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On Possible Indicators of Negative Selection in Germinal Centers

Bertrand Ottino-Loffler, Gabriel Victora

TL;DR

A tractable model of selection is presented, proposed signatures of negative selection are analyzed, and a number of intuitively appealing metrics -- such as preferential ancestry ratios, terminal node counts, and mutation count skewness -- are all ill-suited for detecting selection method.

Abstract

A central feature of vertebrate immune response is affinity maturation, wherein antibody-producing B cells undergo evolutionary selection in microanatomical structures called germinal centers, which form in secondary lymphoid organs upon antigen exposure. While it has been shown that the median B cell affinity dependably increases over the course of maturation, the exact logic behind this evolution remains vague. Three potential selection methods include encouraging the reproduction of high affinity cells (``birth/positive selection''), encouraging cell death in low affinity cells (``death/negative selection''), and adjusting the mutation rate based on cell affinity (``mutational selection''). While all three forms of selection would lead to a net increase in affinity, different selection methods may lead to distinct statistical dynamics. We present a tractable model of selection, and analyze proposed signatures of negative selection. Given the simplicity of the model, such signatures should be stronger here than in real systems. However, we find a number of intuitively appealing metrics -- such as preferential ancestry ratios, terminal node counts, and mutation count skewness -- are all ill-suited for detecting selection method.

On Possible Indicators of Negative Selection in Germinal Centers

TL;DR

A tractable model of selection is presented, proposed signatures of negative selection are analyzed, and a number of intuitively appealing metrics -- such as preferential ancestry ratios, terminal node counts, and mutation count skewness -- are all ill-suited for detecting selection method.

Abstract

A central feature of vertebrate immune response is affinity maturation, wherein antibody-producing B cells undergo evolutionary selection in microanatomical structures called germinal centers, which form in secondary lymphoid organs upon antigen exposure. While it has been shown that the median B cell affinity dependably increases over the course of maturation, the exact logic behind this evolution remains vague. Three potential selection methods include encouraging the reproduction of high affinity cells (``birth/positive selection''), encouraging cell death in low affinity cells (``death/negative selection''), and adjusting the mutation rate based on cell affinity (``mutational selection''). While all three forms of selection would lead to a net increase in affinity, different selection methods may lead to distinct statistical dynamics. We present a tractable model of selection, and analyze proposed signatures of negative selection. Given the simplicity of the model, such signatures should be stronger here than in real systems. However, we find a number of intuitively appealing metrics -- such as preferential ancestry ratios, terminal node counts, and mutation count skewness -- are all ill-suited for detecting selection method.
Paper Structure (13 sections, 4 theorems, 105 equations, 10 figures)

This paper contains 13 sections, 4 theorems, 105 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Model of Generic Selection
  3. Net Selection
  4. The Ancestry Hypothesis
  5. Mutation Count Distribution
  6. Discussion
  7. Asymmetric Mutation Rates
  8. Infinite fitness edge cases
  9. Limit of large positive selection.
  10. Limit of large negative selection.
  11. The Matrix Exponential Solution
  12. Recusion for Mutant Moments
  13. Useful Lemmas

Key Result

Lemma 1

Given the following equation for $z$, we have the solution with

Figures (10)

  • Figure 1: Heat map of $h$ via equation \ref{['eq_h']}. Here, $\alpha_H = 0.196$, $\alpha_L = 0.679$, $\beta_H = 0.504$, and $\beta_L = 0.021$. Axes are plotted according to $1 - 1/r_X$ to span from one to infinity.
  • Figure 2: Heat map of $F_H$ via equation \ref{['eq_fH']}. Here, $\alpha_H = 0.196$, $\alpha_L = 0.679$, $\beta_H = 0.504$, and $\beta_L = 0.021$. Axes are plotted according to $1 - 1/r_X$ to span from one to infinity.
  • Figure 3: Heat map of $F_L$ via equation \ref{['eq_fL']}. Here, $\alpha_H = 0.196$, $\alpha_L = 0.679$, $\beta_H = 0.504$, and $\beta_L = 0.021$. Axes are plotted according to $1 - 1/r_X$ to span from one to infinity.
  • Figure 4: Plot of the final fraction of L cells which are in terminal nodes. Here, $N = 5e3$, $\alpha_H = 0.196$, $\alpha_L = 0.679$, $\beta_H = 0.504$, and $\beta_L = 0.021$. Axes are plotted according to $1 - 1/r_X$ to span from one to infinity. Simulation was ran over $T = 3e5$ steps, for a total of 60 generations. We numerically take $0/0 = 1$.
  • Figure 5: Plot of the final fraction of cells which are in terminal nodes. Here, $N = 5e3$, $\alpha_H = 0.196$, $\alpha_L = 0.679$, $\beta_H = 0.504$, and $\beta_L = 0.021$. Axes are plotted according to $1 - 1/r_X$ to span from one to infinity. Simulation was ran over $T = 3e5$ steps, for a total of 60 generations.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4