  A one-dimensional run-and-tumble particle (RTP) switches randomly between a
left and right moving state of constant speed $v$. This type of motion arises
in a wide range of applications in cell biology, including the unbiased growth
and shrinkage of microtubules or cytonemes, the bidirectional motion of
molecular motors, and the "run-and-tumble" motion of bacteria such as {\em E.
coli}. RTPs are also of more general interest within the non-equilibrium
statistical physics community, both at the single particle level and at the
interacting population level, where it provides a simple example of active
matter. In this paper we use asymptotic methods to calculate the mean first
passage time (MFPT) for a one-dimensional RTP to escape an effective trapping
potential generated by space-dependent switching rates. Such methods are part
of a more general framework for studying metastability in so-called piecewise
deterministic Markov processes (PDMPs), which include the RTP as a special
case.
