```
require(IgGeneUsage)
require(knitr)
require(ggplot2)
require(ggforce)
require(gridExtra)
require(ggrepel)
require(rstan)
require(reshape2)
rstan_options(auto_write = TRUE)
```

Decoding the properties of immune repertoires is key in understanding the
response of adaptive immunity to challenges such as viral infection. One
important property is biases in immunoglobulin (Ig) gene usage between
biological conditions (e.g. healthy vs tumor). Yet, most analyses for
differential gene usage (DGU) are performed qualitatively, or with
inadequate statistical methods. Here we introduce *IgGeneUsage*,
a computational tool for DGU analysis.

The main input of *IgGeneUsage* is a data.frame that has the
following 4 columns:

- sample_id: name of the repertoire (e.g. Patient-1)
- condition: name of the condition to which each repertoire belongs (e.g. healthy or tumor)
- gene_name: gene name (e.g. IGHV1-10 or family TRVB1)
- gene_usage_count: numeric (count) of usage related in sample x gene x condition specified in columns 1-3

The sum of all gene usage counts (column 4) for a given repertoire is equal to the repertoire size (number of cells in the repertoire).

*IgGeneUsage* transforms the provided input in the following
way. Given \(R\) repertoires, each having \(G\) genes, *IgGeneUsage*
generates a gene usage matrix \(Y^{R \times G}\). Row sums in \(Y\) define the
total usage in each repertoire (\(N\)). The design variable \(X\) is set to
\(X = 1\) for repertoires that belong to the first condition, and \(X = -1\)
otherwise.

For the analysis of DGU between two biological conditions, we designed the following Bayesian model (\(M\)) for zero-inflated beta-binomial regression. This model can fit over-dispersed gene usage data. The immune repertoire data is also not exhaustive, which leads to misdetection of genes that are systematically rearranged at low probability. The zero-inflated component of our model accounts for this:

\[\begin{align} p(Y_{ij} \mid M) = \begin{cases} \kappa + (1 - \kappa) \operatorname{BB}\left(0 \mid N_{i}, \theta_{ij}, \phi \right), & \text{if $Y_{ij}$ = 0} \\ (1 - \kappa) \operatorname{BB}\left(Y_{ij} \mid N_{i}, \theta_{ij}, \phi \right), & \text{if $Y_{ij}$ > 0} \end{cases}\\ \theta_{ij}=\operatorname{logit^{-1}}\left(\alpha_{j}+\beta_{ij}X_{i}\right)\\ \beta_{ij}\sim\operatorname{Normal}\left(\gamma_{j},\gamma_{\sigma} \right)\\ \gamma_{j}\sim\operatorname{Normal}\left(\hat{\gamma},\hat{\gamma}_{\sigma} \right) \\ \alpha_{j}\sim\operatorname{Normal}\left(\hat{\alpha},\hat{\alpha}_{\sigma} \right) \\ \hat{\gamma} \sim \operatorname{Normal}\left(0, 5\right) \\ \hat{\alpha} \sim \operatorname{Normal}\left(0, 10\right) \\ \gamma_{\sigma}, \hat{\gamma}_{\sigma}, \hat{\alpha}_{\sigma} \sim \operatorname{Cauchy^{+}}\left(0, 1\right) \\ \phi \sim \operatorname{Exponential}\left(\tau\right) \\ \tau \sim \operatorname{Gamma}\left(3, 0.1\right) \\ \kappa \sim \operatorname{Beta}\left(1, 3\right) \end{align}\]

Model \(M\) legend:

- \(i\) and \(j\): index of different repertoires and genes, respectively
- \(\kappa\): zero-inflation probability
- \(\theta\): probability of gene usage
- \(\phi\): dispersion
- \(\alpha\): intercept/baseline gene usage
- \(\beta\): slope/within-repertoire DGU coefficient
- \(\gamma\), \(\gamma_{\sigma}\): slope/gene-specific DGU coefficient; standard deviation
- \(\hat{\gamma}\), \(\hat{\gamma}_{\sigma}\): mean and standard deviation of the population of gene-specific DGU coefficients
- \(\hat{\alpha}\), \(\hat{\alpha}_{\sigma}\): mean and standard deviation of the population of gene-specific baseline usages
- \(\operatorname{BB}\): beta-binomial probability mass function (pmf)
- \(\operatorname{Normal}\): normal probability density function (pdf)
- \(\operatorname{Cauchy^{+}}\): half-Cauchy pdf
- \(\operatorname{Exponential}\): exponential pdf
- \(\operatorname{Gamma}\): gamma pdf
- \(\operatorname{Beta}\): beta pdf
- \(\operatorname{logit^{-1}}\): inverse logistic function

In the output of *IgGeneUsage*, we report the mean effect
size (\(\gamma\)) and its 95% highest density interval (HDI). Genes with
\(\gamma \neq 0\) (e.g. if 95% HDI of \(\gamma\) excludes 0) are most likely
to experience differential usage. Additionally, we report the probability of
differential gene usage (\(\pi\)):
\[\begin{align}
\pi = 2 \cdot \max\left(\int_{\gamma = -\infty}^{0} p(\gamma)\mathrm{d}\gamma,
\int_{\gamma = 0}^{\infty} p(\gamma)\mathrm{d}\gamma\right) - 1
\end{align}\]
with \(\pi = 1\) for genes with strong differential usage, and \(\pi = 0\) for
genes with negligible differential gene usage. Both metrics are computed based
on the posterior distribution of \(\gamma\), and are thus related. We find \(\pi\)
slightly easier to interpret.

\[\begin{align} p(Y_{ij} \mid M) = \begin{cases} \kappa + (1 - \kappa) \operatorname{BB}\left(0 \mid N_{i}, \theta_{ij}, \phi \right), & \text{if $Y_{ij}$ = 0} \\ (1 - \kappa) \operatorname{BB}\left(Y_{ij} \mid N_{i}, \theta_{ij}, \phi \right), & \text{if $Y_{ij}$ > 0} \end{cases}\\ \theta_{ij}=\operatorname{logit^{-1}}\left(\alpha_{ij}+\beta_{ij}X_{i}\right)\\ \alpha_{ij}\sim\operatorname{Normal}\left(\delta_{j},\delta_{\sigma} \right)\\ \beta_{ij}\sim\operatorname{Normal}\left(\gamma_{j},\gamma_{\sigma} \right)\\ \gamma_{j}\sim\operatorname{Normal}\left(0.0,\hat{\gamma}_{\sigma} \right) \\ \delta_{j}\sim\operatorname{Normal}\left(0.0,\hat{\delta}_{\sigma} \right) \\ \gamma_{\sigma}, \hat{\gamma}_{\sigma}, \delta_{\sigma}, \hat{\delta}_{\sigma} \sim \operatorname{Cauchy^{+}}\left(0, 1\right) \\ \phi \sim \operatorname{Exponential}\left(\tau\right) \\ \tau \sim \operatorname{Gamma}\left(3, 0.1\right) \\ \kappa \sim \operatorname{Beta}\left(1, 3\right) \end{align}\]

*IgGeneUsage* has a couple of built-in Ig gene usage datasets.
Some were obtained from studies and others were simulated.

Lets look into the simulated dataset `d_zibb`

. This dataset was generated by a
zero-inflated beta-binomial (ZIBB) model, and *IgGeneUsage*
was designed to fit ZIBB-distributed data.

```
data("d_zibb", package = "IgGeneUsage")
knitr::kable(head(d_zibb))
```

sample_id | condition | gene_name | gene_usage_count |
---|---|---|---|

S1 | C1 | G1 | 25 |

S1 | C1 | G2 | 123 |

S1 | C1 | G3 | 29 |

S1 | C1 | G4 | 442 |

S1 | C1 | G5 | 60 |

S1 | C1 | G6 | 106 |

We can also visualize `d_zibb`

with *ggplot*:

```
ggplot(data = d_zibb)+
geom_point(aes(x = gene_name, y = gene_usage_count, col = condition),
position = position_dodge(width = .7), shape = 21)+
theme_bw(base_size = 11)+
ylab(label = "Gene usage")+
xlab(label = '')+
theme(legend.position = "top")+
theme(axis.text.x = element_text(angle = 90, hjust = 1, vjust = 0.4))
```

As main input *IgGeneUsage* uses a data.frame formatted as
`d_zibb`

. Other input parameters allow you to configure specific settings
of the *rstan* sampler.

In this example we analyze `d_zibb`

with 3 MCMC chains, 1500 iterations
each including 500 warm-ups using a single CPU core (Hint: for parallel
chain execution set parameter `mcmc.cores`

= 3). We report for each model
parameter its mean and 95% highest density interval (HDIs).

**Important remark:** you should run DGU analyses using default
*IgGeneUsage* parameters. If warnings or errors are reported
with regard to the MCMC sampling, please consult the Stan manual1 https://mc-stan.org/misc/warnings.html and
adjust the inputs accordingly. If the warnings persist, please submit an
issue with a reproducible script at the Bioconductor support site or on
Github2 https://github.com/snaketron/IgGeneUsage/issues.

```
M <- DGU(usage.data = d_zibb, # input data
mcmc.warmup = 500, # how many MCMC warm-ups per chain (default: 500)
mcmc.steps = 1500, # how many MCMC steps per chain (default: 1,500)
mcmc.chains = 3, # how many MCMC chain to run (default: 4)
mcmc.cores = 1, # how many PC cores to use? (e.g. parallel chains)
hdi.level = 0.95, # highest density interval level (de fault: 0.95)
adapt.delta = 0.8, # MCMC target acceptance rate (default: 0.95)
max.treedepth = 10) # tree depth evaluated at each step (default: 12)
```

```
FALSE
FALSE SAMPLING FOR MODEL 'zibb' NOW (CHAIN 1).
FALSE Chain 1:
FALSE Chain 1: Gradient evaluation took 0.000245 seconds
FALSE Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 2.45 seconds.
FALSE Chain 1: Adjust your expectations accordingly!
FALSE Chain 1:
FALSE Chain 1:
FALSE Chain 1: Iteration: 1 / 1500 [ 0%] (Warmup)
FALSE Chain 1: Iteration: 250 / 1500 [ 16%] (Warmup)
FALSE Chain 1: Iteration: 500 / 1500 [ 33%] (Warmup)
FALSE Chain 1: Iteration: 501 / 1500 [ 33%] (Sampling)
FALSE Chain 1: Iteration: 750 / 1500 [ 50%] (Sampling)
FALSE Chain 1: Iteration: 1000 / 1500 [ 66%] (Sampling)
FALSE Chain 1: Iteration: 1250 / 1500 [ 83%] (Sampling)
FALSE Chain 1: Iteration: 1500 / 1500 [100%] (Sampling)
FALSE Chain 1:
FALSE Chain 1: Elapsed Time: 11.5906 seconds (Warm-up)
FALSE Chain 1: 14.2992 seconds (Sampling)
FALSE Chain 1: 25.8898 seconds (Total)
FALSE Chain 1:
FALSE
FALSE SAMPLING FOR MODEL 'zibb' NOW (CHAIN 2).
FALSE Chain 2:
FALSE Chain 2: Gradient evaluation took 0.000217 seconds
FALSE Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 2.17 seconds.
FALSE Chain 2: Adjust your expectations accordingly!
FALSE Chain 2:
FALSE Chain 2:
FALSE Chain 2: Iteration: 1 / 1500 [ 0%] (Warmup)
FALSE Chain 2: Iteration: 250 / 1500 [ 16%] (Warmup)
FALSE Chain 2: Iteration: 500 / 1500 [ 33%] (Warmup)
FALSE Chain 2: Iteration: 501 / 1500 [ 33%] (Sampling)
FALSE Chain 2: Iteration: 750 / 1500 [ 50%] (Sampling)
FALSE Chain 2: Iteration: 1000 / 1500 [ 66%] (Sampling)
FALSE Chain 2: Iteration: 1250 / 1500 [ 83%] (Sampling)
FALSE Chain 2: Iteration: 1500 / 1500 [100%] (Sampling)
FALSE Chain 2:
FALSE Chain 2: Elapsed Time: 12.2258 seconds (Warm-up)
FALSE Chain 2: 7.32203 seconds (Sampling)
FALSE Chain 2: 19.5478 seconds (Total)
FALSE Chain 2:
FALSE
FALSE SAMPLING FOR MODEL 'zibb' NOW (CHAIN 3).
FALSE Chain 3:
FALSE Chain 3: Gradient evaluation took 0.000212 seconds
FALSE Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 2.12 seconds.
FALSE Chain 3: Adjust your expectations accordingly!
FALSE Chain 3:
FALSE Chain 3:
FALSE Chain 3: Iteration: 1 / 1500 [ 0%] (Warmup)
FALSE Chain 3: Iteration: 250 / 1500 [ 16%] (Warmup)
FALSE Chain 3: Iteration: 500 / 1500 [ 33%] (Warmup)
FALSE Chain 3: Iteration: 501 / 1500 [ 33%] (Sampling)
FALSE Chain 3: Iteration: 750 / 1500 [ 50%] (Sampling)
FALSE Chain 3: Iteration: 1000 / 1500 [ 66%] (Sampling)
FALSE Chain 3: Iteration: 1250 / 1500 [ 83%] (Sampling)
FALSE Chain 3: Iteration: 1500 / 1500 [100%] (Sampling)
FALSE Chain 3:
FALSE Chain 3: Elapsed Time: 12.9189 seconds (Warm-up)
FALSE Chain 3: 14.3646 seconds (Sampling)
FALSE Chain 3: 27.2835 seconds (Total)
FALSE Chain 3:
```

The following objects are provided as part of the output of DGU:

`glm.summary`

(main results of*IgGeneUsage*): quantitative DGU summary`test.summary`

: quantitative DGU summary from frequentist methods: Welch’s t-test (T-test) and Wilcoxon signed-rank test (U-test)`glm`

: rstan (‘stanfit’) object of the fitted model \(rightarrow\) used for model checks (see section ‘Model checking’)`ppc.data`

: posterior predictive checks data (see section ‘Model checking’)

`summary(M)`

```
FALSE Length Class Mode
FALSE glm.summary 9 data.frame list
FALSE test.summary 9 data.frame list
FALSE glm 1 stanfit S4
FALSE ppc.data 2 -none- list
FALSE usage.data 9 -none- list
```

**Check your model fit**. For this, you can use the object glm.- Minimal checklist of successful MCMC sampling3 https://mc-stan.org/misc/warnings.html:
- no divergences
- no excessive warnings from rstan
- Rhat < 1.05
- high Neff

- Minimal checklist for valid model:
- posterior predictive checks (PPCs): is model consistent with reality, i.e. is there overlap between simulated and observed data?
- leave-one-out analysis

- Minimal checklist of successful MCMC sampling3 https://mc-stan.org/misc/warnings.html:

- divergences, tree-depth, energy

`rstan::check_hmc_diagnostics(M$glm)`

```
FALSE
FALSE Divergences:
FALSE
FALSE Tree depth:
FALSE
FALSE Energy:
```

- Rhat and Neff

```
gridExtra::grid.arrange(rstan::stan_rhat(object = M$glm),
rstan::stan_ess(object = M$glm),
nrow = 1)
```

The model used by *IgGeneUsage* is generative, i.e. with the
model we can generate usage of each Ig gene in a given repertoire (y-axis).
Error bars show 95% HDI of mean posterior prediction. The predictions can be
compared with the observed data (x-axis). For points near the diagonal
\(\rightarrow\) accurate prediction.

```
ggplot(data = M$ppc.data$ppc.repertoire)+
facet_wrap(facets = ~sample_name, nrow = 3)+
geom_abline(intercept = 0, slope = 1, linetype = "dashed", col = "darkgray")+
geom_errorbar(aes(x = observed_count, y = ppc_mean_count,
ymin = ppc_L_count, ymax = ppc_H_count), col = "darkgray")+
geom_point(aes(x = observed_count, y = ppc_mean_count,
fill = condition), shape = 21, size = 1)+
theme_bw(base_size = 11)+
theme(legend.position = "top")+
scale_x_log10()+
scale_y_log10()+
xlab(label = "Observed usage [counts]")+
ylab(label = "Predicted usage [counts]")+
annotation_logticks(base = 10, sides = "lb")
```

Prediction of generalized gene usage within a biological condition is also possible. We show the predictions (y-axis) of the model, and compare them against the observed mean usage (x-axis). If the points are near the diagonal \(\rightarrow\) accurate prediction. Errors are 95% HDIs of the mean.

```
ggplot(data = M$ppc.data$ppc.gene)+
geom_abline(intercept = 0, slope = 1, linetype = "dashed", col = "darkgray")+
geom_errorbar(aes(x = observed_prop*100, ymin = ppc_L_prop*100,
ymax = ppc_H_prop*100), col = "darkgray")+
geom_point(aes(x = observed_prop*100, y = ppc_mean_prop*100,
col = condition), size = 2)+
theme_bw(base_size = 11)+
theme(legend.position = "top")+
xlab(label = "Observed usage [%]")+
ylab(label = "Predicted usage [%]")+
scale_x_log10()+
scale_y_log10()+
annotation_logticks(base = 10, sides = "lb")
```

Each row of `glm.summary`

summarizes the degree of DGU observed for specific
Igs. Two metrics are reported:

`es`

(also referred to as`\gamma`

): effect size encoded in(parameter \(\gamma\) from model \(M\)) on DGU, where`contrast`

gives the direction of the effect (e.g. tumor - healthy or healthy - tumor)`pmax`

(also referred to as`\pi`

): probability of DGU (parameter \(\pi\) from model \(M\))

For `es`

we also have the mean, median standard error (se), standard
deviation (sd), L (low bound of 95% HDI), H (high bound of 95% HDI)

`kable(x = head(M$glm.summary), row.names = FALSE, digits = 3)`

es_mean | es_mean_se | es_sd | es_median | es_L | es_H | contrast | pmax | gene_name |
---|---|---|---|---|---|---|---|---|

0.218 | 0.004 | 0.241 | 0.209 | -0.247 | 0.692 | C2 - C1 | 0.664 | G1 |

0.001 | 0.006 | 0.485 | -0.007 | -0.959 | 1.006 | C2 - C1 | 0.011 | G10 |

0.295 | 0.002 | 0.177 | 0.291 | -0.048 | 0.647 | C2 - C1 | 0.904 | G11 |

0.098 | 0.003 | 0.207 | 0.094 | -0.306 | 0.521 | C2 - C1 | 0.363 | G12 |

-0.201 | 0.006 | 0.448 | -0.181 | -1.148 | 0.631 | C2 - C1 | 0.330 | G13 |

0.129 | 0.003 | 0.202 | 0.131 | -0.261 | 0.514 | C2 - C1 | 0.471 | G14 |

We know that the values of `\gamma`

and `\pi`

are related to each other.
Lets visualize them for all genes (shown as a point). Names are shown for
genes associated with \(\pi \geq 0.9\). Dashed horizontal line represents
null-effect (\(\gamma = 0\)).

Notice that the gene with \(\pi \approx 1\) also has an effect size whose 95% HDI (error bar) does not overlap the null-effect. The genes with high degree of differential usage are easy to detect with this figure.

```
# format data
stats <- M$glm.summary
stats <- stats[order(abs(stats$es_mean), decreasing = FALSE), ]
stats$gene_fac <- factor(x = stats$gene_name, levels = stats$gene_name)
stats <- merge(x = stats, y = M$test.summary, by = "gene_name")
ggplot(data = stats)+
geom_hline(yintercept = 0, linetype = "dashed", col = "gray")+
geom_errorbar(aes(x = pmax, y = es_mean, ymin = es_L, ymax = es_H),
col = "darkgray")+
geom_point(aes(x = pmax, y = es_mean), col = "darkgray")+
geom_text_repel(data = stats[stats$pmax >= 0.9, ],
aes(x = pmax, y = es_mean, label = gene_fac),
min.segment.length = 0, size = 2.75)+
theme_bw(base_size = 11)+
xlab(label = expression(pi))+
xlim(c(0, 1))
```