Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T07:06:45.217Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiscale modelling of erythrocytes in Stokes flow

Published online by Cambridge University Press:  29 September 2011

Zhangli Peng ,
Robert J. Asaro  and
Qiang Zhu
Zhangli Peng
Affiliation:
Department of Structural Engineering, University of California San Diego, La Jolla, CA 92093, USA
Robert J. Asaro
Affiliation:
Department of Structural Engineering, University of California San Diego, La Jolla, CA 92093, USA
Qiang Zhu*
Affiliation:
Department of Structural Engineering, University of California San Diego, La Jolla, CA 92093, USA
*
†Email address for correspondence: qizhu@ucsd.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

To quantitatively understand the correlation between the molecular structure of an erythrocyte (red blood cell, RBC) and its mechanical response, and to predict mechanically induced structural remodelling in physiological conditions, we developed a computational model by coupling a multiscale approach of RBC membranes with a boundary element method (BEM) for surrounding Stokes flows. The membrane is depicted at three levels: in the whole cell level, a finite element method (FEM) is employed to model the lipid bilayer and the cytoskeleton as two distinct layers of continuum shells. The mechanical properties of the cytoskeleton are obtained from a molecular-detailed model of the junctional complex. The spectrin, a major protein of the cytoskeleton, is simulated using a molecular-based constitutive model. The BEM model is coupled with the FEM model through a staggered coupling algorithm. Using this technique, we first simulated RBC dynamics in capillary flow and found that the protein density variation and bilayer–skeleton interaction forces are much lower than those in micropipette aspiration, and the maximum interaction force occurs at the trailing edge. Then we investigated mechanical responses of RBCs in shear flow during tumbling, tank-treading and swinging motions. The dependencies of tank-treading frequency on the blood plasma viscosity and the membrane viscosity we found match well with benchmark data. The simulation results show that during tank-treading the protein density variation is insignificant for healthy erythrocytes, but significant for cells with a smaller bilayer–skeleton friction coefficient, which may be the case in hereditary spherocytosis.

JFM classification

Biological Fluid Dynamics: Capsule/cell dynamics Micro-/Nano-fluid dynamics: Microfluidics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 686 , 10 November 2011 , pp. 299 - 337
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

Current address: Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

References

1. Abkarian, M., Faivre, M. & Viallat, A. 2007 Swinging of red blood cells under shear flow. Phys. Rev. Lett. 98, 188302.CrossRefGoogle ScholarPubMed
2. Bagchi, P. 2007 Mesoscale simulation of blood flow in small vessels. Biophys. J. 92, 1858–1877.CrossRefGoogle ScholarPubMed
3. Barthès-Biesel, D. 1980 Motion of spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100, 831–853.CrossRefGoogle Scholar
4. Barthès-Biesel, D., Diaz, A. & Dhenin, E. 2002 Effect of constitutive laws for two-dimensional membranes on ow-induced capsule deformation. J. Fluid Mech. 460, 211–222.Google Scholar
5. Barthès-Biesel, D. & Rallison, J. M. 1981 The time-dependent deformation of a capsule freely suspended in a linear shear flow. J. Fluid Mech. 113, 251–267.CrossRefGoogle Scholar
6. Belytschko, T., Liu, W. & Moran, B. 2000 Nonlinear Finite Elements for Continua and Structures. Wiley.Google Scholar
7. Bennett, V. & Stenbuck, P. J. 1979 The membrane attachment protein for spectrin is associated with band 3 in human RBC membranes. Nature 280, 468–473.CrossRefGoogle Scholar
8. Berk, A. A. & Hochmuth, R. M. 1992 Lateral mobility of integral proteins in red blood cell tethers. Biophys. J. 61, 9–18.CrossRefGoogle ScholarPubMed
9. Bruce, L. J., Beckmann, R., Ribeiro, M. L., Peters, L. L., Chasis, J. A., Delaunay, J., Mohandas, N., Anstee, D. J. & Tanner, M. J. 2003 A band 3-based macrocomplex of integral and peripheral proteins in the RBC membrane. Blood 101, 4180–4188.CrossRefGoogle ScholarPubMed
10. Butler, J., Mohandas, N. & Waugh, R. E. 2008 Integral protein linkage and the bilayer-skeletal separation energy in red blood cells. Biophys. J. 95, 1826–1836.CrossRefGoogle ScholarPubMed
11. Chang, S. H. & Low, P. S. 2001 Regulation of the glycophorin c-protein 4.1 membrane-to-skeleton bridge and evaluation of its contribution to erythrocyte membrane stability. J. Biol. Chem. 276, 22223–22230.CrossRefGoogle ScholarPubMed
12. Chien, S. 1987 Red cell deformability and its relevance to blood flow. Annu. Rev. Physiol. 49, 177–192.CrossRefGoogle ScholarPubMed
13. Cohen, C. M., Tyler, J. M. & Branton, D. 1980 Spectrin–actin associations studied by electron microscopy of shadowed preparations. Cell 21, 875–883.Google Scholar
14. Craiem, D. & Magin, R. L. 2010 Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (RBC) membrane mechanics. Phys. Biol. 7, 013001–013003.Google Scholar
15. Dao, M., Li, J. & Suresh, S. 2006 Molecularly based analysis of deformation of spectrin network and human erythocyte. Mater. Sci. Engng C26, 1232–1244.Google Scholar
16. Dao, M., Lim, C. T. & Suresh, S. 2003 Mechanics of the human red blood cell deformed by optical tweezers. J. Mech. Phys. Solids 51, 2259–2280.CrossRefGoogle Scholar
17. Deutsch, S., Tarbell, J. M., Manning, K. B., Rosenberg, G. & Fontaine, A. A. 2006 Experimental fluid mechanics of pulsatile artificial blood pumps. Annu. Rev. Fluid Mech. 38, 65–86.CrossRefGoogle Scholar
18. Discher, D. E., Boal, D. H. & Boey, S. K. 1998 Simulations of the erythrocyte cytoskeleton at large deformation. Biophys. J. 75, 1584–1597.CrossRefGoogle ScholarPubMed
19. Discher, D. E., Mohandas, N. & Evans, E. A. 1994 Molecular maps of red cell deformation: hidden elasticity and in situ connectivity. Science 266, 1032–1035.CrossRefGoogle ScholarPubMed
20. Dodson, W. R. & Dimitrakopoulos, P. 2010 Tank-treading of erythrocytes in strong shear flows via a nonstiff cytoskeleton-based continuum computational modelling. Biophys. J. 99, 2906–2916.CrossRefGoogle Scholar
21. Eggleton, C. D. & Popel, A. S. 1998 Large deformation of red blood cell ghosts in a simple shear flow. Phys. Fluids 10, 1834–1845.CrossRefGoogle Scholar
22. Evans, E. & Hochmuth, R. 1976 Membrane viscoelasticity. Biophys. J. 16, 1–11.CrossRefGoogle ScholarPubMed
23. Evans, E. A. & Skalak, R. 1980 Mechanics and Thermodynamics of Biomembranes. CRC.Google Scholar
24. Fedosov, D. A., Caswell, B. & Karniadakis, G. E. 2010 A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys. J. 98, 2215–2225.CrossRefGoogle ScholarPubMed
25. Fedosov, D. A., Caswell, B., Suresh, S. & Karniadakis, G. E. 2011 Quantifying the biophysical characteristics of plasmodium–falciparum-parasitized red blood cells in microcirculation. Proc. Natl Acad. Sci. USA 108, 35–39.CrossRefGoogle ScholarPubMed
26. Fischer, T. M. 1992 Is the surface area of the red cell membrane skeleton locally conserved. Biophys. J. 61, 298–305.CrossRefGoogle ScholarPubMed
27. Fischer, T. M. 2004 Shape memory of human red blood cells. Biophys. J. 86, 3304–3313.Google Scholar
28. Fischer, T. 2007 Tank-tread frequency of the red cell membrane: dependence on the viscosity of the suspending medium. Biophys. J. 93, 2553–2561.CrossRefGoogle ScholarPubMed
29. Fischer, T. M., Stöhr-Liesen, M. & Schmid-Schönbein, H. 1978 The red cell as a fluid droplet: tank tread-like motion of the human erythrocyte membrane in shear flow. Science 202, 894–896.CrossRefGoogle ScholarPubMed
30. Foessel, E., Walter, J., Salsac, A.-V. & Barthes-Biesel, D. 2011 Influence of internal viscosity on the large deformation and buckling of a spherical capsule in a simple shear flow. J. Fluid Mech. 672, 477–486.Google Scholar
31. Hallquist, J. 1998 LS-DYNA Theoretical Manual. Livermore Software Technology Corporation, Livermore, California.Google Scholar
32. Halpern, D. & Secomb, T. W. 1992 The squeezing of red blood cells through parallel-sided channels with near-minimal widths. J. Fluid Mech. 244, 307–322.Google Scholar
33. Henon, S. 1999 A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers. Biophys. J. 76, 1145–1151.CrossRefGoogle ScholarPubMed
34. Hochareon, P. 2003 Development of particle imaging velocimetry (piv) for wall shear stress estimation within a 50 cc penn state artificial heart ventricular chamber. PhD thesis, The Pennsylvania State University, University Park, PA.Google Scholar
35. Hochmuth, R. M. 1973 Measurement of the elastic modulus for red blood cell membrane using a fluid mechanical technique. Biophys. J. 13, 747–762.Google Scholar
36. Holzapfel, G. A. 2000 Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley.Google Scholar
37. Hughes, T. J. R. & Liu, W. K. 1981a Nonlinear finite-element analysis of shells 1. Two-dimensional shells. Comput. Meth. Appl. Mech. Engng 27, 167–181.CrossRefGoogle Scholar
38. Hughes, T. J. R. & Liu, W. K. 1981b Non-linear finite-element analysis of shells 1. 3-dimensional shells. Comput. Meth. Appl. Mech. Engng 26, 331–362.CrossRefGoogle Scholar
39. Kabaso, D., Shlomovitz, R., Auth, T., Lew, V. & Gov, N. 2010 Curling and local shape changes of red blood cell membranes driven by cytoskeletal reorganization. Biophys. J. 99, 808–816.Google Scholar
40. Kapitza, H., Rüppel, D., Galla, H. & Sackmann, E. 1984 Lateral diffusion of lipids and glycophorin in solid phosphatidylcholine bilayers. the role of structural defects. Biophys. J. 45, 577–587.CrossRefGoogle ScholarPubMed
41. Keller, S. & Skalak, R. 1982 Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech. 120, 27–47.CrossRefGoogle Scholar
42. Kessler, S., Finken, R. & Seifert, U. 2004 Swinging and tumbling of elastic capsules in shear flow. J. Fluid Mech. 605, 207–226.Google Scholar
43. Kleinbongard, P., Schulz, R., Rassaf, T., Lauer, T., Dejam, A., Jax, T., Kumara, I., Gharini, P., Kabanova, S., Ozüyaman, B., Schnürch, H.-G., Gödecke, A., Weber, A.-A., Robenek, M., Robenek, H., Bloch, W., Rösen, P. & Kelm, M. 2009 Red blood cells express a functional endothelial nitric oxide synthase. Blood 107, 2943–2951.CrossRefGoogle Scholar
44. Kodippili, G., Spector, J., Sullivan, C., Kuypers, F., Labotka, R., Gallagher, P., Ritchie, K. & Low, P. 2009 Imaging of the diffusion of single band 3 molecules on normal and mutant erythrocytes. Blood 113, 6237–6245.CrossRefGoogle ScholarPubMed
45. Lac, E. & Barthès-Biesel, D. 2005 Deformation of a capsule in simple shear flow: effect of membrane prestress. Phys. Fluids 17, 072105.CrossRefGoogle Scholar
46. Lac, E., Barthès-Biesel, D., Pelekasis, N. A. & Tsamopoulos, J. 2004 Spherical capsules in three-dimensional unbounded stokes flows: effect of the membrane constitutive law and onset of buckling. J. Fluid Mech. 516, 303–334.CrossRefGoogle Scholar
47. Law, R., Carl, P., Harper, S., Dalhaimer, P., Speicher, D. W. & Discher, D. E. 2003 Cooperativity in forced unfolding of tandem spectrin repeats. Biophys. J. 84, 533–544.CrossRefGoogle ScholarPubMed
48. Le, D. 2010 Subdivision elements for large deformation of liquid capsules enclosed by thin shells. Comput. Meth. Appl. Mech. Engng 199, 2622–2632.Google Scholar
49. Lee, G., Abdi, K., Jiang, Y., Michaely, P., Bennett, V. & Marszalek, P. E. 2006 Nanospring behaviour of ankyrin repeats. Nature 440, 246–249.CrossRefGoogle ScholarPubMed
50. Lefebvre, Y. & Barthès-Biesel, D. 2007 Motion of a capsule in a cylindrical tube: effect of membrane pre-stress. J. Fluid Mech. 589, 157–181.CrossRefGoogle Scholar
51. Li, J., Lykotrafitis, G., Dao, M. & Suresh, S. 2007 Cytoskeletal dynamics of human erythrocyte. Proc. Natl Acad. Sci. USA 104, 4937–4942.CrossRefGoogle ScholarPubMed
52. Lim, G., Wortis, M. & Mukhopadhyay, R. 2002 Stomatocytecdiscocytecechinocyte sequence of the human red blood cell: evidence for the bilayerc couple hypothesis from membrane mechanics. Proc. Natl Acad. Sci. USA 99, 16766–16769.Google Scholar
53. Lubarda, V. 2011 Rate theory of elasticity and viscoelasticity for an erythrocyte membrane. J. Mech. Mater. Struct. (in press).CrossRefGoogle Scholar
54. Malone, J. G. & Johnson, N. L. 1994 A parallel finite-element contact/impact algorithm for nonlinear explicit transient analysis 1. The search algorithm and contact mechanics. Intl J. Numer. Meth. Engng 37, 559–590.Google Scholar
55. McWhirter, J., Noguchi, H. & Gompper, G. 2009 Flow-induced clustering and alignment of vesicles and red blood cells in microcapillaries. Proc. Natl Acad. Sci. USA 106, 6039–6043.Google Scholar
56. Mebius, R. E. & Kraal, G. 2004 Structure and function of the spleen. Nat. Rev. Immunol. 5, 606–616.CrossRefGoogle Scholar
57. Mohandas, N. & Evans, E. 1994 Mechanical properties of the red cell membrane in relation to molecular structure and genetic defects. Annu. Rev. Biophys. Biomol. Struct. 23, 787–818.CrossRefGoogle ScholarPubMed
58. Noguchi, H. & Gompper, G. 2005 Shape transitions of fluid vesicles and red blood cells in capillary flows. Proc. Natl Acad. Sci. USA 102, 14159–14164.Google Scholar
59. Omori, T., Ishikawa, T., Barthès-Biesel, D., Salsac, A.-V., Walter, J., Imai, Y. & Yamaguchi, T. 2011 Comparison between spring network models and continuum constitutive laws: application to the large deformation of a capsule in shear flow. Phys. Rev. E 83, 041918.Google Scholar
60. Otter, W. K. & Shkulipa, S. A. 2007 Intermonolayer friction and surface shear viscosity of lipid bilayer membranes. Biophys. J. 93, 423–433.CrossRefGoogle Scholar
61. Park, Y., Best, C. A., Auth, T., Gov, N. S., Safran, S. A., Popescu, G., Suresh, S. & Feld, M. S. 2009 Metabolic remodelling of the human red blood cell membrane. Proc. Natl Acad. Sci. USA 107, 1289–1294.Google Scholar
62. Peng, Z., Asaro, R. J. & Zhu, Q. 2010 Multiscale simulation of erythrocyte membranes. Phys. Rev. E 81, 031904.CrossRefGoogle ScholarPubMed
63. Perrotta, S., Borriello, A., Scaloni, A., De Franceschi, L., Brunati, A. M., Turrini, F., Nigro, V., del Giudice, E. M., Nobili, B., Conte, M. L., Rossi, F., Iolascon, A., Donella-Deana, A., Zappia, V., Poggi, V., Anong, W., Low, P., Mohandas, N. & Della Ragione, F. 2005 The N-terminal 11 amino acids of human erythrocyte band 3 are critical for aldolase binding and protein phosphorylation: implications for band 3 function. Blood 106 (13), 4359–4366.Google Scholar
64. Pozrikidis, C. 1990 The axisymmetric deformation of a red blood cell in uniaxial straining flow. J. Fluid Mech. 216, 231–254.CrossRefGoogle Scholar
65. Pozrikidis, C. 1992 Boundary integral and singularity methods for linearized viscous flow.Google Scholar
66. Pozrikidis, C. 2001 Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. J. Fluid Mech. 440, 269–291.CrossRefGoogle Scholar
67. Pozrikidis, C. 2003a Modelling and Simulation of Capsules and Biological Cells. Chapman & Hall/CRC.CrossRefGoogle Scholar
68. Pozrikidis, C. 2003b Numerical simulation of the flow-induced deformation of red blood cells. Ann. Biomed. Engng 31, 1194–1205.Google ScholarPubMed
69. Pozrikidis, C. 2005 Axisymmetric motion of a file of red blood cells through capillaries. Phys. Fluids 17, 031503.Google Scholar
70. Pozrikidis, C. 2010 Computational Hydrodynamics of Capsules and Biological Cells. Taylor & Francis.CrossRefGoogle Scholar
71. Puig-De-Morales-Marinkovic, M., Turner, K. T., Butler, J. P., Fredberg, J. J. & Suresh, S. 2007 Viscoelasticity of the human red blood cell. Am. J. Physiol. Cell Physiol. 293, C597–C605.CrossRefGoogle ScholarPubMed
72. Ramanujan, S. & Pozrikidis, C. 1998 Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of capsule viscosity. J. Fluid Mech. 361, 117–143.CrossRefGoogle Scholar
73. Reid, M. E., Takakuwa, Y., Conboy, J., Tchernia, G. & Mohandas, N. 1990 Glycophorin c content of human erythrocyte membrane is regulated by protein 4.1. Blood 75, 2229–2234.Google Scholar
74. Rief, M., Gautel, M., Oesterhelt, F., Fernandez, J. M. & Gaub, H. E. 1997 Reversible unfolding of individual titin immunoglobulin domains by AFM. Science 276, 1109–1112.CrossRefGoogle ScholarPubMed
75. Rief, M., Pascual, J., Saraste, M. & Gaub, H. 1999 Single molecule force spectroscopy of spectrin repeats: low unfolding forces in helix bundles. J. Mol. Biol. 286 (2), 553–561.CrossRefGoogle ScholarPubMed
76. Secomb, T. W., Skalak, R., Ozkaya, N. & Gross, J. F. 1986 Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405–423.Google Scholar
77. Seifert, U., Berndl, K. & Lipowsky, R. 1991 Shape transformations of vesicles: phase diagram for spontaneous-curvature and bilayer-coupling models. Phys. Rev. A 44, 1182–1202.CrossRefGoogle ScholarPubMed
78. Sheetz, M. P., Schindler, M. & Koppel, D. E. 1980 Lateral mobility of integral membrane proteins is increased in spherocytic erythrocytes. Nature 285, 510–551.Google Scholar
79. Skalak, R., Tozeren, A., Zarda, R. P. & Chien, S. 1973 Strain energy function of red blood cell membranes. Biophys. J. 13, 245–264.CrossRefGoogle ScholarPubMed
80. Skotheim, J. M. & Secomb, T. W. 2007 Red blood cells and other nonspherical capsules in shear flow: oscillatory dynamics and the tank-treading-to-tumbling transition. Phys. Rev. Lett. 98, 078301.Google Scholar
81. Sleep, J., Wilson, D., Simmons, R. & Gratzer, W. 1999 Elasticity of the red cell membrane and its relation to hemolytic disorders: an optical tweezers study. Biophys. J. 77, 3085–3095.Google Scholar
82. Stroeva, P., Hoskinsb, P. & Eassona, W. 2007 Distribution of wall shear rate throughout the arterial tree: A case study. Atherosclerosis 191, 276–280.Google Scholar
83. Sung, L. A. & Vera, C. 2003 Protofilament and hexagon: a three-dimensional mechanical model for the junctional complex in the rbc membrane skeleton. Ann. Biomed. Engng 31, 1314–1326.Google Scholar
84. Tran-Son-Tay, R. 1983 A study of the tank-treading motion of red blood cells in shear flow. PhD thesis, Washington University, St. Louis, MO.Google Scholar
85. Tran-Son-Tay, R., Sutera, S. & Rao, P. 1984 Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion. Biophys. J. 46, 65–72.CrossRefGoogle ScholarPubMed
86. Tsubota, K. & Wada, S. 2010 Effect of the natural state of an elastic cellular membrane on tank-treading and tumbling motions of a single red blood cell. Phys. Rev. E 81, 011910.Google Scholar
87. Walensky, L., Mohandas, N. & Lux, S. E. 2003 Disorders of the red blood cell membrane. In Blood: Principles and Practice of Hematology, 2nd edn (ed. R. I. Handin, S. E. Lux & T. P. Stossel), pp. 1709–1858. Lippincott Williams & Wilkins.Google Scholar
88. Walter, J., Salsac, A.-V. & Barthès-Biesel, D. 2011 Ellipsoidal capsules in simple shear flow: prolate versus oblate initial shapes. J. Fluid Mech. 676, 318–347.Google Scholar
89. Walter, J., Salsac, A.-V., Barthès-Biesel, D. & Tallec, P. Le 2010 Coupling of finite element and boundary integral methods for a capsule in a stokes flow. Intl J. Numer. Methods Engng 83, 829–850.CrossRefGoogle Scholar
90. Wan, J., Ristenpart, W. D. & Stone, H. A. 2008 Dynamics of shear-induced ATP release from red blood cells. Proc. Natl Acad. Sci. USA 105, 16432–16437.Google Scholar
91. Waugh, R. & Evans, E. 1979 Thermoelasticity of red blood cell membrane. Biophys. J. 26, 115–131.Google Scholar
92. Waugh, R., Mohandas, N., Jackson, C., Mueller, T., Suzuki, T. & Dale, G. 1992 Rheologic properties of senescent erythrocytes: loss of surface area and volume with red blood cell age. Blood 79, 1351–1358.CrossRefGoogle ScholarPubMed
93. Zhang, J., Johnson, P. C. & Popel, A. S. 2008 Red blood cell aggregation and dissociation in shear flows simulated by lattice Boltzmann method. J. Biomech. 41, 47–55.CrossRefGoogle ScholarPubMed
94. Zhao, H., Isfahani, A. H. G., Olson, L. N. & Freund, J. B. 2010 A spectral boundary integral method for flowing blood cells. J. Comput. Phys. 229, 3726–3744.Google Scholar
95. Zhou, H. & Pozrikidis, C. 1990 Deformation of capsules with incompressible interfaces in simple shear flow. J. Fluid Mech. 283, 175–200.Google Scholar
96. Zhu, Q. & Asaro, R. 2008 Spectrin folding vs. unfolding reactions and RBC membrane stiffness. Biophys. J. 94, 2529–2545.Google Scholar
97. Zhu, Q., Vera, C., Asaro, R., Sche, P. & Sung, L. A. 2007 A hybrid model for erythrocyte membrane: a single unit of protein network coupled with lipid bilayer. Biophys. J. 93, 386–400.CrossRefGoogle ScholarPubMed