A minimal mathematical model describing mass transport in the connecting cilium (CC) of a photoreceptor cell in response to a suddenly increased protein concentration at the base of the CC is developed. Dimensionless governing equations and dimensionless parameters are identified. Analytical solutions are obtained for concentrations of free (diffusion-driven) and motor-driven proteins. The obtained solutions make it possible to predict mass transfer in the CC as a function of two dimensionless transport parameters involved in the model: the diffusivity of free soluble proteins and the transition rate from the diffusion-driven to the motor-driven state. Sensitivities of the obtained solutions to these two parameters are discussed.