)]] = new_ids

paired_ds <- heffron2021_wuhan
colData(paired_ds) <- colData_paired

## calculate p-values
pval_res <- calcCombPValues(
    obj = paired_ds,
    t_paired = TRUE
)

knitr::kable(assay(pval_res[1:3,],"pvalue"))
Sample_39 Sample_23 Sample_56 Sample_53 Sample_43
AAAYYVGYLQPRTFLL 0.4192558 0.8596157 0.3643977 0.5703188 0.5695711
AADLDDFSKQLQQSMS 0.0001467 0.0913935 0.0000501 0.7835355 0.0000709
AAEASKKPRQKRTATK 0.0839504 0.2146309 0.0049659 0.2129732 0.1286111

2 Use of the condition column

col_data <- colData(heffron2021_wuhan)
covid <- which(col_data$visit == "post")

col_data$condition[covid[1:10]] <- "COVID2"

seq_ds <- heffron2021_wuhan
colData(seq_ds) <- col_data

seq_ds_qn <- quantileNormalize(seq_ds)
seq_pval_res <- calcCombPValues(seq_ds_qn)
probe_pval_res <- convertSequenceDSToProbeDS(seq_pval_res)
probe_calls_res <- makeProbeCalls(probe_pval_res)
probe_calls_k_of_n <- getKofN(probe_calls_res)

2.1 Funding

HERON and its developers have been partially supported by funding from the Clinical and Translational Science Award (CTSA) program (ncats.nih.gov/ctsa), through the National Institutes of Health National Center for Advancing Translational Sciences (NCATS), grants UL1TR002373 and KL2TR002374, NIH National Institute of Allergy and Infectious Diseases, 2U19AI104317-06, NIH National Cancer Institute (P30CA14520, P50CA278595, and P01CA250972), startup funds through the University of Wisconsin Department of Obstetrics and Gynecology and the University of Wisconsin-Madison Office of the Chancellor and the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation through the Data Science Initiative award.

2.2 Acknowledgments

We have benefited in the development of HERON from the help and feedback of many individuals, including but not limited to:

The Bioconductor Core Team, Paul Sondel, Anna Hoefges, Amy Erbe Gurel, Jessica Vera, Rene Welch, Ken Lo (Nimble Therapeutics), Brad Garcia (Nimble Therapeutics), Jigar Patel (Nimble Therapeutics), John Tan, Nicholas Mathers, Richard Pinapatti.

2.3 Session Info

sessionInfo()
#> R Under development (unstable) (2023-10-22 r85388)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 22.04.3 LTS
#> 
#> Matrix products: default
#> BLAS:   /home/biocbuild/bbs-3.19-bioc/R/lib/libRblas.so 
#> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
#> 
#> locale:
#>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
#>  [3] LC_TIME=en_GB              LC_COLLATE=C              
#>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
#>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
#>  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
#> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
#> 
#> time zone: America/New_York
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats4    stats     graphics  grDevices utils     datasets  methods  
#> [8] base     
#> 
#> other attached packages:
#>  [1] HERON_1.1.1                 SummarizedExperiment_1.33.0
#>  [3] Biobase_2.63.0              GenomicRanges_1.55.1       
#>  [5] GenomeInfoDb_1.39.0         IRanges_2.37.0             
#>  [7] S4Vectors_0.41.1            BiocGenerics_0.49.0        
#>  [9] MatrixGenerics_1.15.0       matrixStats_1.0.0          
#> 
#> loaded via a namespace (and not attached):
#>  [1] tidyselect_1.2.0        dplyr_1.1.3             bitops_1.0-7           
#>  [4] fastmap_1.1.1           RCurl_1.98-1.12         TH.data_1.1-2          
#>  [7] mathjaxr_1.6-0          digest_0.6.33           lifecycle_1.0.3        
#> [10] sf_1.0-14               cluster_2.1.4           survival_3.5-7         
#> [13] statmod_1.5.0           magrittr_2.0.3          compiler_4.4.0         
#> [16] rlang_1.1.1             sass_0.4.7              tools_4.4.0            
#> [19] utf8_1.2.4              plotrix_3.8-2           yaml_2.3.7             
#> [22] data.table_1.14.8       knitr_1.45              sn_2.1.1               
#> [25] S4Arrays_1.3.0          sp_2.1-1                classInt_0.4-10        
#> [28] mnormt_2.1.1            DelayedArray_0.29.0     multcomp_1.4-25        
#> [31] abind_1.4-5             KernSmooth_2.23-22      numDeriv_2016.8-1.1    
#> [34] grid_4.4.0              fansi_1.0.5             multtest_2.59.0        
#> [37] e1071_1.7-13            MASS_7.3-60.1           cli_3.6.1              
#> [40] mvtnorm_1.2-3           rmarkdown_2.25          crayon_1.5.2           
#> [43] generics_0.1.3          metap_1.9               spdep_1.2-8            
#> [46] DBI_1.1.3               cachem_1.0.8            proxy_0.4-27           
#> [49] zlibbioc_1.49.0         splines_4.4.0           s2_1.1.4               
#> [52] XVector_0.43.0          vctrs_0.6.4             boot_1.3-28.1          
#> [55] Matrix_1.6-1.1          sandwich_3.0-2          jsonlite_1.8.7         
#> [58] spData_2.3.0            limma_3.59.1            jquerylib_0.1.4        
#> [61] units_0.8-4             glue_1.6.2              codetools_0.2-19       
#> [64] deldir_1.0-9            tibble_3.2.1            pillar_1.9.0           
#> [67] htmltools_0.5.6.1       GenomeInfoDbData_1.2.11 R6_2.5.1               
#> [70] wk_0.9.0                Rdpack_2.5              evaluate_0.22          
#> [73] lattice_0.22-5          rbibutils_2.2.16        qqconf_1.3.2           
#> [76] TFisher_0.2.0           bslib_0.5.1             class_7.3-22           
#> [79] mutoss_0.1-13           Rcpp_1.0.11             SparseArray_1.3.0      
#> [82] xfun_0.40               pkgconfig_2.0.3         zoo_1.8-12