Kinetic Avidity Under Rebinding and Dissociation

Why Binding Kinetics Matter for Avidity

A multivalent antibody can remain associated with a target pair after one arm dissociates, because its other arm still holds the molecule nearby. The free arm may rebind before the remaining arm dissociates. This sequence can extend the lifetime of the whole complex even though neither arm’s intrinsic dissociation rate has changed. The effect is kinetic: it concerns how long the complex persists, not simply the equilibrium occupancy described by equilibrium avidity.

To quantify that persistence, follow the complex through its binding states. The lifetime considered here begins with both arms bound and ends when both arms have dissociated. A fully dissociated molecule is treated as lost; rebinding from the surrounding solution is excluded. This endpoint is appropriate, for example, when free antibody is rapidly diluted or washed away.

Definition: In a two-arm sequential binding model, \(D\) denotes the doubly bound state, \(A\) denotes the state in which only arm A remains bound, and \(B\) denotes the state in which only arm B remains bound. The fully dissociated state \(U\) is absorbing. An arm’s dissociation takes a bound complex to a state with one fewer bound arm; from a singly bound state, the detached arm may rebind at an effective first-order rate.

The effective rebinding rate summarizes return to the bound state while the other arm keeps the molecule associated. It is not automatically equal to an intrinsic association constant: it depends on the local conditions represented by the model. The model below does not resolve those local conditions in detail.

State Model and Assumptions

Begin with symmetric arms. Each bound arm dissociates at rate \(d\), and a detached arm rebinds from a singly bound state at rate \(r\). Thus \(D\) moves to \(A\) or \(B\), each at rate \(d\); either singly bound state returns to \(D\) at rate \(r\) or reaches \(U\) at rate \(d\). Rates have units of inverse time.

D \xrightarrow{d} A,\qquad D \xrightarrow{d} B,\qquad A \xrightarrow{r} D,\qquad B \xrightarrow{r} D,\qquad A \xrightarrow{d} U,\qquad B \xrightarrow{d} U

The interpretation requires several assumptions: transitions are memoryless with constant rates; the two arms have the same dissociation rate; the effective rebinding rate is the same from either singly bound state; and no new association occurs after reaching \(U\). The model also omits internalization, degradation, and other routes of complex removal. If these processes matter, they must be added as states or competing transitions rather than silently folded into the arm dissociation rate.

Let \(T_D\) and \(T_S\) be the expected remaining times until \(U\), starting in \(D\) and in either singly bound state, respectively. In state \(D\), either arm can dissociate, so the total rate of leaving \(D\) is \(2d\). In state \(S\), the total rate of leaving is \(r+d\). Conditioning on the first transition gives the mean-lifetime equations.

T_D=\frac{1}{2d}+T_S,\qquad T_S=\frac{1}{r+d}+\frac{r}{r+d}T_D

The first term in each equation is the expected waiting time for a transition out of the current state. After leaving \(D\), the complex is singly bound, so its expected remaining time is \(T_S\). From \(S\), rebinding occurs with probability \(r/(r+d)\) and returns the system to \(D\); dissociation of the remaining arm occurs with probability \(d/(r+d)\) and ends the lifetime. These probabilities follow from competing constant-rate transitions.

Deriving the Mean Lifetime

Substitute the expression for \(T_S\) into the equation for \(T_D\), then collect terms containing \(T_D\). This gives a closed-form mean lifetime from the doubly bound state.

\begin{aligned} T_D &=\frac{1}{2d}+\frac{1}{r+d}+\frac{r}{r+d}T_D\\ \left(1-\frac{r}{r+d}\right)T_D &=\frac{1}{2d}+\frac{1}{r+d}\\ \frac{d}{r+d}T_D &=\frac{1}{2d}+\frac{1}{r+d}\\ T_D &=\frac{r+3d}{2d^2} \end{aligned}

When rebinding is absent (\(r=0\)), the result reduces to \(T_D=3/(2d)\). This also follows directly: the expected wait in \(D\) is \(1/(2d)\), and after the first arm dissociates the remaining arm has expected lifetime \(1/d\). When \(r\) increases, \(T_D\) increases because rebinding creates additional visits to \(D\) before eventual escape. The expression has units of time: its numerator has units of inverse time, and its denominator has units of inverse time squared.

Key result: Under the symmetric, constant-rate model, the mean time from the doubly bound state to full dissociation is \(T_D=(r+3d)/(2d^2)\). The no-rebinding case is \(T_D=3/(2d)\). These are mean lifetimes, not guarantees that each individual complex persists for that duration.

Worked Comparisons

Worked Example: No Rebinding After Dissociation

Consider a hypothetical complex whose two arms each dissociate at \(d=0.20\ \text{min}^{-1}\). Suppose conditions prevent a detached arm from rebinding, so \(r=0\). Starting from \(D\), the mean wait until the first arm dissociates is \(1/(2d)\), because either arm can dissociate.

Substitute and calculate.

T_D=\frac{3}{2d} =\frac{3}{2(0.20\ \text{min}^{-1})} =7.50\ \text{min}

The first dissociation occurs after an expected \(1/(0.40\ \text{min}^{-1})=2.50\) minutes. The remaining arm then takes an expected \(1/(0.20\ \text{min}^{-1})=5.00\) minutes to dissociate, giving \(2.50+5.00=7.50\) minutes. This checks the formula against the sequential waiting-time interpretation.

Worked Example: Rebinding Extends the Mean Lifetime

Keep the same hypothetical dissociation rate, \(d=0.20\ \text{min}^{-1}\), but now let the effective rebinding rate be \(r=1.80\ \text{min}^{-1}\). The probability that rebinding occurs before the remaining arm dissociates from a singly bound state is \(r/(r+d)\).

Calculate the return probability and mean lifetime.

\frac{r}{r+d} =\frac{1.80}{1.80+0.20} =0.900

T_D=\frac{r+3d}{2d^2} =\frac{1.80+3(0.20)}{2(0.20)^2} =\frac{2.40}{0.0800} =30.0\ \text{min}

Under these assumed rates, the predicted mean lifetime is 30.0 minutes, compared with 7.50 minutes without rebinding. This fourfold difference is a model-based comparison, not evidence that either rate applies to a particular therapeutic. A high probability of return during each singly bound episode can produce repeated returns to \(D\) before full dissociation.

Unequal Arms

The symmetric formula is useful when arms are exchangeable, but the same first-transition reasoning accommodates unequal rates. Let \(d_A\) and \(d_B\) be the dissociation rates of arms A and B. Let \(r_A\) be the rebinding rate of arm A from state \(B\), and \(r_B\) the rebinding rate of arm B from state \(A\). Denote the mean remaining lifetimes in these states by \(T_D,T_A,T_B\). Conditioning on the first transition gives:

\begin{aligned} T_D&=\frac{1}{d_A+d_B} +\frac{d_B}{d_A+d_B}T_A +\frac{d_A}{d_A+d_B}T_B\\ T_A&=\frac{1}{r_B+d_A} +\frac{r_B}{r_B+d_A}T_D\\ T_B&=\frac{1}{r_A+d_B} +\frac{r_A}{r_A+d_B}T_D \end{aligned}

For example, \(D\) reaches \(A\) when arm B dissociates, which occurs with probability \(d_B/(d_A+d_B)\) at the first transition. From \(A\), arm B rebinds at rate \(r_B\), while arm A’s dissociation at rate \(d_A\) ends the complex lifetime. Solving these equations gives the mean lifetime without assuming that the arms behave identically.

Worked Example: Unequal Dissociation and Rebinding Rates

Use hypothetical rates \(d_A=0.10\ \text{min}^{-1}\), \(d_B=0.30\ \text{min}^{-1}\), \(r_A=0.40\ \text{min}^{-1}\), and \(r_B=0.90\ \text{min}^{-1}\). The state equations become:

T_D=2.50+0.75T_A+0.25T_B,\qquad T_A=1.00+0.90T_D,\qquad T_B=1.42857+0.571429T_D

Substitute the last two equations into the first:

\begin{aligned} T_D &=2.50+0.75(1.00+0.90T_D) +0.25(1.42857+0.571429T_D)\\ T_D&=3.60714+0.817857T_D\\ T_D&=\frac{3.60714}{1-0.817857} =19.8\ \text{min} \end{aligned}

For comparison, set both rebinding rates to zero while retaining the same dissociation rates. The initial expected wait is \(1/(0.10+0.30)=2.50\) minutes. The first transition leads to \(A\) with probability \(0.75\), followed by an expected \(1/d_A=10.0\) minutes; it leads to \(B\) with probability \(0.25\), followed by \(1/d_B=3.333\) minutes. Therefore:

T_D=2.50+0.75(10.0)+0.25(3.333) =10.8\ \text{min}

With the specified unequal rebinding rates, the calculated mean is 19.8 minutes rather than 10.8 minutes. The increase follows from the specified return transitions; it does not imply that either arm’s intrinsic dissociation rate changed.

Diagnostics, Alternatives, and Limitations

The explicit state model is useful when the sequence of transitions matters. A single effective dissociation rate is simpler, but it compresses the process into one summary. For example, \(1/T_D\) is the reciprocal of the mean lifetime and has units of inverse time, but it is not generally equal to either arm’s dissociation rate. Nor does it guarantee that the probability of remaining bound declines as a single exponential: a sequence of states can produce a non-exponential lifetime distribution even when each transition has a constant rate.

Equilibrium avidity and kinetic lifetime answer related but different questions. An equilibrium summary describes occupancy under specified binding conditions; the state model here describes time to full dissociation under its stated transition rules. Equilibrium behavior alone does not determine a unique set of transition rates. When observed data contain time-course information, comparing predicted and observed dissociation trajectories can therefore be more informative than matching one equilibrium measure.

Check whether the model is adequate before interpreting its lifetime estimate. Useful warning signs include systematic mismatches between predicted and observed dissociation time courses, different apparent behavior across assay conditions, or estimates that change greatly when the assumed rebinding rate is slightly varied. Confirm that the observation conditions actually suppress association from solution if \(U\) is treated as absorbing. If free antibody remains available, association from \(U\) may need to be included. Likewise, if molecules disappear through internalization or degradation during the measurement, a model ending only at arm dissociation may not represent the observed endpoint.

The calculation also assumes that effective rates stay constant and that singly bound states can be represented by a small number of compartments. If the data cannot distinguish \(d\) from \(r\), a precise-looking mean lifetime does not resolve that identifiability problem. The result is conditional on the model and its inputs; it is not a direct measurement of a universal complex lifetime.

Confusing return probability with rebinding rate. The probability \(r/(r+d)\) depends on both competing rates; the rate \(r\) alone is not a probability.
Calling \(1/T_D\) the arm dissociation rate. It is a reciprocal mean lifetime for the whole modeled process, not an intrinsic arm-specific rate.
Assuming no rebinding means no sequential dissociation. The first arm may dissociate while the second still holds the complex; no rebinding only removes the return transition.

Key takeaway: Kinetic avidity emerges when dissociation of one arm leaves a route back to the doubly bound state. State transitions make that route explicit and allow mean lifetimes to be compared under rebinding and no-rebinding conditions.

Check Your Understanding

Answer each question, then check the brief solutions.

For symmetric arms with \(d=0.25\ \text{min}^{-1}\) and \(r=0\), what is the mean lifetime from \(D\)?
If \(r=3d\), what is the probability of rebinding before dissociation of the remaining arm from a singly bound state?
Why is \(1/T_D\) not generally an arm’s intrinsic dissociation rate?
In the unequal-arm model, which transition takes the system from \(D\) to \(A\), and what is its rate?
What assumption must be reconsidered if free antibody in solution can re-associate after the complex reaches \(U\)?

Solutions. (1) \(T_D=3/(2d)=3/[2(0.25\ \text{min}^{-1})]=6.00\ \text{min}\). (2) \(r/(r+d)=3d/(4d)=0.750\). (3) \(T_D\) summarizes time through multiple sequential states and possible returns; its reciprocal is an overall summary, not a single arm’s transition rate. (4) Dissociation of arm B takes \(D\) to \(A\), at rate \(d_B\). (5) The absorbing-state assumption for \(U\) fails; association from \(U\) must be modeled if it is relevant.