Frequency domain analysis of noise in autoregulated gene circuits

Abstract

We describe a frequency domain technique for the analysis of intrinsic noise within negatively autoregulated gene circuits. This approach is based on the transfer function around the feedback loop (loop transmission) and the equivalent noise bandwidth of the system. The loop transmission, T, is shown to be a determining factor of the dynamics and the noise behavior of autoregulated gene circuits, and this T-based technique provides a simple and flexible method for the analysis of noise arising from any source within the gene circuit. We show that negative feedback not only reduces the variance of the noise in the protein concentration, but also shifts this noise to higher frequencies where it may have a negligible effect on the noise behavior of following gene circuits within a cascade. This predicted effect is demonstrated through the exact stochastic simulation of a two-gene cascade. The analysis elucidates important aspects of gene circuit structure that control functionality, and may provide some insights into selective pressures leading to this structure. The resulting analytical relationships have a simple form, making them especially useful as synthetic gene circuit design equations. With the exception of the linearization of Hill kinetics, this technique is general and may be applied to the analysis or design of networks of higher complexity. This utility is demonstrated through the exact stochastic simulation of an autoregulated two-gene cascade operating near instability.

Fig 1.

Model of single gene expression. (a) mRNA molecules are synthesized from the template DNA strand at rate k_R, and proteins are translated at a rate of k_P off of each mRNA molecule. γ_R and γ_P are the decay rates for mRNA and protein respectively. (b) The same as a except that the protein is negatively autoregulated such that k_R = k_Rmax/[1 + (p/k_d)ⁿ], where k_Rmax is the maximum rate of transcription, k_d is the concentration of protein where k_{R =} k_Rmax/2, and n is the Hill coefficient.

Fig 2.

Cascaded gene circuits. (a) Schematic representation of a two-gene cascade where the expression of the second gene is positively regulated by the concentration of the first protein such that k_R2 = k_Rmax2/[1 + (k_d2/p₁)ⁿ]. The first gene circuit is negatively autoregulated as described in Fig. 1b. (b) Graphical representation of the change in the p₁ noise PSD caused by feedback in the p₁ circuit. If properly filtered by the second gene circuit in the cascade, the noise between the unregulated and regulated noise bandwidths may be rejected.

Fig 3.

Calculated and simulated ratio of p₂ noise strengths (p₁ circuit: without feedback (i.e., k_R1 = constant)/with feedback) of the gene cascade in Fig. 2a. This ratio is plotted both for total p₂ noise and with the p₂ intrinsic noise removed, and shows that feedback may decrease the effect of p₁ noise by as much as (1 + |T(0)|)². The calculations were performed by using Eq. 16 for the regulated p₁ noise strength and Eq. 9 for the intrinsic p₂ and unregulated p₁ noise strengths. The total p₂ noise strength was found by adding the intrinsic p₂ noise strength to the p₁ noise strength multiplied by the noise power gain, (dp₂/dp₁)²|p₁ = 〈p₁〉, and modifying the noise bandwidth to reflect the additional p₂ pole. The parameters for this circuit are as follows: b₁ = 8, b₂ = 4, γ_p1 = 0.0001925/s, γ_R1 = 0.00289/s, γ_R2 = 0.00578/s, k_d1 = 800, k_d2 = 600, k_rmax1 = 0.0231/s, k_rmax2 = 0.05/s, n₁ = n₂ = 7. For the no feedback case, k_R1 was set to a constant value of 0.0167/s, giving the same value of 〈p₁〉, 700, for both feedback and no feedback cases. 〈p₂〉 varied with the value of γ_p2, but was the same for feedback and no feedback cases. The intrinsic noise of p₂ was determined in simulation by setting k_R2 = 0.0367/s whereas all other parameters remained unchanged. The simulation method is described in Appendix.

Fig 4.

Global feedback in a gene cascade. (a) Schematic representation of a two-gene cascade with both single-gene circuits within the feedback path such that k_R1 = k_Rmax1/[1 + (p₂/k_d1)ⁿ] and k_R2 = k_Rmax2/[1 + (k_d2/p₁)ⁿ]. (b) The calculated and simulated noise PSD for the circuit in a both with and without (i.e., k_R1 = constant) feedback. The calculations were performed by using Eq. A-1 and A-2 from Appendix. The parameters for this circuit are as follows: k_rmax1 = k_rmax2 = 0.0231/s, b₁ = b₂ = 13, γ_p1 = γ_p2 = 0.0001925/s, γ_R1 = γ_R2 = 0.00578/s, k_d1 = k_d2 = 350, n₁ = n₂ = 7. For the no feedback case, k_R1 was set to a constant value of 0.00449/s, giving the same value of 〈p₂〉, 490, for both feedback and no feedback cases, allowing direct comparison of the PSDs. The simulation method is described in Appendix.

Fig 5.

Partitioning of a circuit to allow analysis by using the feedback formalism presented here. For a noise source at any location within the circuit, the transfer functions A(f) and β(f) are defined. A(f) is the transfer function from the noise source location to the output (i.e., location where noise behavior is to be calculated) and β(f) is the transfer function from this output back to the noise source location.