Each column is for a specific difference between the competing targets, Δ = *s*_{A}−*s*_{B}, given in Å. The rate production of the correct product *R*_{A} is a sum of a Gaussian centered at *d* = 0, which arises from the specific binding, and a uniform contribution due to non-specific binding (top row, blue). Similarly, the incorrect rate production, *R*_{B}, is composed of a Gaussian centered at *d* = Δ and a uniform non-specific contribution (top row, red). The specificity, ξ, is the ratio between the correct and incorrect production rates and therefore depends on the location and width of the recognition windows (bottom row). (A) If the competing targets differ in structure, Δ = 3 Å, the windows of recognition partly overlap and the resulting specificity is optimal at a *nonzero mismatch*. (B) For Δ = 0, both *R*_{A} and *R*_{B} are centered around zero mismatch and the resulting specificity is approximately a rectangular window of width, *d*_{1/2}≈*k*^{−1/2}((*N−m*)ε−*f*−log(αr)). (C) If the competing targets differ much, Δ≫*d*_{1/2}, the recognition windows do not overlap and the specificity is again optimal for zero mismatch. The parameters of the plot are *N* = 15, *m* = 2, ε = 2 *k*_{B}*T*, *r* = 0.1, *g* = 1 Å, *k* = 1 *k*_{B}*T*/Å^{2}, *f* = 15 *k*_{B}*T*.

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