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c7 = 2 (RA +iRB + \+ 7)+2 V(RA + IRB + 1 + 7)2 - 4l{RA + RB + 1) (74)

In a similar fashion, we define coefficients Cm and Cm, akin to the coefficients Am of the solution to the minimal model (7), that describe the ultímate linear growth of the histogram bins: F^(m, t) - Cmt, and similarly for Fsim, t). The form of the coefficients is very similar to the minimal model's Am:

The normalized probability distribution corresponding to this limit can be found using the same normalization identity that was helpful in deriving the probability distribution in the minimal model (Appendix H):

= 1 A fr 1 + 1 B TT »7 ~ C1 + 1 Ra + Rb ¿I ci + i + 1 ci + 7 Ra + Rb C1 + j(i + 1)

We have also briefly considered the case of more than two duplication types. When there is no introduction of new folds into the genome, the same argument behind equations (72) and (73) generalizes: the sub-histogram for each duplication type is exponential. Furthermore, we have confirmed numerically that the terminal distribution is not dramatically affected by selection pressure, even when there are several families with significandy different rates of duplication. One particular example, involving a four duplication types appears in Figure 10. In this rather extreme case, types "B", "C" and "D" are 4.0, 8.0 and 16.0 times more likely to be duplicated than type "A". The total rate of new fold acquisition is the same for both genomes.

Appendix H: A Useful Normalization Identity

A series whose terms zm, m =1, I,--- are defined by a recursion relation:

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