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This result will be important in fitting actual genomic data to the models.

Appendix F: Perturbation Theory Approximation for the Extended Model

As before, relate time and the number of genes through <j)(t):

This extends the previous definition (36); the variable u is still defined as before: u = log (pit).

Recall that when Q = 0 and R > 0 the long-term behavior of F(m, t) is determined by the coefficients Am, as shown in equation (30). Assume that the large-time solution in the presence of gene deletion is determined by new coefficients Bm:

F(m, t) —> Bm<j>(t) = Bm exp (u) as t —» oo (50)

Substituting this ansatz into the fundamental equations (10) leads to:

(1 + R - Q)B1 = RN0 - (1 + Q)Bi + 2QB2 (1 + R- Q)Bm = (m- l)5m_! - (1 + Q)mBm + Q(m + 1 )Bm+1

Motivated by the numerical results, we will develop the perturbation around a new variable ym:

that relates Bm to the Q = 0 solution (Am) as closely as possible. Using the explicit form of Am from (30) in (51) leads to:

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