Frequency domain chemical Langevin analysis of stochasticity in gene transcriptional regulation.

We present a frequency domain Langevin approach for stochastic analysis that remains valid for many important gene circuit elements even as molecular populations approach zero. We begin by considering the case of low-rate transcription and show that the previously reported shot noise representation is exact at all mRNA population levels for a constant transcription rate. Next, we consider transcriptional control through protein-DNA interactions at an operator site within the gene promoter region. This analysis results in expressions for the dynamics and noise behavior of this important gene sub-circuit, including the spectral density of the intrinsic operator noise and the processing of extrinsic noise by this transcriptional regulation system. This analysis shows that mRNA synthesis noise is composed of wideband shot noise and band-limited operator binding generated noise components. We find that the bandwidth of operator noise and its ultimate effect on total mRNA and protein noise is controlled by operator binding and unbinding dynamics. The most substantial impact of the operator noise is seen at transcription rates just above basal expression. This analysis captures the full behavior of this transcriptional regulation system, and points to potentially serious flaws in simplified mathematical relationships often used to model transcriptional regulation.