= Solution
With zero initial <messenger RNA> and <protein concentration>, solving the two linear <ordinary differential equations> in sequence gives
$$
\frac{\mathbb E X_P(t)}{X_{P,*}}=1-\frac{\alpha_R e^{-\alpha_Pt}-\alpha_Pe^{-\alpha_Rt}}{\alpha_R-\alpha_P}
\simeq1-e^{-\alpha_Pt}.
$$
The last expression applies after the short <messenger RNA> transient when $\alpha_R\gg\alpha_P$. All three mean curves have this shape. At a fixed <steady state>, a larger <translational burst size> requires fewer <transcription> events: $\beta_R=\alpha_PX_{P,*}/b$. Thus $b=400$ gives infrequent large upward jumps followed by <protein degradation>, whereas $b=3$ gives many smaller jumps and a smoother trace.
Noise depends on molecule counts. If the fixed cell <volume> is $V$, write $\bar N=VX_{P,*}$. In the instantaneous-burst limit, independent <Poisson process> arrivals of geometrically distributed bursts, with count arrival rate $\lambda=V\beta_R$, and unit <protein degradation> give
$$
\boxed{\operatorname{var}N=\bar N(1+b),\qquad \mathrm{CV}=\sqrt{\frac{1+b}{VX_{P,*}}}.}
$$
Indeed, the <moment equations> are $d\mathbb EN/dt=\lambda b-\alpha_P\mathbb EN$ and $d\operatorname{var}N/dt=\lambda b(1+2b)+\alpha_P\mathbb EN-2\alpha_P\operatorname{var}N$. With zero initial counts the <variance> becomes $\bar N[(1+b)-e^{-\alpha_Pt}-be^{-2\alpha_Pt}]$. These equations explain both the approach to the <steady state> and the scatter about it.
For illustration only, $V=1\,\mu\mathrm m^3$ gives $\bar N=10^4$ at the higher <concentration>: the two values of the <coefficient of variation> are $0.020$ and $0.200$. At the lower <concentration>, $\bar N=100$ and $b=3$ gives $0.200$. Decreasing the <concentration> by a factor of 100 therefore increases relative fluctuations tenfold, despite keeping small bursts. Absolute fluctuation sizes cannot be inferred without a cell <volume>. The original sketches below stipulate that <volume>; they are model realizations, not source data.<Transient protein-count variance in the instantaneous-burst model>[Derivation of the transient and stationary variance]{ref}
\Image[/past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2015-expression-noise.png]
{title=<Translational burst size>, relative expression fluctuations, and paired <fluorescent reporters>}
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