Time-triggered stochastic hybrid systems with two timer-dependent resets

We analyze a class of time-triggered stochastic hybrid systems where the state-space evolves as per a linear time-invariant dynamical system. This continuous time evolution is interspersed with two kinds of stochastic resets. The first reset occurs based on an internal timer that measures the time elapsed since it last occurred. Whenever the first reset occurs the states-space undergoes a random jump and the timer is reset to zero. The second reset occurs based on an arbitrary timer-depended rate, and whenever this reset fires, the state-space is changed based on a given random map. For this class of systems, we provide exact conditions that lead to finite statistic moments, and the corresponding exact analytical expressions for the first two moments. This framework is applied to study random fluctuations in the concentration of a protein in a growing cell. In the context of this example, the timer denotes the time elapsed since the cell was born, and the cell division event (first reset) is triggered based on a timer-dependent rate. The second reset corresponds to synthesis of the protein in stochastic bursts, and finally, during cell growth protein concentration continuously decrease due to dilution. Our analysis provides closed-form formulas for the noise in the protein concentration and leads to a striking result - for a constant (timer-independent) protein synthesis rate the noise in the protein concentration is invariant of the noise in the cell-cycle time. Finally, we provide a rigorous framework for investigating protein noise levels for different forms of time-dependent synthesis rates, as is the case for cell-cycle regulated genes inside the cell.