Using noise to probe and characterize gene circuits

Abstract

Stochastic fluctuations (or "noise") in the single-cell populations of molecular species are shaped by the structure and biokinetic rates of the underlying gene circuit. The structure of the noise is summarized by its autocorrelation function. In this article, we introduce the noise regulatory vector as a generalized framework for making inferences concerning the structure and biokinetic rates of a gene circuit from its noise autocorrelation function. Although most previous studies have focused primarily on the magnitude component of the noise (given by the zero-lag autocorrelation function), our approach also considers the correlation component, which encodes additional information concerning the circuit. Theoretical analyses and simulations of various gene circuits show that the noise regulatory vector is characteristic of the composition of the circuit. Although a particular noise regulatory vector does not map uniquely to a single underlying circuit, it does suggest possible candidate circuits, while excluding others, thereby demonstrating the probative value of noise in gene circuit analysis.

Fig. 1.

Noise regulatory vector and its application in the analysis of gene circuits. (A) The noise regulatory vector for an uncharacterized gene circuit is determined by comparison of its experimental noise structure to the noise structure of an assumed model. The vector points toward a family of gene circuits that includes the true gene circuit, and away from inappropriate models. (B) The assumed structure of the a priori model is a constitutive transcription-translation circuit. (C) Time series and autocorrelation functions for two stochastic protein populations characterized by identical 〈p〉 and CV² but different half correlation times τ_1/2.

Fig. 2.

Relationship of 3D noise map and noise regulatory vectors, ΔN⃗_r. (A) Noise map for 2,920 ORFs in S. cerevisiae. A subset of 25 proteins that were both included in the Bar-Even et al. (2) study and present in our compiled database (shown in red) are observed to scatter about the Bar-Even (2) noise model CV² = 1,200/〈(p〉 (red line), suggesting a similarity in measured noise trends [Bar-Even (2)] and those inferred from literature data (blue points). (B) Graphical definition of the bias line and ΔN⃗_r. The bias line represents the behavior of the a priori model. To determine N⃗Δ_r for a protein with measured coordinates on the noise map (filled circle), one first locates the bias point (open circle) by projecting vertically to the bias line. ΔN⃗_r is defined by the 2D vector connecting the bias point to the measured point in the log(CV²)−log(τ_1/2) plane.

Fig. 3.

Noise regulatory vectors for the gene circuit in Fig. 1B as affected by slow operator kinetics. Points indicate ΔN⃗_rT for G(t) values of (starting from and moving in the direction of the red arrow) 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5 (red), 0.6, 0.7. 0.8, 0.9, 0.95, and 0.99. The effect of operator kinetics is captured in the ratios κ₁ = k_r/γ_p and κ₂ = k_r/α₀.

Fig. 4.

Noise regulatory vectors for autoregulation. Points indicate ΔN⃗_rT at gene activation levels of (starting from and moving in the direction of the red arrow) 0.02, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7. 0.8, 0.9, and 0.95. (A) Negative autoregulation. The effect of decreasing κ₂ is to increase Δlog(τ_1/2) when κ₁ 1 and to increase both Δlog(CV²) and Δlog(τ_1/2) when κ₁ < 1. (B) Positive autoregulation. The effect of multimerization and regimes of mono- and bistability.

Fig. 5.

Summary of noise vector domains for various regulatory motifs. (−ar, negative autoregulation; +ar, positive autoregulation; sk, slow gene activation kinetics). Bold font denotes domains of primary influence.