# 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:

- Calculating of the nearest neighbor distances for each sequence.
- 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 AIRR Rearrangement database is included in the `alakazam`

package.
Calculating the nearest neighbor distances requires the following
fields (columns) to be present in the table:

`sequence_id`

`v_call`

`j_call`

`junction`

`junction_length`

```
# Subset example data to one sample
library(shazam)
data(ExampleDb, package="alakazam")
```

## Calculating nearest neighbor distances (heavy chain sequences)¶

By default, `distToNearest`

, the function for calculating distance between every
sequence and its nearest neighbor, assumes that it is running under non-single-cell
mode and that every input sequence is a heavy chain sequence and will be used for
calculation. It 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 provided 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, sequenceColumn="junction",
vCallColumn="v_call_genotyped", jCallColumn="j_call",
model="ham", normalize="len", nproc=1)
# Use genotyped V assignments, a 5-mer model and no normalization
dist_s5f <- distToNearest(ExampleDb, sequenceColumn="junction",
vCallColumn="v_call_genotyped", jCallColumn="j_call",
model="hh_s5f", normalize="none", nproc=1)
```

## Calculating nearest neighbor distances (single-cell paired heavy and light chain sequences)¶

The `distToNearest`

function also supports running under single-cell mode where an input
`Example10x`

containing single-cell paired IGH:IGK/IGL, TRB:TRA, or TRD:TRG chain sequences are
supplied. In this case, by default, cells are first divided into partitions containing
the same heavy/long chain (IGH, TRB, TRD) V gene and J gene (and if specified, junction
length), and the same light/short chain (IGK, IGL, TRA, TRG) V gene and J gene (and if
specified, junction length). Then, only the heavy chain sequences are used for
calculating the nearest neighbor distances.

Under the single-cell mode, each row of the input `Example10x`

should represent a sequence/chain.
Sequences/chains from the same cell are linked by a cell ID in a `cellIdColumn`

column.
Note that a cell should have exactly one `IGH`

sequence (BCR) or `TRB`

/`TRD`

(TCR).
The values in the `locusColumn`

column must be one of `IGH`

, `IGI`

, `IGK`

, or `IGL`

(BCR)
or `TRA`

, `TRB`

, `TRD`

, or `TRG`

(TCR). To invoke the single-cell mode, `cellIdColumn`

must be specified and `locusColumn`

must be correct.

There is a choice of whether grouping should be done as a one-stage process or a
two-stage process. This can be specified via `VJthenLen`

. In the one-stage process
(`VJthenLen=FALSE`

), cells are divided into partitions containing same heavy/long chain V
gene, J gene, and junction length (V-J-length combination), and the same light chain
V-J-length combination. In the two-stage process (`VJthenLen=TRUE`

), cells are first
divided by heavy/long chain V gene and J gene (V-J combination), and light/short
chain V-J combination; and then by the corresponding junction lengths.

There is also a choice of whether grouping should be done using `IGH`

(BCR)
or `TRB/TRD`

(TCR) sequences only, or using both `IGH`

and `IGK`

/`IGL`

(BCR) or
`TRB`

/`TRD`

and `TRA`

/`TRG`

(TCR) sequences. This is governed by `onlyHeavy`

.

```
# Single-cell mode
# Group cells in a one-stage process (VJthenLen=FALSE) and using
# both heavy and light chain sequences (onlyHeavy=FALSE)
data(Example10x, package="alakazam")
dist_sc <- distToNearest(Example10x, cellIdColumn="cell_id", locusColumn="locus",
VJthenLen=FALSE, onlyHeavy=FALSE)
```

Regardless of whether grouping was done using only the heavy chain sequences, or both heavy
and light chain sequences, only heavy chain sequences will be used for calculating the
nearest neighbor distances. Hence, under the single-cell mode, rows in the returned
`data.frame`

corresponding to light chain sequences will have `NA`

in the `dist_nearest`

field.

## 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/Gaussian 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
library(ggplot2)
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(p1)
```

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(p2)
```

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")
```

```
# Print threshold
print(output)
```

```
## [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 Gaussian and upper gamma distribution),
`"gamma-norm"`

(lower gamm and upper Gaussian 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")
```

```
# Print threshold
print(output)
```

```
## [1] 0.1221371
```

**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_id"`

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_id"`

. This will reproduce previous results for sample
`-1h`

and add results for sample `+7d`

.

```
dist_fields <- distToNearest(ExampleDb, model="ham", normalize="len",
fields="sample_id", 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_id ~ ., scales="free_y")
plot(p4)
```

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, sequenceColumn="junction",
vCallColumn="v_call_genotyped", jCallColumn="j_call",
model="ham", first=FALSE,
normalize="len", cross="sample_id", 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_id ~ ., scales="free_y")
plot(p5)
```

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
library(dplyr)
library(alakazam)
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
ungroup() %>%
group_by(GROUP_ID) %>% # Group by GROUP_ID
distinct(junction) %>% # Vount unique junctions per group
summarize(SIZE=n()) %>% # Get the size of the group
arrange(desc(SIZE)) %>% # Sort by decreasing size
select(SIZE) %>%
top_n(10) # Filter to the top 10
```

```
## Selecting by SIZE
```

```
top_10_sizes
```

```
## # A tibble: 10 x 1
## SIZE
## <int>
## 1 89
## 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. Subsampling to 30 sequence with real data is unwise
dist <- distToNearest(ExampleDb, sequenceColumn="junction",
vCallColumn="v_call_genotyped", jCallColumn="j_call",
model="ham",
first=FALSE, normalize="len",
subsample=30)
```