An exact steady-state solution of the stochastic equations governing the behavior of a gene regulated by a self-generated proteomic atmosphere is presented. The solutions depend on an adiabaticity parameter measuring the relative rate of DNA-protein unbinding and protein degradation. The steady-state solution reveals deviations from the commonly used Ackers et al approximation based on the equilibrium law of mass action, allowing anticooperative behavior in the "nonadiabatic" limit of slow binding and unbinding rates. Noise from binding and unbinding events dominates the shot noise of protein synthesis and degradation up to quite high values of the adiabaticity parameter.

The exact time-dependent solution for the stochastic equations governing the behavior of a binary self-regulating gene is presented. Using the generating function technique to rephrase the master equations in terms of partial differential equations, we show that the model is totally integrable and the analytical solutions are the celebrated confluent Heun functions. Self-regulation plays a major role in the control of gene expression, and it is remarkable that such a microscopic model is completely integrable in terms of well-known complex functions.

Mathematical models, as instruments for understanding the workings of nature, are a traditional tool of physics, but they also play an ever increasing role in biology--in the description of fundamental processes as well as that of complex systems. In this review, the authors discuss two examples of the application of group theoretical methods, which constitute the mathematical discipline for a quantitative description of the idea of symmetry, to genetics. The first one appears, in the form of a pseudo-orthogonal (Lorentz like) symmetry, in the stochastic modelling of what may be regarded as the simplest possible example of a genetic network and, hopefully, a building block for more complicated ones: a single self-interacting or externally regulated gene with only two possible states: 'on' and 'off'. The second is the algebraic approach to the evolution of the genetic code, according to which the current code results from a dynamical symmetry breaking process, starting out from an initial state of complete symmetry and ending in the presently observed final state of low symmetry. In both cases, symmetry plays a decisive role: in the first, it is a characteristic feature of the dynamics of the gene switch and its decay to equilibrium, whereas in the second, it provides the guidelines for the evolution of the coding rules.

Intrinsic transcriptional noise induced by operator fluctuations is investigated with a simple spin like stochastic model. The effects of transcriptional fluctuations in protein synthesis is probed by coupling transcription and translation by an amplificative interaction. In the presence of repression a new term contributes to the noise which depends on the rate of mRNA production. If the switching time is small compared with the mRNA life time the noise is also small. In general the dumping of protein production by a repressive agent occurs linearly but the fluctuations can show a maxima at intermediate repression. Thediscrepancy between the switching time, the mRNA degradation and protein degradation is crucial for the repressive control in translation without large fluctuations. The noise profiles obtained here are in quantitative agreement with recent experiments.