This thesis is motivated by two associated obstacles we face for the solution and analysis of master equation models of gene transcription. First, the master equation – a differential-difference equation that describes the time evolution of the probability distribution of a discrete Markov process – is difficult to solve and few approaches for solution are known, particularly for non-stationary systems. Second, we lack a general framework for solving master equations that promotes explicit comprehension of how extrinsic processes and variation affect the system, and physical intuition for the solutions and their properties. We address the second obstacle by deriving the exact solution of the master equation under general time-dependent assumptions for transcription and degradation rates. With this analytical solution we obtain the general properties of a broad class of gene transcription models, within which solutions and properties of specific models may be placed and understood. Furthermore, there naturally emerges a decoupling of the discrete component of the solution, common to all transcription models of this kind, and the continuous, model-specific component that describes uncertainty of the parameters and extrinsic variation. Thus we also address the first obstacle, since to solve a model within this framework one needs only the probability density for the extrinsic component, which may be non-stationary. We detail its physical interpretations, and methods to calculate its probability density. Specific models are then addressed. In particular we solve for classes of multistate models, where the gene cycles stochastically between discrete states. We use the insights
gained from these approaches to deduce properties of several other models. Finally, we introduce a quantitative characterisation of timescales for multistate models, to delineate “fast” and “slow” switching regimes. We have thus demonstrated the power of the obtained general solution for analytically predicting gene transcription in non-stationary conditions.