Distance to nearest neighbor

Estimating the optimal distance threshold for partitioning clonally related sequences is accomplished by calculating the distance from each sequence in the data set to its nearest neighbor and finding the break point in the resulting bi-modal distribution that separates clonally related from unrelated sequences. This is done via the following steps:

  1. Calculating of the nearest neighbor distances for each sequence.
  2. Generating a histogram of the nearest neighbor distances followed by either manual inspect for the threshold separating the two modes or automated threshold detection.

Example data

A small example Change-O database is included in the alakazam package. Calculating the nearest neighbor distances requires the following fields (columns) to be present in the Change-O database:

  • V_CALL
  • J_CALL
# Subset example data to one sample
data(ExampleDb, package="alakazam")

Calculating nearest neighbor distances

The function for calculating distance between every sequence and its nearest neighbor takes a few parameters to adjust how the distance is measured. If a genotype has been inferred using the methods in the tigger package, and a V_CALL_GENOTYPED field has been added to the database, then this column may be used instead of the default V_CALL column by specifying the vCallColumn argument. This will allows the more accurate V call from tigger to be used for grouping of the sequences. Furthermore, for more leniency toward ambiguous V(D)J segment calls, the parameter first can be set to FALSE. Setting first=FALSE will use the union of all possible genes to group sequences, rather than the first gene in the field. The model parameter determines which underlying SHM model is used to calculate the distance. The default model is single nucleotide Hamming distance with gaps considered as a match to any nucleotide (ham). Other options include a human Ig-specific single nucleotide model similar to a transition/transversion model (hh_s1f) and the corresponding 5-mer context model from Yaari et al, 2013 (hh_s5f), an analogous pair of mouse specific models from Cui et al, 2016 (mk_rs1nf and mk_rs5nf), and amino acid Hamming distance (aa).

Note: Human and mouse distance measures that are backward compatible with SHazaM v0.1.4 and Change-O v0.3.3 are also provide as hs1f_compat and m1n_compat, respectively.

For models that are not symmetric (e.g., distance from A to B is not equal to the distance from B to A), there is a symmetry parameter that allows the user to specify whether the average or minimum of the two distances is used to determine the overall distance.

# Use nucleotide Hamming distance and normalize by junction length
dist_ham <- distToNearest(ExampleDb, vCallColumn="V_CALL_GENOTYPED", 
                          model="ham", normalize="len", nproc=1)

# Use genotyped V assignments, a 5-mer model and no normalization
dist_s5f <- distToNearest(ExampleDb, vCallColumn="V_CALL_GENOTYPED", 
                          model="hh_s5f", normalize="none", nproc=1)

Using nearest neighbor distances to determine clonal assignment thresholds

The primary use of the distance to nearest calculation in SHazaM is to determine the optimal threshold for clonal assignment using the DefineClones tool in Change-O. Defining a threshold relies on distinguishing clonally related sequences (represented by sequences with close neighbors) from singletons (sequences without close neighbors), which show up as two modes in a nearest neighbor distance histogram.

Thresholds may be manually determined by inspection of the nearest neighbor histograms or by using one of the automated threshold detection algorithms provided by the findThreshold function. The available methods are density (smoothed density) and gmm (gamma/Guassian mixture model), and are chosen via the method parameter of findThreshold.

Threshold determination by manual inspection

Manual threshold detection simply involves generating a histrogram for the values in the DIST_NEAREST column of the distToNearest output and selecting a suitable value within the valley between the two modes.

# Generate Hamming distance histogram
p1 <- ggplot(subset(dist_ham, !is.na(DIST_NEAREST)),
             aes(x=DIST_NEAREST)) + 
    theme_bw() + 
    xlab("Hamming distance") + 
    ylab("Count") +
    scale_x_continuous(breaks=seq(0, 1, 0.1)) +
    geom_histogram(color="white", binwidth=0.02) +
    geom_vline(xintercept=0.12, color="firebrick", linetype=2)

plot of chunk DistToNearest-Vignette-3

By manual inspection, the length normalized ham model distance threshold would be set to a value near 0.12 in the above example.

# Generate HH_S5F distance histogram
p2 <- ggplot(subset(dist_s5f, !is.na(DIST_NEAREST)),
             aes(x=DIST_NEAREST)) + 
    theme_bw() + 
    xlab("HH_S5F distance") + 
    ylab("Count") +
    scale_x_continuous(breaks=seq(0, 50, 5)) +
    geom_histogram(color="white", binwidth=1) +
    geom_vline(xintercept=7, color="firebrick", linetype=2)

plot of chunk DistToNearest-Vignette-4

In this example, the unnormalized hh_s5f model distance threshold would be set to a value near 7.

Automated threshold detection via smoothed density

The density method will look for the minimum in the valley between two modes of a smoothed distribution based on the input vector (distances), which will generally be the DIST_NEAREST column from the distToNearest output. Below is an example of using the density method for threshold detection.

# Find threshold using density method
output <- findThreshold(dist_ham$DIST_NEAREST, method="density")
threshold <- output@threshold

# Plot distance histogram, density estimate and optimum threshold
plot(output, title="Density Method")

plot of chunk DistToNearest-Vignette-5

# Print threshold
## [1] 0.1738391

Automated threshold detection via a mixture model

The findThreshold function includes approaches for automatically determining a clonal assignment threshold. The "gmm" method (gamma/Gaussian mixture method) of findThreshold (method="gmm") performs a maximum-likelihood fitting procedure over the distance-to-nearest distribution using one of four combinations of univariate density distribution functions: "norm-norm" (two Gaussian distributions), "norm-gamma" (lower Guassian and upper gamma distribution), "gamma-norm" (lower gamm and upper Guassian distribution), and "gamma-gamma" (two gamma distributions). By default, the threshold will be selected by calculating the distance at which the average of sensitivity and specificity reaches its maximum (cutoff="optimal"). Alternative threshold selection criteria are also providing, including the curve intersection (cutoff="intersect"), user defined sensitivity (cutoff="user", sen=x), or user defined specificity (cutoff="user", spc=x)

In the example below the mixture model method (method="gmm") is used to find the optimal threshold for separating clonally related sequences by fitting two gamma distributions (model="gamma-gamma"). The red dashed-line shown in figure below defines the distance where the average of the sensitivity and specificity reaches its maximum.

# Find threshold using gmm method
output <- findThreshold(dist_ham$DIST_NEAREST, method="gmm", model="gamma-gamma")

# Plot distance histogram, Gaussian fits, and optimum threshold
plot(output, binwidth=0.02, title="GMM Method: gamma-gamma")

plot of chunk DistToNearest-Vignette-6

# Print threshold
## [1] 0.1216784

Note: The shape of histogram plotted by plotGmmThreshold is governed by the binwidth parameter. Meaning, any change in bin size will change the form of the distribution, while the gmm method is completely bin size independent and only engages the real input data.

Calculating nearest neighbor distances independently for subsets of data

The fields argument to distToNearest will split the input data.frame into groups based on values in the specified fields (columns) and will treat them independently. For example, if the input data has multiple samples, then fields="SAMPLE" would allow each sample to be analyzed separately.

In the previous examples we used a subset of the original example data. In the following example, we will use the two available samples, -1h and +7d, and will set fields="SAMPLE". This will reproduce previous results for sample -1h and add results for sample +7d.

dist_fields <- distToNearest(ExampleDb, model="ham", normalize="len", 
                             fields="SAMPLE", nproc=1)

We can plot the nearest neighbor distances for the two samples:

# Generate grouped histograms
p4 <- ggplot(subset(dist_fields, !is.na(DIST_NEAREST)), 
             aes(x=DIST_NEAREST)) + 
    theme_bw() + 
    xlab("Grouped Hamming distance") + 
    ylab("Count") +
    geom_histogram(color="white", binwidth=0.02) +
    geom_vline(xintercept=0.12, color="firebrick", linetype=2) +
    facet_grid(SAMPLE ~ ., scales="free_y")

plot of chunk DistToNearest-Vignette-7

In this case, the threshold selected for -1h seems to work well for +7d as well.

Calculating nearest neighbor distances across groups rather than within a groups

Specifying the cross argument to distToNearest forces distance calculations to be performed across groups, such that the nearest neighbor of each sequence will always be a sequence in a different group. In the following example we set cross="SAMPLE", which will group the data into -1h and +7d sample subsets. Thus, nearest neighbor distances for sequences in sample -1h will be restricted to the closest sequence in sample +7d and vice versa.

dist_cross <- distToNearest(ExampleDb, model="ham", first=FALSE, 
                            normalize="len", cross="SAMPLE", nproc=1)
# Generate cross sample histograms
p5 <- ggplot(subset(dist_cross, !is.na(CROSS_DIST_NEAREST)), 
             aes(x=CROSS_DIST_NEAREST)) + 
    theme_bw() + 
    xlab("Cross-sample Hamming distance") + 
    ylab("Count") +
    geom_histogram(color="white", binwidth=0.02) +
    geom_vline(xintercept=0.12, color="firebrick", linetype=2) +
    facet_grid(SAMPLE ~ ., scales="free_y")

plot of chunk DistToNearest-Vignette-8

This can provide a sense of overlap between samples or a way to compare within-sample variation to cross-sample variation.

Speeding up pairwise-distance-matrix calculations with subsampling

The subsample option in distToNearest allows to speed up calculations and reduce memory usage.

If there are very large groups of sequences that share V call, J call and junction length, distToNearest will need a lot of memory and it will take a long time to calculate all the distances. Without subsampling, in a large group of n=70,000 sequences distToNearest calculates a n*n distance matrix. With subsampling, e.g. to s=15,000, the distance matrix for the same group has size s*n, and for each sequence in db, the distance value is calculated by comparing the sequence to the subsampled sequences from the same V-J-junction length group.

# Explore V-J-junction length groups sizes to use subsample
# Show the size of the largest groups
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
##     filter, lag
## The following objects are masked from 'package:base':
##     intersect, setdiff, setequal, union
top_10_sizes <- ExampleDb %>%
     group_by(JUNCTION_LENGTH) %>% # group by junction length
     do(alakazam::groupGenes(., first=TRUE)) %>% # group by V and J call
     mutate(GROUP_ID=paste(JUNCTION_LENGTH,VJ_GROUP, sep="_")) %>% # Create group ids based on junction length and VJ calls
     ungroup() %>%
     group_by(GROUP_ID) %>% # group by GROUP_ID
     distinct(JUNCTION) %>% # for each group, we want to count unique junctions, so keep distinct
     summarize(SIZE=n()) %>% # get the size of the group, the number of sequences
     arrange(desc(SIZE)) %>% # sort by decreasing size
     select(SIZE) %>% 
     top_n(10) # show the top 10
## Selecting by SIZE
## # A tibble: 10 x 1
##     SIZE
##    <int>
##  1    90
##  2    37
##  3    36
##  4    34
##  5    33
##  6    33
##  7    32
##  8    26
##  9    25
## 10    25
# Use 30 to subsample
# NOTE. This is a toy example. To use 30 with real data 
# is probably not a good choice.
dist <- distToNearest(ExampleDb, 
                      first=FALSE, normalize="len",
                      subsample = 30)