Extremal events dictate population growth rate inference (2501.08404v1)

Abstract: Recent methods have been developed to map single-cell lineage statistics to population growth. Because population growth selects for exponentially rare phenotypes, these methods inherently depend on sampling large deviations from finite data, which introduces systematic errors. A comprehensive understanding of these errors in the context of finite data remains elusive. To address this gap, we study the error in growth rate estimates across different models. We show that under the usual bias-variance decomposition, the bias can be decomposed into a finite-time bias and nonlinear averaging bias. We demonstrate that finite-time bias, which dominates at short times, can be mitigated by fitting its monotonic behavior. In contrast, at longer times, nonlinear averaging bias becomes the predominant source of error, leading to a phase transition. This transition can be understood through the Random Energy Model, a mean-field model of disordered systems, where a few lineages dominate the estimator. Applying these methods to experimental data demonstrates that correcting for biases in lineage-based approaches yields consistent results for the long-term growth rate across multiple methods and enables the reverse-engineering of dynamic models. This new framework provides a quantitative understanding of growth rate estimators, clarifies the conditions under which they can be effectively applied to finite data, and introduces model-free approaches for studying the connections between physiology and cell growth.

• The paper demonstrates that rare extremal events introduce systematic biases in estimating population growth rates from single-cell lineage data.
• It applies a novel finite-size scaling methodology, rooted in statistical physics and the Random Energy Model, to ensure monotonic convergence of growth estimators.
• Simulation results show that while both FTE and FDE estimators improve over time, FDE consistently yields lower finite-time bias, informing better experimental design.

Extremal Events and Their Role in Population Growth Rate Inference

The paper "Extremal events dictate population growth rate inference" by Trevor GrandPre, Ethan Levien, and Ariel Amir explores the challenges associated with estimating population growth rates from single-cell lineage data. The authors focus on the inherent errors and biases that emerge when using finite data sets for such inferences, particularly emphasizing the significance of extremal statistical events. Their comprehensive analysis reveals how these rare events impact the accuracy of population growth estimators and proposes methodologies to mitigate the biases associated with finite sampling.

The problem at the heart of this research is the translation of single-cell lineage statistics into reliable estimates of population growth rates. Recent advancements have provided a theoretical framework for this task; however, these methods inherently hinge on sampling large deviations from finite datasets, which introduces systematic biases. The authors identified two primary sources of bias: finite-time bias and nonlinear averaging bias. Finite-time bias dominates during short observational periods, while nonlinear averaging bias becomes more prominent over extended durations as few lineages increasingly dominate the estimators, leading to non-monotonic convergence behavior.

The paper introduces a novel methodology employing finite-size scaling to eliminate finite-time biases. Finite-size scaling, a principle rooted in statistical physics, is applied here to achieve a monotonic convergence of growth rate estimates with time, thus reducing estimation errors. The team further demonstrates that such biases, when related to nonlinear averaging, are fundamentally connected to the Random Energy Model (REM), a mean-field model from disordered systems physics. Under the lens of REM, the growth rate estimators undergo a phase transition akin to the well-known phenomena in physical systems, where the dominance of a few extremal lineages marks a "frozen" phase akin to low-temperature regimes in statistical mechanics. This transition can be quantitatively described using the REM framework, providing insights into the conditions that lead to poor convergence of estimators.

Two specific estimators are scrutinized in this context: the fixed-time ensemble (FTE) and the fixed-divisions ensemble (FDE). Both are examined through a classical bias-variance decomposition. The nonlinear averaging bias, a significant contributor to estimation error, is closely aligned with the quenched free energy concept in disordered systems, emphasizing the necessity of understanding extremal statistics for accurate inference.

Simulation results validate that for both FTE and FDE, the total error from asymptotic growth rate estimation decreases over time, with variance and finite-time bias being primary contributors at shorter times and nonlinear bias dominating at longer time intervals. Importantly, FDE is shown to have a consistently smaller finite-time bias than FTE across various scenarios, a finding bolstered by analytical derivations through a von Foerster equation approach that underscores the absence of finite-time bias for uncorrelated generation times.

This research holds significant implications for experimental design and theoretical understanding. By enhancing the accuracy of growth rate inference, it aids in bridging experimental observations with underlying physiological processes and evolutionary dynamics. Furthermore, the REM analogy not only clarifies the conditions under which estimation errors proliferate but also provides a practical framework to gauge the influence of extremal events, which is critical for data-limited biological investigations.

Future research can build on these findings by exploring the implications of these biases in diverse biological systems and extending the theoretical models to incorporate additional complexities of real-world data. Additionally, leveraging insights from other fields that deal with rare event sampling and estimation, such as in ATM networks or thermodynamic inference, may also prove fruitful in honing these inference methodologies.

The work of GrandPre, Levien, and Amir highlights the necessity of recognizing and addressing the intricacies of extremal event sampling within finite data for accurate population growth rate estimation, presenting both a significant challenge and an opportunity for advance in the field of theoretical biology.