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Open Access Correction

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

Kunihiko Kaneko

Author affiliations

Department of Basic Science, Univ. of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan

Citation and License

BMC Evolutionary Biology 2012, 12:240  doi:10.1186/1471-2148-12-240

The electronic version of this article is the complete one and can be found online at: http://www.biomedcentral.com/1471-2148/12/240


Received:24 May 2012
Accepted:3 December 2012
Published:9 December 2012

© 2012 Kaneko; licensee BioMed Central Ltd.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Correction

Although the simulation data as well as the conclusion on the proportionality between Vip(i) and Vg(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 Vg(i). Finiteness in the number of sampling L will generally cause a bias of the order of Vip(i)/L, in the estimate of the variance Vg(i). The too good proportionality between Vip(i) and Vg(i) for large σ, shown in Figure two (a)(b) of [1] (especially for small Vg(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 Vip(i) and Vg(i), albeit not so sharp, holds, as already observed in the region with larger Vg(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 Vg(i)/Vip(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 = Vg(i)/Vip(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 Vg(i) ≲ 0.001 is due to the above bias, while the discussion therein concerns with the approach of Vg(i) to Vip(i) at larger Vg(i), which is not affected by the bias here.

thumbnailFigure 1. Relationship between Vg(i) and Vip(i). As described in the Method section of [1], Vip(i) was computed as the variance of the distribution of Sign(xi) over L runs for an identical genotype, while Vg(i) was computed as a variance of the distribution of <a onClick="popup('http://www.biomedcentral.com/1471-2148/12/240/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/12/240/mathml/M1">View MathML</a> over N individuals, where <a onClick="popup('http://www.biomedcentral.com/1471-2148/12/240/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/12/240/mathml/M2">View MathML</a> 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 (Vg(i) and Vip(i)) for all genes i over 55-65th generations, where we have plotted only those genes with Vg(i) > .0002, as the those with smaller than that may have little accuracy in estimating Vg(i).

To sum up, the main claim of [1], i.e., proportionality between Vip(i) and Vg(i) is valid, but the value of the proportion coefficient ρ = Vg(i)/Vip(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 <a onClick="popup('http://www.biomedcentral.com/1471-2148/12/240/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/12/240/mathml/M3">View MathML</a>: In other words, a state with an intermediate expression level (i.e., smaller <a onClick="popup('http://www.biomedcentral.com/1471-2148/12/240/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/12/240/mathml/M4">View MathML</a>) can be more easily switched on or off, both by noise and also by mutation, and hence the variances generally increase as <a onClick="popup('http://www.biomedcentral.com/1471-2148/12/240/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/12/240/mathml/M5">View MathML</a> approaches 0. Still, some correlation between Vip(i) and Vg(i) remains even after removing this correlation through <a onClick="popup('http://www.biomedcentral.com/1471-2148/12/240/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2148/12/240/mathml/M6">View MathML</a>.

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. PubMed Abstract | BioMed Central Full Text | PubMed Central Full Text OpenURL