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

*BMC Evolutionary Biology*
**volume 12**, Article number: 240 (2012)

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 $\overline{\mathit{\text{Sign}}(x(i))}$: 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.

## References

- 1.
Kaneko K: Proportionality between variances in gene expression induced by noise and mutation: consequence of evolutionary robustness. BMC Evol Biol. 2011, 11: 27-10.1186/1471-2148-11-27.

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The online version of the original article can be found at 10.1186/1471-2148-11-27

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Kaneko, K. Erratum to: Proportionality between variances in gene expression induced by noise and mutation: consequence of evolutionary robustness.
*BMC Evol Biol* **12, **240 (2012). https://doi.org/10.1186/1471-2148-12-240

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