Artigo Acesso aberto Revisado por pares

The Prolactin Receptor and Severely Truncated Erythropoietin Receptors Support Differentiation of Erythroid Progenitors

1997; Elsevier BV; Volume: 272; Issue: 22 Linguagem: Inglês

10.1074/jbc.272.22.14009

ISSN

1083-351X

Autores

Merav Socolovsky, Isabelle Dusanter‐Fourt, Harvey F. Lodish,

Tópico(s)

Blood disorders and treatments

Resumo

Activation of the erythropoietin receptor is essential for the survival, proliferation, and differentiation of erythroid progenitors. To understand the role of erythropoietin receptor (EpoR) activation in erythroid differentiation, we infected primary erythroid progenitors with high-titer retrovirus encoding the non-hematopoietic prolactin receptor. The infected progenitors responded to prolactin in the absence of Epo by generating fully differentiated erythroid colonies. Therefore, differentiation of erythroid progenitors does not require an intracellular signal generated uniquely by the EpoR; the EpoR does not have an instructive role in erythroid differentiation. We also infected primary erythroid progenitors with retrovirus encoding chimeric receptors containing the extracellular domain of PrlR and the intracellular domain of either the wild-type or truncated EpoRs. A chimeric receptor containing only the membrane-proximal 136 amino acids of the EpoR cytoplasmic domain efficiently supported prolactin-dependent differentiation of erythroid progenitors. Substitution of the single cytoplasmic domain tyrosine in this receptor with phenylalanine (Y343F) eliminated its ability to support differentiation. The minimal EpoR cytoplasmic domain required for erythroid differentiation is therefore the same as that previously reported to be sufficient to support cell proliferation (D'Andrea, A. D., Yoshimura, A., Youssoufian, H., Zon, L. I., Koo, J. W., and Lodish, H. F. (1991) Mol. Cell. Biol. 11, 1980–1987; Miura, O., D'Andrea, A. D., Kabat, D., and Ihle, J. N. (1991) Mol. Cell. Biol. 11, 4895–4902; He, T.-C., Jiang, N., Zhuang, H., Quelle, D. E., and Wojchowski, D. M. (1994) J. Biol. Chem. 269, 18291–18294). Activation of the erythropoietin receptor is essential for the survival, proliferation, and differentiation of erythroid progenitors. To understand the role of erythropoietin receptor (EpoR) activation in erythroid differentiation, we infected primary erythroid progenitors with high-titer retrovirus encoding the non-hematopoietic prolactin receptor. The infected progenitors responded to prolactin in the absence of Epo by generating fully differentiated erythroid colonies. Therefore, differentiation of erythroid progenitors does not require an intracellular signal generated uniquely by the EpoR; the EpoR does not have an instructive role in erythroid differentiation. We also infected primary erythroid progenitors with retrovirus encoding chimeric receptors containing the extracellular domain of PrlR and the intracellular domain of either the wild-type or truncated EpoRs. A chimeric receptor containing only the membrane-proximal 136 amino acids of the EpoR cytoplasmic domain efficiently supported prolactin-dependent differentiation of erythroid progenitors. Substitution of the single cytoplasmic domain tyrosine in this receptor with phenylalanine (Y343F) eliminated its ability to support differentiation. The minimal EpoR cytoplasmic domain required for erythroid differentiation is therefore the same as that previously reported to be sufficient to support cell proliferation (D'Andrea, A. D., Yoshimura, A., Youssoufian, H., Zon, L. I., Koo, J. W., and Lodish, H. F. (1991) Mol. Cell. Biol. 11, 1980–1987; Miura, O., D'Andrea, A. D., Kabat, D., and Ihle, J. N. (1991) Mol. Cell. Biol. 11, 4895–4902; He, T.-C., Jiang, N., Zhuang, H., Quelle, D. E., and Wojchowski, D. M. (1994) J. Biol. Chem. 269, 18291–18294). The EpoR 1The abbreviations used are:EpoerythropoietinEpoRerythropoietin receptorPrlprolactinPrlRprolactin receptorBFU-eburst-forming unit erythroidCFU-ecolony-forming unit erythroidIMDMIscove's modified Dulbecco's mediumIGF-Iinsulin-like growth factor 1FACSfluorescence-activated cell sortingMuLvmurine leukemia virus belongs to a large family of cytokine receptors, many of which are required for the proliferation and differentiation of hematopoietic as well as other cell types (4Bazan J.F. Proc. Natl. Acad. Sci. U. S. A. 1990; 87: 6934-6938Crossref PubMed Scopus (1869) Google Scholar, 5Ihle J.N. Witthuhn B. Tang B. Yi T. Quelle F.W. Bailliere's Clin. Hematol. 1994; 7: 17-48Abstract Full Text PDF PubMed Scopus (33) Google Scholar, 6Ihle J.N. Witthuhn B.A. Quelle F.W. Yamamoto K. Silvennoinen O. Annu. Rev. Immunol. 1995; 13: 369-398Crossref PubMed Scopus (545) Google Scholar, 7Watowich S.S. Wu H. Socolovksy M. Klingmuller U. Constantinescu S.N. Lodish H.F. Annu. Rev. Cell. Dev. Biol. 1996; 12: 91-128Crossref PubMed Scopus (158) Google Scholar). Throughout life, eight different hematopoietic lineages arise from pluripotent stem cells in the bone marrow (8Metcalf D. Nature. 1989; 339: 27-30Crossref PubMed Scopus (861) Google Scholar, 9Dexter T.M. Spooncer E. Annu. Rev. Cell Biol. 1987; 3: 423-441Crossref PubMed Scopus (173) Google Scholar). The exact role of growth factors in this process is not clear and has been described broadly by two alternative hypotheses. The stochastic hypothesis suggests that commitment of a progenitor to a particular lineage is a stochastic event, subsequent to which cell differentiation proceeds along a predetermined program; growth factors are merely required to ensure the survival and proliferation of committed progenitors (10Korn A.P. Henkelman R.M. Ottensmeyer F.P. Till J.E. Exp. Hematol. 1973; 1: 362-375PubMed Google Scholar, 11Till J.E. McCulloch E.A. Siminovitch L. Proc. Natl. Acad. Sci. U. S. A. 1964; 51: 29-36Crossref PubMed Scopus (609) Google Scholar, 12Nakahata T. Gross A.J. Ogawa M. J. Cell. Physiol. 1982; 113: 455-458Crossref PubMed Scopus (141) Google Scholar, 13Suda T. Suda J. Ogawa M. Proc. Natl. Acad. Sci. U. S. A. 1984; 81: 2520-2524Crossref PubMed Scopus (159) Google Scholar, 14Fairbairn L.J. Cowling G.J. Reipert B.M. Dexter T.M. Cell. 1993; 74: 823-832Abstract Full Text PDF PubMed Scopus (346) Google Scholar). The contrasting inductive, or instructive, hypothesis attributes to growth factors a direct role in cell differentiation, predicting that growth factor receptors transduce signals that uniquely specify the differentiation outcome of a progenitor (15Borzillo G.V. Ashmun R.A. Sherr C.J. Mol. Cell. Biol. 1990; 10: 2703-2714Crossref PubMed Scopus (101) Google Scholar, 16Metcalf D. Proc. Natl. Acad. Sci. U. S. A. 1991; 88: 11310-11314Crossref PubMed Scopus (94) Google Scholar, 17Fukunaga R. Ishizaka-Ikeda E. Nagata S. Cell. 1993; 74: 1079-1087Abstract Full Text PDF PubMed Scopus (227) Google Scholar). A number of hybrid hypotheses have also been proposed, where, for example, committed progenitors arise stochastically, but their subsequent differentiation and acquisition of the mature phenotype are uniquely induced by lineage-specific growth factors (18Just U. Stocking C. Spooncer E. Dexter T.M. Ostertag W. Cell. 1991; 64: 1163-1173Abstract Full Text PDF PubMed Scopus (73) Google Scholar).Although Epo is essential for the production of red blood cells, it is not thought to play a role in the generation of committed erythroid progenitors from pluripotent progenitors: expression of recombinant EpoR does not bias the lineage commitment of pluripotent hematopoietic progenitors (19Pharr P.N. Hankins D. Hofbauer A. Lodish H.F. Longmore G.D. Proc. Natl. Acad. Sci. U. S. A. 1993; 90: 938-942Crossref PubMed Scopus (49) Google Scholar, 20Pharr P.N. Ogawa M. Hofbauer A. Longmore G.D. Proc. Natl. Acad. Sci. U. S. A. 1994; 91: 7482-7486Crossref PubMed Scopus (45) Google Scholar), and normal numbers of committed erythroid BFU-e and CFU-e progenitors are found in the fetal livers of EpoR−/− mutant mice (21Wu H. Liu X. Jaenisch R. Lodish H.F. Cell. 1995; 83: 59-67Abstract Full Text PDF PubMed Scopus (848) Google Scholar). However, there is a unique requirement for EpoR activation during the subsequent proliferation and terminal differentiation of committed erythroid progenitors: the EpoR−/− CFU-e and BFU-e progenitors fail to give rise to mature erythrocytes unless EpoR is expressed by retroviral infection (21Wu H. Liu X. Jaenisch R. Lodish H.F. Cell. 1995; 83: 59-67Abstract Full Text PDF PubMed Scopus (848) Google Scholar); and in vitro, other growth factors only partially substitute for Epo (22Correa P.N. Axelrad A.A. Blood. 1991; 78: 2823-2833Crossref PubMed Google Scholar, 23Boyer S.H. Bishop T.R. Rogers O.C. Noyes A.N. Frelin L.P. Hobbs S. Blood. 1992; 80: 2503-2512Crossref PubMed Google Scholar, 24Kieran M.W. Perkins A. Orkin S. Zon L. Proc. Natl. Acad. Sci. U. S. A. 1996; 93: 9126-9131Crossref PubMed Scopus (145) Google Scholar). It is not known whether EpoR activation is essential at this stage of erythroid differentiation because of an EpoR-unique instructive signal or whether it is simply required for functions that are not unique to EpoR, such as its proliferative and anti-apoptotic effects.Some evidence for the capability of EpoR to promote the erythroid phenotype comes from the ability of Epo to induce surface expression of glycophorin (25Jubinsky P.T. Nathan D.G. Wilson D.J. Sieff C.A. Blood. 1993; 81: 587-591Crossref PubMed Google Scholar) and transcription of the β-globin gene (26Liboi E. Carrol M. D'Anderea A.D. Mathey-Prevot B. Proc. Natl. Acad. Sci. U. S. A. 1993; 90: 11351-11355Crossref PubMed Scopus (82) Google Scholar, 27Chiba T. Nagata Y. Machide M. Kishi A. Amanuma H. Sugiyama M. Todokoro D. Nature. 1993; 362: 646-648Crossref PubMed Scopus (53) Google Scholar) in pre-B Ba/F3 cells expressing a transfected EpoR. However, the uncertain lineage commitment of many cell lines and their incomplete differentiation response makes them less suitable for the study of signaling in differentiation. EpoR-mediated signaling for proliferation can be studied in a number of interleukin-3-dependent cell lines, where heterologous expression of EpoR allows Epo to support cell proliferation (1D'Andrea A.D. Yoshimura A. Youssoufian H. Zon L.I. Koo J.W. Lodish H.F. Mol. Cell. Biol. 1991; 11: 1980-1987Crossref PubMed Scopus (223) Google Scholar, 2Miura O. D'Andrea A.D. Kabat D. Ihle J.N. Mol. Cell. Biol. 1991; 11: 4895-4902Crossref PubMed Scopus (184) Google Scholar). Only the membrane proximal ∼120 amino acids is essential for this function (1D'Andrea A.D. Yoshimura A. Youssoufian H. Zon L.I. Koo J.W. Lodish H.F. Mol. Cell. Biol. 1991; 11: 1980-1987Crossref PubMed Scopus (223) Google Scholar, 2Miura O. D'Andrea A.D. Kabat D. Ihle J.N. Mol. Cell. Biol. 1991; 11: 4895-4902Crossref PubMed Scopus (184) Google Scholar, 3He T.-C. Jiang N. Zhuang H. Quelle D.E. Wojchowski D.M. J. Biol. Chem. 1994; 269: 18291-18294Abstract Full Text PDF PubMed Google Scholar). Similarly truncated mutants of other cytokine receptors are also able to support mitogenic signaling in such cells (17Fukunaga R. Ishizaka-Ikeda E. Nagata S. Cell. 1993; 74: 1079-1087Abstract Full Text PDF PubMed Scopus (227) Google Scholar, 28Hatakeyama M. Mori H. Doi T. Taniguchi T. Cell. 1989; 59: 837-845Abstract Full Text PDF PubMed Scopus (301) Google Scholar, 29Porteu F. Rouyez M.C. Cocault L. Benit L. Charon M. Picard F. Gisselbrecht S. Souyri M. Dusanter-Fourt I. Mol. Cell. Biol. 1996; 16: 2473-2482Crossref PubMed Google Scholar, 30Murakami M. Narazaki M. Hibi M. Yawata H. Yasukawa K. Hamaguchi M. Taga T. Kishimoto T. Proc. Natl. Acad. Sci. U. S. A. 1991; 88(24): 11349-11353Crossref Scopus (486) Google Scholar). Since the greatest homology between cytokine receptors is found in the Box 1 and Box 2 domains of their membrane-proximal regions (30Murakami M. Narazaki M. Hibi M. Yawata H. Yasukawa K. Hamaguchi M. Taga T. Kishimoto T. Proc. Natl. Acad. Sci. U. S. A. 1991; 88(24): 11349-11353Crossref Scopus (486) Google Scholar) (see Fig. 1), it might be expected that this region would generate signals for functions common to all these receptors such as cell survival and proliferation and that the divergent membrane-distal regions would endow the specificity of signaling presumed unique to each receptor.We therefore examined whether the distal half of the EpoR cytoplasmic domain is essential for differentiation of primary fetal liver erythroid progenitors. We also examined whether the entirety of the cytosolic domain of the EpoR can be replaced with the corresponding segment of a different receptor; we chose the prolactin receptor, which plays no role in hematopoiesis, but belongs to the same subfamily of cytokine receptors as EpoR, and shares many of its signaling molecules. The EpoR 1The abbreviations used are:EpoerythropoietinEpoRerythropoietin receptorPrlprolactinPrlRprolactin receptorBFU-eburst-forming unit erythroidCFU-ecolony-forming unit erythroidIMDMIscove's modified Dulbecco's mediumIGF-Iinsulin-like growth factor 1FACSfluorescence-activated cell sortingMuLvmurine leukemia virus belongs to a large family of cytokine receptors, many of which are required for the proliferation and differentiation of hematopoietic as well as other cell types (4Bazan J.F. Proc. Natl. Acad. Sci. U. S. A. 1990; 87: 6934-6938Crossref PubMed Scopus (1869) Google Scholar, 5Ihle J.N. Witthuhn B. Tang B. Yi T. Quelle F.W. Bailliere's Clin. Hematol. 1994; 7: 17-48Abstract Full Text PDF PubMed Scopus (33) Google Scholar, 6Ihle J.N. Witthuhn B.A. Quelle F.W. Yamamoto K. Silvennoinen O. Annu. Rev. Immunol. 1995; 13: 369-398Crossref PubMed Scopus (545) Google Scholar, 7Watowich S.S. Wu H. Socolovksy M. Klingmuller U. Constantinescu S.N. Lodish H.F. Annu. Rev. Cell. Dev. Biol. 1996; 12: 91-128Crossref PubMed Scopus (158) Google Scholar). Throughout life, eight different hematopoietic lineages arise from pluripotent stem cells in the bone marrow (8Metcalf D. Nature. 1989; 339: 27-30Crossref PubMed Scopus (861) Google Scholar, 9Dexter T.M. Spooncer E. Annu. Rev. Cell Biol. 1987; 3: 423-441Crossref PubMed Scopus (173) Google Scholar). The exact role of growth factors in this process is not clear and has been described broadly by two alternative hypotheses. The stochastic hypothesis suggests that commitment of a progenitor to a particular lineage is a stochastic event, subsequent to which cell differentiation proceeds along a predetermined program; growth factors are merely required to ensure the survival and proliferation of committed progenitors (10Korn A.P. Henkelman R.M. Ottensmeyer F.P. Till J.E. Exp. Hematol. 1973; 1: 362-375PubMed Google Scholar, 11Till J.E. McCulloch E.A. Siminovitch L. Proc. Natl. Acad. Sci. U. S. A. 1964; 51: 29-36Crossref PubMed Scopus (609) Google Scholar, 12Nakahata T. Gross A.J. Ogawa M. J. Cell. Physiol. 1982; 113: 455-458Crossref PubMed Scopus (141) Google Scholar, 13Suda T. Suda J. Ogawa M. Proc. Natl. Acad. Sci. U. S. A. 1984; 81: 2520-2524Crossref PubMed Scopus (159) Google Scholar, 14Fairbairn L.J. Cowling G.J. Reipert B.M. Dexter T.M. Cell. 1993; 74: 823-832Abstract Full Text PDF PubMed Scopus (346) Google Scholar). The contrasting inductive, or instructive, hypothesis attributes to growth factors a direct role in cell differentiation, predicting that growth factor receptors transduce signals that uniquely specify the differentiation outcome of a progenitor (15Borzillo G.V. Ashmun R.A. Sherr C.J. Mol. Cell. Biol. 1990; 10: 2703-2714Crossref PubMed Scopus (101) Google Scholar, 16Metcalf D. Proc. Natl. Acad. Sci. U. S. A. 1991; 88: 11310-11314Crossref PubMed Scopus (94) Google Scholar, 17Fukunaga R. Ishizaka-Ikeda E. Nagata S. Cell. 1993; 74: 1079-1087Abstract Full Text PDF PubMed Scopus (227) Google Scholar). A number of hybrid hypotheses have also been proposed, where, for example, committed progenitors arise stochastically, but their subsequent differentiation and acquisition of the mature phenotype are uniquely induced by lineage-specific growth factors (18Just U. Stocking C. Spooncer E. Dexter T.M. Ostertag W. Cell. 1991; 64: 1163-1173Abstract Full Text PDF PubMed Scopus (73) Google Scholar). erythropoietin erythropoietin receptor prolactin prolactin receptor burst-forming unit erythroid colony-forming unit erythroid Iscove's modified Dulbecco's medium insulin-like growth factor 1 fluorescence-activated cell sorting murine leukemia virus Although Epo is essential for the production of red blood cells, it is not thought to play a role in the generation of committed erythroid progenitors from pluripotent progenitors: expression of recombinant EpoR does not bias the lineage commitment of pluripotent hematopoietic progenitors (19Pharr P.N. Hankins D. Hofbauer A. Lodish H.F. Longmore G.D. Proc. Natl. Acad. Sci. U. S. A. 1993; 90: 938-942Crossref PubMed Scopus (49) Google Scholar, 20Pharr P.N. Ogawa M. Hofbauer A. Longmore G.D. Proc. Natl. Acad. Sci. U. S. A. 1994; 91: 7482-7486Crossref PubMed Scopus (45) Google Scholar), and normal numbers of committed erythroid BFU-e and CFU-e progenitors are found in the fetal livers of EpoR−/− mutant mice (21Wu H. Liu X. Jaenisch R. Lodish H.F. Cell. 1995; 83: 59-67Abstract Full Text PDF PubMed Scopus (848) Google Scholar). However, there is a unique requirement for EpoR activation during the subsequent proliferation and terminal differentiation of committed erythroid progenitors: the EpoR−/− CFU-e and BFU-e progenitors fail to give rise to mature erythrocytes unless EpoR is expressed by retroviral infection (21Wu H. Liu X. Jaenisch R. Lodish H.F. Cell. 1995; 83: 59-67Abstract Full Text PDF PubMed Scopus (848) Google Scholar); and in vitro, other growth factors only partially substitute for Epo (22Correa P.N. Axelrad A.A. Blood. 1991; 78: 2823-2833Crossref PubMed Google Scholar, 23Boyer S.H. Bishop T.R. Rogers O.C. Noyes A.N. Frelin L.P. Hobbs S. Blood. 1992; 80: 2503-2512Crossref PubMed Google Scholar, 24Kieran M.W. Perkins A. Orkin S. Zon L. Proc. Natl. Acad. Sci. U. S. A. 1996; 93: 9126-9131Crossref PubMed Scopus (145) Google Scholar). It is not known whether EpoR activation is essential at this stage of erythroid differentiation because of an EpoR-unique instructive signal or whether it is simply required for functions that are not unique to EpoR, such as its proliferative and anti-apoptotic effects. Some evidence for the capability of EpoR to promote the erythroid phenotype comes from the ability of Epo to induce surface expression of glycophorin (25Jubinsky P.T. Nathan D.G. Wilson D.J. Sieff C.A. Blood. 1993; 81: 587-591Crossref PubMed Google Scholar) and transcription of the β-globin gene (26Liboi E. Carrol M. D'Anderea A.D. Mathey-Prevot B. Proc. Natl. Acad. Sci. U. S. A. 1993; 90: 11351-11355Crossref PubMed Scopus (82) Google Scholar, 27Chiba T. Nagata Y. Machide M. Kishi A. Amanuma H. Sugiyama M. Todokoro D. Nature. 1993; 362: 646-648Crossref PubMed Scopus (53) Google Scholar) in pre-B Ba/F3 cells expressing a transfected EpoR. However, the uncertain lineage commitment of many cell lines and their incomplete differentiation response makes them less suitable for the study of signaling in differentiation. EpoR-mediated signaling for proliferation can be studied in a number of interleukin-3-dependent cell lines, where heterologous expression of EpoR allows Epo to support cell proliferation (1D'Andrea A.D. Yoshimura A. Youssoufian H. Zon L.I. Koo J.W. Lodish H.F. Mol. Cell. Biol. 1991; 11: 1980-1987Crossref PubMed Scopus (223) Google Scholar, 2Miura O. D'Andrea A.D. Kabat D. Ihle J.N. Mol. Cell. Biol. 1991; 11: 4895-4902Crossref PubMed Scopus (184) Google Scholar). Only the membrane proximal ∼120 amino acids is essential for this function (1D'Andrea A.D. Yoshimura A. Youssoufian H. Zon L.I. Koo J.W. Lodish H.F. Mol. Cell. Biol. 1991; 11: 1980-1987Crossref PubMed Scopus (223) Google Scholar, 2Miura O. D'Andrea A.D. Kabat D. Ihle J.N. Mol. Cell. Biol. 1991; 11: 4895-4902Crossref PubMed Scopus (184) Google Scholar, 3He T.-C. Jiang N. Zhuang H. Quelle D.E. Wojchowski D.M. J. Biol. Chem. 1994; 269: 18291-18294Abstract Full Text PDF PubMed Google Scholar). Similarly truncated mutants of other cytokine receptors are also able to support mitogenic signaling in such cells (17Fukunaga R. Ishizaka-Ikeda E. Nagata S. Cell. 1993; 74: 1079-1087Abstract Full Text PDF PubMed Scopus (227) Google Scholar, 28Hatakeyama M. Mori H. Doi T. Taniguchi T. Cell. 1989; 59: 837-845Abstract Full Text PDF PubMed Scopus (301) Google Scholar, 29Porteu F. Rouyez M.C. Cocault L. Benit L. Charon M. Picard F. Gisselbrecht S. Souyri M. Dusanter-Fourt I. Mol. Cell. Biol. 1996; 16: 2473-2482Crossref PubMed Google Scholar, 30Murakami M. Narazaki M. Hibi M. Yawata H. Yasukawa K. Hamaguchi M. Taga T. Kishimoto T. Proc. Natl. Acad. Sci. U. S. A. 1991; 88(24): 11349-11353Crossref Scopus (486) Google Scholar). Since the greatest homology between cytokine receptors is found in the Box 1 and Box 2 domains of their membrane-proximal regions (30Murakami M. Narazaki M. Hibi M. Yawata H. Yasukawa K. Hamaguchi M. Taga T. Kishimoto T. Proc. Natl. Acad. Sci. U. S. A. 1991; 88(24): 11349-11353Crossref Scopus (486) Google Scholar) (see Fig. 1), it might be expected that this region would generate signals for functions common to all these receptors such as cell survival and proliferation and that the divergent membrane-distal regions would endow the specificity of signaling presumed unique to each receptor. We therefore examined whether the distal half of the EpoR cytoplasmic domain is essential for differentiation of primary fetal liver erythroid progenitors. We also examined whether the entirety of the cytosolic domain of the EpoR can be replaced with the corresponding segment of a different receptor; we chose the prolactin receptor, which plays no role in hematopoiesis, but belongs to the same subfamily of cytokine receptors as EpoR, and shares many of its signaling molecules. We thank D. Sjolly (Viagene, Inc.) for the 293 gag/pol line; David A. Saunders (Purdue University) forpenv.min; Jean Djiane for the M110 monoclonal antibody; Glen Paradis for assistance with FACS analysis; and Dan Ory, Stefan Constantinescu, Ralph Lin, Carlos Rodriguez, and Svetlana Bergelson for helpful discussion.

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