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Large Deviations Analysis for Stochastic Models of Bacterial Evolution

Robert Azencott, Brett Geiger, Ilya Timofeyev

Abstract

Radical shifts in the genetic composition of large cell populations are rare events with quite low probabilities, which direct numerical simulations generally fail to evaluate accurately. In this paper, we develop a theoretical large deviations framework for a class of Markov chains modeling the genetic evolution of bacteria such as E. coli. In particular, we develop the cost function for discrete-time Markov chains which describe the daily evolution of histograms of bacterial populations. We also develop explicit formulas that can be used to numerically quantify the most likely evolutionary trajectories connecting an initial histogram and the target histogram.

Large Deviations Analysis for Stochastic Models of Bacterial Evolution

Abstract

Radical shifts in the genetic composition of large cell populations are rare events with quite low probabilities, which direct numerical simulations generally fail to evaluate accurately. In this paper, we develop a theoretical large deviations framework for a class of Markov chains modeling the genetic evolution of bacteria such as E. coli. In particular, we develop the cost function for discrete-time Markov chains which describe the daily evolution of histograms of bacterial populations. We also develop explicit formulas that can be used to numerically quantify the most likely evolutionary trajectories connecting an initial histogram and the target histogram.

Paper Structure

This paper contains 33 sections, 27 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Stochastic model for bacterial evolution experiments
  3. The space of genetic histograms
  4. Path Space of histograms sequences
  5. Phase 1 - Deterministic growth
  6. Phase 2 - Random mutations
  7. Random Mutation matrices and mutation rates
  8. Mutations constraint set $K(H)$
  9. The Poisson companion matrices $Z$
  10. Population histogram after mutations
  11. Phase 3 - Random selection
  12. Markov chain dynamics in the space of histograms
  13. Large deviations for daily cycles
  14. Random mutations matrices and products of Poisson distributions
  15. Proof of Theorem \ref{['companionTH']}
  16. ...and 18 more sections

Key Result

Lemma 2.10

Fix $a>0$. For $N > \frac{g^2}{a F_1}$, any $H \in \mathcal{H}$ with $b(H) \geq a$, and any $r \in K(H)$, there is an $N$-rational matrix $s = \{ s_{j,k} \}$ such that

Theorems & Definitions (80)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9: Constraint Sets
  • Lemma 2.10
  • ...and 70 more