arXiv subjects

Brett Geiger

Publications and source records attributed to Brett Geiger.

4 recordsLinked to original sources

arXiv2023

Rare Events Analysis and Computation for Stochastic Evolution of Bacterial Populations

In this paper, we develop a computational approach for computing most likely trajectories describing rare events that correspond to the emergence of non-dominant genotypes. This work is based on the large deviations approach for discrete Markov chains describing the genetic evolution of large bacterial populations. We demonstrate that a gradient descent algorithm developed in this paper results in the fast and accurate computation of most-likely trajectories for a large number of bacterial genotypes. We supplement our analysis with extensive numerical simulations demonstrating the computational advantage of the designed gradient descent algorithm over other, more simplified, approaches.

q-bio.PE↗

arXiv2018

Large Deviations Analysis for Stochastic Models of Bacterial Evolution

Radical shifts in the genetic composition of large cell populations are rare events with quite low probabilities that direct numerical simulations generally fail to evaluate accurately. In this paper, we develop a theoretical large-deviation framework for a class of Markov chains modeling the genetic evolution of bacteria such as E. coli in ``locked-box'' laboratory experiments. In particular, we develop the cost function for discrete-time Markov chains that describe the daily evolution of histograms of bacterial populations. We obtain explicit formulas for the cost function for interior histograms. 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.

math.PR↗

arXiv2018

Nonstationary open dynamical systems

This paper studies nonstationary open dynamical systems from the statistical viewpoint. By open, we mean that trajectories may escape through holes in the phase space. By nonstationary, we mean that the dynamical model itself (as well as the holes) may vary with time. Unlike the setting of random dynamical systems, no assumptions are made about the statistics of this variability. We formulate general results that yield conditional memory loss (an analog of decay of correlations) at exponential rates for nonstationary open dynamical systems. We apply our results to a class of nonstationary piecewise-smooth expanding systems in higher dimension with moving holes.

math.DS↗

arXiv2016

Large deviations for Gaussian diffusions with delay

Dynamical systems driven by nonlinear delay SDEs with small noise can exhibit important rare events on long timescales. When there is no delay, classical large deviations theory quantifies rare events such as escapes from metastable fixed points. Near such fixed points, one can approximate nonlinear delay SDEs by linear delay SDEs. Here, we develop a fully explicit large deviations framework for (necessarily Gaussian) processes $X_t$ driven by linear delay SDEs with small diffusion coefficients. Our approach enables fast numerical computation of the action functional controlling rare events for $X_t$ and of the most likely paths transiting from $X_0 = p$ to $X_T=q$. Via linear noise local approximations, we can then compute most likely routes of escape from metastable states for nonlinear delay SDEs. We apply our methodology to the detailed dynamics of a genetic regulatory circuit, namely the co-repressive toggle switch, which may be described by a nonlinear chemical Langevin SDE with delay.

math.PR↗