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The mean formation time E(T 1, n)) in years is calculated using the formula (A.6.4) and the current empirical estimates of the gene duplication rate [ 24].
For the linear BDIM, the value of the mean formation time from an essential singleton is given by formula M 1)(1, n) = 1/ λ M 1) n, where M 1) n, the mean formation time in 1/ λ units is calculated using the formula (A.4.6) The transition from the 1/ λ time units to years is considered in s.6 of the Mathematical Appendix [see Additional file 1 ].
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On the contrary, the mean formation times in years for the quadratic and cubic models are of the same order (Tables 2 and 3).
Below we examine only the mean family formation time from an essential singleton (model (1.3)).
Thus, as in the case of rational BDIMs, increase of the degree of logistic BDIMs under a fixed value of average duplication rate r du cannot yield mean family formation times < 1011 years.
Specifically, with the empirical estimates of the duplication rates used above for the linear BDIM, the quadratic model gives the mean family formation times ~1011 yrs.
The mean family formation times given by the linear BDIM would become realistic only if the recent analyses underestimated the gene duplication rate by a factor of ~10, which does not seem plausible.
Figures 19 and 20 show the dependence of the mean and the coefficients of variation of family formation time on family size for the cubic BDIM.
Thus, the actual formation time for the largest family could differ from the mean value by several orders of magnitude with a high probability.
Due to the large variance of the family formation time estimates (see the detailed discussion above), this value is likely to be much less than the mean.
"It's pair formation time for these guys," he said.
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