Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation

Abstract

Many regulatory molecules are present in low copy numbers per cell so that significant random fluctuations emerge spontaneously. Because cell viability depends on precise regulation of key events, such signal noise has been thought to impose a threat that cells must carefully eliminate. However, the precision of control is also greatly affected by the regulatory mechanisms' capacity for sensitivity amplification. Here we show that even if signal noise reduces the capacity for sensitivity amplification of threshold mechanisms, the effect on realistic regulatory kinetics can be the opposite: stochastic focusing (SF). SF particularly exploits tails of probability distributions and can be formulated as conventional multistep sensitivity amplification where signal noise provides the degrees of freedom. When signal fluctuations are sufficiently rapid, effects of time correlations in signal-dependent rates are negligible and SF works just like conventional sensitivity amplification. This means that, quite counterintuitively, signal noise can reduce the uncertainty in regulated processes. SF is exemplified by standard hyperbolic inhibition, and all probability distributions for signal noise are first derived from underlying chemical master equations. The negative binomial is suggested as a paradigmatic distribution for intracellular kinetics, applicable to stochastic gene expression as well as simple systems with Michaelis-Menten degradation or positive feedback. SF resembles stochastic resonance in that noise facilitates signal detection in nonlinear systems, but stochastic resonance is related to how noise in threshold systems allows for detection of subthreshold signals and SF describes how fluctuations can make a gradual response mechanism work more like a threshold mechanism.

Figure 1

(Upper) Number of product molecules as a function of time when product formation is inhibited by a noisy or noise-free signal (see main text). Product half-life is ln(2) time units, k = 10⁴ and 〈q〉 = 1%. To keep the same 〈q〉 before the shift for noisy and noise-free signals, the value of Kv, the inhibition constant multiplied by the reaction volume, is different in the two cases. After five time units the signal time average 〈n〉 = 10 shifts to 〈n〉 = 5 due to a 2-fold reduction in k_s from 10 k_d to 5 k_d. For slow fluctuations, k_d = 100. When k_s and k_d are 10 times higher, slow and rapid signal fluctuations give rise to almost indistinguishable processes for product formation (not shown). Rapid signal fluctuations correspond to insignificant time correlations. (Lower) Stationary signal distributions (Poissonian) before and after the shift in conditions in Upper. 〈q〉 (Eq. 2) is calculated by using the same Kv as in Upper.

Figure 2

〈q〉 (Eq. 2) for noisy (solid lines) and noise-free (dotted lines) signals as a function of the average number of signal molecules 〈n〉 in log-log scale. Kv is the inhibition constant multiplied by the reaction volume. For a mathematical specification of the distributions, see Eq. A3. For NB, the value of 〈n〉 is changed by changing λ for fixed ρ = 10/11. N in the lower two graphs is the upper limit in the number of molecules.

Figure 3

Amplification factors (–3) (the slope of the curves in Fig. 2 for other values of Kv) as functions of 〈q〉 in lin-log scale. The upper curve corresponds to noise-free signals.

Figure 4

(Upper) The number of signal molecules as a function of time. The synthesis rate constant decreases exponentially (Eq. A7). The quasistationary distribution of molecules is NB and the underlying random process is given by gene expression in Appendix. (Lower) The probability density for the switching delay time. The two curves for noise-free response corresponds to the minimal dispersion possible when all parameters can be chosen freely (Eq. A8), for the same initial intensity (dashed line) and the same average delay (dotted line) as the response (solid line) to the noisy signal, respectively. The noisy signal was NB-distributed with ρ = 10/11, as in Upper, Kv = 1 and assumed to fluctuate so rapidly that correlations are insignificant. The choice of underlying process does not change the result. See Appendix for a mathematical and numerical specification.