Table 1 Summary of statistics for three different cases of burst in gene expression.

From: Efficient and flexible implementation of Langevin simulation for gene burst production

 

mRNA burst (y = m)

both bursts (y = p)

protein burst (y = p)

condition

γ g  ≥ 10 γ m

γ g  ≥ 10 γ m  ≥ 100 γ p

γ m  ≥ 10 γ p

simulated subjects

m(t), p(t)

p(t)

g(t), p(t)

burst event distribution:

\({\bar{e}}_{y}(\tau )\)

k g Ï„

\({k}_{g}\tau {\bar{b}}_{m}\) ‡

g(t)k m Ï„

\({\sigma }_{ey}^{2}(\tau )\)

k g Ï„

\({k}_{g}\tau {\bar{b}}_{m}(2{\bar{b}}_{m}+1)\) ‡

g(t)k m Ï„

burst size† distribution:

\({\bar{b}}_{y}\)

\({\bar{b}}_{m}\)

\({\bar{b}}_{p}\)

\({\bar{b}}_{p}\)

\({\sigma }_{by}^{2}\)

\({\bar{b}}_{m}^{2}+{\bar{b}}_{m}\)

\({\bar{b}}_{p}^{2}+{\bar{b}}_{p}\)

\({\bar{b}}_{p}^{2}+{\bar{b}}_{p}\)

burst production in Langevin equation:

\({{\rm{\Delta }}}_{y}(\tau )\) by equation (10)

\({k}_{g}\tau {\bar{b}}_{m}\)‡

\({k}_{g}\tau {\bar{b}}_{m}{\bar{b}}_{p}\)

\(g(t){k}_{m}\tau {\bar{b}}_{p}\)

\({\sigma }_{{{\rm{\Delta }}}_{y}}^{2}(\tau )\) by equation (11)

\({k}_{g}\tau {\bar{b}}_{m}(2{\bar{b}}_{m}+1)\)‡

\({k}_{g}\tau {\bar{b}}_{m}{\bar{b}}_{p}(2{\bar{b}}_{m}{\bar{b}}_{p}+2{\bar{b}}_{p}+1)\)

\(g(t){k}_{m}\tau {\bar{b}}_{p}(2{\bar{b}}_{p}+1)\)

steady-state distribution: exact expression*

\(\bar{m}\,=\) \(\bar{g}\frac{{k}_{m}}{{\gamma }_{m}}\)

same as exact

—

—

\({\sigma }_{m,ss}^{2}\,=\) \(\bar{m}({F}_{1}{\bar{b}}_{m}+1)\)

\(\bar{m}({\bar{b}}_{m}+1)\) §

—

—

\(\bar{p}\,=\) \(\bar{m}\frac{{k}_{p}}{{\gamma }_{p}}\)

same as exact

same as exact

same as exact

\({\sigma }_{p,ss}^{2}\,=\) \(\bar{p}({F}_{0}{\bar{b}}_{m}{\bar{b}}_{p0}+{\bar{b}}_{p0}+1)\)

\(\bar{p}({\bar{b}}_{m}{\bar{b}}_{p0}+{\bar{b}}_{p0}+1)\) ¶

\(\bar{p}({\bar{b}}_{m}{\bar{b}}_{p}+{\bar{b}}_{p}+1)\) ||

\(\bar{p}({F}_{2}{\bar{b}}_{m}{\bar{b}}_{p}+{\bar{b}}_{p}+1)\) #

  1. †Definition for burst sizes are: \({\bar{b}}_{m}=\frac{{k}_{m}}{{\gamma }_{g}+{k}_{g}}\), \({\bar{b}}_{p0}=\frac{{k}_{p}}{{\gamma }_{m}+{\gamma }_{p}}\), \({\bar{b}}_{p}=\frac{{k}_{p}}{{\gamma }_{m}}\).
  2. ‡The mean and variance of mRNA production with mRNA burst are the mean and variance of the burst events of protein in the case of both bursts.
  3. *Definitions for the fractions are: \({F}_{0}=\frac{{\gamma }_{g}({\gamma }_{g}+{\gamma }_{m}+{\gamma }_{p}+{k}_{g})}{({\gamma }_{g}+{\gamma }_{m}+{k}_{g})({\gamma }_{g}+{\gamma }_{p}+{k}_{g})}\), \({F}_{1}=\frac{{\gamma }_{g}}{({\gamma }_{g}+{\gamma }_{m}+{k}_{g})}\), \({F}_{2}=\frac{{\gamma }_{g}}{({\gamma }_{g}+{\gamma }_{p}+{k}_{g})}\).
  4. §With the conditions of \({\gamma }_{g}\gg {\gamma }_{m}\) and \({\gamma }_{g}\gg {k}_{g}\), \({F}_{1}\) replaced by \(1\) with mRNA burst.
  5. ¶With the conditions of \({\gamma }_{g}\gg {\gamma }_{m}\) and \({\gamma }_{g}\gg {k}_{g}\), \({F}_{0}\) replaced by \(1\) with mRNA burst.
  6. ||With the conditions of \({\gamma }_{g}\gg {\gamma }_{m}\) and \({\gamma }_{g}\gg {k}_{g}\), \({F}_{0}\) replaced by \(1\) with both bursts; with \({\gamma }_{m}\gg {\gamma }_{p}\), \({\bar{b}}_{p0}\) replaced by \({\bar{b}}_{p}\).
  7. #With the conditions of \({\gamma }_{m}\gg {\gamma }_{p}\), \({F}_{0}\) replaced by \({F}_{2}\) with protein burst; and with \({\gamma }_{m}\gg {\gamma }_{p}\), \({\bar{b}}_{p0}\) replaced by \({\bar{b}}_{p}\).
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