Counter-intuitive stochastic behavior of simple gene circuits with negative feedback

Abstract

It has often been taken for granted that negative feedback loops in gene regulation work as homeostatic control mechanisms. If one increases the regulation strength a less noisy signal is to be expected. However, recent theoretical studies have reported the exact contrary, counter-intuitive observation, which has left a question mark over the relationship between negative feedback loops and noise. We explore and systematically analyze several minimal models of gene regulation, where a transcriptional repressor negatively regulates its own expression. For models including a quasi-steady-state assumption, we identify processes that buffer noise change (RNA polymerase binding) or accentuate it (repressor dimerization) alongside increasing feedback strength. Moreover, we show that lumping together transcription and translation in simplified models clearly underestimates the impact of negative feedback strength on the system's noise. In contrast, in systems without a quasi-steady-state assumption, noise always increases with negative feedback strength. Hence, subtle mathematical properties and model assumptions yield different types of noise profiles and, by consequence, previous studies have simultaneously reported decrease, increase or persistence of noise levels with increasing feedback. We discuss our findings in terms of separation of timescales and time correlations between molecular species distributions, extending current theoretical findings on the topic and allowing us to propose what we believe new ways to better characterize noise.

Figure 1

The basic modules of gene regulation modeling including (A) RNA polymerase binding to the gene, (B) making a clear distinction between mRNA transcription and protein translation, and (C) including repressor dimerization.

Figure 2

(A and B) Feedback-dependent noise of the RNAP + DM module. (C–H) Correlation behavior of the TT module. Initial TF levels were fixed by tuning the (A) RNA polymerase binding rate (k₅), (B) TF dimerization rate (k₉), (C and F) mRNA degradation rate (k₇), (D and G) the protein translation rate (k₈), and (E and H) the protein degradation rate (k₄), respectively. Colors show the log₁₀(CV) of protein numbers (A–E) and time correlations between mRNA and TF (F–H). White crosses indicate cases where all kinetic parameters are within typical biological ranges.

Figure 3

Protein and mRNA time courses in the TT module, portraying TF (A and B) multimodal behavior and (C and D) bust-like synthesis, using a feedback of α = 10¹³ in parameter set 2 and initial TF level of 100 molecules. FSP analysis (E), with fixed feedback α = 10¹⁰ (solid line) and α = 10¹⁵ (dotted line). Probabilities are evaluated for times between 10² and 10¹⁵ s, time represented in log scale. Labels refer to the probability of having 0 (cross), 1 (circle), and 2 (dot) molecules of mRNA, respectively.

Figure 4

Feedback-dependent noise (CV) and correlation behavior of the TT module, without a QSS assumption. All labels are identical to Fig. 2, C–H.

Figure 5

Summary of observed CV behavior as negative feedback strength is increased. Observations were classified according to the model and whether a QSS assumption was included or not. Tuned kinetic rates are specified in each box; arrows indicate the form of dependency. Round/squared brackets show positive/negative time correlation of TF and mRNA time courses, double brackets portray mild correlations. The lower panel shows rate parameterization behavior with increasing feedback.