Noise-limited frequency signal transmission in gene circuits

Abstract

To maintain normal physiology, cells must properly process diverse signals arising from changes in temperature, pH, nutrient concentrations, and other factors. Many physiological processes are controlled by temporal aspects of oscillating signals; that is, these signals can encode information in the frequency domain. By modeling simple gene circuits, we analyze the impact of cellular noise on the fidelity and speed of frequency-signal transmission. We find that transmission of frequency signals is "all-or-none", limited by a critical frequency (f(c)). Signals with frequencies <f(c) are transmitted with high fidelity, whereas those with frequencies >f(c) are severely corrupted or completely lost in transmission. We argue that f(c) is an intrinsic property of a gene circuit and it varies with circuit parameters and additional feedback or feedforward regulation. Our results may have implications for understanding signal processing in natural biological networks and for engineering synthetic gene circuits.

FIGURE 1

Analysis of frequency signals with noise. (A) A one-stage gene circuit where the output protein P is controlled by a transcription activator, A. (B) An oscillatory input signal can generate an output signal with oscillations compounded with noise. The mean and standard deviation of the output signal of the linearized model can be analytically computed. Here, we define the mean value as the oscillatory component and the standard deviation as the noise component. Alternatively, the stochastic simulations of the output signal for the nonlinear system can be analyzed by the FFT method to obtain its dominant frequency (see Methods for more details).

FIGURE 2

Critical frequency for the one-stage gene circuit. (A) The amplitude of output oscillations decreased with f_in. f_c was calculated as the intersection between the “average noise level” curve and the “oscillation amplitude” curve. (B) Calculations of f_out for varying f_in using stochastic simulations. (C) Fraction of stochastic simulations that generated correct f_out (i.e., where f_out = f_in).

FIGURE 3

Transmission of a multiplexed signal. (A) A multiplexed input signal. (B) The corresponding output signal computed by stochastic simulation. (C) Power spectra of the input signal. (D) Power spectra of the output signal. Power spectra of the output signal indicate that all three frequencies were transmitted with complete fidelity. Even though power spectra decreased when the input frequency increased, they were still at least 10-fold higher than the power spectra of background noise. Three frequencies (0.005/min, 0.0067/min, and 0.01/min) were multiplexed in a composite signal with an amplitude of five molecules for each input frequency.

FIGURE 4

Effects of mRNA or protein dynamics on f_c demonstrate the dependence of f_c on the speed of mRNA dynamics and the speed of protein dynamics. The speed of mRNA or protein dynamics was modulated by proportionally changing the translation rate constant, k_p, and the mRNA or protein decay rate constant (g_m or g_p), keeping other parameters constant. It was normalized with respect to the base case (g_m = 0.2/min, g_p = 0.02/min). The color bar represents log₁₀(f_c).

FIGURE 5

Effects of circuit architectures on f_c. (A) Dependence of f_c on negative feedback and positive feedback. A high 1/K_d value corresponds to stronger feedback regulation. Cooperativity of feedback regulation (n) was fixed at 2. (B) Dependence of f_c on feedforward rate constant (normalized with respect to the base value of 0.1/min). See Supplementary Material for description of models.