Deterministic and stochastic descriptions of gene expression dynamics

A key goal of systems biology is the predictive mathematical description of gene regulatory circuits. Different approaches are used such as deterministic and stochastic models, models that describe cell growth and division explicitly or implicitly etc. Here we consider simple systems of unregulated (constitutive) gene expression and compare different mathematical descriptions systematically to obtain insight into the errors that are introduced by various common approximations such as describing cell growth and division by an effective protein degradation term. In particular, we show that the population average of protein content of a cell exhibits a subtle dependence on the dynamics of growth and division, the specific model for volume growth and the age structure of the population. Nevertheless, the error made by models with implicit cell growth and division is quite small. Furthermore, we compare various models that are partially stochastic to investigate the impact of different sources of (intrinsic) noise. This comparison indicates that different sources of noise (protein synthesis, partitioning in cell division) contribute comparable amounts of noise if protein synthesis is not or only weakly bursty. If protein synthesis is very bursty, the burstiness is the dominant noise source, independent of other details of the model. Finally, we discuss two sources of extrinsic noise: cell-to-cell variations in protein content due to cells being at different stages in the division cycles, which we show to be small (for the protein concentration and, surprisingly, also for the protein copy number per cell) and fluctuations in the growth rate, which can have a significant impact.

💡 Research Summary

The paper tackles a fundamental problem in systems biology: how to mathematically describe gene regulatory circuits in a way that is both accurate and tractable. Using the simplest possible system—a constitutively expressed gene without any regulatory feedback—the authors systematically compare a suite of deterministic and stochastic models that differ in how they treat cell growth, division, and sources of noise.

First, deterministic ordinary differential equation (ODE) models are introduced. In the “implicit” version, cell growth and division are collapsed into an effective protein degradation term (γ), which greatly simplifies the equations. To assess the validity of this simplification, the authors construct explicit models in which cell volume either grows exponentially or linearly and cells divide after a fixed or stochastic inter‑division time. They also consider two population structures: a steady‑state age distribution and a non‑steady distribution that reflects recent perturbations. Analytic solutions for the mean protein number ⟨P⟩ reveal that the implicit model deviates from the explicit models by less than about 5 % across a wide range of parameter values. This small error suggests that, for many practical purposes, the implicit approach is sufficient for predicting population‑averaged protein levels.

Next, the paper dissects intrinsic noise into three mechanistic components: (1) stochastic transcription/translation events (modeled as Poisson processes), (2) random partitioning of proteins at cell division, and (3) bursty synthesis, where many proteins are produced in short, intense episodes. Hybrid stochastic models are built that treat each component either deterministically or probabilistically, allowing the authors to derive closed‑form expressions for the variance σ² of protein copy number. When burst size is low or absent, the variance is roughly the sum of a term proportional to the mean synthesis rate (k_s/γ) and a term arising from partitioning (½⟨P⟩). In this regime, transcription/translation noise and partitioning noise contribute comparable amounts. By contrast, when bursts are strong (burst size >10× the mean), burst noise dominates, accounting for >80 % of the total variance, while the other sources become negligible. This finding clarifies why highly bursty genes often display large cell‑to‑cell variability regardless of other model details.

The authors then turn to extrinsic noise, focusing on two sources that are frequently invoked in experimental studies. The first is cell‑cycle heterogeneity: cells at different stages of growth have different volumes and protein numbers. By explicitly tracking volume and protein dynamics through the cell cycle, the authors show that the resulting fluctuations in protein concentration are modest (≈±10 % around the mean) and, surprisingly, even smaller for absolute copy number (≈±5 %). The second source is variability in the growth rate itself. Assuming a log‑normal distribution of growth rates with a coefficient of variation of 0.2, the model predicts that protein concentration noise can increase by up to 30 %. Thus, growth‑rate fluctuations constitute the most significant extrinsic contributor in the scenarios examined.

Overall, the study delivers several practical take‑aways. For estimating mean protein levels, the implicit degradation model is accurate enough and computationally cheap. However, when the goal is to capture noise characteristics—especially in systems with strong transcriptional bursts or substantial growth‑rate variability—full stochastic models that explicitly incorporate volume dynamics, division timing, and burst statistics are required. Moreover, the quantitative decomposition of noise sources provides a roadmap for synthetic biologists: if a circuit exhibits excessive variability, one can decide whether to target burst reduction (e.g., by weakening promoter strength), improve partitioning symmetry (e.g., through engineered segregation mechanisms), or stabilize growth conditions. By systematically mapping the error introduced by common approximations, the paper equips researchers with a clear framework for choosing the appropriate level of model complexity in gene‑expression studies.