  Using analytical calculations and computer simulations we consider both the
lateral diffusion of a membrane protein and the fluctuation spectrum of the
membrane in which the protein is embedded. The membrane protein interacts with
the membrane shape through its spontaneous curvature and bending rigidity. The
lateral motion of the protein may be viewed as diffusion in an effective
potential, hence, the effective mobility is always reduced compared to the case
of free diffusion. Using a rigorous path-integral approach we derive an
analytical expression for the effective diffusion coefficient for small ratios
of temperature and bending rigidity, which is the biologically relevant limit.
Simulations show very good quantitative agreement with our analytical result.
The analysis of the correlation functions contributing to the diffusion
coefficient shows that the correlations between the stochastic force of the
protein and the response in the membrane shape are responsible for the
reduction.
  Our quantitative analysis of the membrane height correlation spectrum shows
an influence of the protein-membrane interaction causing a distinctly altered
wave-vector dependence compared to a free membrane. Furthermore, the time
correlations exhibit the two relevant timescales of the system: that of
membrane fluctuations and that of lateral protein diffusion with the latter
typically much longer than the former. We argue that the analysis of the
long-time decay of membrane height correlations can thus provide a new means to
determine the effective diffusion coefficient of proteins in the membrane.
