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Comment on a suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degradation by heterobifunctional ligands

Open AccessPublished:March 01, 2021DOI:https://doi.org/10.1016/j.jbc.2021.100331
      We are writing to express serious concerns about the recent paper published in JBC (https://doi.org/10.1074/jbc.RA120.014715) by Bomie Han entitled “A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degradation by heterobifunctional ligands.” (
      • Han B.
      A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degradation by heterobifunctional ligands.
      ) We believe that the author has incorrectly interpreted previous literature and misrepresented the scope and novelty of his contribution. While much of this work is novel and interesting, we believe claims of “universality” and “exactness” are misleading and critical mathematical and historical context is needed to frame this contribution appropriately.
      Specifically, the author claims to have derived an “exact” solution to the cooperative ternary equilibrium, but has actually rederived an approximate solution, which was solved by Alan Perelson in 1980 (equation 33 in reference 2) (
      • Perelson A.S.
      Receptor clustering on a cell-surface .2. Theory of receptor cross-linking by ligands bearing 2 chemically distinct functional-groups.
      ). Indeed, our work from 2013 does provide exact mathematical solutions that describe both cooperative (equation 6 in reference 3) and noncooperative (equation 1 in reference 3) equilibria (
      • Douglass Jr., E.F.
      • Miller C.J.
      • Sparer G.
      • Shapiro H.
      • Spiegel D.A.
      A comprehensive mathematical model for three-body binding equilibria.
      ), but Han misstates this fact in his manuscript. In addition, several other equations—such as the equation for Emax—are presented as novel but were in fact first derived by Perelson in 1980 (equation 46 in reference 2) (
      • Perelson A.S.
      Receptor clustering on a cell-surface .2. Theory of receptor cross-linking by ligands bearing 2 chemically distinct functional-groups.
      ).
      Critically, both Perelson’s and Han’s equations model a subset of ternary complex behavior represented by Quadrant 1 in Figure 4 of our published work (
      • Douglass Jr., E.F.
      • Miller C.J.
      • Sparer G.
      • Shapiro H.
      • Spiegel D.A.
      A comprehensive mathematical model for three-body binding equilibria.
      ). As a result, Han’s model is restricted to the subset of experimental conditions wherein [L]t = [L]; in other words, his findings are relevant only when the bifunctional ligand is in large excess over all other components. Obviously, this simplification does not always apply. Importantly, attempts by other researchers to employ Han’s models to systems that do not fulfill the [L]t = [L] approximation will lead to incorrect results.
      Approximating the concentrations of free species to solve ternary complex equilibrium models has a rich history dating back to the 1950s as detailed by Segel (
      • Segel I.H.
      Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady State Enzyme Systems.
      ). Proper historical context, therefore, is critical to establishing the novelty of this contribution. The author seems to overlook the seminal work of several other authors who have used similar free-concentration approximations to solve ternary complex equilibria, including:
      • [L]t = [L] by Perelson (
        • Perelson A.S.
        Receptor clustering on a cell-surface .2. Theory of receptor cross-linking by ligands bearing 2 chemically distinct functional-groups.
        )
      • [P]t = [P] by Delean et al. (
        • Delean A.
        • Stadel J.M.
        • Lefkowitz R.J.
        A ternary complex model explains the agonist-specific binding-properties of the adenylate cyclase-coupled beta-adrenergic-receptor.
        )
      • [P]t = [P] & [L]t = [L] by Hogg and Jackson (
        • Hogg P.J.
        • Jackson C.M.
        Heparin promotes the binding of thrombin to fibrin polymer - quantitative characterization of a thrombin-fibrin polymer-heparin ternary complex.
        )
      • [P]t = [P] & [E]t = [E] by Segel (
        • Segel I.H.
        Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady State Enzyme Systems.
        )
      Citing these precedents is further critical for establishing the utility of the mathematics, as each model is optimized for a specific experimental system (e.g., DeLean’s model is optimal for modeling GCPR ternary complex signaling and is most utilized within that field (
      • Delean A.
      • Stadel J.M.
      • Lefkowitz R.J.
      A ternary complex model explains the agonist-specific binding-properties of the adenylate cyclase-coupled beta-adrenergic-receptor.
      )).
      Overall, we believe that, in its current form, this manuscript is misleading and could hamper readers’ understanding of how bifunctional ligands, ternary complexes, and degraders behave. We believe these issues can be addressed by removing claims of universal scope and adding critical historical and mathematical context to appropriately frame the novelty and scope of this contribution.

      References

        • Han B.
        A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degradation by heterobifunctional ligands.
        J. Biol. Chem. 2020; 295: 15280-15291
        • Perelson A.S.
        Receptor clustering on a cell-surface .2. Theory of receptor cross-linking by ligands bearing 2 chemically distinct functional-groups.
        Math. Biosci. 1980; 49: 87-110
        • Douglass Jr., E.F.
        • Miller C.J.
        • Sparer G.
        • Shapiro H.
        • Spiegel D.A.
        A comprehensive mathematical model for three-body binding equilibria.
        J. Am. Chem. Soc. 2013; 135: 6092-6099
        • Segel I.H.
        Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady State Enzyme Systems.
        Wiley, New York, NY1975
        • Delean A.
        • Stadel J.M.
        • Lefkowitz R.J.
        A ternary complex model explains the agonist-specific binding-properties of the adenylate cyclase-coupled beta-adrenergic-receptor.
        J. Biol. Chem. 1980; 255: 7108-7117
        • Hogg P.J.
        • Jackson C.M.
        Heparin promotes the binding of thrombin to fibrin polymer - quantitative characterization of a thrombin-fibrin polymer-heparin ternary complex.
        J. Biol. Chem. 1990; 265: 241-247

      Linked Article

      • Reply to Douglass and Spiegel: A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degredation by heterobifunctional ligands
        Journal of Biological ChemistryVol. 296
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          Calculating equilibrium concentration of a ternary complex for a given total ligand concentration (1, 2) and predicting the ternary complex concentration in equilibrium with a given free ligand concentration (3) are two totally different questions that lead to clearly distinct mathematical solutions. Even though an exact solution to one question can be an approximate answer to the other, the latter approach (3) is consistent with a long-established tradition of analyzing equilibrium behavior of binding reactions: the equilibrium dissociation constants, Kds, that are used in both approaches are defined by the free, not total, ligand concentration at equilibrium, and the universally adopted equation for a bimolecular binding reaction, B = Bmax ∗ [L]/([L] + Kd), is also a function of free ligand concentration at equilibrium.
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