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.

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
: In other words, a state with an intermediate expression level (i.e., smaller
) can be more easily switched on or off, both by noise and also by mutation, and hence the variances generally increase as
approaches 0. Still, some correlation between V_{
ip
}(i) and V_{
g
}(i) remains even after removing this correlation through
.

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

Authors’ Affiliations

(1)

Department of Basic Science, Univ. of Tokyo

References

Kaneko K: Proportionality between variances in gene expression induced by noise and mutation: consequence of evolutionary robustness.BMC Evol Biol 2011, 11:27.PubMedView Article

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