+
Differential expression analysis
+
The following functions are from packagetidybulk, taking a tbl input (with at least three columns for sample, feature and transcript abundance) or SummarizedExperiment as input.
+
+
+keep_abundant() returns a consistent object (to the input) with additional columns for the statistics from the hypothesis test.scale_abundance() scales transcript abundance compansating for sequencing depth.reduce_dimensions() calculates the reduced dimensional space of the transcript abundance.pivot_sample() returns a tbl with only sample-related columns.test_differential_abundance() performs differential transcription testing using edgeR quasi-likelihood (QLT), edgeR likelihood-ratio (LR), limma-voom, limma-voom-with-quality-weights or DESeq2.pivot_transcript() returns a tbl with only transcript-related columns.
+
+
Counts normalisation
+
Visualise the difference in abundance densities before and after scaling by plotting:
+
+
+- Raw counts density
+- Raw counts scaled density
+- Quantile normalised density
+
+
counts |> keep_abundant(factor_of_interest = disease) |>
+ scale_abundance(.subset_for_scaling = TRUE) |>
+ pivot_longer(
+ cols = c("counts", "quantile_normalised", "counts_scaled"),
+ names_to = "source",
+ values_to = "abundance"
+) |>
+
+ # Plotting
+ ggplot(aes(x = abundance + 1, color = sample_)) +
+ geom_density() +
+ facet_wrap( ~ source) +
+ scale_x_log10() +
+ custom_theme
+
## tidybulk says: the sample with largest library size GSE122960___GSM3489183___macrophage was chosen as reference for scaling
+
## tidySummarizedExperiment says: A data frame is returned for independent data analysis.
+
+
In the curated atlas, the distribution of the scaled counts and quantile normalised counts are not very different to each other. Comparing to raw counts, both scaling and quantile normalising make the distribution similar across samples.
+
+
+
Dimensionality reduction PCA visualisation
+
Idiopathic pulmonary fibrosis (IPF) shows some degree of separation from normal in the sample, especially noticeable in certain cell types. This separation could indicate that the disease state influences the gene expression profile of these cells.
+
+
Scaled raw counts PCA
+
# Get principal components
+counts_scal_PCA <-
+ counts |>
+ keep_abundant(factor_of_interest = disease) |>
+ scale_abundance() |>
+ reduce_dimensions(method = "PCA")
+
## tidybulk says: the sample with largest library size GSE122960___GSM3489183___macrophage was chosen as reference for scaling
+
## Getting the 500 most variable genes
+
## Fraction of variance explained by the selected principal components
+
## # A tibble: 2 × 2
+## `Fraction of variance` PC
+## <dbl> <int>
+## 1 0.450 1
+## 2 0.0990 2
+
## tidybulk says: to access the raw results do `attr(..., "internals")$PCA`
+
counts_scal_PCA |>
+ pivot_sample() |>
+ ggplot(aes(
+ x = PC1,
+ y = PC2,
+ colour = cell_type_harmonised,
+ shape = disease
+ )) +
+ geom_point() +
+ custom_theme
+
+
The PCA plot indicates that same cell types tend to cluster together.
+
+
+
+
Differential expression models
+
de_all <- counts_scal_PCA |>
+
+ # edgeR QLT
+ test_differential_abundance(
+ ~ disease + age_days + sex,
+ method = "edgeR_quasi_likelihood",
+ prefix = "edgerQLT_"
+ ) |>
+
+ # edgeR LRT
+ test_differential_abundance(
+ ~ disease + age_days + sex,
+ method = "edgeR_likelihood_ratio",
+ prefix = "edgerLR_"
+ ) |>
+
+ # limma-voom
+ test_differential_abundance(
+ ~ disease + age_days + sex,
+ method = "limma_voom",
+ prefix = "voom_"
+ ) |>
+
+ # DESeq2
+ test_differential_abundance(
+ ~ disease + age_days + sex,
+ method = "deseq2",
+ prefix = "deseq2_"
+ )
+
## =====================================
+## tidybulk says: All testing methods use raw counts, irrespective of if scale_abundance
+## or adjust_abundance have been calculated. Therefore, it is essential to add covariates
+## such as batch effects (if applicable) in the formula.
+## =====================================
+## tidybulk says: The design column names are "(Intercept), diseasenormal, age_days, sexmale"
+##
+## tidybulk says: to access the raw results (fitted GLM) do `attr(..., "internals")$edgeR_quasi_likelihood`
+## tidybulk says: The design column names are "(Intercept), diseasenormal, age_days, sexmale"
+##
+## tidybulk says: to access the raw results (fitted GLM) do `attr(..., "internals")$edgeR_likelihood_ratio`
+## tidybulk says: The design column names are "(Intercept), diseasenormal, age_days, sexmale"
+##
+## tidybulk says: to access the raw results (fitted GLM) do `attr(..., "internals")$limma_voom`
+## converting counts to integer mode
+##
+## the design formula contains one or more numeric variables with integer values,
+## specifying a model with increasing fold change for higher values.
+## did you mean for this to be a factor? if so, first convert
+## this variable to a factor using the factor() function
+##
+## the design formula contains one or more numeric variables that have mean or
+## standard deviation larger than 5 (an arbitrary threshold to trigger this message).
+## Including numeric variables with large mean can induce collinearity with the intercept.
+## Users should center and scale numeric variables in the design to improve GLM convergence.
+##
+## Note: levels of factors in the design contain characters other than
+## letters, numbers, '_' and '.'. It is recommended (but not required) to use
+## only letters, numbers, and delimiters '_' or '.', as these are safe characters
+## for column names in R. [This is a message, not a warning or an error]
+##
+## estimating size factors
+##
+## Note: levels of factors in the design contain characters other than
+## letters, numbers, '_' and '.'. It is recommended (but not required) to use
+## only letters, numbers, and delimiters '_' or '.', as these are safe characters
+## for column names in R. [This is a message, not a warning or an error]
+##
+## estimating dispersions
+##
+## gene-wise dispersion estimates
+##
+## mean-dispersion relationship
+##
+## Note: levels of factors in the design contain characters other than
+## letters, numbers, '_' and '.'. It is recommended (but not required) to use
+## only letters, numbers, and delimiters '_' or '.', as these are safe characters
+## for column names in R. [This is a message, not a warning or an error]
+##
+## final dispersion estimates
+##
+## fitting model and testing
+##
+## Note: levels of factors in the design contain characters other than
+## letters, numbers, '_' and '.'. It is recommended (but not required) to use
+## only letters, numbers, and delimiters '_' or '.', as these are safe characters
+## for column names in R. [This is a message, not a warning or an error]
+##
+## -- replacing outliers and refitting for 125 genes
+## -- DESeq argument 'minReplicatesForReplace' = 7
+## -- original counts are preserved in counts(dds)
+##
+## estimating dispersions
+##
+## fitting model and testing
+##
+## Note: levels of factors in the design contain characters other than
+## letters, numbers, '_' and '.'. It is recommended (but not required) to use
+## only letters, numbers, and delimiters '_' or '.', as these are safe characters
+## for column names in R. [This is a message, not a warning or an error]
+##
+## tidybulk says: to access the raw results (fitted GLM) do `attr(..., "internals")$deseq2`
+## This message is displayed once per session.
+
+
+
Comparison of methods
+
The comparison help assess how similarly each pair of tests ranks the significance of gene expression differences.
+
The correlation coefficients between edgeR QLT and edgeR LRT, suggest a very high degree of agreement, which is expected as these methods are variations of each other under the edgeR framework. Meanwhile, correlations involving limma-voom are generally lower, indicating differing sensitivities or assumptions in the statistical models used by limma compared to edgeR or DESeq2.
+
de_all |>
+ pivot_transcript() |>
+ select(edgerQLT_PValue, edgerLR_PValue, voom_P.Value, deseq2_pvalue, .feature) |>
+ ggpairs(1:4,
+ mapping = aes(alpha = 0.2),
+ lower = list(continuous = wrap("points", size = 0.4, alpha = 0.2)
+ )
+ )
+
+
+
+
Volcano plots
+
# run default edgeR method
+counts_de <- counts_scal_PCA |>
+ test_differential_abundance(~ disease + age_days + sex )
+
## tidybulk says: The design column names are "(Intercept), diseasenormal, age_days, sexmale"
+
## tidybulk says: to access the raw results (fitted GLM) do `attr(...,
+## "internals")$edgeR_quasi_likelihood`
+
topgenes_symbols <- counts_de |>
+ pivot_transcript() |>
+ arrange(PValue) |>
+ head(6) |> pull(.feature)
+topgenes_symbols
+
## [1] "RPS26" "FOSB" "IGHG4" "FOS" "BEST1" "NFKBIZ"
+
counts_de |>
+ pivot_transcript() |>
+
+ # Subset data
+ mutate(significant = FDR < 0.05 & abs(logFC) >= 2) |>
+ mutate(.feature = ifelse(.feature %in% topgenes_symbols, as.character(.feature), "")) |>
+
+ # Plot
+ ggplot(aes(x = logFC, y = PValue, label = .feature)) +
+ geom_point(aes(color = significant, size = significant, alpha = significant)) +
+ geom_text_repel() +
+
+ # Custom scales
+ custom_theme +
+ scale_y_continuous(trans = trans_reverser("log10")) +
+ scale_color_manual(values = c("black", "#e11f28")) +
+ scale_size_discrete(range = c(0, 2))
+
## Scale for colour is already present.
+## Adding another scale for colour, which will replace the existing scale.
+
## Warning: Using size for a discrete variable is not advised.
+
## Warning: Using alpha for a discrete variable is not advised.
+
+
In the volcano plot, top genes are annotated. The IGHG4 gene shows both significant changes in expression and high fold changes.
+
+
+
Analysis of a single gene across the curated data
+
+
Collect pseudobulk counts for TGFB and MUC5B genes
+
my_cache <- "/vast/scratch/users/shen.m/CuratedAtlasQueryR/pseudobulk"
+metadata <- get_metadata(cache_directory = my_cache)
+metadata
+
MUC5B_merged_counts <- metadata |>
+ get_pseudobulk(cache_directory = my_cache, assays = c("counts", "quantile_normalised"), feature = "MUC5B")
+
MUC5B_merged_counts
+
## # A SummarizedExperiment-tibble abstraction: 133,133 × 52
+## # [90mFeatures=1 | Samples=133133 | Assays=counts, quantile_normalised[0m
+## .feature .sample counts quantile_normalised sample_ cell_type_harmonised
+## <chr> <chr> <dbl> <dbl> <chr> <chr>
+## 1 MUC5B 689e2fe4f2c… 5.00 13.6 689e2f… basal_cell
+## 2 MUC5B 689e2fe4f2c… 10.0 280. 689e2f… luminal_cell_of_pro…
+## 3 MUC5B 689e2fe4f2c… 1.00 353. 689e2f… epithelial_cell
+## 4 MUC5B 689e2fe4f2c… 5.00 1303. 689e2f… secretory_cell
+## 5 MUC5B 19d3f767883… 20.0 1170. 19d3f7… secretory_cell
+## 6 MUC5B 19d3f767883… 0 0 19d3f7… luminal_cell_of_pro…
+## 7 MUC5B 19d3f767883… 0 0 19d3f7… epithelial_cell
+## 8 MUC5B 19d3f767883… 0 0 19d3f7… basal_cell
+## 9 MUC5B 19d3f767883… 0 0 19d3f7… neuroendocrine_cell
+## 10 MUC5B 4164e2a99c2… 1.00 0 4164e2… basal_cell
+## 11 MUC5B 4164e2a99c2… 0 0 4164e2… epithelial_cell
+## 12 MUC5B 4164e2a99c2… 30.0 17148. 4164e2… luminal_cell_of_pro…
+## 13 MUC5B 4164e2a99c2… 0 0 4164e2… secretory_cell
+## 14 MUC5B 4164e2a99c2… 0 0 4164e2… neuroendocrine_cell
+## 15 MUC5B bd54ab01e2d… 0 0 bd54ab… basal_cell
+## 16 MUC5B bd54ab01e2d… 0 0 bd54ab… epithelial_cell
+## 17 MUC5B bd54ab01e2d… 1.00 116. bd54ab… secretory_cell
+## 18 MUC5B bd54ab01e2d… 0 0 bd54ab… luminal_cell_of_pro…
+## 19 MUC5B bd54ab01e2d… 0 0 bd54ab… neuroendocrine_cell
+## 20 MUC5B 9092e556e98… 0 0 9092e5… basal_cell
+## # ℹ 46 more variables: X_sample_name <chr>, assay <chr>,
+## # assay_ontology_term_id <chr>, development_stage <chr>,
+## # development_stage_ontology_term_id <chr>, disease <chr>,
+## # disease_ontology_term_id <chr>, ethnicity <chr>,
+## # ethnicity_ontology_term_id <chr>, experiment___ <chr>, file_id <chr>,
+## # is_primary_data_x <chr>, organism <chr>, organism_ontology_term_id <chr>,
+## # sample_placeholder <chr>, sex <chr>, sex_ontology_term_id <chr>, …
+
TGFB1_merged_counts <- metadata |>
+ get_pseudobulk(cache_directory = my_cache, assays = c("counts", "quantile_normalised"), feature = "TGFB1")
+
TGFB1_merged_counts
+
## # A SummarizedExperiment-tibble abstraction: 133,133 × 52
+## # [90mFeatures=1 | Samples=133133 | Assays=counts, quantile_normalised[0m
+## .feature .sample counts quantile_normalised sample_ cell_type_harmonised
+## <chr> <chr> <dbl> <dbl> <chr> <chr>
+## 1 TGFB1 689e2fe4f2c… 568. 12932. 689e2f… basal_cell
+## 2 TGFB1 689e2fe4f2c… 54.0 3038. 689e2f… luminal_cell_of_pro…
+## 3 TGFB1 689e2fe4f2c… 20.0 11881. 689e2f… epithelial_cell
+## 4 TGFB1 689e2fe4f2c… 12.0 3282. 689e2f… secretory_cell
+## 5 TGFB1 19d3f767883… 69.0 4600. 19d3f7… secretory_cell
+## 6 TGFB1 19d3f767883… 13.0 2144. 19d3f7… luminal_cell_of_pro…
+## 7 TGFB1 19d3f767883… 166. 14872. 19d3f7… epithelial_cell
+## 8 TGFB1 19d3f767883… 247. 15054. 19d3f7… basal_cell
+## 9 TGFB1 19d3f767883… 0 0 19d3f7… neuroendocrine_cell
+## 10 TGFB1 4164e2a99c2… 446. 10388. 4164e2… basal_cell
+## 11 TGFB1 4164e2a99c2… 17.0 6691. 4164e2… epithelial_cell
+## 12 TGFB1 4164e2a99c2… 1.00 310. 4164e2… luminal_cell_of_pro…
+## 13 TGFB1 4164e2a99c2… 3.00 2979. 4164e2… secretory_cell
+## 14 TGFB1 4164e2a99c2… 0 0 4164e2… neuroendocrine_cell
+## 15 TGFB1 bd54ab01e2d… 544. 15553. bd54ab… basal_cell
+## 16 TGFB1 bd54ab01e2d… 270. 11032. bd54ab… epithelial_cell
+## 17 TGFB1 bd54ab01e2d… 10.0 3354. bd54ab… secretory_cell
+## 18 TGFB1 bd54ab01e2d… 0 0 bd54ab… luminal_cell_of_pro…
+## 19 TGFB1 bd54ab01e2d… 1.00 16705. bd54ab… neuroendocrine_cell
+## 20 TGFB1 9092e556e98… 281. 7546. 9092e5… basal_cell
+## # ℹ 46 more variables: X_sample_name <chr>, assay <chr>,
+## # assay_ontology_term_id <chr>, development_stage <chr>,
+## # development_stage_ontology_term_id <chr>, disease <chr>,
+## # disease_ontology_term_id <chr>, ethnicity <chr>,
+## # ethnicity_ontology_term_id <chr>, experiment___ <chr>, file_id <chr>,
+## # is_primary_data_x <chr>, organism <chr>, organism_ontology_term_id <chr>,
+## # sample_placeholder <chr>, sex <chr>, sex_ontology_term_id <chr>, …
+
The query illustrates the expression of a single gene across 133,133 pseudo samples, encompassing 156 cell types and 45 organs.
+
+
+
Visualising genes expression in cells and tissues
+
+
MUC5B gene
+
MUC5B (Mucin 5B) is known to play a crucial role in the production of mucus, which is essential for protecting and lubricating the surfaces of the respiratory, digestive, and reproductive systems.
+* High expression in Esophagus and Trachea tissue is consistent with the role of MUC5B in mucus production, which is essential for protecting the respiratory tract by clearing pathogens and particulates.
+* Goblet and Secretory cells are known for their role in mucus secretion, which aligns with the high expression of MUC5B.
+
MUC5B_merged_counts |>
+ filter(quantile_normalised != 0) |>
+ group_by(tissue_harmonised) |>
+ mutate(median_expression = median(quantile_normalised)) |>
+ ungroup() |>
+ ggplot(aes(
+ x = fct_reorder(tissue_harmonised, median_expression, .desc = TRUE),
+ y = quantile_normalised,
+ fill = tissue_harmonised
+ )) +
+ geom_boxplot(outlier.shape = NA) +
+ scale_y_log10() +
+ custom_theme +
+ guides(fill = guide_legend(ncol = 1)) +
+ ggtitle("MUC5B expression across tissues")
+
## tidySummarizedExperiment says: A data frame is returned for independent data analysis.
+
+
MUC5B_merged_counts |>
+ filter(quantile_normalised != 0) |>
+ group_by(cell_type_harmonised) |>
+ mutate(median_expression = median(quantile_normalised)) |>
+ ungroup() |>
+ ggplot(aes(x = fct_reorder(cell_type_harmonised, median_expression, .desc = TRUE),
+ y = quantile_normalised,
+ fill = cell_type_harmonised)) +
+ geom_boxplot(outlier.shape = NA) +
+ scale_y_log10() +
+ custom_theme +
+ guides(fill = guide_legend(ncol = 2)) +
+ theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 1)) +
+ ggtitle("MUC5B expression across cell types")
+
## tidySummarizedExperiment says: A data frame is returned for independent data analysis.
+
+
+
+
TGFB1 gene
+
TGFB1 (Transforming Growth Factor Beta 1) play a crucial role in various cellular processes, including cell growth, differentiation, apoptosis, and immune response. The variability in its expression across different cell types highlights its diverse roles in different cellular contexts.
+The boxplots indicate an overview of TGFB1 gene expression across various cell types and tissues:
+* High expression in Mucosa and Skeletal Muscle suggests that the TGFB1 gene is important for maintaining structural integrity and responding to injuries.
+* High expression in Megakaryocytes and Platelets suggests that the TGFB1 gene is essential for blood clotting and wound healing.
+
TGFB1_merged_counts |>
+ filter(quantile_normalised != 0) |>
+ group_by(tissue_harmonised) |>
+ mutate(median_expression = median(quantile_normalised)) |>
+ ungroup() |>
+ ggplot(aes(
+ x = fct_reorder(tissue_harmonised, median_expression, .desc = TRUE),
+ y = quantile_normalised,
+ fill = tissue_harmonised
+ )) +
+ geom_boxplot(outlier.shape = NA) +
+ scale_y_log10() +
+ custom_theme +
+ guides(fill = guide_legend(ncol = 1)) +
+ ggtitle("TGFB1 expression across tissues")
+
## tidySummarizedExperiment says: A data frame is returned for independent data analysis.
+
+
TGFB1_merged_counts |>
+ filter(quantile_normalised != 0) |>
+ group_by(cell_type_harmonised) |>
+ mutate(median_expression = median(quantile_normalised)) |>
+ ungroup() |>
+ ggplot(aes(x = fct_reorder(cell_type_harmonised, median_expression, .desc = TRUE),
+ y = quantile_normalised,
+ fill = cell_type_harmonised)) +
+ geom_boxplot(outlier.shape = NA) +
+ scale_y_log10() +
+ custom_theme +
+ guides(fill = guide_legend(ncol = 3)) +
+ theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 1)) +
+ ggtitle("TGFB1 expression across cell types")
+
## tidySummarizedExperiment says: A data frame is returned for independent data analysis.
+
+
+
+
+
Signature calculation across the curated atlas
+
meta <- metadata |> filter(tissue_harmonised == "lung" &
+ cell_type_harmonised == "cd8 tem")
+sig_calc_pseudobulk <- meta |> get_pseudobulk(cache_directory = my_cache, assays = "counts")
+
sig_calc_pseudobulk
+
## # A SummarizedExperiment-tibble abstraction: 17,027,630 × 51
+## # [90mFeatures=36229 | Samples=470 | Assays=counts[0m
+## .feature .sample counts cell_type_harmonised file_id sample_ sex age_days
+## <chr> <chr> <dbl> <chr> <chr> <chr> <chr> <dbl>
+## 1 A1BG d30b7689… 150. cd8 tem 07beec… d30b76… male 13140
+## 2 A1BG-AS1 d30b7689… 8.94 cd8 tem 07beec… d30b76… male 13140
+## 3 A1CF d30b7689… 0 cd8 tem 07beec… d30b76… male 13140
+## 4 A2M d30b7689… 12.3 cd8 tem 07beec… d30b76… male 13140
+## 5 A2M-AS1 d30b7689… 0 cd8 tem 07beec… d30b76… male 13140
+## 6 A2ML1 d30b7689… 3.99 cd8 tem 07beec… d30b76… male 13140
+## 7 A2MP1 d30b7689… 0 cd8 tem 07beec… d30b76… male 13140
+## 8 A3GALT2 d30b7689… 0 cd8 tem 07beec… d30b76… male 13140
+## 9 A4GALT d30b7689… 7.86 cd8 tem 07beec… d30b76… male 13140
+## 10 A4GNT d30b7689… 0 cd8 tem 07beec… d30b76… male 13140
+## # ℹ 40 more rows
+## # ℹ 43 more variables: ethnicity <chr>, assay <chr>, organism <chr>,
+## # tissue <chr>, tissue_harmonised <chr>, disease <chr>, dataset_id <chr>,
+## # collection_id <chr>, cell_count <int>, is_primary_data_x <chr>,
+## # X_sample_name <chr>, assay_ontology_term_id <chr>, development_stage <chr>,
+## # development_stage_ontology_term_id <chr>, disease_ontology_term_id <chr>,
+## # ethnicity_ontology_term_id <chr>, experiment___ <chr>, …
+
# load calculated signature
+scores <- read.table("/Users/shen.m/Documents/GitHub/tidyomicsBlog/blog/content/post/2024-05-17-CuratedAtalasQueryR-pseudobulk/data/ResExhMarkers_CD8_CLs_LCM.txt", header = TRUE, sep = "\t")
+scores |> head()
+
## Res_Pct Exh_Pct Blood_Pct gene CLsPass LcmPass CancerPass Marker
+## 1 12.40547 11.80152 10.246697 ZFP36 FALSE FALSE FALSE Res
+## 2 11.96635 11.79224 9.788263 CD69 TRUE FALSE TRUE Res
+## 3 11.91051 11.84514 11.436771 EIF1 FALSE FALSE FALSE Res
+## 4 11.82417 10.70216 7.373063 FOS FALSE FALSE FALSE Res
+## 5 11.15705 11.07872 10.402976 KLF6 FALSE FALSE FALSE Res
+## 6 10.98749 10.97823 8.834414 TNFAIP3 FALSE FALSE FALSE Res
+
The signature represents T cell exhaustion, referenced in the paper: https://bmcbioinformatics.biomedcentral.com/articles/10.1186/s12859-018-2435-4
+
my_matrix <- sig_calc_pseudobulk@assays@data$counts
+
+rownames(my_matrix) <- rownames(sig_calc_pseudobulk)
+colnames(my_matrix) <- colnames(sig_calc_pseudobulk)
+
+my_matrix[rowSums(my_matrix) != 0 |
+ rownames(my_matrix) %in% scores$gene, , drop = FALSE] |>
+ as.matrix() |>
+ rankGenes() |>
+ simpleScore(
+ upSet = scores$gene,
+ knownDirection = T,
+ centerScore = F
+ ) |>
+ as_tibble(rownames = "sample_") |> left_join(
+ sig_calc_pseudobulk |> tidybulk::select(sample_, disease, cell_type_harmonised, sample_identifier),
+ by = c("sample_" = "sample_identifier")
+ ) |> ggplot(aes(x = disease, y = TotalScore, fill = disease)) + geom_jitter() + custom_theme
+
## tidySummarizedExperiment says: Key columns are missing. A data frame is returned for independent data analysis.
+
+
This plot visualises the total score of a signature that represents T cell exhaustion across different lung-related diseases and conditions. Sarcoidosis shows more concentrated and relatively higher score ranges, indicating less variability among the samples compared to other diseases. This suggests a more uniform expression of the signature being analysed within these specific conditions.
+