We investigate the dynamical effects of non-Gaussian asymmetric stable L\'evy fluctuations on the evolution of the transcription factor activator in a genetic regulation system . The noisy fluctuations arise from the synthesis reaction rate . We compute two deterministic quantities, the mean first exit time (MFET) and the first escape probability (FEP), in order to examine the likelihood for transcriptions: The mean time scale for the system exits the low concentration state (the longer the exit time, the less likely for transcription) and the switch probability from low concentration states to high concentration states (corresponding to likelihood for transcription) . By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that the fluctuations in the synthesis reaction rate lead to peculiar transitions to high concentrations and thus to possible transcriptions, such as realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index \beta, causing a bifurcation for the likelihood of transcription at the critical non-Gaussianity index value \alpha=1 (i.e., beyond which the likelihood for transcription suddenly increases ), and achieving a turning point at the threshold value \beta \approx 0.55 (i.e., beyond which the likelihood for transcription  reversed for \alpha values). The  bifurcation and turning point phenomena do not occur in the symmetric noise case (\beta =0). We conduct a series of numerical experiments about `regulating' the likelihood of gene transcription by tuning asymmetric stable L\'evy noise indexes. These offer insights for possible ways of achieving gene regulation in experimental research.
This work is devoted to investigating the evolution of concentration in a genetic regulation system , when the synthesis reaction rate is under additive and multiplicative asymmetric stable L\'evy fluctuations . By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index \beta, causing a stochastic bifurcation at the non-Gaussianity index value \alpha=1 (i.e., it is a separating point or line for the likelihood for transcription ), and achieving a turning point at the threshold value \beta \approx -0.5 (i.e., beyond which the likelihood for transcription suddenly reversed for \alpha values). The stochastic bifurcation and turning point phenomena do not occur in the symmetric noise case (\beta =0). While in the multiplicative noise case, non-Gaussianity index value \alpha=1 is a separating point or line for both the mean first exit time (MFET) and the first escape probability (FEP). We also investigate the noise enhanced stability phenomenon. Additionally, we are able to specify the regions in the whole parameter space for the asymmetric noise, in which we attain desired likelihood for transcription. We have conducted a series of numerical experiments in `regulating' the likelihood of gene transcription by tuning asymmetric stable L\'evy noise indexes. This work offers insights for possible ways of achieving gene regulation in experimental research.
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