How input fluctuations reshape the dynamics of a biological switching system

Abstract

An important task in quantitative biology is to understand the role of stochasticity in biochemical regulation. Here, as an extension of our recent work [Phys. Rev. Lett. 107, 148101 (2011)], we study how input fluctuations affect the stochastic dynamics of a simple biological switch. In our model, the on transition rate of the switch is directly regulated by a noisy input signal, which is described as a non-negative mean-reverting diffusion process. This continuous process can be a good approximation of the discrete birth-death process and is much more analytically tractable. Within this setup, we apply the Feynman-Kac theorem to investigate the statistical features of the output switching dynamics. Consistent with our previous findings, the input noise is found to effectively suppress the input-dependent transitions. We show analytically that this effect becomes significant when the input signal fluctuates greatly in amplitude and reverts slowly to its mean.

FIG. 1

Illustration of our model: X(t) represents the input signal which fluctuates around a mean level over time; Y(t) records the switch process which flips between the off (Y = 0) and on (Y = 1) states with transition rates k_onX(t) and k_off. τ̃ is the dwell time in the off state.

FIG. 2

The slow switch case. Here we use λ = 10, k_on = 0.02, and k_off = 0.1 (thus K_d = 5). (a) P (τ̃ > t) vs t for μ = 3 and θ = 0.005, 0.5, and 2.0, which are achieved by choosing σ = 5, 50, and 100. Symbols represent simulation results, while lines denote exp(−k̃_onμt). (b) ${\stackrel{\sim }{\sigma }}_Y^2-{\sigma }_Y^2$ vs ${\sigma }_X^2$, where different values of ${\sigma }_X^2$ are obtained by tuning σ. (c) ${\stackrel{\sim }{\sigma }}_Y^2$ and ${\sigma }_Y^2$ vs μ/K_d with fixed θ = 0.50. (d) 𝒯̃_Y − 𝒯_Y vs ${\sigma }_X^2$.

FIG. 3

The fast switch case. Here we choose λ = 0.0125, μ = 5, k_off = 0.1, k_on = 0.02, and θ = 10 (such that σ ≃ 0.28). (a) Sample autocorrelation function (ACF) of successive off-state time intervals. (b) Sample ACF of a simulated sample path of Y(t) in the semilogarithmic scale. (c) Distribution of the off-state intervals, P(τ̃ > t). Symbols are from Monte Carlo simulations and the solid blue line is the semianalytical prediction described in the main text. (d) Input distributions conditioned on the output.

FIG. 4

Comparison of the Poisson, “shifted” Gamma, Gaussian, and Gamma distributions, all of which have the same mean μ = 3 and the same variance ${\sigma }_X^2=\mu$.