  We study the functional characteristics of a two-gene motif consisting of a
double positive feedback loop and an autoregulatory negative feedback loop. The
motif appears in the gene regulatory network controlling the functional
activity of pancreatic $\beta$-cells. The model exhibits bistability and
hysteresis in appropriate parameter regions. The two stable steady states
correspond to low (OFF state) and high (ON state) protein levels respectively.
Using a deterministic approach, we show that the region of bistability
increases in extent when the copy number of one of the genes is reduced from
two to one. The negative feedback loop has the effect of reducing the size of
the bistable region. Loss of a gene copy, brought about by mutations, hampers
the normal functioning of the $\beta$-cells giving rise to the genetic
disorder, maturity-onset diabetes of the young (MODY). The diabetic phenotype
makes its appearance when a sizable fraction of the $\beta$-cells is in the OFF
state. Using stochastic simulation techniques, we show that, on reduction of
the gene copy number, there is a transition from the monostable ON to the ON
state in the bistable region of the parameter space. Fluctuations in the
protein levels, arising due to the stochastic nature of gene expression, can
give rise to transitions between the ON and OFF states. We show that as the
strength of autorepression increases, the ON$\to$OFF state transitions become
less probable whereas the reverse transitions are more probable. The
implications of the results in the context of the occurrence of MODY are
pointed out..
