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We study the most probable trajectories of the concentration evolution forthe transcription factor activator in a genetic regulation system, withnon-Gaussian stable L\'evy noise in the synthesis reaction rate taking intoaccount. We calculate the most probable trajectory by spatially maximizing theprobability density of the system path, i.e., the solution of the associatednonlocal Fokker-Planck equation. We especially examine those most probabletrajectories from low concentration state to high concentration state (i.e.,the likely transcription regime) for certain parameters, in order to gaininsights into the transcription processes and the tipping time for thetranscription likely to occur. This enables us: (i) to visualize the progressof concentration evolution (i.e., observe whether the system enters thetranscription regime within a given time period); (ii) to predict or avoidcertain transcriptions via selecting specific noise parameters in particularregions in the parameter space. Moreover, we have found some peculiar orcounter-intuitive phenomena in this gene model system, including (a) a smallernoise intensity may trigger the transcription process, while a larger noiseintensity can not, under the same asymmetric L\'evy noise. This phenomenon doesnot occur in the case of symmetric L\'evy noise; (b) the symmetric L\'evymotion always induces transition to high concentration, but certain asymmetricL\'evy motions do not trigger the switch to transcription. These findingsprovide insights for further experimental research, in order to achieve or toavoid specific gene transcriptions, with possible relevance for medicaladvances.

Most Probable Evolution Trajectories in a Genetic Regulatory System Excited by Stable L\'evy Noise