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The effects of carrying capacity of environment $K$ for degradation (the $K$effect for short) on the constitutive gene expression and a simple geneticregulation system, are investigated by employing a stochastic Langevin equationcombined with the corresponding Fokker-Planck equation for the two stochasticsystems subjected to internal and external noises. This $K$ effectcharacterizes the limited degradation ability of the environment for RNA orproteins, such as insufficient catabolic enzymes. The $K$ effect couldsignificantly change the distribution of mRNA copy-number in constitutive geneexpression, and interestingly, it leads to the Fano factor slightly larger than1 if only the internal noise exists. Therefore, that the recent experimentalmeasurements suggests the Fano factor deviates from 1 slightly (Science {\bf346} (2014) 1533), probably originates from the $K$ effect. The $K$ effects onthe steady and transient properties of genetic regulation system, have beeninvestigated in detail. It could enhance the mean first passage timesignificantly especially when the noises are weak and reduce thesignal-to-noise ratio in stochastic resonance substantially.

Effects of restrained degradation on gene expression and regulation