Equilibrium binding models#

bindcurve includes mass-balance models for direct and competitive binding. Unlike the empirical IC₅₀ model, these models assign physical meaning to the concentrations and fitted dissociation constants.

The notation used below is:

Symbol

Code

Meaning

\(R\), \(R_T\)

R, RT

free and total receptor

\(L^*\), \(L_T^*\)

Ls, LsT

free and total labeled ligand (tracer)

\(L\), \(L_T\)

L, LT

free and total unlabeled ligand (competitor)

\(RL^*\)

RLs

receptor-tracer complex

\(RL\)

RL

receptor-competitor complex

\(RLL^*\)

RLLs

ternary receptor-competitor-tracer complex

\(K_d^*\)

Kds

tracer dissociation constant

\(K_d\)

Kd

competitor dissociation constant

\(K_{d3}\)

Kd3

tracer dissociation constant from \(RL\)

\(N^*\), \(N\)

Ns, N

linear nonspecific-binding factors

All concentration-like quantities must use the same numerical unit. The models assume equilibrium, a single receptor population, ideal dilute-solution behavior, and the stated binding stoichiometry.

The physical tracer-bound fraction, \(F_b^*\), is mapped to the measured response with

\[ y = y_{\min} + (y_{\max}-y_{\min})F_b^*. \]

This assumes that the measured response is affine in the modeled bound fraction. The fitted \(y_{\min}\) and \(y_{\max}\) correspond to hypothetical \(F_b^*=0\) and \(F_b^*=1\) states; for finite-concentration competitive models they need not equal the smallest and largest responses actually observed in a titration.

Note

For fluorescence-polarization assays, unequal fluorescence intensities of the free and bound tracer can make the relationship between anisotropy and bound fraction nonlinear. The present models do not include that additional photophysical correction.

Direct binding#

The direct system contains receptor and tracer,

\[ R + L^* \rightleftharpoons RL^*, \qquad K_d^* = \frac{[R][L^*]}{[RL^*]}. \]

For specific binding,

\[ F_b^* = \frac{[RL^*]}{[L_T^*]} = \frac{[R]}{K_d^*+[R]}. \]

Simplified direct binding#

Model: dir_simple

The simplified model treats the concentration axis as free receptor:

\[ F_b^* = \frac{[R]}{K_d^*+[R]}. \]

When experimental input is total receptor, this amounts to the approximation \([R]\approx[R_T]\). It is appropriate only when tracer depletion and receptor occupancy do not materially separate free from total receptor. Its midpoint is \([R]=K_d^*\).

Depletion-aware direct binding#

Model: dir_specific

The exact one-site mass balances are

\[ [L_T^*]=[L^*]+[RL^*], \qquad [R_T]=[R]+[RL^*]. \]

Eliminating the other species gives a quadratic in free receptor,

\[ [R]^2+a[R]+b=0, \]

with

\[ a=K_d^*+[L_T^*]-[R_T], \qquad b=-K_d^*[R_T]. \]

The physical solution is the non-negative root

\[ [R]=\frac{-a+\sqrt{a^2-4b}}{2}. \]

bindcurve evaluates this root in a rationalized form when needed to avoid loss of precision. The complex concentration and observable are then

\[ [RL^*]=[R_T]-[R], \qquad F_b^*=\frac{[RL^*]}{[L_T^*]}. \]

Direct binding with nonspecific tracer immobilization#

Model: dir_total

Nonspecific tracer immobilization is represented as a nonsaturable amount proportional to free tracer,

\[ [L^*]_{\mathrm{NS}}=N^*[L^*], \qquad N^*\ge0. \]

The tracer balance becomes

\[ [L_T^*]=(1+N^*)[L^*]+[RL^*], \]

while the receptor balance remains

\[ [R_T]=[R]+[RL^*]. \]

Free receptor again satisfies a quadratic, now with

\[ a=(1+N^*)K_d^*+[L_T^*]-[R_T], \qquad b=-(1+N^*)K_d^*[R_T]. \]

The physical total bound fraction is

\[ F_{b,\mathrm{total}}^* =\frac{[RL^*]+N^*[L^*]}{[L_T^*]}. \]

At zero receptor its nonspecific baseline is \(F_0=N^*/(1+N^*)\). Because ymin represents that fitted baseline, dir_total maps the baseline-normalized fraction

\[ \frac{F_{b,\mathrm{total}}^*-F_0}{1-F_0} =\frac{[RL^*]}{[L_T^*]} \]

to the response. The component API exposes both the specific and total bound fractions. The factor \((1+N^*)\) changes the mass balance and the apparent location of the curve; it does not mean that the microscopic \(K_d^*\) has changed or that total binding is simply \((1+N^*)\) times specific binding.

Complete competitive binding: three states#

Models: comp_3st_specific, comp_3st_total

In the complete-competition model, tracer and competitor bind mutually exclusively:

\[ R+L^*\rightleftharpoons RL^*, \qquad R+L\rightleftharpoons RL. \]

The independent equilibria are

\[ K_d^*=\frac{[R][L^*]}{[RL^*]}, \qquad K_d=\frac{[R][L]}{[RL]}. \]

For specific binding, conservation of mass requires

\[\begin{split} \begin{aligned} [L_T^*]&=[L^*]+[RL^*],\\ [L_T]&=[L]+[RL],\\ [R_T]&=[R]+[RL^*]+[RL]. \end{aligned} \end{split}\]

Consequently,

\[ [RL^*]=\frac{[L_T^*][R]}{K_d^*+[R]}, \qquad [RL]=\frac{[L_T][R]}{K_d+[R]}, \]

and the measured tracer fraction is

\[ F_b^*=\frac{[RL^*]}{[L_T^*]} =\frac{[R]}{K_d^*+[R]}. \]

In particular, the denominator contains the tracer dissociation constant \(K_d^*\), not the competitor dissociation constant \(K_d\).

Substitution into the receptor balance gives

\[ [R]^3+a[R]^2+b[R]+c=0, \]

where

\[\begin{split} \begin{aligned} a={}&K_d^*+K_d+[L_T^*]+[L_T]-[R_T],\\ b={}&K_d^*([L_T]-[R_T]) +K_d([L_T^*]-[R_T])+K_d^*K_d,\\ c={}&-K_d^*K_d[R_T]. \end{aligned} \end{split}\]

Although this cubic has a closed-form physical root, bindcurve solves the monotonic receptor mass balance directly on \(0\le[R]\le[R_T]\). This avoids the branch selection and numerical cancellation problems of the explicit cubic formula.

For comp_3st_total, nonspecific competitor immobilization is

\[ [L]_{\mathrm{NS}}=N[L] \]

and the competitor balance becomes

\[ [L_T]=(1+N)[L]+[RL]. \]

Within the receptor mass balance this is algebraically equivalent to replacing \(K_d\) by \((1+N)K_d\). The parameter \(K_d\) nevertheless remains the microscopic specific dissociation constant; \((1+N)K_d\) is an effective term created by the additional ligand sink.

Incomplete competitive binding: four states#

Models: comp_4st_specific, comp_4st_total

The four-state model allows tracer and competitor to occupy the receptor simultaneously:

\[\begin{split} \begin{aligned} R+L^*&\rightleftharpoons RL^*,\\ R+L&\rightleftharpoons RL,\\ RL+L^*&\rightleftharpoons RLL^*,\\ RL^*+L&\rightleftharpoons RLL^*. \end{aligned} \end{split}\]

bindcurve parameterizes the first three equilibria as

\[ K_d^*=\frac{[R][L^*]}{[RL^*]}, \qquad K_d=\frac{[R][L]}{[RL]}, \qquad K_{d3}=\frac{[RL][L^*]}{[RLL^*]}. \]

The fourth constant is not an independent parameter. Thermodynamic consistency around the binding cycle requires

\[ K_{d4}=\frac{[RL^*][L]}{[RLL^*]} =\frac{K_dK_{d3}}{K_d^*}. \]

The three mass balances are

\[\begin{split} \begin{aligned} [L_T^*]&=[L^*]+[RL^*]+[RLL^*],\\ [L_T]&=[L]+[RL]+[RLL^*],\\ [R_T]&=[R]+[RL^*]+[RL]+[RLL^*]. \end{aligned} \end{split}\]

Given the free concentrations,

\[\begin{split} \begin{aligned} [RL^*]&=\frac{[R][L^*]}{K_d^*},\\ [RL]&=\frac{[R][L]}{K_d},\\ [RLL^*]&=\frac{[R][L][L^*]}{K_dK_{d3}}. \end{aligned} \end{split}\]

The observable must count tracer in both receptor-bound species:

\[ F_b^*=\frac{[RL^*]+[RLL^*]}{[L_T^*]}. \]

Eliminating the free ligands generally produces a quintic polynomial in true free receptor. bindcurve finds candidate roots in the physical interval \(0\le[R]\le[R_T]\), reconstructs all species to enforce non-negativity and the mass balances, and falls back to direct numerical solution of the receptor balance when necessary.

The limiting behavior gives useful checks on the model:

  • At \([L_T]=0\), it reduces to direct tracer binding with \(K_d^*\).

  • At very large \([L_T]\), receptor is predominantly competitor-bound and tracer binding approaches a direct-binding curve governed by \(K_{d3}\):

    \[ \lim_{[L_T]\to\infty}F_b^* =\frac{K_{d3}+[L_T^*]+[R_T] -\sqrt{(K_{d3}+[L_T^*]+[R_T])^2-4[L_T^*][R_T]}} {2[L_T^*]}. \]

  • If \(K_{d3}=K_d^*\), competitor occupancy does not alter tracer affinity and \(F_b^*\) is independent of competitor concentration.

  • If \(K_{d3}\to\infty\), the ternary state is suppressed and the model tends toward complete three-state competition.

Thus \(K_{d3}<K_d^*\) describes favorable coupling of tracer binding to the competitor-bound receptor, whereas \(K_{d3}>K_d^*\) describes unfavorable coupling. A finite \(K_{d3}\) can produce a nonzero high-competitor plateau.

For comp_4st_total, nonspecific competitor immobilization adds \(N[L]\) to the competitor balance:

\[ [L_T]=(1+N)[L]+[RL]+[RLL^*]. \]

As in the three-state total model, the observable equilibrium can be solved with the effective term \((1+N)K_d\), while the reported \(K_d\) remains the microscopic competitor dissociation constant.

References#