A comprehensive mathematical model for three-body binding equilibria

Abstract

Three-component systems are often more complex than their two-component counterparts. Although the reversible association of three components in solution is critical for a vast array of chemical and biological processes, no general physical picture of such systems has emerged. Here we have developed a general, comprehensive framework for understanding ternary complex equilibria, which relates directly to familiar concepts such as EC50 and IC50 from simpler (binary complex) equilibria. Importantly, application of our model to data from the published literature has enabled us to achieve new insights into complex systems ranging from coagulation to therapeutic dosing regimens for monoclonal antibodies. We also provide an Excel spreadsheet to assist readers in both conceptualizing and applying our models. Overall, our analysis has the potential to render complex three-component systems--which have previously been characterized as "analytically intractable"--readily comprehensible to theoreticians and experimentalists alike.

Figure 1

A. Structure of ternary complex and definition of terminology for interacting components. B. Graph depicting ternary complex concentration ([ABC]) as a function of total bridging species ([B]_t) and illustrating autoinhibition. This curve can be understood in terms of critical points: the maximum concentration of ternary complex that can be achieved ([ABC]_max), the concentration of [B]_t required to achieve [ABC]_max ([B]_t,max), and the [B]_t values that elicit half-maximal [ABC] formation on the left (TF₅₀) and right (TI₅₀) sides of the curve. C. Thermodynamic cycle for the reversible formation of ternary complexes. D. Illustrations and mathematical definitions for positive and negative cooperativity.

Figure 2

A. A non-cooperative, bell-shaped ternary complex dose-response curve (black curve) can be explained by a sigmoidal “formation” term (red curve) and a reversed sigmoidal “autoinhibition” term (blue curve). When both of these curves are plateaued at [B]_t,max, the left side of the ternary complex curve (black curve) is described by the A–B binding event (TF₅₀), whereas the right side is described by the B–C binding event (TI₅₀). B. Non-cooperative systems can be divided into four quadrants, which enable simplification of [ABC]_max, [B]_t,max, TF₅₀, and TI₅₀ equations. Regarding the height (y-axis, or [ABC]_max), ternary complex does not form appreciably in Quadrants I and II but forms quantitatively in Quadrants III and IV. The width and position (x-axis, or TF₅₀, TI₅₀, and [B]_t,max) of these curves are controlled by either the binding constants (Quadrant I), the terminal species concentrations (Quadrant IV), or a combination of both (Quadrants II and III). (Red indicates A–B binding event; blue indicates B–C binding event)

Figure 3

Non-Cooperative Ternary Complexes in the Literature. For each example, elements of the relevant ternary complex are defined by the keys located below each graph. A. The dose-response curves of mAbs mediating immune responses can be best explained via Quadrant I (Figure 2B) as observed in the anti-renal carcinoma mAb 3F2 (Data from Ref. , binding constants from Ref. 37). B. Antibody-Recruiting Molecules targeting Prostate cancer (ARM-Ps) also exhibit bell-shaped dose-response curves characterized by Quadrant III (Figure 2B). Even as the resolvability assumption breaks down, Quadrant III closely approximates the system’s behavior. Dashed lines represent the predicted ternary complex curves, and solid lines represent the two φ terms; φ_AB is a constant, whereas φ_BC changes with increasing [C]_t, which is 10 nM (green curve), 40 nM (yellow curve), and 160 nM (red curve). C. Differences between the in vitro and in vivo potency and efficacy of heparin can be explained by the fact the former is a Quadrant I system (predicted curve in red, data from Ref. 2), whereas the latter is a Quadrant III system (predicted curve in black, clinical dose estimates from Ref. 47). (ADCC: antibody-dependent cell-mediated cytotoxicity; mAb: monoclonal antibody; DNP: dinitrophenyl)

Figure 4

Cooperative perturbation (axes into the page) on the quadrant framework presented in Figure 2B. The black curves in each of these plots correspond to a non-cooperative reference curve equivalent to those presented in Figure 2B. The purple curves show the effect of cooperativity when α > α_crit (width perturbation) and the orange curves show the effect of cooperativity when α < α_crit (height perturbation). Cooperative TF₅₀ and TI₅₀ expressions are presented in boxes and represent the [B]_t value for the point. Overall, Quadrants I & II do not form appreciable ternary complex when α = 1 and the effect of positive cooperativity is to first increase the height and then increase the width; Quadrants III & IV form quantitative ternary complex when α = 1 and positive cooperativity predominantly increases the width of these curves. (TPF = [ABC]_max/[L]_t. The Quadrant II α > α_crit TF₅₀ shown above is correct when A is the limiting reagent; otherwise it equals eq S181)

Figure 5

Examples of Quadrant III Cooperative Ternary Complexes. A. Supermolecular Assembly: the effects of negative cooperativity can be understood as a reduction in the height (eq 7) and width (eq S205) of a Quadrant III non-cooperative curve (data from Ref. 4). B. Antigen presentation: the effects of positive cooperativity can be understood as an increase in the height (eq 7) and/or width (eqs S191 and S192) of a Quadrant III non-cooperative curve (data from Ref. 20). The TF50 and TI50 expressions simplify for each quadrant, enabling conceptualization of the width as a perturbation on the non-cooperative reference curve.