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A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degredation by heterobifunctional ligands
Open AccessPublished:February 23, 2021DOI:https://doi.org/10.1016/j.jbc.2021.100330
      Calculating equilibrium concentration of a ternary complex for a given total ligand concentration (
      • Douglass Jr., E.F.
      • Miller C.J.
      • Sparer G.
      • Shapiro H.
      • Spiegel D.A.
      A comprehensive mathematical model for three-body binding equilibria.
      ,
      • Perelson A.S.
      Receptor clustering on a cell-surface. II. Theory of receptor cross-linking by ligands bearing two chemically distinct functional groups.
      ) and predicting the ternary complex concentration in equilibrium with a given free ligand concentration (
      • Han B.
      A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degradation by heterobifunctional ligands.
      ) 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 (
      • Han B.
      A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degradation by heterobifunctional ligands.
      ) 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.
      The solutions that I offered (
      • Han B.
      A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degradation by heterobifunctional ligands.
      ) are: (i) “exact” because the exact ternary complex concentration can be calculated by the provided mathematical equation for any free ligand concentration at equilibrium; (ii) “universal” because the same mathematical equations apply to all ternary complex systems regardless of the value of cooperativity; and (iii) “original” because this is the first time such solutions are reported. Previous reports are either based on total ligand concentration (
      • Douglass Jr., E.F.
      • Miller C.J.
      • Sparer G.
      • Shapiro H.
      • Spiegel D.A.
      A comprehensive mathematical model for three-body binding equilibria.
      ,
      • Perelson A.S.
      Receptor clustering on a cell-surface. II. Theory of receptor cross-linking by ligands bearing two chemically distinct functional groups.
      ) or with an assumption that ligand depletion can be ignored (
      • Perelson A.S.
      Receptor clustering on a cell-surface. II. Theory of receptor cross-linking by ligands bearing two chemically distinct functional groups.
      ). Other important outcomes of my paper include: (i) seamless expansion of the solution to accommodate modified equilibria, as shown in Figure 3 (
      • Han B.
      A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degradation by heterobifunctional ligands.
      ); and (ii) intuitive understanding of the complex systems with one set of equations.

      Conflict of interest

      The authors declare that they have no conflicts of interest with the contents of this article.

      References

        • 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
        • Perelson A.S.
        Receptor clustering on a cell-surface. II. Theory of receptor cross-linking by ligands bearing two chemically distinct functional groups.
        Math. Biosci. 1980; 49: 87-110
        • 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

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