Fig. 3. The regeneration process.

Gene duplication (A) or recombination (B) generates a starting condition for the search process, at rate w. (C) From the starting condition, we require k mutational steps, each at rate u, to reach the target sequence, which encodes a new function. At each step, there is the possibility to receive inactivating mutations, at rate v, which destroy the search. The frequency of the wild type is denoted by x₀. The frequencies of the intermediate steps in the search process are denoted by x_i. At steady state and assuming neutrality, we have the following frequencies: ${\mathit{x}}_0=\frac{\mathit{v}}{\mathit{v}+\mathit{w}}$ and ${\mathit{x}}_{\mathit{i}}=\left(\frac{\mathit{w}}{\mathit{u}}\right)\left(\frac{\mathit{v}}{\mathit{v}+\mathit{w}}\right){\left(\frac{\mathit{u}}{\mathit{v}+\mathit{u}}\right)}^{\mathit{i}}$. Let us consider a numerical example: w = 10⁻⁷, u = 10⁻⁹, v = 10⁻⁷ per cell division. Then, cells that have made as many as 10 steps toward the target have a frequency of about 5 × 10⁻¹⁹ and are present on a planetary scale with a total cell number of the order on 10³⁰.