Figure 6

Mathematical model of random flipping. Flipping was modeled as a Markov Chain in which each of the possible eight starting permutations is a state (shown in parenthesis in the graph). At each step (Number of flips (k)) in a random walk on the graph in Figure 1b, the probability of a plasmid being properly sorted (% Plasmids that solved the problem) after k flips is calculated as the number of paths of length k from the initial state to the solution state, divided by the total number of paths of length k from the initial state to any state. Starting permutations that show equivalent behavior can be grouped into three families (distinguished by color in the graph). The families are distinguishable for up to 4 flips; at 5 flips and beyond they reach a state of equilibrium at 25% plasmids solved.