2012, 9(3): 663-683. doi: 10.3934/mbe.2012.9.663

A minimal mathematical model for the initial molecular interactions of death receptor signalling

1. 

Institute of Applied Analysis and Numerical Simulation, Univ. of Stuttgart, Pfa enwaldring 57, 70569 Stuttgart, Germany

2. 

Institute of Cell Biology and Immunology, Univ. of Stuttgart, Allmandring 31, 70569 Stuttgart, Germany, Germany

3. 

Institute of Computational and Applied Mathematics, WWU Münster, Einsteinstr. 62, 48149 Münster, Germany

Received  August 2011 Revised  May 2012 Published  July 2012

Tumor necrosis factor (TNF) is the name giving member of a large cytokine family mirrored by a respective cell membrane receptor super family. TNF itself is a strong proinflammatory regulator of the innate immune system, but has been also recognized as a major factor in progression of autoimmune diseases. A subgroup of the TNF ligand family, including TNF, signals via so-called death receptors, capable to induce a major form of programmed cell death, called apoptosis. Typical for most members of the whole family, death ligands form homotrimeric proteins, capable to bind up to three of their respective receptor molecules. But also unligated receptors occur on the cell surface as homomultimers due to a homophilic interaction domain. Based on these two interaction motivs (ligand/receptor and receptor/receptor) formation of large ligand/receptor clusters can be postulated which have been also observed experimentally. We use here a mass action kinetics approach to establish an ordinary differential equations model describing the dynamics of primary ligand/receptor complex formation as a basis for further clustering on the cell membrane. Based on available experimental data we develop our model in a way that not only ligand/receptor, but also homophilic receptor interaction is encompassed. The model allows formation of two distict primary ligand/receptor complexes in a ligand concentration dependent manner. At extremely high ligand concentrations the system is dominated by ligated receptor homodimers.
Citation: Christian Winkel, Simon Neumann, Christina Surulescu, Peter Scheurich. A minimal mathematical model for the initial molecular interactions of death receptor signalling. Mathematical Biosciences & Engineering, 2012, 9 (3) : 663-683. doi: 10.3934/mbe.2012.9.663
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B. B. Aggarwal, Signalling pathways of the TNF superfamily: A double-edged sword, Nature Reviews Immunology, 3 (2003), 745-756. doi: 10.1038/nri1184.

[2]

Hyun-Jung An, Young Jin Kim, Dong Hyun Song, Beom Suk Park, Ho Min Kim, Ju Dong Lee, Sang-Gi Paik, Jie-Oh Lee and Hayyoung Lee, Crystallographic and mutational analysis of the CD40-CD154 complex and its implications for receptor activation, Journal of Biological Chemistry, 286 (2011), 11226-11235. doi: 10.1074/jbc.M110.208215.

[3]

David W. Banner, Allan D'Arcy, Wolfgang Janes, Reiner Gentz, Hans-Joachim Schoenfeld, Clemens Broger, Hansruedi Loetscher and Werner Lesslauer, Crystal structure of the soluble human 55 kd TNF receptor-human TNF[beta] complex: Implications for TNF receptor activation, Cell, 73 (1993), 431-445. doi: 10.1016/0092-8674(93)90132-A.

[4]

D. Berg, M. Lehne, N. Müller, D. Siegmund, S. Münkel, W. Sebald, K. Pfizenmaier and H. Wajant, Enforced covalent trimerization increases the activity of the TNF ligand family members TRAIL and CD95L, Cell Death and Differentiation, 14 (2007), 2021-2034. doi: 10.1038/sj.cdd.4402213.

[5]

Verena Boschert, Anja Krippner-Heidenreich, Marcus Branschädel, Jessica Tepperink, Andrew Aird and Peter Scheurich, Single chain TNF derivatives with individually mutated receptor binding sites reveal differential stoichiometry of ligand receptor complex formation for TNFR1 and TNFR2, Cellular Signalling, 22 (2010), 1088-1096. doi: 10.1016/j.cellsig.2010.02.011.

[6]

Marcus Branschädel, Andrew Aird, Andrea Zappe, Carsten Tietz, Anja Krippner-Heidenreich and Peter Scheurich, Dual function of cysteine rich domain (CRD) 1 of TNF receptor type 1: Conformational stabilization of CRD2 and control of receptor responsiveness, Cellular Signalling, 22 (2010), 404-414. doi: 10.1016/j.cellsig.2009.10.011.

[7]

Francis Ka-Ming Chan, Hyung J. Chun, Lixin Zheng, Richard M. Siegel, Kimmie L. Bui and Michael J. Lenardo, A domain in TNF receptors that mediates ligand-independent receptor assembly and signaling, Science, 288 (2000), 2351-2354. doi: 10.1126/science.288.5475.2351.

[8]

Lauren Clancy, Karen Mruk, Kristina Archer, Melissa Woelfel, Juthathip Mongkolsapaya, Gavin Screaton, Michael J. Lenardo and Francis Ka-Ming Chan, Preligand assembly domain-mediated ligand-independent association between TRAIL receptor 4 (TR4) and TR2 regulates TRAIL-induced apoptosis, Proceedings of the National Academy of Sciences of the USA, 102 (2005), 18099-18104. doi: 10.1073/pnas.0507329102.

[9]

M. A. Degli-Esposti, M. C. Dougall, P. J. Smolak, J. Y. Waugh, C. A. Smith and R. G. Goodwin, The novel receptor TRAIL-R4 induces NF-kappaB and protects against TRAIL-mediated apoptosis, yet retains an incomplete death domain, Immunity, 7 (1997), 813-820. doi: 10.1016/S1074-7613(00)80399-4.

[10]

John G. Emery, Peter McDonnell, Michael Brigham Burke, Keith C. Deen, Sally Lyn, Carol Silverman, Edward Dul, Edward R. Appelbaum, Chris Eichman, Rocco DiPrinzio, Robert A. Dodds, Ian E. James, Martin Rosenberg, John C. Lee and Peter R. Young, Osteoprotegerin is a receptor for the cytotoxic ligand TRAIL, Journal of Biological Chemistry, 273 (1998), 14363-14367. doi: 10.1074/jbc.273.23.14363.

[11]

Matthias Grell, Eleni Douni, Harald Wajant, Matthias Löhden, Matthias Clauss, Beate Maxeiner, Spiros Georgopoulos, Werner Lesslauer, George Kollias, Klaus Pfizenmaier and Peter Scheurich, The transmembrane form of tumor necrosis factor is the prime activating ligand of the 80 kDa tumor necrosis factor receptor, Cell, 83 (1995), 793-802. doi: 10.1016/0092-8674(95)90192-2.

[12]

Matthias Grell, Harald Wajant, Gudrun Zimmermann and Peter Scheurich, The type 1 receptor (CD120a) is the high-affinity receptor for soluble tumor necrosis factor, Proceedings of the National Academy of Sciences of the United States of America, 95 (1998), 570-575. doi: 10.1073/pnas.95.2.570.

[13]

F. Henkler, E. Behrle, K. M. Dennehy, A. Wicovsky, N. Peters, C. Warnke, K. Pfizenmaier and H. Wajant, The extracellular domains of Fasl and Fas are sufficient for the formation of supramolecular FasL-Fas clusters of high stability, Journal of Cell Biology, 168 (2005), 1087-1098. doi: 10.1083/jcb.200501048.

[14]

N. Holler, A. Tardivel, M. Kovacsovics-Bankowski, S. Hertig, O. Gaide, F. Martinon, A. Tinel, D. Deperthes, S. Calderara, T. Schulthess, J. Engel, P. Schneider and E. Tschopp, Two adjacent trimeric Fas ligands are required for Fas signaling and formation of a death-inducing signaling complex, Molecular And Cellular Biology, 23 (2003), 1428-1440. doi: 10.1128/MCB.23.4.1428-1440.2003.

[15]

Sarah G. Hymowitz, Hans W. Christinger, Germaine Fuh, Mark Ultsch, Mark O'Connell, Robert F. Kelley, Avi Ashkenazi and Abraham M. de Vos, Triggering cell death: The crystal structure of apo2l/TRAIL in a complex with death receptor 5, Molecular Cell, 4 (1999), 563-571. doi: 10.1016/S1097-2765(00)80207-5.

[16]

Sarah G. Hymowitz, Mark P. O'Connell, Mark H. Ultsch, Amy Hurst, Klara Totpal, Avi Ashkenazi, Abraham M. de Vos and Robert F. Kelley, A unique zinc-binding site revealed by a high-resolution x-ray structure of homotrimeric Apo2L/TRAIL, Biochemistry, 39 (2000), 633-640. doi: 10.1021/bi992242l.

[17]

A. Krippner-Heidenreich, F. Tübing, S. Bryde, S. Willi, G. Zimmermann and P. Scheurich, Control of receptor-induced signaling complex formation by the kinetics of ligand/receptor interaction, Journal of Biological Chemistry, 277 (2002), 44155-44163. doi: 10.1074/jbc.M207399200.

[18]

F. C. Kull, S. Jacobs and P. Cuatrecasas, Cellular receptor for 125I-labeled tumor necrosis factor: Specific binding, affinity labeling, and relationship to sensitivity, Proceedings of the National Academy of Sciences of the United States of America, 82 (1985), 5756-5760. doi: 10.1073/pnas.82.17.5756.

[19]

R. Lai and T. L. Jackson, A mathematical model of receptor-mediated apoptosis: Dying to know why FasL is a trimer, Mathematical Biosciences and Engineering, 1 (2004), 325-338.

[20]

H. W. Lee, S. H. Lee, Y. W. Ryu, M. H. Kwon and Y. S. Kim, Homomeric and heteromeric interactions of the extracellular domains of death receptors and death decoy receptors, Biochemical and Biophysical Research Communications, 330 (2005), 1205-1212. doi: 10.1016/j.bbrc.2005.03.101.

[21]

Frank Mühlenbeck, Pascal Schneider, Jean-Luc Bodmer, Ralph Schwenzer, Angelika Hauser, Gisela Schubert, Peter Scheurich, Dieter Moosmayer, Jürg Tschopp and Harald Wajant, The tumor necrosis factor-related apoptosis-inducing ligand receptors TRAIL-R1 and TRAIL-R2 have distinct cross-linking requirements for Initiation of apoptosis and are non-redundant in JNK activation, Journal of Biological Chemistry, 275 (2000), 32208-32213. doi: 10.1074/jbc.M000482200.

[22]

Y. Mukai, T. Nakamura, M. Yoshikawa, Y. Yoshioka, S. Tsunoda, S. Nakagawa, Y. Yamagata and Y. Tsutsumi, Solution of the structure of the TNF-TNFR2 complex, Science Signaling, 3 (2010), ra83. doi: 10.1126/scisignal.2000954.

[23]

James H. Naismith, Tracey Q. Devine, Barbara J. Brandhuber and Stephen R. Sprang, Crystallographic evidence for dimerization of unliganded tumor necrosis factor receptor, Journal of Biological Chemistry, 270 (1995), 13303-13307. doi: 10.1074/jbc.270.22.13303.

[24]

James H. Naismith, Tracey Q. Devine, Tadahiko Kohno and Stephen R. Sprang, Structures of the extracellular domain of the type I tumor necrosis factor receptor, Structure, 4 (1996), 1251-1262. doi: 10.1016/S0969-2126(96)00134-7.

[25]

P. Schneider, J. L. Bodmer, N. Holler, C. Mattmann, P. Scuderi, A. Terskikh, M. C. Peitsch and J. Tschopp, Characterization of Fas (Apo-1, CD95)-Fas ligand interaction, Journal of Biological Chemistry, 272 (1997), 18827-18833. doi: 10.1074/jbc.272.30.18827.

[26]

P. Schneider, N. Holler, J. L. Bodmer, M. Hahne, K. Frei, A. Fontana and J. Tschopp, Conversion of membrane-bound Fas ligand to its soluble form is associated with down regulation of its proapoptotic activity and loss of liver toxicity, Journal of Experimental Medicine, 187 (1998), 1205-1213. doi: 10.1084/jem.187.8.1205.

[27]

Richard M. Siegel, John K. Frederiksen, David A. Zacharias, Francis Ka-Ming Chan, Michele Johnson, David Lynch, Roger Y. Tsien and Michael J. Lenardo, Fas preassociation required for apoptosis signaling and dominant inhibition by pathogenic mutations, Science, 288 (2000), 2354-2357. doi: 10.1126/science.288.5475.2354.

[28]

Richard M. Siegel, Jagan R. Muppidi, Malabika Sarker, Adrian Lobito, Melinda Jen, David Martin, Stephen E. Straus and Michael J. Lenardo, SPOTS: Signaling protein oligomeric transduction structures are early mediators of death receptor-induced apoptosis at the plasma membrane, The Journal of Cell Biology, 167 (2004), 735-744. doi: 10.1083/jcb.200406101.

[29]

Steven H. Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering,'' $1^{st}$ edition, Perseus Books, Cambridge, Massachusetts, 1994.

[30]

Alemseged Truneh, Sunita Sharma, Carol Silverman, Sanjay Khandekar, Manjula P. Reddy, Keith C. Deen, Megan M. McLaughlin, Srinivasa M. Srinivasula, George P. Livi, Lisa A. Marshall, Emad S. Alnemri, William V. Williams and Michael L. Doyle, Temperature-sensitive differential affinity of TRAIL for its receptors. DR5 is the highest affinity receptor Journal of Biological Chemistry, 275 (2000), 23319-23325. doi: 10.1074/jbc.M910438199.

[31]

A. Waage, P. Brandtzaeg, A. Halstensen, P. Kierulf and T. Espevik, The complex pattern of cytokines in serum from patients with meningococcal septic shock. association between interleukin 6, interleukin 1, and fatal outcome, Journal of Experimental Medicine, 169 (1989), 333-338. doi: 10.1084/jem.169.1.333.

[32]

Harald Wajant, Dieter Moosmayer, Thomas Wüest, Till Bartke, Elke Gerlach, Ulrike Schönherr, Nathalie Peters, Peter Scheurich and Klaus Pfizenmaier, Differential activation of TRAIL-R1 and -2 by soluble and membrane TRAIL allows selective surface antigen-directed activation of TRAIL-R2 by a soluble TRAIL derivative, Oncogene, 20 (2001), 4101-4106. doi: 10.1038/sj.onc.1204558.

[33]

Wolfgang Walter, "Gewöhnliche Differentialgleichungen. Eine Einführung,'' $6^{th}$ edition, Springer-Verlag, Berlin, 1996.

[34]

Liwei Wang, Jin Kuk Yang, Venkataraman Kabaleeswaran, Amanda J. Rice, Anthony C. Cruz, Ah Young Park, Qian Yin, Ermelinda Damko, Se Bok Jang, Stefan Raunser, Carol V. Robinson, Richard M. Siegel, Thomas Walz and Hao Wu, The Fas-FADD death domain complex structure reveals the basis of DISC assembly and disease mutations, Nature Structural & Molecular Biology, 17 (2010), 1324-1329. doi: 10.1038/nsmb.1920.

show all references

References:
[1]

B. B. Aggarwal, Signalling pathways of the TNF superfamily: A double-edged sword, Nature Reviews Immunology, 3 (2003), 745-756. doi: 10.1038/nri1184.

[2]

Hyun-Jung An, Young Jin Kim, Dong Hyun Song, Beom Suk Park, Ho Min Kim, Ju Dong Lee, Sang-Gi Paik, Jie-Oh Lee and Hayyoung Lee, Crystallographic and mutational analysis of the CD40-CD154 complex and its implications for receptor activation, Journal of Biological Chemistry, 286 (2011), 11226-11235. doi: 10.1074/jbc.M110.208215.

[3]

David W. Banner, Allan D'Arcy, Wolfgang Janes, Reiner Gentz, Hans-Joachim Schoenfeld, Clemens Broger, Hansruedi Loetscher and Werner Lesslauer, Crystal structure of the soluble human 55 kd TNF receptor-human TNF[beta] complex: Implications for TNF receptor activation, Cell, 73 (1993), 431-445. doi: 10.1016/0092-8674(93)90132-A.

[4]

D. Berg, M. Lehne, N. Müller, D. Siegmund, S. Münkel, W. Sebald, K. Pfizenmaier and H. Wajant, Enforced covalent trimerization increases the activity of the TNF ligand family members TRAIL and CD95L, Cell Death and Differentiation, 14 (2007), 2021-2034. doi: 10.1038/sj.cdd.4402213.

[5]

Verena Boschert, Anja Krippner-Heidenreich, Marcus Branschädel, Jessica Tepperink, Andrew Aird and Peter Scheurich, Single chain TNF derivatives with individually mutated receptor binding sites reveal differential stoichiometry of ligand receptor complex formation for TNFR1 and TNFR2, Cellular Signalling, 22 (2010), 1088-1096. doi: 10.1016/j.cellsig.2010.02.011.

[6]

Marcus Branschädel, Andrew Aird, Andrea Zappe, Carsten Tietz, Anja Krippner-Heidenreich and Peter Scheurich, Dual function of cysteine rich domain (CRD) 1 of TNF receptor type 1: Conformational stabilization of CRD2 and control of receptor responsiveness, Cellular Signalling, 22 (2010), 404-414. doi: 10.1016/j.cellsig.2009.10.011.

[7]

Francis Ka-Ming Chan, Hyung J. Chun, Lixin Zheng, Richard M. Siegel, Kimmie L. Bui and Michael J. Lenardo, A domain in TNF receptors that mediates ligand-independent receptor assembly and signaling, Science, 288 (2000), 2351-2354. doi: 10.1126/science.288.5475.2351.

[8]

Lauren Clancy, Karen Mruk, Kristina Archer, Melissa Woelfel, Juthathip Mongkolsapaya, Gavin Screaton, Michael J. Lenardo and Francis Ka-Ming Chan, Preligand assembly domain-mediated ligand-independent association between TRAIL receptor 4 (TR4) and TR2 regulates TRAIL-induced apoptosis, Proceedings of the National Academy of Sciences of the USA, 102 (2005), 18099-18104. doi: 10.1073/pnas.0507329102.

[9]

M. A. Degli-Esposti, M. C. Dougall, P. J. Smolak, J. Y. Waugh, C. A. Smith and R. G. Goodwin, The novel receptor TRAIL-R4 induces NF-kappaB and protects against TRAIL-mediated apoptosis, yet retains an incomplete death domain, Immunity, 7 (1997), 813-820. doi: 10.1016/S1074-7613(00)80399-4.

[10]

John G. Emery, Peter McDonnell, Michael Brigham Burke, Keith C. Deen, Sally Lyn, Carol Silverman, Edward Dul, Edward R. Appelbaum, Chris Eichman, Rocco DiPrinzio, Robert A. Dodds, Ian E. James, Martin Rosenberg, John C. Lee and Peter R. Young, Osteoprotegerin is a receptor for the cytotoxic ligand TRAIL, Journal of Biological Chemistry, 273 (1998), 14363-14367. doi: 10.1074/jbc.273.23.14363.

[11]

Matthias Grell, Eleni Douni, Harald Wajant, Matthias Löhden, Matthias Clauss, Beate Maxeiner, Spiros Georgopoulos, Werner Lesslauer, George Kollias, Klaus Pfizenmaier and Peter Scheurich, The transmembrane form of tumor necrosis factor is the prime activating ligand of the 80 kDa tumor necrosis factor receptor, Cell, 83 (1995), 793-802. doi: 10.1016/0092-8674(95)90192-2.

[12]

Matthias Grell, Harald Wajant, Gudrun Zimmermann and Peter Scheurich, The type 1 receptor (CD120a) is the high-affinity receptor for soluble tumor necrosis factor, Proceedings of the National Academy of Sciences of the United States of America, 95 (1998), 570-575. doi: 10.1073/pnas.95.2.570.

[13]

F. Henkler, E. Behrle, K. M. Dennehy, A. Wicovsky, N. Peters, C. Warnke, K. Pfizenmaier and H. Wajant, The extracellular domains of Fasl and Fas are sufficient for the formation of supramolecular FasL-Fas clusters of high stability, Journal of Cell Biology, 168 (2005), 1087-1098. doi: 10.1083/jcb.200501048.

[14]

N. Holler, A. Tardivel, M. Kovacsovics-Bankowski, S. Hertig, O. Gaide, F. Martinon, A. Tinel, D. Deperthes, S. Calderara, T. Schulthess, J. Engel, P. Schneider and E. Tschopp, Two adjacent trimeric Fas ligands are required for Fas signaling and formation of a death-inducing signaling complex, Molecular And Cellular Biology, 23 (2003), 1428-1440. doi: 10.1128/MCB.23.4.1428-1440.2003.

[15]

Sarah G. Hymowitz, Hans W. Christinger, Germaine Fuh, Mark Ultsch, Mark O'Connell, Robert F. Kelley, Avi Ashkenazi and Abraham M. de Vos, Triggering cell death: The crystal structure of apo2l/TRAIL in a complex with death receptor 5, Molecular Cell, 4 (1999), 563-571. doi: 10.1016/S1097-2765(00)80207-5.

[16]

Sarah G. Hymowitz, Mark P. O'Connell, Mark H. Ultsch, Amy Hurst, Klara Totpal, Avi Ashkenazi, Abraham M. de Vos and Robert F. Kelley, A unique zinc-binding site revealed by a high-resolution x-ray structure of homotrimeric Apo2L/TRAIL, Biochemistry, 39 (2000), 633-640. doi: 10.1021/bi992242l.

[17]

A. Krippner-Heidenreich, F. Tübing, S. Bryde, S. Willi, G. Zimmermann and P. Scheurich, Control of receptor-induced signaling complex formation by the kinetics of ligand/receptor interaction, Journal of Biological Chemistry, 277 (2002), 44155-44163. doi: 10.1074/jbc.M207399200.

[18]

F. C. Kull, S. Jacobs and P. Cuatrecasas, Cellular receptor for 125I-labeled tumor necrosis factor: Specific binding, affinity labeling, and relationship to sensitivity, Proceedings of the National Academy of Sciences of the United States of America, 82 (1985), 5756-5760. doi: 10.1073/pnas.82.17.5756.

[19]

R. Lai and T. L. Jackson, A mathematical model of receptor-mediated apoptosis: Dying to know why FasL is a trimer, Mathematical Biosciences and Engineering, 1 (2004), 325-338.

[20]

H. W. Lee, S. H. Lee, Y. W. Ryu, M. H. Kwon and Y. S. Kim, Homomeric and heteromeric interactions of the extracellular domains of death receptors and death decoy receptors, Biochemical and Biophysical Research Communications, 330 (2005), 1205-1212. doi: 10.1016/j.bbrc.2005.03.101.

[21]

Frank Mühlenbeck, Pascal Schneider, Jean-Luc Bodmer, Ralph Schwenzer, Angelika Hauser, Gisela Schubert, Peter Scheurich, Dieter Moosmayer, Jürg Tschopp and Harald Wajant, The tumor necrosis factor-related apoptosis-inducing ligand receptors TRAIL-R1 and TRAIL-R2 have distinct cross-linking requirements for Initiation of apoptosis and are non-redundant in JNK activation, Journal of Biological Chemistry, 275 (2000), 32208-32213. doi: 10.1074/jbc.M000482200.

[22]

Y. Mukai, T. Nakamura, M. Yoshikawa, Y. Yoshioka, S. Tsunoda, S. Nakagawa, Y. Yamagata and Y. Tsutsumi, Solution of the structure of the TNF-TNFR2 complex, Science Signaling, 3 (2010), ra83. doi: 10.1126/scisignal.2000954.

[23]

James H. Naismith, Tracey Q. Devine, Barbara J. Brandhuber and Stephen R. Sprang, Crystallographic evidence for dimerization of unliganded tumor necrosis factor receptor, Journal of Biological Chemistry, 270 (1995), 13303-13307. doi: 10.1074/jbc.270.22.13303.

[24]

James H. Naismith, Tracey Q. Devine, Tadahiko Kohno and Stephen R. Sprang, Structures of the extracellular domain of the type I tumor necrosis factor receptor, Structure, 4 (1996), 1251-1262. doi: 10.1016/S0969-2126(96)00134-7.

[25]

P. Schneider, J. L. Bodmer, N. Holler, C. Mattmann, P. Scuderi, A. Terskikh, M. C. Peitsch and J. Tschopp, Characterization of Fas (Apo-1, CD95)-Fas ligand interaction, Journal of Biological Chemistry, 272 (1997), 18827-18833. doi: 10.1074/jbc.272.30.18827.

[26]

P. Schneider, N. Holler, J. L. Bodmer, M. Hahne, K. Frei, A. Fontana and J. Tschopp, Conversion of membrane-bound Fas ligand to its soluble form is associated with down regulation of its proapoptotic activity and loss of liver toxicity, Journal of Experimental Medicine, 187 (1998), 1205-1213. doi: 10.1084/jem.187.8.1205.

[27]

Richard M. Siegel, John K. Frederiksen, David A. Zacharias, Francis Ka-Ming Chan, Michele Johnson, David Lynch, Roger Y. Tsien and Michael J. Lenardo, Fas preassociation required for apoptosis signaling and dominant inhibition by pathogenic mutations, Science, 288 (2000), 2354-2357. doi: 10.1126/science.288.5475.2354.

[28]

Richard M. Siegel, Jagan R. Muppidi, Malabika Sarker, Adrian Lobito, Melinda Jen, David Martin, Stephen E. Straus and Michael J. Lenardo, SPOTS: Signaling protein oligomeric transduction structures are early mediators of death receptor-induced apoptosis at the plasma membrane, The Journal of Cell Biology, 167 (2004), 735-744. doi: 10.1083/jcb.200406101.

[29]

Steven H. Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering,'' $1^{st}$ edition, Perseus Books, Cambridge, Massachusetts, 1994.

[30]

Alemseged Truneh, Sunita Sharma, Carol Silverman, Sanjay Khandekar, Manjula P. Reddy, Keith C. Deen, Megan M. McLaughlin, Srinivasa M. Srinivasula, George P. Livi, Lisa A. Marshall, Emad S. Alnemri, William V. Williams and Michael L. Doyle, Temperature-sensitive differential affinity of TRAIL for its receptors. DR5 is the highest affinity receptor Journal of Biological Chemistry, 275 (2000), 23319-23325. doi: 10.1074/jbc.M910438199.

[31]

A. Waage, P. Brandtzaeg, A. Halstensen, P. Kierulf and T. Espevik, The complex pattern of cytokines in serum from patients with meningococcal septic shock. association between interleukin 6, interleukin 1, and fatal outcome, Journal of Experimental Medicine, 169 (1989), 333-338. doi: 10.1084/jem.169.1.333.

[32]

Harald Wajant, Dieter Moosmayer, Thomas Wüest, Till Bartke, Elke Gerlach, Ulrike Schönherr, Nathalie Peters, Peter Scheurich and Klaus Pfizenmaier, Differential activation of TRAIL-R1 and -2 by soluble and membrane TRAIL allows selective surface antigen-directed activation of TRAIL-R2 by a soluble TRAIL derivative, Oncogene, 20 (2001), 4101-4106. doi: 10.1038/sj.onc.1204558.

[33]

Wolfgang Walter, "Gewöhnliche Differentialgleichungen. Eine Einführung,'' $6^{th}$ edition, Springer-Verlag, Berlin, 1996.

[34]

Liwei Wang, Jin Kuk Yang, Venkataraman Kabaleeswaran, Amanda J. Rice, Anthony C. Cruz, Ah Young Park, Qian Yin, Ermelinda Damko, Se Bok Jang, Stefan Raunser, Carol V. Robinson, Richard M. Siegel, Thomas Walz and Hao Wu, The Fas-FADD death domain complex structure reveals the basis of DISC assembly and disease mutations, Nature Structural & Molecular Biology, 17 (2010), 1324-1329. doi: 10.1038/nsmb.1920.

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