The graph of the chain network indicates that all species are coupled in the indicated chain pattern and that all other possible couplings are ignored. The path from 2 to N + 3 is not necessarily straight in the protein and so the distance R merely represents the total distance travelled. Note that despite the barrier there is a strong driving force from the source to sink-the overall route is up 0.1V, down -0.2V, fixed potential through the relays, and then down -0.6V at the last step upon reaching the surface site. For sake of gauging hopping advantage a simple parametric study was undertaken where r 12 = 5 Å is fixed and, for the moment, the overall distance from species 2 to species N + 3 (the sink) is fixed as well (e.g., 60 Å for Figure ). Uniform spacing is used in the path from 2 to N + 3. Since the transport time from source to sink (actually, its reciprocal) can be used to measure hopping advantage, the time τ for species N + 3 to reach 50% of its equilibrium concentration was computed. Given that the process is quite spontaneous downhill and with the initial concentration [1] t=0 = 1.0M assigned, we have [N + 3] t=∞ = 1.0M so we need the time for N + 3 to reach 0.5M within desirable numerical accuracy (for this task a standard bisection algorithm was applied to the concentration vector in MATLAB). A direct measure of hopping advantage was defined using a standard p-value analogous to the pH scale used in analytical chemistry. Specifically: