Erratum to: Proportionality between variances in gene expression induced by noise and mutation: consequence of evolutionary robustness

Although the simulation data as well as the conclusion on the proportionality between V _ip (i) and V _g (i) in the work [1] is correct, interpretation of some data therein should be corrected. As the sampling number (L = 200) to measure the average gene expression level is not large enough, there is a bias in the estimate in V _g (i). Finiteness in the number of sampling L will generally cause a bias of the order of V _ip (i)/L, in the estimate of the variance V _g (i). The too good proportionality between V _ip (i) and V _g (i) for large σ, shown in Figure two (a)(b) of [1] (especially for small V _g (i)), is due to this artifact. Accordingly, the sharp peak at ∼1/L = 1/200 in Figure three of [1] is due to this insufficiency by the sample number.

Still, the proportionality between the two variances V _ip (i) and V _g (i), albeit not so sharp, holds, as already observed in the region with larger V _g (i) in [1]. We have simulated the model with a larger number of samples, i.e., N = L = 1000. As is shown in Figure 1, the proportionality is well discernible, where the proportion coefficient V _g (i)/V _ip (i) decreased with the increase in the noise level σ, which was already observed in the broad peak beyond 1/L in Figure three of [1]. This broad peak beyond 1/L in Figure three of [1] was found to be sharper as N was increased, from 200 to 1000. This peak indeed corresponds to the proportion coefficient extracted from Figure 1 in the present Correction. As the noise level σ was increased, the peak position ρ = V _g (i)/V _ip (i) decreased. Hence for larger σ, larger L is needed to get reliable estimate in the proportion coefficient. As for Figure five and Figure six of [1], the sharp proportionality for V _g (i) ≲ 0.001 is due to the above bias, while the discussion therein concerns with the approach of V _g (i) to V _ip (i) at larger V _g (i), which is not affected by the bias here.

Figure 1

Relationship between V _g ( i ) and V _ip ( i ). As described in the Method section of [1], V _ip (i) was computed as the variance of the distribution of Sign(x _i ) over L runs for an identical genotype, while V _g (i) was computed as a variance of the distribution of $\left(\overline{\mathit{\text{Sign}}(x_i)}\right)$ over N individuals, where $\overline{\mathit{\text{Sign}}(x_i)}$ was the mean over L runs. Here we adopted N = L = 1000, instead of 200 in [1]. σ = 0.09 (blue *) and 0.03 (red +). The plot of (V _g (i) and V _ip (i)) for all genes i over 55-65th generations, where we have plotted only those genes with V _g (i) > .0002, as the those with smaller than that may have little accuracy in estimating V _g (i).

To sum up, the main claim of [1], i.e., proportionality between V _ip (i) and V _g (i) is valid, but the value of the proportion coefficient ρ = V _g (i)/V _ip (i) should be corrected. It decreases with the noise level, in contrast to the discussion in [1] for large σ. Major factor on this proportionality is attributed to the correlation of each variance with the average value $\overline{\mathit{\text{Sign}}(x(i))}$: In other words, a state with an intermediate expression level (i.e., smaller $\left|\overline{\mathit{\text{Sign}}(x(i))}\right|$) can be more easily switched on or off, both by noise and also by mutation, and hence the variances generally increase as $\left|\overline{\mathit{\text{Sign}}(x(i))}\right|$ approaches 0. Still, some correlation between V _ip (i) and V _g (i) remains even after removing this correlation through $\overline{\mathit{\text{Sign}}(x(i))}$.

I regret any inconvenience that misintepretation of the data with an insufficient sample size may have caused.