Mitochondria are the semi-autonomous organelles that are responsible for generating the majority of the energy required by mammalian cells under normal conditions. They are only semi-autonomous because the replication, transcription and translation of the DNA molecules within the mitochondrion, mtDNA, are ultimately controlled by the cell nucleus. We present a series of three models of the nuclear control of mtDNA replication, with an increasing complexity in the role of mtDNA mutations in the models. We solve these deterministic models exactly, and compare these solutions to the results of stochastic simulations of the same systems. We use the steady states of the deterministic model to explain behaviors, such as threshold effects and mitochondrial proliferation, that are often seen in the cells of patients affected by mitochondrial diseases and that also occur with age. The parameters of these models illustrate the dual control of mitochondria by both the nuclear and mitochondrial DNA.