State-Based Models of Multispecific Binding

Why Represent Binding as States?

A bispecific antibody can bind two different antigens, which we will call \(X\) and \(Y\). Tracking only “free antibody” and “bound antibody” loses important information: an antibody bound to \(X\) alone is not in the same state as one bound to both \(X\) and \(Y\). Those states have different available binding sites and different routes into and out of them.

A state-based model makes these distinctions explicit. It is a set of species and transitions: each association or dissociation moves antibody between states, while the concentrations of free antigens determine the association rates. The model below describes binding kinetics only; processes such as internalization or antibody distribution would require additional states or fluxes.

The model assumes one available binding site for each antigen on each antibody, binding of \(X\) and \(Y\) to their respective sites, and no direct interaction between the antigens. It also assumes a well-mixed system in which mass-action rates are appropriate. For the binding-only equations, synthesis, degradation, internalization, and transport are omitted. These assumptions describe a minimal network, not every possible bispecific construct or biological environment.

Transitions Between the Four Antibody States

There are two ways to form the doubly bound state. Free antibody can bind \(X\) and then \(Y\), or it can bind \(Y\) and then \(X\). Dissociation reverses each of those steps. The reaction network is:

Here \(k_{X,\mathrm{on}}\) and \(k_{Y,\mathrm{on}}\) are association rate constants, with units of concentration\(^{-1}\) time\(^{-1}\); the dissociation rate constants \(k_{X,\mathrm{off}}\) and \(k_{Y,\mathrm{off}}\) have units of time\(^{-1}\). Thus an association flux is proportional to both reactant concentrations. For example, \(k_{X,\mathrm{on}}AX\) means \(k_{X,\mathrm{on}}\times[A]\times[X]\), not the product of a rate constant and the state label alone.

This simplest version uses the same \(X\)- and \(Y\)-specific rate constants whether the other site is unoccupied or occupied. That is a modeling assumption: if binding in the doubly bound context has different kinetics, separate state-specific rate constants are needed. The equations below use the simpler assumption so that each transition has a clearly defined flux.

From Transition Fluxes to Differential Equations

The concentration rate for a state is the sum of fluxes entering it minus the sum of fluxes leaving it. For instance, \(AX\) is formed when \(A\) binds \(X\), and is depleted when \(AX\) loses \(X\) or binds free \(Y\). It is also formed when \(AXY\) loses \(Y\). Applying the same accounting to all four antibody states gives:

In these equations, a juxtaposition such as \(AX\) in a rate term means the product of the concentrations of free \(A\) and free \(X\); the state \(AX\) in a derivative or dissociation term means the concentration of the \(X\)-bound antibody. To make that distinction explicit when calculating, one may write concentrations in square brackets, such as \([A]\), \([X]\), and \([AX]\). The equations use \(A\), \(X\), and the state labels as shorthand for those concentrations.

The terms involving \(AXY\) are especially informative. Its formation has two incoming fluxes, one from each singly bound state. Its loss has two outgoing fluxes: dissociation of \(X\) returns antibody to \(AY\), while dissociation of \(Y\) returns it to \(AX\). The destination depends on which antigen leaves.

Worked Example: Listing the States and Their Transitions

Worked Example: Calculating Instantaneous Rates

A Short Derivation and a Bookkeeping Check

The equation for \(AXY\) follows directly by counting its incoming and outgoing transitions. Its incoming transitions are the association of \(Y\) with \(AX\) and the association of \(X\) with \(AY\). Its outgoing transitions are dissociation of \(X\) and dissociation of \(Y\). Under the mass-action assumptions, the four fluxes are respectively \(k_{Y,\mathrm{on}}[AX][Y]\), \(k_{X,\mathrm{on}}[AY][X]\), \(k_{X,\mathrm{off}}[AXY]\), and \(k_{Y,\mathrm{off}}[AXY]\). Subtracting outgoing from incoming fluxes gives:

A useful check for a binding-only model is that each antibody transition transfers antibody between states rather than creating or destroying it. Adding the four antibody-state equations therefore cancels every transition flux:

This check can reveal an omitted or incorrectly signed transition. It applies only to the stated binding-only system; adding processes such as internalization or degradation changes the total-antibody balance.

Worked Example: A Short-Euler Update

Model Choice, Diagnostics, and Limitations

An explicit state model is useful when the distinction among free, singly bound, and doubly bound antibody matters to the question—for example, when a measurement distinguishes complexes or when a downstream process acts on a particular bound state. A lumped “bound antibody” variable can be simpler, but it combines \(AX\), \(AY\), and \(AXY\), obscuring which antigen is bound and which transitions are possible. An equilibrium occupancy description is another modeling choice, but it does not represent these time-dependent association and dissociation fluxes in the same explicit way.

Before interpreting model outputs, check that each transition has a source state, a destination state, a rate law with the right units, and a reverse transition if dissociation is part of the model. Useful warning signs include:

• A concentration becomes negative during a simulation. This can indicate an integration step that is too large or an implementation problem.
• The sum of antibody-state concentrations changes in a binding-only model. Recheck signs, missing transitions, and whether all state equations use the same flux definitions.
• A second-antigen association term does not depend on the concentration of the free antigen. For a bimolecular mass-action association, it should.
• The doubly bound state has only one formation route in a model that allows either antigen to bind first. The second route may have been omitted.

The model is deliberately limited. It does not account for target production or removal, antibody clearance, internalization, spatial separation, multiple copies of an antigen, or context-dependent binding rates. These processes should not be added implicitly by changing a binding constant; they require explicit model structure or justified parameter choices. A model should be no more detailed than the data and scientific question can support, but sufficiently detailed to preserve the states needed for the intended interpretation.