Deriving Equilibrium Avidity From Sequential Binding

Why Sequential Binding Changes Apparent Affinity

A two-arm antibody can bind through one arm first and then capture a second target with its remaining arm. The second interaction is not an independent encounter from bulk solution: after the first arm binds, the other arm is tethered near the surface. Its local opportunity to bind can therefore be much greater than the target concentration in the surrounding solution. This is the equilibrium basis of avidity in the sequential-binding model.

The important distinction is between intrinsic affinity, which describes one binding arm and its target, and apparent affinity, which summarizes the stability of the antibody’s target-associated states. Avidity does not change the intrinsic dissociation constant of either arm. It changes the relative abundance of singly and doubly bound states.

For identical arms binding identical targets, write both monovalent dissociation constants as \(K_d\). The second arm’s equilibrium ratio of doubly bound to singly bound antibody is \(C_{\mathrm{eff}}/K_d\). Thus, when \(C_{\mathrm{eff}}\) is large relative to \(K_d\), the doubly bound state can be strongly favored after initial capture. This is an equilibrium statement; it does not by itself describe how rapidly the second bond forms or how long a molecule remains bound during a particular observation.

Derivation for Sequential Binding

Consider a specified first-capture route. Let \(c\) be the concentration of targets available for that initial interaction, \(A\) be free antibody, \(A_1\) be antibody bound through the first arm, and \(A_2\) be antibody bound through both arms. At equilibrium, the first step gives \( [A_1]/[A]=c/K_{d,1}\). Once \(A_1\) has formed, the second target is encountered at effective local concentration \(C_{\mathrm{eff}}\), so \( [A_2]/[A_1]=C_{\mathrm{eff}}/K_{d,2}\). Combining the two ratios gives the complete bound-to-free ratio for this route.

The first term inside the parentheses accounts for singly bound antibody; the second accounts for the doubly bound state. Equating the first and last expressions and solving for the apparent dissociation constant yields the sequential-binding result.

For identical arms, this reduces to \(K_{d,\mathrm{app}}=K_d/(1+C_{\mathrm{eff}}/K_d)=K_d^2/(K_d+C_{\mathrm{eff}})\). The avidity enhancement relative to the specified monovalent first-capture route is \(K_{d,1}/K_{d,\mathrm{app}}=1+C_{\mathrm{eff}}/K_{d,2}\). A smaller \(K_{d,\mathrm{app}}\) means stronger apparent equilibrium binding. When \(C_{\mathrm{eff}}\) is much greater than \(K_{d,2}\), the expression approaches \(K_{d,1}K_{d,2}/C_{\mathrm{eff}}\); for identical arms, it approaches \(K_d^2/C_{\mathrm{eff}}\). When \(C_{\mathrm{eff}}\) is negligible, the apparent value approaches \(K_{d,1}\), as expected when the second arm rarely binds.

If both arms are equivalent, either arm can make the first contact. Counting both routes introduces a statistical factor of two in the first-capture association. That factor is not the stabilization due to the second bond. The formula above is normalized to one specified first-capture route; comparisons with a monovalent reference should use consistent conventions for the number of available first-binding arms. In a full state model, count each distinct state and its multiplicity explicitly rather than adding or removing a factor of two informally.

Worked Calculations

How Target Density Enters

Target density matters through the number of accessible targets within reach of the unbound arm, and therefore through \(C_{\mathrm{eff}}\). A simple geometric approximation is

Here \(\rho_s\) is the surface density expressed in amount per unit area, \(A_{\mathrm{cap}}\) is an effective capture area, \(V_{\mathrm{eff}}\) is an effective volume in liters, and \(q\) is a dimensionless factor for accessibility and orientation. The units reduce to amount per volume. This is a model for local availability, not a universal conversion from surface density to molarity: linker length, epitope orientation, membrane organization, and steric obstruction can all change the effective geometry.

At low target density, and with geometry held fixed, increasing density will often increase \(C_{\mathrm{eff}}\) approximately in proportion. At higher density, simply extending that linear relation may be inappropriate. For example, if potential targets are randomly distributed, the probability of at least one target in a capture area is \(1-\exp(-\rho_N A_{\mathrm{cap}})\), where \(\rho_N\) is number density. This probability approaches one rather than increasing without limit. A refined model can account for this saturation and for the possibility that some nearby targets cannot be reached.

Diagnostics, Alternatives, and Limits

The sequential model is useful because it separates the contribution of the first capture from the added stabilization of the second bond. A fuller state-based binding model, as used for multispecific binding networks, can represent explicit free, singly bound, and doubly bound states and their separate target balances. That is preferable when target depletion, competing partners, or several binding routes are important. The compact apparent-\(K_d\) expression is a reduction of those state relationships under restrictive assumptions, not a replacement for them in every setting. An independent-arm model that omits the doubly bound state is a useful limiting comparison when second-arm capture is negligible, but it cannot represent avidity stabilization.

Several checks help determine whether the reduction is plausible. The calculated \(C_{\mathrm{eff}}\) should have concentration units and should be tied to a stated geometry or calibration. The apparent value should approach the first-arm \(K_d\) as \(C_{\mathrm{eff}}\) approaches zero, and should decrease as \(C_{\mathrm{eff}}\) increases while the other quantities remain fixed. If a fitted apparent affinity changes with target density, that pattern may be consistent with avidity, but it does not establish that density is the only cause: changes in target accessibility, clustering, or assay conditions can also alter the effective local opportunity. If data require different apparent values across conditions, inspect those assumptions rather than treating a single fitted \(K_d\) as an intrinsic arm property.

The formula also has limits. It presumes equilibrium; an observed binding signal collected before equilibrium need not reflect the predicted state ratios. It summarizes a particular sequential route and does not automatically account for all first-arm alternatives, heterogeneous target populations, steric constraints, or depletion of local targets. Nor does equilibrium avidity alone specify dissociation rates or rebinding during wash steps; those are distinct kinetic questions. For interpretation, report the monovalent \(K_d\), the assumed or estimated \(C_{\mathrm{eff}}\), the route and multiplicity convention, and the conditions under which the apparent value is defined.

Common Pitfall

A common mistake is to report \(K_d^2/C_{\mathrm{eff}}\) as the exact apparent dissociation constant at every target density. That expression is only the high-local-concentration approximation for identical arms when \(C_{\mathrm{eff}}\) is much larger than \(K_d\). The expression derived here is \(K_d^2/(K_d+C_{\mathrm{eff}})\), which retains the singly bound contribution and approaches \(K_d\), not infinity, as local target availability vanishes. Another mistake is to treat the two-arm statistical factor as the avidity gain. Count equivalent first-binding routes consistently, but distinguish that multiplicity from the stabilization contributed by the second bond.