  The effects of carrying capacity of environment $K$ for degradation (the $K$
effect for short) on the constitutive gene expression and a simple genetic
regulation system, are investigated by employing a stochastic Langevin equation
combined with the corresponding Fokker-Planck equation for the two stochastic
systems subjected to internal and external noises. This $K$ effect
characterizes the limited degradation ability of the environment for RNA or
proteins, such as insufficient catabolic enzymes. The $K$ effect could
significantly change the distribution of mRNA copy-number in constitutive gene
expression, and interestingly, it leads to the Fano factor slightly larger than
1 if only the internal noise exists. Therefore, that the recent experimental
measurements suggests the Fano factor deviates from 1 slightly (Science {\bf
346} (2014) 1533), probably originates from the $K$ effect. The $K$ effects on
the steady and transient properties of genetic regulation system, have been
investigated in detail. It could enhance the mean first passage time
significantly especially when the noises are weak and reduce the
signal-to-noise ratio in stochastic resonance substantially.

Biological Physics