(A) Lock and key. No conformational changes occur upon binding. The ligand (white) and the target (green) have complementary structures. (B) Induced fit. The target changes its conformation due to the interaction with the ligand. (C) Pre-existing equilibrium model. The native state is actually an ensemble of conformations, that is deformations may occur even before binding. The ligand selectively binds the matching target within this ensemble of fluctuating conformations.

Both the ligand (white) and the target (green) are interconverting between an ensemble of conformations denoted by indices, a_i and A_i, respectively. All the different conformations may interact and as a result, a variety of complexes is formed. In some of them the target and the ligand are perfectly matched, for example a_iA_i and a_jA_j, and in some there is only partial fit, for example a_iA_j and a_jA_i. The rate of product formation depends on the concentrations of the complexes, which depend on K_ij, and on the functionality of each complex, which depends on the turnover numbers, υ_ij. In a similar fashion, the different ligand conformations, a_i, may interact with competing target conformations B_i and thus catalyze incorrect product.

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_BT, r = 0.1, g = 1 Å, k = 1 k_BT/Ų, f = 15 k_BT.

Color bar shows log of specificity. In two dimensions, the ligand structure may stretch or shrink along x and y. The mismatches along these axes are d_x and d_y. The gray circle denotes zero mismatch. In the presence of multiple competitors (green crosses), the optimal mismatch (black X) is nonzero and depends on the structure of the various competitors. Competitors that slightly differ from the correct target have a “window of recognition” which overlaps with correct target recognition window. As a result, the specificity is maximal for a non-zero mismatch. The parameters of the plot are N = 15, m = 2, ε = 2 k_BT, r = 0.1, g = 1 Å, k = 1 k_BT/Ų, f = 15 k_BT.

The ligand (white) is interconverting within an ensemble of conformations while interacting with two rigid competitors, A and B (green and orange), characterized by different structures. Non-specific binding energy may lead to the formation of functional complexes in which the target and the ligand are not exactly matched. The unmatched complexes may also be functional but their product formation rates, ν_um, may differ from these of the matched complexes, ν_m. The specificity of the ligand, that is its ability to discriminate between A and B, depends on the ligand flexibility, the structural mismatch between its native state and the correct target and on the structural difference between the competing targets.

Conformational changes occur upon binding of a ligand (white) and a target (green). In the native state of the ligand, the binding sites are equally spaced and positioned at x_i⁰ (i = 0,1,2…N−1) and the total length of the binding domain is l₀. The ligand is interacting with a rigid target on which N complementary binding sites are equally spaced and positioned at y_i  = y₀+i·s/(N−1) where s is the length of the target binding domain. The ligand may undergo a conformational change to fit the target. Since we consider only the lowest mode motion, the ligand may only stretch or expand uniformly. Thus, its binding sites are displaced to x_i = x₀+i·l/(N−1) and the total length changes from l₀ to l.

Colors denote various values of rigidity k (in units of k_BT/Ų, legend). (A) For targets that differ by Δ = 3 Å the specificity is optimal at a nonzero mismatch. As the ligand becomes more rigid the optimal mismatch tends to zero as d₀∼k^−1/2. (B) For competing targets with similar structure, Δ = 0, the specificity resembles a rectangular window centered on zero mismatch. The width of this window also decreases as k^−1/2. (C) The specificity when only matched complexes are functional, r = 0, increases exponentially with the mismatch as ξ∼exp(k·Δ·d). The parameters of the plot are N = 15, m = 2, ε = 2 k_BT, r = 0.1, g = 1 Å, f = 15 k_BT.

Colors denote various values of rigidity k (in units of k_BT/Ų, legend). The lengths of the target binding domain, s_A and s_B are fluctuating according to a Gaussian noise with variances σ_A and σ_B. (A, B) For competing targets of different structure, Δ = 3 Å, similarly to noiseless targets, the specificity is optimal at a nonzero mismatch. (C, D) Competing targets of similar structure, Δ = 0. The specificity has an extremum at d = 0, but whether this point is maximum or minimum depends on the noise. For a noisier correct target, σ_A>σ_B, an optimal ligand has a nonzero mismatch. The parameters of the plot are: N = 15, m = 2, ε = 2 k_BT, r = 0.1, g = 1 Å, σ_A,B = 0.5, 0.6 Å, f = 15 k_BT.