A stochastic two-state signalling module with negative feedback

Motivated by the negative feedback calcium exerts on the gating dynamics of a calcium-conducting ion channel in olfactory receptor neurons, we develop an abstract two-state (open/closed) signalling module with negative feedback. The coupling between the gating dynamics of the channel and the conducted ion makes the effective dynamics of the channel non-Markovian and difficult to treat in a Langevin-approach. We make use of two different techniques to describe the stochastic dynamics of the module. First, we calculate the steady state probability distribution using a Master/Fokker-Planck-type equation. Second, a path-integral formulation based on the temporal statistics of the channel state-flips is developed to calculate dynamical properties of the module. The feedback effect is built into the model in a systematic way in the form of a weak perturbation. Analytic results are obtained for the open probability of the channel as well as the auto-correlation and response functions (both for the discrete channel variable and the continuous calcium concentration). Monte Carlo simulations are performed which support the analytical predictions in the weak feedback limit and provide results beyond linear perturbation theory.