# Substitution and per-residue selection in B cell affinity maturation

## Jenner’s 1796 vaccine

Where are we 200 years later?

## RV144 HIV trial: 2003-2009

• 26,676 volunteers enrolled
• 16,395 volunteers randomized
• 125 infections
• \$105,000,000 and 6 years (!!)

Prospective studies are expensive, slow, and entail complex moral issues. This does not lend itself to rapid vaccine development.

How might we guide vaccine development without disease exposure?

## Vaccines manipulate the adaptive immune system

Antibody-making B cells: a key part of adaptive immunity.

What can we learn from B cells without battle-testing them?

Biological background

Outline

## Part 1 of talk: find appropriate substitution models

These are needed for likelihood-based phylogenetic inference.

## Goal 2: understand how we might manipulate immune repertoire with interventions

Which sites can be changed?

Part 2 of talk: natural selection inference

## The data: sequences from the CDR3 locus

Plenty: a total of about 15 million unique 130nt sequences from memory B cell populations of three healthy individuals A, B, and C.

Part 1

## Goal 1: Understand determinants of molecular evolution

Investigate overall mutation patterns of the B cell repertoire.

Models like this are used throughout phylogenetic inference.

## Not all BCRs share ancestry

In fact, most don’t.

This is different from traditional phylogenetics.

## Use two taxon “trees” for model fitting

#### But: we know ancestral state within V, D, J.

Our “trees” have an observed read on the bottom and the corresponding “ancestral” germline sequence on top, connected by a branch, representing some amount of divergence.

## Collection of two taxon “trees” for model fitting

We will test various models for the V, D, and J segments to select an appropriate evolutionary model for somatic hypermutation to eventually use in phylogenetic inference.

First question: do segments evolve differently?

## Model testing results

Identical model ranking across individuals (using AIC / BIC).

## Branch length distribution under this best model

• D segments evolve substantially faster than V
• J segments evolve more slowly than V
• Individual A has a higher mutational load.

## Estimates of the mutational process are quite consistent between individuals

(each point is a single entry for one of the matrices for a pair of individuals.)

## (Important) aside: productive versus out-of-frame receptors

Each cell may carry two IGH alleles, but only one is expressed.

## Next: what determines mutational processes of different IGHV genes?

Subdivide V genes:

• by individual (A, B, and C)
• by gene (V1-18, V2-3, etc)
• by productive / unproductive status

and fit each subset separately.

## Principal components analysis of individual IGHV GTR matrices

Inspired by the work of (Kosakovsky Pond et al, 2010) on evolutionary “fingerprinting.”

## Branch length differences between productive, unproductive

Unproductive rearrangements are more likely to be either: unchanged from germline, or more divergent.

Part 2

## Goal 2: what if we want to mutate specific residues in an antibody. Is that allowed?

We can’t answer that directly, but we can look across the repertoire at which sites have tolerated change.

## This is (natural) selection inference

$\omega \equiv \frac{dN}{dS} \equiv \frac{\hbox{rate of non-synonymous substitution}}{\hbox{rate of synonymous substitution}}$

We want to estimate this value for each site:

## Challenges

• strange mutational process
• millions of unique sequences
(rules out otherwise lovely tools like PAML, HYPHY):

“FUBAR [HYPHY] allows us to analyze larger data sets than other methods: We illustrate this on a large inﬂuenza hemagglutinin data set (3,142 sequences)” – Murrell et. al 2013

## Strange mutational process

Per-site inference is made difficult by a complicated mutation process

We can use this to tell us about the neutral mutation process.

## $$\omega_l$$ is a ratio of rates in terms of observed neutral process

• $$\lambda_l^{(N-I)}$$: nonsynonymous in-frame rate for site $$l$$
• $$\lambda_l^{(N-O)}$$: nonsynonymous out-of-frame rate for site $$l$$
• $$\lambda_l^{(S-I)}$$: synonymous in-frame rate for site $$l$$
• $$\lambda_l^{(S-O)}$$: synonymous out-of-frame rate for site $$l$$

$\omega_l = \frac{\lambda_l^{(N-I)} / \lambda_l^{(N-O)}}{\lambda_l^{(S-I)} / \lambda_l^{(S-O)}}$

## Stabilize with empirical Bayes regularization

Say we are doing a per-county smoking survey.

Use all of the data to fit prior distribution of smoking prevalence, then given observations obtain per-county posterior.

## Stabilize with empirical Bayes regularization

Assume that $$\lambda_l$$, the substitution rate at site $$l$$, comes from a Gamma distribution with shape $$\alpha$$ and rate $$\beta$$:

$\lambda_l \sim \mathrm{Gamma}(\alpha, \beta).$

Model total substitution counts (sampled via stochastic mapping) for a site as Poisson with rate $$\lambda_l$$:

$C_l \sim \mathrm{Poisson}(\lambda_l),$

Fit $$\hat \alpha$$ and $$\hat \beta$$ to all data, then draw rates $$\lambda_l$$ from the posterior:

$\lambda_l \mid C_l \sim \mathrm{Gamma}(C_l + \hat \alpha, 1 + \hat \beta).$

We extended this regularization to case of non-constant coverage.

## Estimating selection coefficient $$\omega_l$$

• $$\lambda_l^{(N-I)}$$: nonsynonymous in-frame rate for site $$l$$
• $$\lambda_l^{(N-O)}$$: nonsynonymous out-of-frame rate for site $$l$$
• $$\lambda_l^{(S-I)}$$: synonymous in-frame rate for site $$l$$
• $$\lambda_l^{(S-O)}$$: synonymous out-of-frame rate for site $$l$$

$\omega_l = \frac{\lambda_l^{(N-I)} / \lambda_l^{(N-O)}}{\lambda_l^{(S-I)} / \lambda_l^{(S-O)}}$

## Wrap-up of part 2

For more details, paper is up on arXiv.

## Thank you

• Connor McCoy, Trevor Bedford, Vladimir Minin, Harlan Robins.
• Molecular work done by Paul Lindau in Phil Greenberg’s lab.

• W. M. Keck Foundation
• University of Washington Center for AIDS Research (CFAR)
• University of Washington eScience Institute
• National Science Foundation and National Institute of Health

We have a postdoc opening to work on molecular evolution methods for HIV vaccine experimental design, and probably another for B cell work.

## Sequence counts

status A B C
functional 4,139,983 4,861,800 3,748,306
out-of-frame 533,919 794,845 558,246
stop 104,525 169,423 112,901

## Correlation between sequence and GTR matrix

Each dot is a pair of genes.

## Random facts

• Mean length of D segment in individual A’s naive repertoire is 16.61.
• Subject A’s naive sequences were 37% CDR3
• Divergence between the various germ-line V genes:
> summary(dist.dna(allele_01, pairwise.deletion=TRUE, model='raw'))
Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
0.003846 0.201300 0.344600 0.304700 0.384900 0.539500