Noisy signaling through promoter logic gates.

We study the influence of noisy transcription factor signals on cis-regulatory promoter elements. These elements process the probability of binary binding events analogous to computer logic gates. At equilibrium, this probability is given by the so-called input function. We show that transcription factor noise causes deviations from the equilibrium value due to the nonlinearity of the input function. For a single binding site, the correction is always negative resulting in an occupancy below the mean-field level. Yet for more complex promoters it depends on the correlation of the transcription factor signals and the geometry of the input function. We present explicit solutions for the basic types of AND and OR gates. The correction size varies among these different types of gates and signal types, mainly being larger in AND gates and for correlated fluctuations. In all cases we find excellent agreement between the analytical results and numerical simulations. We also study the E. coli Lac operon as an example of an AND NOR gate. We present a consistent mathematical method that allows one to separate different sources of noise and quantifies their effect on promoter occupation. A surprising result of our analysis is that Poissonian molecular fluctuations, in contrast to external fluctuations, do no contribute to the correction.