A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degradation by heterobifunctional ligands
2020; Elsevier BV; Volume: 295; Issue: 45 Linguagem: Inglês
10.1074/jbc.ra120.014715
ISSN1083-351X
Autores Tópico(s)Chronic Lymphocytic Leukemia Research
ResumoSmall molecule–induced targeted protein degradation by heterobifunctional ligands or molecular glues represents a new modality in drug development, allowing development of therapeutic agents for targets previously considered undruggable. Successful target engagement requires the formation of a ternary complex (TC) when the ligand brings its target protein in contact with an E3 ubiquitin ligase. Unlike traditional drugs, where target engagement can be described by a simple bimolecular equilibrium equation, similar mathematical tools are currently not available to describe TC formation in a universal manner. This current limitation substantially increases the challenges of developing drugs with targeted protein degradation mechanism. In this article, I provide a full, exact, and universal mathematical description of the TC system at equilibrium for the first time. I have also constructed a comprehensive suite of mathematical tools for quantitative measurement of target engagement and equilibrium constants from experimental data. Mechanistic explanations are provided for many common challenges associated with developing this type of therapeutic agent. Insights from these analyses provide testable hypotheses and grant direction to drug development efforts in this promising area. The mathematical and analytical tools described in this article may also have broader applications in other areas of biology and chemistry in which ternary complexes are observed. Small molecule–induced targeted protein degradation by heterobifunctional ligands or molecular glues represents a new modality in drug development, allowing development of therapeutic agents for targets previously considered undruggable. Successful target engagement requires the formation of a ternary complex (TC) when the ligand brings its target protein in contact with an E3 ubiquitin ligase. Unlike traditional drugs, where target engagement can be described by a simple bimolecular equilibrium equation, similar mathematical tools are currently not available to describe TC formation in a universal manner. This current limitation substantially increases the challenges of developing drugs with targeted protein degradation mechanism. In this article, I provide a full, exact, and universal mathematical description of the TC system at equilibrium for the first time. I have also constructed a comprehensive suite of mathematical tools for quantitative measurement of target engagement and equilibrium constants from experimental data. Mechanistic explanations are provided for many common challenges associated with developing this type of therapeutic agent. Insights from these analyses provide testable hypotheses and grant direction to drug development efforts in this promising area. The mathematical and analytical tools described in this article may also have broader applications in other areas of biology and chemistry in which ternary complexes are observed. One of the reasons conventional drug development approaches fail to yield a therapeutic agent is a lack of expected in vivo efficacy despite a high degree of target validation by gene knockout or knockdown. In those cases, it is often concluded that the target protein has a scaffolding function in addition to the enzyme activity that was inhibited by the drug candidate molecule (1Rauch J. Volinsky N. Romano D. Kolch W. The secret life of kinases: functions beyond catalysis.Cell. Commun. Signal. 2011; 9 (22035226): 2310.1186/1478-811X-9-23Crossref PubMed Scopus (105) Google Scholar). Targeted protein degradation has drawn a lot of attention in recent years partly because of its potential to remove the entire protein and reproduce the gene knockout or knockdown phenotypes. Targeting a specific protein for degradation is initiated by recruiting the target protein into a ternary complex with an E3 ubiquitin ligase using a ligand that can bind simultaneously to both (2Sakamoto K.M. Kim K.B. Kumagai A. Mercurio F. Crews C.M. Deshaies R.J. PROTACs: chimeric molecules that target proteins to the Skp1–cullin–F-box complex for ubiquitination and degradation.Proc. Natl. Acad. Sci. U.S.A. 2001; 98 (11438690): 8554-855910.1073/pnas.141230798Crossref PubMed Scopus (684) Google Scholar, 3Lai A. Crews C. Induced protein degradation: an emerging drug discovery paradigm.Nat. Rev. Drug Discov. 2017; 16 (27885283): 101-11410.1038/nrd.2016.211Crossref PubMed Scopus (493) Google Scholar). Once a ternary complex is formed, endogenous E2 ubiquitin ligases transfer the ubiquitin to the target protein in a target-oblivious manner as long as the target protein is oriented in such a way that a surface-exposed lysine side chain is available for ubiquitin conjugation (4Deshaies R.J. Joazeiro C.A.P. RING domain E3 ubiquitin ligases.Annu. Rev. Biochem. 2009; 78 (19489725): 399-43410.1146/annurev.biochem.78.101807.093809Crossref PubMed Scopus (1679) Google Scholar). Certain E3 ligases are known to generate a growing chain of ubiquitin conjugation through Lys48 of the target-conjugated ubiquitin as an acceptor for addition of another ubiquitin. These Lys48-polyubiquitinated proteins are recognized by the cellular proteasome complex and get degraded (5Chau V. Tobias J.W. Bachmair A. Marriott D. Ecker D.J. Gonda D.K. Varshavsky A. A multiubiquitin chain is confined to specific lysine in a targeted short-lived protein.Science. 1989; 243 (2538923): 1576-158310.1126/science.2538923Crossref PubMed Scopus (1056) Google Scholar, 6Lu Y. Lee B.H. King R.W. Finley D. Kirschner M.W. Substrate degradation by the proteasome: a single-molecule kinetic analysis.Science. 2015; 348 (25859050): 125083410.1126/science.1250834Crossref PubMed Scopus (122) Google Scholar). There are two different types of ligands that are often utilized for targeted protein degradation: heterobifunctional ligands that are often called PROTAC (proteolysis targeting chimera) (2Sakamoto K.M. Kim K.B. Kumagai A. Mercurio F. Crews C.M. Deshaies R.J. PROTACs: chimeric molecules that target proteins to the Skp1–cullin–F-box complex for ubiquitination and degradation.Proc. Natl. Acad. Sci. U.S.A. 2001; 98 (11438690): 8554-855910.1073/pnas.141230798Crossref PubMed Scopus (684) Google Scholar) and molecular glues (7Tan X. Calderon-Villalobos L.I.A. Sharon M. Zheng C. Robinson C.V. Estelle M. Zheng N. Mechanism of auxin perception by the TIR1 ubiquitin ligase.Nature. 2007; 446 (17410169): 640-64510.1038/nature05731Crossref PubMed Scopus (1000) Google Scholar). A heterobifunctional ligand has a ligand for the target protein connected through a linker to another ligand for an E3 ubiquitin ligase. Because of the modular nature of heterobifunctional ligands, they are often the method of choice. The heterobifunctional ligands can be readily constructed from a collection of existing ligands for the target protein and a list of known ligands for different E3 ligases (8Paiva S.L. Crews C.M. Targeted protein degradation: elements of PROTAC design.Curr. Opin. Chem. Biol. 2019; 50 (31004963): 111-11910.1016/j.cbpa.2019.02.022Crossref PubMed Scopus (122) Google Scholar). Small molecule ligands have been characterized for MDM2 (9Schneekloth A.R. Pucheault M. Tae H.S. Crews C.M. Targeted intracellular protein degradation induced by a small molecule: En route to chemical proteomics.Bioorg. Med. Chem. Lett. 2008; 18 (18752944): 5904-590810.1016/j.bmcl.2008.07.114Crossref PubMed Scopus (249) Google Scholar, 10Hines J. Lartigue S. Dong H. Qian Y. Crews C.M. MDM2-recruiting PROTAC offers superior, synergistic antiproliferative activity via simultaneous degradation of BRD4 and stabilization of p53.Cancer Res. 2019; 79 (30385614): 251-26210.1158/0008-5472.CAN-18-2918Crossref PubMed Scopus (97) Google Scholar), IAP (11Itoh Y. Ishikawa M. Naito M. Hashimoto Y. Protein knockdown using methyl bestatin-ligand hybrid molecules: design and synthesis of inducers of ubiquitination-mediated degradation of cellular retinoic acid-binding proteins.J. Am. Chem. Soc. 2010; 132 (20369832): 5820-582610.1021/ja100691pCrossref PubMed Scopus (193) Google Scholar, 12Ohoka N. Okuhira K. Ito M. Nagai K. Shibata N. Hattori T. Ujikawa O. Shimokawa K. Sano O. Koyama R. Fujita H. Teratani M. Matsumoto H. Imaeda Y. Nara H. et al.In vivo knockdown of pathogenic proteins via specific and nongenetic inhibitor of apoptosis protein (IAP)–dependent protein erasers (SNIPERs).J. Biol. Chem. 2017; 292 (28154167): 4556-457010.1074/jbc.M116.768853Abstract Full Text Full Text PDF PubMed Scopus (113) Google Scholar), CRBN (13Krönke J. Udeshi N.D. Narla A. Grauman P. Hurst S.N. McConkey M. Svinkina T. Heckl D. Comer E. Li X. Ciarlo C. Hartman E. Munshi N. Schenone M. Schreiber S.L. et al.Lenalidomide causes selective degradation of IKZF1 and IKZF3 in multiple myeloma cells.Science. 2014; 343 (24292625): 301-30510.1126/science.1244851Crossref PubMed Scopus (857) Google Scholar, 14Ito T. Ando H. Suzuki T. Ogura T. Hotta K. Imamura Y. Yamaguchi Y. Handa H. Identification of a primary target of thalidomide teratogenicity.Science. 2010; 327 (20223979): 1345-135010.1126/science.1177319Crossref PubMed Scopus (1069) Google Scholar), and VHL (15Buckley D.L. Molle I.V. Gareiss P.C. Tae H.S. Michel J. Noblin D.J. Jorgensen W.L. Ciulli A. Crews C.M. Targeting the von Hippel–Lindau E3 ubiquitin ligase using small molecules to disrupt the VHL/HIF-1α interaction.J. Am. Chem. Soc. 2012; 134 (22369643): 4465-446810.1021/ja209924vCrossref PubMed Scopus (236) Google Scholar, 16Galdeano C. Gadd M.S. Soares P. Scaffidi S. Molle I.V. Birced I. Hewitt S. Dias D.M. Ciulli A. Structure-guided design and optimization of small molecules targeting the protein–protein interaction between the von Hippel–Lindau (VHL) E3 ubiquitin ligase and the hypoxia inducible factor (HIF) α subunit with in vitro nanomolar affinities.J. Med. Chem. 2014; 57 (25166285): 8657-866310.1021/jm5011258Crossref PubMed Scopus (161) Google Scholar, 17Soares P. Gadd M.S. Frost J. Galdeano C. Ellis L. Epemolu O. Rocha S. Read K.D. Ciulli A. Group-based optimization of potent and cell-active inhibitors of the von Hippel–Lindau (VHL) E3 ubiquitin ligase: structure–activity relationships leading to the chemical probe (2S,4R)-1-((S)-2-(1-cyanocyclopropanecarboxamido)-3,3-dimethylbutanoyl)-4-hydroxy-N-(4-(4-methylthiazol-5-yl)benzyl)pyrrolidine-2-carboxamide (VH298).J. Med. Chem. 2018; 61 (28853884): 599-61810.1021/acs.jmedchem.7b00675PubMed Google Scholar, 18Lucas X. Molle I.V. Ciulli A. Surface probing by fragment-based screening and computational methods identifies ligandable pockets on the von Hippel–Lindau (VHL) E3 ubiquitin ligase.J. Med. Chem. 2018; 61 (30040896): 7387-739310.1021/acs.jmedchem.8b00842Crossref PubMed Scopus (12) Google Scholar). Despite the simplicity in the concept, optimizing the linker type, length, and attachment point to the existing ligand is not a trivial process. All these factors play a role in the overall affinity and efficiency of the ligand for inducing ternary complex formation. Although a large number of the linkers in the literature have a flexible structure in solution, many known X-ray crystal structures of the ternary complex show a tight folding of the linker in such a way to accommodate protein–protein interactions at the interface (19Gadd M.S. Testa A. Lucas X. Chan K.-H. Chen W. Lamont D.J. Zengerle M. Ciulli A. Structural basis of PROTAC cooperative recognition for selective protein degradation.Nat. Chem. Biol. 2017; 13 (28288108): 514-52110.1038/nchembio.2329Crossref PubMed Scopus (356) Google Scholar). In this case, the two ligand groups on the heterobifunctional ligand do not act independently of each other. Binding of one end of the ligand to the target protein would cause a large change in the affinity of the other end toward the E3 ligase or vice versa. Such positive or negative cooperativity in two binding events can be critically affected by small changes in the linker length or the structure and by the attachment points of the linker to each of the two ligand groups. Optimizing each of these three components often requires extensive efforts and currently relies largely on empirical outcomes (8Paiva S.L. Crews C.M. Targeted protein degradation: elements of PROTAC design.Curr. Opin. Chem. Biol. 2019; 50 (31004963): 111-11910.1016/j.cbpa.2019.02.022Crossref PubMed Scopus (122) Google Scholar). Objective and quantitative understanding of biochemical properties of ligands during the early phase of the development will be very helpful in guiding the direction of the SAR efforts. A second molecular construct capable of ternary complex formation is known as "molecular glue" (20Neklesa T.K. Winkler J.D. Crews C.M. Targeted protein degradation by PROTACs.Pharmacol. Therapeut. 2017; 174 (28223226): 138-14410.1016/j.pharmthera.2017.02.027Crossref PubMed Scopus (195) Google Scholar). Molecular glues are smaller (typically <500 Da) than heterobifunctional ligands (mostly ∼1000 Da) (21Che Y. Gilbert A.M. Shanmugasundaram V. Noe M.C. Inducing protein–protein interactions with molecular glues.Bioorg. Med. Chem. Lett. 2018; 28 (29980357): 2585-259210.1016/j.bmcl.2018.04.046Crossref PubMed Scopus (25) Google Scholar) because dual binding functionality is built into a single pharmacophore without the use of a linker, and they typically show no measurable "hook effect" within a large concentration range, unlike the heterobifunctional ligands. Despite their desirable drug-like properties, molecular glues are difficult to identify and even harder to design in a rational manner, even though progress has been made recently in this direction (22Simonetta K.R. Taygerly J. Boyle K. Basham S.E. Padovani C. Lou Y. Cummins T.J. Yung S.L. von Soly S.K. Kayser F. Kuriyan J. Rape M. Cardozo M. Gallop M.A. Bence N.F. et al.Prospective discovery of small molecule enhancers of an E3 ligase–substrate interaction.Nat. Commun. 2019; 10 (30926793): 140210.1038/s41467-019-09358-9Crossref PubMed Scopus (42) Google Scholar). Because they lack a modular structure, it is difficult to predict which E3 ligase is most likely to yield a glue-like ligand for the target protein of interest. As a result, heterobifunctional ligands tend to be the preferred method of choice. For targeted protein degradation, ternary complex formation can be considered equivalent to target engagement in the traditional sense of drug action. Although ternary complex formation may not guarantee subsequent polyubiquitination and degradation of the target protein (23Ishoey M. Chorn S. Singh N. Jaeger M.G. Brand M. Paulk J. Bauer S. Erb M.A. Parapatics K. Müller A.C. Bennett K.L. Ecker G.F. Bradner J.E. Winter G.E. Translation termination factor GSPT1 is a phenotypically relevant off-target of heterobifunctional phthalimide degraders.ACS Chem. Biol. 2018; 13 (29356495): 553-56010.1021/acschembio.7b00969Crossref PubMed Scopus (57) Google Scholar, 24Nowak R.P. DeAngelo S.L. Buckley D. He Z. Donovan K.A. An J. Safaee N. Jedrychowski M.P. Ponthier C.M. Ishoey M. Zhang T. Mancias J.D. Gray N.S. Bradner J.E. Fischer E.S. et al.Plasticity in binding confers selectivity in ligand-induced protein degradation.Nat. Chem. Biol. 2018; 14 (29892083): 706-71410.1038/s41589-018-0055-yCrossref PubMed Scopus (164) Google Scholar), none of these would occur without the initial ternary complex. Therefore, screening and characterization of ligands often start with an in vitro measurement of ternary complex formation. For traditional drugs that form a binary complex with the target protein, the dose-response curve of the target engagement shows a sigmoidal curve on the semi-log plot, reaching a plateau at sufficiently high ligand concentrations. This behavior is elegantly described by a simple mathematical equation of [B] = [B]max × [L]/([L] + Kd), where [B] is the ligand-bound concentration of the target protein, [B]max is the total target protein concentration, [L] is the free ligand concentration at equilibrium, and Kd is the equilibrium dissociation constant of the protein–ligand binary complex. In this binary complex system, two easily measured parameters, [B]/[B]max and Kd, have good predictive value for drug efficacy and potency, respectively. The higher fractional target occupancy, the higher biological response to drug, or efficacy, is expected. The lower the Kd value is, the drug is expected to be more potent or elicit the same biological responses at lower drug concentrations when everything else is equal. In comparison, the dose-response curve of the ternary complex shows the hook effect, or a bell-shaped curve reaching a maximum at certain ligand concentration but falling back down to baseline level at sufficiently high concentrations (12Ohoka N. Okuhira K. Ito M. Nagai K. Shibata N. Hattori T. Ujikawa O. Shimokawa K. Sano O. Koyama R. Fujita H. Teratani M. Matsumoto H. Imaeda Y. Nara H. et al.In vivo knockdown of pathogenic proteins via specific and nongenetic inhibitor of apoptosis protein (IAP)–dependent protein erasers (SNIPERs).J. Biol. Chem. 2017; 292 (28154167): 4556-457010.1074/jbc.M116.768853Abstract Full Text Full Text PDF PubMed Scopus (113) Google Scholar, 25An S. Fu L. Small-molecule PROTACs: An emerging and promising approach for the development of targeted therapy drugs.EBioMedicine. 2018; 36 (30224312): 553-56210.1016/j.ebiom.2018.09.005Abstract Full Text Full Text PDF PubMed Scopus (111) Google Scholar, 26Buckley D.L. Raina K. Darricarrere N. Hines J. Gustafson J.L. Smith I.E. Miah A.H. Harling J.D. Crews C.M. HaloPROTACS: use of small molecule PROTACs to induce degradation of HaloTag fusion proteins.ACS Chem. Biol. 2015; 10 (26070106): 1831-183710.1021/acschembio.5b00442Crossref PubMed Scopus (180) Google Scholar, 27Bondeson D.P. Mares A. Smith I.E.D. Ko E. Campos S. Miah A.H. Mulholland K.E. Routly N. Buckley D.L. Gustafson J.L. Zinn N. Grandi P. Shimamura S. Bergamini G. Faelth-Savitski M. et al.Catalytic in vivo protein knockdown by small-molecule PROTACs.Nat. Chem. Biol. 2015; 11 (26075522): 611-61710.1038/nchembio.1858Crossref PubMed Scopus (493) Google Scholar, 28Burslem G.M. Smith B.E. Lai A.C. Jaime-Figueroa S. McQuaid D.C. Bondeson D.P. Toure M. Dong H. Qian Y. Wang J. Crew A.P. Hines J. Crews C.M. The advantages of targeted protein degradation over inhibition: an RTK case study.Cell Chem. Biol. 2018; 25 (29129716): 67-7710.1016/j.chembiol.2017.09.009Abstract Full Text Full Text PDF PubMed Scopus (204) Google Scholar, 29Winter G.E. Buckley D.L. Paulk J. Roberts J.M. Souza A. Dhe-Paganon S. Bradner J.E. Phthalimide conjugation as a strategy for in vivo target protein degradation.Science. 2015; 348 (25999370): 1376-138110.1126/science.aab1433Crossref PubMed Scopus (740) Google Scholar, 30Lu J. Qian Y. Altieri M. Dong H. Wang J. Raina K. Hines J. Winkler J.D. Crew A.P. Coleman K. Crews C.M. Hijacking the E3 ubiquitin ligase Cereblon to efficiently target BRD4.Chem. Biol. 2015; 22 (26051217): 755-76310.1016/j.chembiol.2015.05.009Abstract Full Text Full Text PDF PubMed Scopus (511) Google Scholar, 31Schiedel M. Herp D. Hammelmann S. Swyter S. Lehotzky A. Robaa D. Oláh J. Ovádi J. Sippl W. Jung M. Chemically induced degradation of sirtuin 2 (Sirt2) by a proteolysis targeting chimera (PROTAC) based on sirtuin rearranging ligands (SirReals).J. Med. Chem. 2018; 61 (28379698): 482-49110.1021/acs.jmedchem.6b01872Crossref PubMed Scopus (132) Google Scholar, 32Olson C.M. Jiang B. Erb M.A. Liang Y. Doctor Z.M. Zhang Z. Zhang T. Kwiatkowski N. Boukhali M. Green J.L. Haas W. Nomanbhoy T. Fischer E.S. Young R.A. Bradner J.E. et al.Pharmacological perturbation of CDK9 using selective CDK9 inhibition or degradation.Nat. Chem. Biol. 2018; 14 (29251720): 163-17010.1038/nchembio.2538Crossref PubMed Scopus (191) Google Scholar). Because of this biphasic response, careful dose titration is required for successful degradation of the target using the heterobifunctional ligands. In addition, there are currently no easily measurable biochemical properties that can address the potential efficacy and potency of these molecules as a drug, adding to the long list of challenges in developing therapeutic agents in this mechanism. Currently, no mathematical equation is available to describe the hook effect and full equilibrium binding characteristics of the ternary complex in a universal manner despite widespread occurrence of the ternary complex in multiple scientific disciplines (reviewed in Ref. 33Douglass 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 (23544844): 6092-609910.1021/ja311795dCrossref PubMed Scopus (147) Google Scholar). It was even demonstrated that solving an exact algebraic equation for the ternary complex as a function of total ligand concentration is mathematically "unsolvable" when the system has cooperativity (33Douglass 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 (23544844): 6092-609910.1021/ja311795dCrossref PubMed Scopus (147) Google Scholar). Analytical solution could be obtained only for a noncooperative equilibrium system (33Douglass 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 (23544844): 6092-609910.1021/ja311795dCrossref PubMed Scopus (147) Google Scholar). Cooperative interaction among the components within the ternary complex induced by the heterobifunctional ligands or molecular glue ligands, however, is considered a critical component of efficient target engagement or target degradation (19Gadd M.S. Testa A. Lucas X. Chan K.-H. Chen W. Lamont D.J. Zengerle M. Ciulli A. Structural basis of PROTAC cooperative recognition for selective protein degradation.Nat. Chem. Biol. 2017; 13 (28288108): 514-52110.1038/nchembio.2329Crossref PubMed Scopus (356) Google Scholar, 34Zorba A. Nguyen C. Xu Y. Starr J. Borzilleri K. Smith J. Zhu H. Farley K.A. Ding W. Schiemer J. et al.Delineating the role of cooperativity in the design of potent PROTACs for BTK.Proc. Natl. Acad. Sci. U.S.A. 2017; 115 (30012605): E7285-E7292Crossref Scopus (136) Google Scholar). Lack of proper mathematical and analytical tools to directly address such interaction makes it difficult to relate the experimentally measured data to the equilibrium constants or biochemical properties of the ligand. When the desired outcome of efficient target protein degradation is not achieved, it is difficult to sort out where the problem is and how to fix it. A universal mathematical description of the ternary complex system of all types that can connect the experimentally measured data to the biochemical properties of the complex such as equilibrium constants, potency, and efficacy is sorely desired. In this article, I provide an exact and universal mathematical description of the ternary complex system at equilibrium and its variations that are commonly found in biological systems. This was made possible by solving the mathematical relationships among different components in terms of free ligand concentration at equilibrium rather than total ligand concentration or the initial ligand concentration. Although free ligand concentration is usually not directly measurable, the binary equilibrium equation of [B] = [B]max × [L]/([L] + Kd) is also written in terms of free ligand concentration at equilibrium. This binary equilibrium equation has been universally adopted by scientists of all fields for many decades, and its impact on pharmacology and drug discovery is immeasurable. A similar mathematical equation for the ternary complex that works universally will be extremely valuable. Using mathematical modeling of the system, mechanistic understandings could be obtained for many commonly encountered challenges during development of reagents for targeted protein degradation. Finally, analytical tools were developed that can extract information on potency and efficacy for target engagement, as well as equilibrium constants from the experimental dose-response data. The suite of mathematical tools provided in this article will be helpful in advancing this exciting field of targeted protein degradation and any other discipline involving a ternary complex. An equilibrium binding of a heterobifunctional ligand (L) with its target protein (P) and an E3 ligase (E) shown in Fig. 1A can be completely described by three independent equilibrium constants, KP1, KE1, and α, as defined by Equations 1 – 1, 1 – 2, and 1 – 3 in Fig. 1B. Note that KP1 and KE1 are binary equilibrium dissociation constants, whereas the third parameter, α, is the ratio between the ternary equilibrium dissociation constants (KP2 and KE2) and the corresponding binary equilibrium dissociation constants. As such, α is considered a cooperativity factor by many in the field (19Gadd M.S. Testa A. Lucas X. Chan K.-H. Chen W. Lamont D.J. Zengerle M. Ciulli A. Structural basis of PROTAC cooperative recognition for selective protein degradation.Nat. Chem. Biol. 2017; 13 (28288108): 514-52110.1038/nchembio.2329Crossref PubMed Scopus (356) Google Scholar, 33Douglass 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 (23544844): 6092-609910.1021/ja311795dCrossref PubMed Scopus (147) Google Scholar). When α is greater than 1, there is a positive cooperativity, whereas a value less than 1 indicates negative cooperativity between the first and second binding events on the same path within the equilibrium diagram. A value of 1 indicates no cooperativity. Universal mathematical equations for this ternary complex system at equilibrium were solved for the first time and are described in Fig. 1B. In short, the concentration of the ternary complex, [PLE], at equilibrium as a function of the free ligand concentration, [L], is given by PLE=fL−f2L−4Pt⋅Et/2 wherefL =Pt+Et+1α⋅LL+KP1L+KE1 where [Pt] and [Et] are total concentrations for the target protein and E3 ligase, respectively. Mathematical equations for the concentration of other species in this diagram and full derivation of these equations are provided in File S1A. Douglas et al. (33Douglass 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 (23544844): 6092-609910.1021/ja311795dCrossref PubMed Scopus (147) Google Scholar) have previously described a mathematical equation for a ternary complex system, but their solution was limited to a noncooperative system. They have proven that an analytical solution did not exist for a system with a cooperativity. Considering that most ternary complex systems involving heterobifunctional ligands or molecular glues have high degree of cooperativity, a universal solution for the ternary complex system regardless of the cooperativity was critically needed. The equations in this article were solved with the cooperativity factor α built into the model, and the solutions apply universally to all ternary complex systems regardless of α. The key difference between the two mathematical approaches is that the previous work (33Douglass 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 (23544844): 6092-609910.1021/ja311795dCrossref PubMed Scopus (147) Google Scholar) used total ligand concentration, whereas the current work described the system in terms of the free ligand concentration at equilibrium. An additional benefit of the current work is that these equations are easy to modify to accommodate variations in the system that occur frequently in biological systems as described in detail below. There is a value in being able to calculate the ternary complex concentration in terms of the total ligand concentration because the free ligand concentration is usually not known because of ligand depletion. Although a universal analytical solution does not exist for the concentration of the ternary complex as a function of the total ligand concentration (33Douglass 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 (23544844): 6092-609910.1021/ja311795dCrossref PubMed Scopus (147) Google Scholar), numeric solution can be easily obtained using the mathematical equations provided in this article. This method is explained in File S1B, and a template is provided with step-by-step instructions in an Excel file in supporting information (BHan_PLEcalc_v1.2_200727.xlsx). The familiar equation for the binary complex system, [B] = [B]max × [L]/([L] + Kd), is also written as a function of the free ligand concentration at equilibrium, and use of total ligand concentration in this equation causes overestimation of the concentration of the bound ligand caused by ligand depletion. The numeric method described in this article can be also used for the binary complex system with a simple modification. The mathematical equations for the two binary complexes in this system, PL and EL, have forms similar to that for a simple binary complex system (Equations 1–8 and 1–10 in File S1A), and the saturation binding curves adopt a sigmoidal shape (open symbols with dotted lines in Fig. 1C) in a manner similar to the simple binary complex systems. The equation for the ternary complex, PLE, takes up a more complicated form (Equations 2 – 1a and 2 – 1b in Fig. 1B) and produces a symmetrical bell-shaped curve (filled circles with a solid line in Fig. 1C) on the semi-log scale, reproducing the well-known hook effect of the heterobifunctional ligands. Unlike the binary complexes (EL and PL) that reach saturated binding at sufficiently high concentrations of the ligand, the ternary complex (PLE) reach
Referência(s)