  We study the most probable trajectories of the concentration evolution for
the transcription factor activator in a genetic regulation system, with
non-Gaussian stable L\'evy noise in the synthesis reaction rate taking into
account. We calculate the most probable trajectory by spatially maximizing the
probability density of the system path, i.e., the solution of the associated
nonlocal Fokker-Planck equation. We especially examine those most probable
trajectories from low concentration state to high concentration state (i.e.,
the likely transcription regime) for certain parameters, in order to gain
insights into the transcription processes and the tipping time for the
transcription likely to occur. This enables us: (i) to visualize the progress
of concentration evolution (i.e., observe whether the system enters the
transcription regime within a given time period); (ii) to predict or avoid
certain transcriptions via selecting specific noise parameters in particular
regions in the parameter space. Moreover, we have found some peculiar or
counter-intuitive phenomena in this gene model system, including (a) a smaller
noise intensity may trigger the transcription process, while a larger noise
intensity can not, under the same asymmetric L\'evy noise. This phenomenon does
not occur in the case of symmetric L\'evy noise; (b) the symmetric L\'evy
motion always induces transition to high concentration, but certain asymmetric
L\'evy motions do not trigger the switch to transcription. These findings
provide insights for further experimental research, in order to achieve or to
avoid specific gene transcriptions, with possible relevance for medical
advances.
