A general model for predicting the binding affinity of reversibly and irreversibly dimerized ligands

Empirical data has shown that bivalent inhibitors can bind a given target protein significantly better than their monomeric counterparts. However, predicting the corresponding theoretical fold improvements has been challenging. The current work builds off the reacted-site probability approach to provide a straightforward baseline reference model for predicting fold-improvements in effective affinity of dimerized ligands over their monomeric counterparts. For the more familiar irreversibly linked bivalents, the model predicts a weak dependence on tether length and a scaling of the effective affinity with the 3/2 power of the monomer’s affinity. For the previously untreated case of the emerging technology of reversibly linking dimers, the effective affinity is also significantly improved over the affinity of the non-dimerizing monomers. The model is related back to experimental quantities, such as EC50s, and the approaches to fully characterize the system given the assumptions of the model. Because of the predicted significant potency gains, both irreversibly and reversibly linked bivalent ligands offer the potential to be a disruptive technology in pharmaceutical research.

The definition of molarity is moles of a substance per L. The one free monomer from the bivalent ligand is ~1.66x10 -24 moles. The volume of the sampled sphere is with R p in angstroms. With 10 10 angstroms in every meter, 10 6 cm 3 in every m 3 , and 10 3 cm 3 in every L, S7 becomes approximately 4.18879x10 -27 R p 3 . Taking the ratio produces the right hand side of the defining equation 27.
Putting the solution into standard form and simplifying gives equation 28.
Rearranging so that the square root is on one side, we get 0, Squaring and consolidating terms yields The two roots are L 0,eff =0 and 2K D . The zero is a false root.

Derivation of equation 31:
Case 1: Unbound target binding unbound ligand . (S13) Case 2: Unbound target binding singly bound ligand . (S14) Case 3: Target singly bound with singly bound ligand binding unbound ligand . (S15) Case 4: Target singly bound with singly bound ligand binding singly bound ligand . (S16) Case 5: Target singly bound with doubly bound ligand binding unbound ligand . (S17) Case 4: Target singly bound with doubly bound ligand binding singly bound ligand . (S18) Defining R' as the fraction of ligand sites bound and noting the parallel between two ligand sites and two target sites, we coopt equation S5 to read Since total sites bound from both sides must be equal, Substituting equations S30 and S29 into any of equations S25-S28 gives

Derivation of equation 35:
By definition, K 2 is which when rearranged gives where the second equality comes from substitution of equation S5. Since L 0 >>T 0 , equation S23 may be approximated as Substituting equations 25 and 26 into S24 yields At R=½, equation S25 becomes Substitution of equation 32 into equation S26 produces equation 35.

Derivation of equation 39:
From equation 38, cancel terms and place all but square-root terms on the right hand side of the equation to get √1 + 4 0, ⁄ = 1 + 4 0, .
Squaring both sides and placing in quadratic form yields The roots are 0 (disallowed) and 1/(

Derivation of equation 40:
Taking the limit of equation 28 as L 0,eff becomes much greater than K D yields Substituting S29 into equation 35 produces Simplifying equation S30 gives The right hand side of equation S31 is equivalent to the right hand side of equation 40.

Derivation of equation 52:
After cancellation of equivalent terms on the left and right hand sides of equation 51, one gets Substitution of equations 46-49 into S32 gives or equivalently, Solving equation S34 for either F L,M or F L,D yields the same result as equation 52.

Derivation of equation 54:
Starting with the immediate equality on the right hand side of equation 53 and solving for F L yields The amount of ligand bound to the target in the 1:1 state must equal the number of target sites occupied in that state, which is stated mathematically as L 0 D 21 =2T 0 F 21 , which when substituted into S37 and simplifying gives which is equation 54.
The left hand side in parentheses is just F LD , while the right hand side in parentheses is F LM 2 . Solving for K Dim gives equation 55.
As L 0 goes to 0, the only way F DL remains finite for a finite K Dim is if the negative root is selected. Factoring out K Dim /4L 0 from the root and simplifying produces equation 56.

Confirmation that dimer limit is obtained when L 0 >>K Dim :
Substituting equations 42, 43, 44, 46, 47, and 52 into equation 51 and cancelling terms yields Note that the factor of 2 in equation 32 is absorbed into the L 0 term when describing the current system since the concentration of dimer is half that of the monomer.

Derivation of equation 58:
Substituting equations 45, 47, and 56 into equation 57 and cancelling terms yields Since L 0 D 21 =2T 0 F 21 , in the limit of L 0 >>T 0 , D 21 <<1, permitting equation S51 to be rewritten as Equation S52 can be recast as a quadratic in F 21 , whose physically relevant root (taken by observing the desired behavior at R=1) is which is equation 58.

Derivation of effective concentrations for dimerizing moieties both bound to a single target:
Both moieties are treated as being in spheres of radius R p /2 whose centers are separated by a distance R s . The volume of overlap between spheres is given by while the volume of each sphere is The effective concentration is given by the number of moieties per total sampled volume, or (S57) Clearly, when there is no overlap, the far right hand side of equation S57 becomes 1/V sph , while in the other limit of complete overlap (V o =V sph ), the far right hand side becomes 2/V sph . Since the denominator is linear in V o , the multiplicative factor to 1/V sph can only vary between 1 and 2.

Derivation of equation 61:
The fraction of targets with two monomers bound is given by the fraction of targets with two sites occupied, scaled by fraction of monomer at each site, namely, from equation 45, Similarly, the fraction of targets with one site containing an otherwise unbound dimer and the other site unoccupied is, from equation 44,

Derivation of equation 63:
Equation S48 is the appropriate expression for K D , and substitution of F 21 is required. In order to simplify the readability, we evaluate the denominator on the right hand side of equation S48 Substituting equations S67-S69 into equation S66 yields