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A Continuum Model for the Unfolding of von Willebrand Factor

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Abstract

von Willebrand Factor is a mechano-sensitive protein circulating in blood that mediates platelet adhesion to subendothelial collagen and platelet aggregation at high shear rates. Its hemostatic function and thrombogenic effect, as well as susceptibility to enzymatic cleavage, are regulated by a conformational change from a collapsed globular state to a stretched state. Therefore, it is essential to account for the conformation of the vWF multimers when modeling vWF-mediated thrombosis or vWF degradation. We introduce a continuum model of vWF unfolding that is developed within the framework of our multi-constituent model of platelet-mediated thrombosis. The model considers two interconvertible vWF species corresponding to the collapsed and stretched conformational states. vWF unfolding takes place via two regimes: tumbling in simple shear and strong unfolding in flows with dominant extensional component. These two regimes were demonstrated in a Couette flow between parallel plates and an extensional flow in a cross-slot geometry. The vWF unfolding model was then verified in several microfluidic systems designed for inducing high-shear vWF-mediated thrombosis and screening for von Willebrand Disease. The model predicted high concentration of stretched vWF in key regions where occlusive thrombosis was observed experimentally. Strong unfolding caused by the extensional flow was limited to the center axis or middle plane of the channels, whereas vWF unfolding near the channel walls relied upon the shear tumbling mechanism. The continuum model of vWF unfolding presented in this work can be employed in numerical simulations of vWF-mediated thrombosis or vWF degradation in complex geometries. However, extending the model to 3-D arbitrary flows and turbulent flows will pose considerable challenges.

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Figure 1

Adapted from Woo and Shaqfeh,72 2003, with the permission of AIP Publishing.

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References

  1. Alexander-Katz, A., and R. R. Netz. Surface-enhanced unfolding of collapsed polymers in shear flow. EPL. 80:18001, 2007.

    Article  CAS  Google Scholar 

  2. Alexander-Katz, A., and R. R. Netz. Dynamics and instabilities of collapsed polymers in shear flow. Macromolecules. 41:3363–3374, 2008.

    Article  CAS  Google Scholar 

  3. Alexander-Katz, A., M. F. Schneider, S. W. Schneider, A. Wixforth, and R. R. Netz. Shear-flow-induced unfolding of polymeric globules. Phys. Rev. Lett. 97:1–4, 2006.

    Article  CAS  Google Scholar 

  4. Babcock, H. P., R. E. Teixeira, J. S. Hur, E. S. G. Shaqfeh, and S. Chu. Visualization of molecular fluctuations near the critical point of the coil−stretch transition in polymer elongation. Macromolecules. 36:4544–4548, 2003.

    Article  CAS  Google Scholar 

  5. Bark, D. L., A. N. Para, and D. N. Ku. Correlation of thrombosis growth rate to pathological wall shear rate during platelet accumulation. Biotechnol. Bioeng. 109:2642–2650, 2012.

    Article  CAS  PubMed  Google Scholar 

  6. Bartoli, C. R., D. J. Restle, D. M. Zhang, M. A. Acker, and P. Atluri. Pathologic von Willebrand factor degradation with a left ventricular assist device occurs via two distinct mechanisms: Mechanical demolition and enzymatic cleavage. J. Thorac. Cardiovasc. Surg. 149:281–289, 2015.

    Article  CAS  PubMed  Google Scholar 

  7. Baumann Kreuziger, L., M. S. Slaughter, K. Sundareswaran, and A. E. Mast. Clinical relevance of histopathologic analysis of heartmate II thrombi. ASAIO J. 2018. https://doi.org/10.1097/MAT.0000000000000759.

    Article  PubMed  Google Scholar 

  8. Blackburn, H. M., N. N. Mansour, and B. J. Cantwell. Topology of fine-scale motions in turbulent channel flow. J. Fluid Mech. 310:269–292, 1996.

    Article  CAS  Google Scholar 

  9. Bortot, M., K. Ashworth, A. Sharifi, F. Walker, N. C. Crawford, K. B. Neeves, D. Bark, and J. Di Paola. Turbulent flow promotes cleavage of VWF (von Willebrand factor) by ADAMTS13 (a disintegrin and metalloproteinase with a thrombospondin type-1 motif, member 13). Arterioscler. Thromb. Vasc. Biol. 39:1831–1842, 2019.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  10. Casa, L. D. C., D. H. Deaton, and D. N. Ku. Role of high shear rate in thrombosis. J. Vasc. Surg. 61:1068–1080, 2015.

    Article  PubMed  Google Scholar 

  11. Chong, M. S., A. E. Perry, and B. J. Cantwell. A general classification of three-dimensional flow fields. Phys. Fluids A. 2:765–777, 1990.

    Article  Google Scholar 

  12. Chorin, A. J. Vorticity and Turbulence. New York: Springer, 1994.

    Book  Google Scholar 

  13. Coghill, P. A., S. Kanchi, Z. J. Azartash-Namin, J. W. Long, and T. A. Snyder. Benchtop von Willebrand factor testing. ASAIO J. 2018. https://doi.org/10.1097/MAT.0000000000000849.

    Article  Google Scholar 

  14. Danish, M., and C. Meneveau. Multiscale analysis of the invariants of the velocity gradient tensor in isotropic turbulence. Phys. Rev. Fluids. 3:1–22, 2018.

    Article  Google Scholar 

  15. De Gennes, P. G. Coil-stretch transition of dilute flexible polymers under ultrahigh velocity gradients. J. Chem. Phys. 5030:5030–5042, 1974.

    Article  Google Scholar 

  16. Dong, C., S. Kania, M. Morabito, X. F. Zhang, W. Im, A. Oztekin, X. Cheng, and E. B. Webb. A mechano-reactive coarse-grained model of the blood-clotting agent von Willebrand factor. J. Chem. Phys. 151:124905, 2019.

    Article  PubMed  CAS  Google Scholar 

  17. Dunlap, P. N., and L. G. Leal. Dilute polystyrene solutions in extensional flows: Birefringence and flow modification. J. Nonnewton. Fluid Mech. 23:5–48, 1987.

    Article  CAS  Google Scholar 

  18. Faghih, M. M., and M. K. Sharp. Evaluation of energy dissipation rate as a predictor of mechanical blood damage. Artif. Organs. 43:666–676, 2019.

    Article  PubMed  Google Scholar 

  19. Favaloro, E. J. Clinical utility of the PFA-100. Semin. Thromb. Hemost. 34:709–733, 2008.

    Article  CAS  PubMed  Google Scholar 

  20. Fu, H., Y. Jiang, D. Yang, F. Scheiflinger, W. P. Wong, and T. A. Springer. Flow-induced elongation of von Willebrand factor precedes tension-dependent activation. Nat. Commun. 8:1–12, 2017.

    Article  CAS  Google Scholar 

  21. Fuller, G. G., and L. G. Leal. Flow birefringence of dilute polymer solutions in two-dimensional flows. Rheol. Acta. 19:580–600, 1980.

    Article  CAS  Google Scholar 

  22. Harrison, P., M. Robinson, R. Liesner, K. Khair, H. Cohen, I. Mackie, and S. Machin. The PFA-100®: a potential rapid screening tool for the assessment of platelet dysfunction. Clin. Lab. Haematol. 24:225–232, 2002.

    Article  PubMed  Google Scholar 

  23. Haward, S. J. Microfluidic extensional rheometry using stagnation point flow. Biomicrofluidics. 10:043401, 2016.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  24. Hund, S. J., J. F. Antaki, and M. Massoudi. On the representation of turbulent stresses for computing blood damage. Int. J. Eng. Sci. 48:1325–1331, 2010.

    Article  PubMed  PubMed Central  Google Scholar 

  25. Hur, J. S., E. S. G. Shaqfeh, H. P. Babcock, and S. Chu. Dynamics and configurational fluctuations of single DNA molecules in linear mixed flows. Phys. Rev. 66:3–6, 2002.

    Google Scholar 

  26. Hur, J. S., E. S. G. Shaqfeh, H. P. Babcock, D. E. Smith, and S. Chu. Dynamics of dilute and semidilute DNA solutions in the start-up of shear flow. J. Rheol. 45:421–450, 2001.

    Article  CAS  Google Scholar 

  27. Jendrejack, R. M., J. J. De Pablo, and M. D. Graham. Stochastic simulations of DNA in flow: Dynamics and the effects of hydrodynamic interactions. J. Chem. Phys. 116:7752–7759, 2002.

    Article  CAS  Google Scholar 

  28. Kania, S., A. Oztekin, X. Cheng, X. F. Zhang, and E. Webb. Predicting pathological von Willebrand factor unraveling in elongational flow. Biophys. J. 2021. https://doi.org/10.1016/j.bpj.2021.03.008.

    Article  PubMed  PubMed Central  Google Scholar 

  29. Kim, D., C. Bresette, Z. Liu, and D. N. Ku. Occlusive thrombosis in arteries. APL Bioeng. 3:041502, 2019.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  30. Konnigk, L., B. Torner, M. Bruschewski, S. Grundmann, and F. H. Wurm. Equivalent scalar stress formulation taking into account non-resolved turbulent scales. Cardiovasc. Eng. Technol. 2021. https://doi.org/10.1007/s13239-021-00526-x.

    Article  PubMed  PubMed Central  Google Scholar 

  31. Kragh, T., M. Napoleone, M. A. Fallah, H. Gritsch, M. F. Schneider, and A. J. Reininger. High shear dependent von willebrand factor self-assembly fostered by platelet interaction and controlled by ADAMTS13. Thromb. Res. 133:1079–1087, 2014.

    Article  CAS  PubMed  Google Scholar 

  32. Kundu, S. K., E. J. Heilmann, R. Sio, C. Garcia, R. M. Davidson, and R. A. Ostgaard. Description of an in vitro platelet function analyzer - PFA-100®. Semin. Thromb. Hemost. 21:106–112, 1995.

    Article  PubMed  Google Scholar 

  33. Larson, R. G. The rheology of dilute solutions of flexible polymers: Progress and problems. J. Rheol. 49:1–70, 2005.

    Article  CAS  Google Scholar 

  34. Lippok, S., T. Obser, J. P. Müller, V. K. Stierle, M. Benoit, U. Budde, R. Schneppenheim, and J. O. Rädler. Exponential size distribution of von Willebrand factor. Biophys. J. 105:1208–1216, 2013.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  35. Lippok, S., M. Radtke, T. Obser, L. Kleemeier, R. Schneppenheim, U. Budde, R. R. Netz, and J. O. Rädler. Shear-induced unfolding and enzymatic cleavage of full-length VWF multimers. Biophys. J. 110:545–554, 2016.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  36. Liu, Z. L., D. N. Ku, and C. K. Aidun. Mechanobiology of shear-induced platelet aggregation leading to occlusive arterial thrombosis: A multiscale in silico analysis. J. Biomech. 120:110349, 2021.

    Article  PubMed  Google Scholar 

  37. Liu, Z., Y. Zhu, J. R. Clausen, J. B. Lechman, R. R. Rao, and C. K. Aidun. Multiscale method based on coupled lattice-Boltzmann and Langevin-dynamics for direct simulation of nanoscale particle/polymer suspensions in complex flows. Int. J. Numer. Methods Fluids. 91:228–246, 2019.

    Article  CAS  Google Scholar 

  38. Lumley, J. L. Drag reduction by additives. Annu. Rev. Fluid Mech. 1:367–384, 1969.

    Article  CAS  Google Scholar 

  39. Nascimbene, A., S. Neelamegham, O. H. Frazier, J. L. Moake, and J.-F. Dong. Acquired von Willebrand syndrome associated with left ventricular assist device. Blood. 127:3133–3141, 2016.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  40. Nesbitt, W. S., E. Westein, F. J. Tovar-Lopez, E. Tolouei, A. Mitchell, J. Fu, J. Carberry, A. Fouras, and S. P. Jackson. A shear gradient–dependent platelet aggregation mechanism drives thrombus formation. Nat. Med. 15:665–673, 2009.

    Article  CAS  PubMed  Google Scholar 

  41. Ng, R.C.-Y., and L. G. Leal. Concentration effects on birefringence and flow modification of semidilute polymer solutions in extensional flows. J. Rheol. 37:443–468, 1993.

    Article  CAS  Google Scholar 

  42. Ouyang, W., W. Wei, X. Cheng, X. F. Zhang, E. B. Webb, and A. Oztekin. Flow-induced conformational change of von Willebrand Factor multimer: Results from a molecular mechanics informed model. J. Nonnewton. Fluid Mech. 217:58–67, 2015.

    Article  CAS  Google Scholar 

  43. Ozturk, M., E. A. O’Rear, and D. V. Papavassiliou. Hemolysis related to turbulent eddy size distributions using comparisons of experiments to computations. Artif. Organs. 39:E227–E239, 2015.

    Article  CAS  PubMed  Google Scholar 

  44. Para, A. N., and D. N. Ku. A low-volume, single pass in-vitro system of high shear thrombosis in a stenosis. Thromb. Res. 131:418–424, 2013.

    Article  CAS  PubMed  Google Scholar 

  45. Perkins, T. T. Single polymer dynamics in an elongational flow. Science. 276:2016–2021, 1997.

    Article  CAS  PubMed  Google Scholar 

  46. Peterson, D. M., N. A. Stathopoulos, T. D. Giorgio, J. D. Hellums, and J. L. Moake. Shear-induced platelet aggregation requires von Willebrand factor and platelet membrane glycoproteins Ib and IIb-IIIa. Blood. 69:625–628, 1987.

    Article  CAS  PubMed  Google Scholar 

  47. Proudfoot, A. G., S. J. Davidson, and M. Strueber. von Willebrand factor disruption and continuous-flow circulatory devices. J. Heart Lung Transpl. 36:1155–1163, 2017.

    Article  Google Scholar 

  48. Rauch, A., S. Susen, and B. Zieger. Acquired von Willebrand syndrome in patients with ventricular assist device. Front. Med. 6:1–9, 2019.

    Article  Google Scholar 

  49. Ruggeri, Z. M. Platelet adhesion under flow. Microcirculation. 16:58–83, 2009.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  50. Ruggeri, Z. M., J. N. Orje, R. Habermann, A. B. Federici, and A. J. Reininger. Activation-independent platelet adhesion and aggregation under elevated shear stress. Blood. 108:1903–1910, 2006.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  51. Savage, B., F. Almus-Jacobs, and Z. M. Ruggeri. Specific synergy of multiple substrate–receptor interactions in platelet thrombus formation under flow. Cell. 94:657–666, 1998.

    Article  CAS  PubMed  Google Scholar 

  52. Savage, B., E. Saldívar, and Z. M. Ruggeri. Initiation of platelet adhesion by arrest onto fibrinogen or translocation on von Willebrand factor. Cell. 84:289–297, 1996.

    Article  CAS  PubMed  Google Scholar 

  53. Schneider, S. W., S. Nuschele, A. Wixforth, C. Gorzelanny, A. Alexander-Katz, R. R. Netz, and M. F. Schneider. Shear-induced unfolding triggers adhesion of von Willebrand factor fibers. Proc. Natl. Acad. Sci. 104:7899–7903, 2007.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  54. Schroeder, C. M., H. P. Babcock, E. S. G. Shaqfeh, and S. Chu. Observation of polymer conformation hysteresis in extensional flow. Science. 301:1515–1519, 2003.

    Article  CAS  PubMed  Google Scholar 

  55. Schroeder, C. M., R. E. Teixeira, E. S. G. Shaqfeh, and S. Chu. Characteristic periodic motion of polymers in shear flow. Phys. Rev. Lett. 95:1–4, 2005.

    Article  CAS  Google Scholar 

  56. Shankaran, H., and S. Neelamegham. Hydrodynamic forces applied on intercellular bonds, soluble molecules, and cell-surface receptors. Biophys. J. 86:576–588, 2004.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  57. Shaqfeh, E. S. G. The dynamics of single-molecule DNA in flow. J. Nonnewton. Fluid Mech. 130:1–28, 2005.

    Article  CAS  Google Scholar 

  58. Sharifi, A., and D. Bark. Mechanical forces impacting cleavage of Von Willebrand factor in laminar and turbulent blood flow. Fluids. 6:67, 2021.

    Article  CAS  Google Scholar 

  59. Siemens Healthineers AG. PFA-100 System. 2021. https://www.siemens-healthineers.com/en-us/hemostasis/systems/pfa-100

  60. Sing, C. E., and A. Alexander-Katz. Elongational flow induces the unfolding of von willebrand factor at physiological flow rates. Biophys. J. 98:L35–L37, 2010.

    Article  PubMed  PubMed Central  Google Scholar 

  61. Sing, C. E., and A. Alexander-Katz. Globule−stretch transitions of collapsed polymers in elongational flow fields. Macromolecules. 43:3532–3541, 2010.

    Article  CAS  Google Scholar 

  62. Sing, C. E., and A. Alexander-Katz. Dynamics of collapsed polymers under the simultaneous influence of elongational and shear flows. J. Chem. Phys. 135:014902, 2011.

    Article  PubMed  CAS  Google Scholar 

  63. Smith, D. E. Single-polymer dynamics in steady shear flow. Science. 283:1724–1727, 1999.

    Article  CAS  PubMed  Google Scholar 

  64. Somani, S., E. S. G. Shaqfeh, and J. R. Prakash. Effect of solvent quality on the coil−stretch transition. Macromolecules. 43:10679–10691, 2010.

    Article  CAS  Google Scholar 

  65. Tadmor, E. B., R. E. Miller, and R. S. Elliott. Continuum Mechanics and Thermodynamics. Cambridge: Cambridge University Press, 2011.

    Book  Google Scholar 

  66. Tennekes, H., and J. L. Lumley. A First Course in Turbulence. Cambridge: The MIT Press, 1972.

    Book  Google Scholar 

  67. Terrapon, V. E., Y. Dubief, P. Moin, E. S. G. Shaqfeh, and S. K. Lele. Simulated polymer stretch in a turbulent flow using Brownian dynamics. J. Fluid Mech. 504:61–71, 2004.

    Article  CAS  Google Scholar 

  68. Tovar-Lopez, F. J., G. Rosengarten, E. Westein, K. Khoshmanesh, S. P. Jackson, A. Mitchell, and W. S. Nesbitt. A microfluidics device to monitor platelet aggregation dynamics in response to strain rate micro-gradients in flowing blood. Lab Chip. 10:291–302, 2010.

    Article  CAS  PubMed  Google Scholar 

  69. Vincentelli, A., S. Susen, T. Le Tourneau, I. Six, O. Fabre, F. Juthier, A. Bauters, C. Decoene, J. Goudemand, A. Prat, and B. Jude. Acquired von Willebrand syndrome in aortic stenosis. N. Engl. J. Med. 349:343–349, 2003.

    Article  PubMed  Google Scholar 

  70. von Springer, T. A. Willebrand factor, Jedi knight of the bloodstream. Blood. 124:1412–1425, 2014.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  71. Westein, E., A. D. van der Meer, M. J. E. Kuijpers, J.-P. Frimat, A. van den Berg, and J. W. M. Heemskerk. Atherosclerotic geometries exacerbate pathological thrombus formation poststenosis in a von Willebrand factor-dependent manner. Proc. Natl. Acad. Sci. 110:1357–1362, 2013.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  72. Woo, N. J., and E. S. G. Shaqfeh. The configurational phase transitions of flexible polymers in planar mixed flows near simple shear. J. Chem. Phys. 119:2908–2914, 2003.

    Article  CAS  Google Scholar 

  73. Wu, P., Q. Gao, and P. L. Hsu. On the representation of effective stress for computing hemolysis. Biomech. Model. Mechanobiol. 18:665–679, 2019.

    Article  CAS  PubMed  Google Scholar 

  74. Wu, W.-T., M. A. Jamiolkowski, W. R. Wagner, N. Aubry, M. Massoudi, and J. F. Antaki. Multi-constituent simulation of thrombus deposition. Sci. Rep. 7:42720, 2017.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  75. Wu, W. T., F. Yang, J. Wu, N. Aubry, M. Massoudi, and J. F. Antaki. High fidelity computational simulation of thrombus formation in Thoratec HeartMate II continuous flow ventricular assist device. Sci. Rep. 6:1–11, 2016.

    Article  CAS  Google Scholar 

  76. Zhang, X., K. Halvorsen, C.-Z. Zhang, W. P. Wong, and T. A. Springer. Mechanoenzymatic cleavage of the ultralarge vascular protein von Willebrand factor. Science. 324:1330–1334, 2009.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  77. Zhussupbekov, M., W.-T. Wu, M. A. Jamiolkowski, M. Massoudi, and J. F. Antaki. Influence of shear rate and surface chemistry on thrombus formation in micro-crevice. J. Biomech. 2021. https://doi.org/10.1016/j.jbiomech.2021.110397.

    Article  PubMed  PubMed Central  Google Scholar 

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Acknowledgments

Author Wei-Tao Wu thanks the support of the Grant NSFC 11802135. This work was supported by the National Institute of Health Grant R01HL089456. Authors are grateful to Dr. Mahdi Esmaily Moghadam (Sibley School of Mechanical and Aerospace Engineering, Cornell University) for valuable discussions on turbulent flows.

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Authors declare no conflicts of interest.

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Authors and Affiliations

  1. Department of Biomedical Engineering, Cornell University, Ithaca, NY, USA

    Mansur Zhussupbekov, Rodrigo Méndez Rojano & James F. Antaki

  2. Department of Aerospace Science and Technology, Nanjing University of Science and Technology, Nanjing, China

    Wei-Tao Wu

  3. U. S. Department of Energy, National Energy Technology Laboratory (NETL), Pittsburgh, PA, USA

    Mehrdad Massoudi

  4. Meinig School of Biomedical Engineering, Cornell University, Weill Hall 109, Ithaca, NY, 14853, USA

    James F. Antaki

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  1. Mansur Zhussupbekov
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Zhussupbekov, M., Méndez Rojano, R., Wu, WT. et al. A Continuum Model for the Unfolding of von Willebrand Factor. Ann Biomed Eng 49, 2646–2658 (2021). https://doi.org/10.1007/s10439-021-02845-5

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