Biomechanics and Modeling in Mechanobiology

, Volume 17, Issue 1, pp 223–234 | Cite as

Effects of membrane deformability and bond formation/dissociation rates on adhesion dynamics of a spherical capsule in shear flow

  • Ziying Zhang
  • Jun Du
  • Zhengying Wei
  • Zhen Wang
  • Minghui Li
Original Paper


Cellular adhesion plays a critical role in biological systems and biomedical applications. Cell deformation and biophysical properties of adhesion molecules are of significance for the adhesion behavior. In the present work, dynamic adhesion of a deformable capsule to a planar substrate, in a linear shear flow, is numerically simulated to investigate the combined influence of membrane deformability (quantified by the capillary number) and bond formation/dissociation rates on the adhesion behavior. The computational model is based on the immersed boundary-lattice Boltzmann method for the capsule–fluid interaction and a probabilistic adhesion model for the capsule–substrate interaction. Three distinct adhesion states, detachment, rolling adhesion and firm adhesion, are identified and presented in a state diagram as a function of capillary number and bond dissociation rate. The impact of bond formation rate on the state diagram is further investigated. Results show that the critical bond dissociation rate for the transition of rolling or firm adhesion to detachment is strongly related to the capsule deformability. At the rolling-adhesion state, smaller off rates are needed for larger capillary number to increase the rolling velocity and detach the capsule. In contrast, the critical off rate for firm-to-detach transition slightly increases with the capillary number. With smaller on rate, the effect of capsule deformability on the critical off rates is more pronounced and capsules with moderate deformability are prone to detach by the shear flow. Further increasing of on rate leads to large expansion of both rolling-adhesion and firm-adhesion regions. Even capsules with relatively large deformability can maintain stable rolling adhesion at certain off rate.


Lattice Boltzmann method Immersed boundary method Receptor–ligand kinetics Deformable capsule Adhesion state 



This study was funded by the Natural Science Foundation of China under Grant No. 31370944.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Abkarian M, Viallat A (2005) Dynamics of vesicles in a wall-bounded shear flow. Biophys J 89:1055–1066. doi: 10.1529/biophysj.104.056036 CrossRefGoogle Scholar
  2. Abkarian M, Lartigue C, Viallat A (2002) Tank treading and unbinding of deformable vesicles in shear flow: determination of the lift force. Phys Rev Lett 88:068103. doi: 10.1103/PhysRevLett.88.068103 CrossRefGoogle Scholar
  3. Aidun CK, Clausen JR (2010) Lattice-Boltzmann method for complex flows. Annu Rev Fluid Mech 42:439–472. doi: 10.1146/annurev-fluid-121108-145519 MathSciNetCrossRefMATHGoogle Scholar
  4. Alon R, Ley K (2008) Cells on the run: shear-regulated integrin activation in leukocyte rolling and arrest on endothelial cells. Curr Opin Cell Biol 20:525–532. doi: 10.1016/ CrossRefGoogle Scholar
  5. Alon R, Hammer DA, Springer TA (1995) Lifetime of the P-selectin-carbohydrate bond and its response to tensile force in hydrodynamic flow. Nature 374:539–542. doi: 10.1038/374539a0 CrossRefGoogle Scholar
  6. Alon R, Chen S, Puri KD, Finger EB, Springer TA (1997) The kinetics of L-selectin tethers and the mechanics of selectin-mediated rolling. J Cell Biol 138:1169–1180. doi: 10.1083/jcb.138.5.1169 CrossRefGoogle Scholar
  7. Balsara HD, Banton RJ, Eggleton CD (2016) Investigating the effects of membrane deformability on artificial capsule adhesion to the functionalized surface. Biomech Model Mechanobiol 15:1055–1068. doi: 10.1007/s10237-015-0742-5 CrossRefGoogle Scholar
  8. Barthes-Biesel D, Diaz A, Dhenin E (2002) Effect of constitutive laws for two-dimensional membranes on flow-induced capsule deformation. J Fluid Mech. doi: 10.1017/s0022112002008352 MATHGoogle Scholar
  9. Bhatia SK, King MR, Hammer DA (2003) The state diagram for cell adhesion mediated by two receptors. Biophys J 84:2671–2690. doi: 10.1016/S0006-3495(03)75073-5 CrossRefGoogle Scholar
  10. Cantat I, Misbah C (1999) Lift force and dynamical unbinding of adhering vesicles under shear flow. Phys Rev Lett 83:880–883CrossRefGoogle Scholar
  11. Chang KC, Tees DF, Hammer DA (2000) The state diagram for cell adhesion under flow: leukocyte rolling and firm adhesion. Proc Nat Acad Sci USA 97:11262–11267. doi: 10.1073/pnas.200240897 CrossRefGoogle Scholar
  12. Chang WC, Lee LP, Liepmann D (2005) Biomimetic technique for adhesion-based collection and separation of cells in a microfluidic channel. Lab Chip 5:64–73. doi: 10.1039/b400455h CrossRefGoogle Scholar
  13. Chen S, Springer TA (1999) An automatic braking system that stabilizes leukocyte rolling by an increase in selectin bond number with shear. J Cell Biol 144:185–200. doi: 10.1083/jcb.144.1.185 CrossRefGoogle Scholar
  14. Choi S, Karp JM, Karnik R (2012) Cell sorting by deterministic cell rolling. Lab Chip 12:1427–1430. doi: 10.1039/C2LC21225K CrossRefGoogle Scholar
  15. Devarajan PV, Jain S (2015) Targeted drug delivery: concepts and design. Springer, New YorkGoogle Scholar
  16. d’Humières D (2002) Multiple-relaxation-time lattice Boltzmann models in three dimensions. Philos Trans Roy Soc Lond A Math Phys Eng Sci 360:437–451. doi: 10.1098/rsta.2001.0955
  17. Didar TF, Tabrizian M (2010) Adhesion based detection, sorting and enrichment of cells in microfluidic Lab-on-Chip devices. Lab Chip 10:3043–3053. doi: 10.1039/c0lc00130a CrossRefGoogle Scholar
  18. Doddi SK, Bagchi P (2008) Lateral migration of a capsule in a plane Poiseuille flow in a channel. Int J Multiph Flow 34:966–986. doi: 10.1016/j.ijmultiphaseflow.2008.03.002
  19. Dong C, Cao J, Struble EJ, Lipowsky HH (1999) Mechanics of leukocyte deformation and adhesion to endothelium in shear flow. Ann Biomed Eng 27:298–312. doi: 10.1114/1.143 CrossRefGoogle Scholar
  20. Dore M, Korthuis R, Granger D, Entman M, Smith C (1993) P-selectin mediates spontaneous leukocyte rolling in vivo. Blood 82:1308–1316Google Scholar
  21. Fedosov DA, Caswell B, Karniadakis GE (2011) Wall shear stress-based model for adhesive dynamics of red blood cells in malaria. Biophys J 100:2084–2093. doi: 10.1016/j.bpj.2011.03.027 CrossRefGoogle Scholar
  22. Feng Z-G, Michaelides EE (2004) The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems. J Comput Phys 195:602–628. doi: 10.1016/ CrossRefMATHGoogle Scholar
  23. Fritz J, Katopodis AG, Kolbinger F, Anselmetti D (1998) Force-mediated kinetics of single P-selectin/ligand complexes observed by atomic force microscopy. Proc Nat Acad Sci 95:12283–12288. doi: 10.1073/pnas.95.21.12283 CrossRefGoogle Scholar
  24. Gholami B, Comerford A, Ellero M (2015) SPH simulations of WBC adhesion to the endothelium: the role of haemodynamics and endothelial binding kinetics. Biomech Model Mechanobiol 14:1317–1333. doi: 10.1007/s10237-015-0676-y CrossRefGoogle Scholar
  25. Goldman AJ, Cox RG, Brenner H (1967) Slow viscous motion of a sphere parallel to a plane wall—II Couette flow. Chem Eng Sci 22:653–660. doi: 10.1016/0009-2509(67)80048-4 CrossRefGoogle Scholar
  26. Guo Z, Shu C (2013) Lattice Boltzmann method and its applications in engineering. World Scientific Publishing, SingaporeCrossRefMATHGoogle Scholar
  27. Hammer DA (2014) Adhesive dynamics. J Biomech Eng 136:021006. doi: 10.1115/1.4026402 CrossRefGoogle Scholar
  28. Hammer DA, Apte SM (1992) Simulation of cell rolling and adhesion on surfaces in shear flow: general results and analysis of selectin-mediated neutrophil adhesion. Biophys J 63:35–57. doi: 10.1016/S0006-3495(92)81577-1 CrossRefGoogle Scholar
  29. Jadhav S, Eggleton CD, Konstantopoulos K (2005) A 3-D computational model predicts that cell deformation affects selectin-mediated leukocyte rolling. Biophys J 88:96–104. doi: 10.1529/biophysj.104.051029 CrossRefGoogle Scholar
  30. Karp JM, Leng Teo GS (2009) Mesenchymal stem cell homing: the devil is in the details. Cell Stem Cell 4:206–216. doi: 10.1016/j.stem.2009.02.001 CrossRefGoogle Scholar
  31. Khalili AA, Ahmad MR (2015) A review of cell adhesion studies for biomedical and biological applications. Int J Mol Sci 16:18149–18184. doi: 10.3390/ijms160818149 CrossRefGoogle Scholar
  32. Khismatullin DB, Truskey GA (2012) Leukocyte rolling on P-selectin: a three-dimensional numerical study of the effect of cytoplasmic viscosity. Biophys J 102:1757–1766. doi: 10.1016/j.bpj.2012.03.018 CrossRefGoogle Scholar
  33. Korn CB, Schwarz US (2008) Dynamic states of cells adhering in shear flow: from slipping to rolling. Phys Rev E 77:041904. doi: 10.1103/PhysRevE.77.041904 CrossRefGoogle Scholar
  34. Krüger T (2011) Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method. Comput Math Appl 61:3485–3505. doi: 10.1016/j.camwa.2010.03.057 MathSciNetCrossRefMATHGoogle Scholar
  35. Krüger T (2014) Deformability-based red blood cell separation in deterministic lateral displacement devices—a simulation study. Biomicrofluidics 8:054114. doi: 10.1063/1.4897913 CrossRefGoogle Scholar
  36. Ladd AJC, Verberg R (2001) Lattice-Boltzmann simulations of particle-fluid suspensions. J Stat Phys 104:1191. doi: 10.1023/A:1010414013942 MathSciNetCrossRefMATHGoogle Scholar
  37. Lawrence MB, Springer TA (1991) Leukocytes roll on a selectin at physiologic flow rates: distinction from and prerequisite for adhesion through integrins. Cell 65:859–873. doi: 10.1016/0092-8674(91)90393-D CrossRefGoogle Scholar
  38. Li J, Dao M, Lim CT, Suresh S (2005) Spectrin-level modeling of the cytoskeleton and optical tweezers stretching of the erythrocyte. Biophys J 88:3707–3719. doi: 10.1529/biophysj.104.047332
  39. Lu J, Han H, Shi B, Guo Z (2012) Immersed boundary lattice Boltzmann model based on multiple relaxation times. Phys Rev E. doi: 10.1103/PhysRevE.85.016711 Google Scholar
  40. Luo ZY, Bai BF (2016) State diagram for adhesion dynamics of deformable capsules under shear flow. Soft Matter 12:6918–6925. doi: 10.1039/c6sm01697a CrossRefGoogle Scholar
  41. Luo LS, Liao W, Chen X, Peng Y, Zhang W (2011) Numerics of the lattice Boltzmann method: effects of collision models on the lattice Boltzmann simulations. Phys Rev E 83:056710. doi: 10.1103/PhysRevE.83.056710 CrossRefGoogle Scholar
  42. Luo ZY, Wang SQ, He L, Xu F, Bai BF (2013) Inertia-dependent dynamics of three-dimensional vesicles and red blood cells in shear flow. Soft Matter 9:9651. doi: 10.1039/c3sm51823j CrossRefGoogle Scholar
  43. McEver RP, Zhu C (2010) Rolling cell adhesion. Annu Rev Cell Dev Biol 26:363–396. doi: 10.1146/annurev.cellbio.042308.113238 CrossRefGoogle Scholar
  44. Ni H, Freedman J (2003) Platelets in hemostasis and thrombosis: role of integrins and their ligands. Transfus Apher Sci 28:257–264. doi: 10.1016/S1473-0502(03)00044-2 CrossRefGoogle Scholar
  45. Pan C, Luo L-S, Miller CT (2006) An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Comput Fluids 35:898–909. doi: 10.1016/j.compfluid.2005.03.008 CrossRefMATHGoogle Scholar
  46. Pappu V, Bagchi P (2008) 3D computational modeling and simulation of leukocyte rolling adhesion and deformation. Comput Biol Med 38:738–753. doi: 10.1016/j.compbiomed.2008.04.002 CrossRefGoogle Scholar
  47. Peskin CS (2003) The immersed boundary method. Acta Numer. doi: 10.1017/s0962492902000077 MATHGoogle Scholar
  48. Ramesh KV, Thaokar R, Prakash JR, Prabhakar R (2015) Significance of thermal fluctuations and hydrodynamic interactions in receptor-ligand-mediated adhesive dynamics of a spherical particle in wall-bound shear flow. Phys Rev E 91:022302. doi: 10.1103/PhysRevE.91.022302 CrossRefGoogle Scholar
  49. Seifert U (1997) Configurations of fluid membranes and vesicles. Adv Phys 46:13–137. doi: 10.1080/00018739700101488 CrossRefGoogle Scholar
  50. Skalak R, Tozeren A, Zarda RP, Chien S (1973) Strain energy function of red blood cell membranes. Biophys J 13:245–264. doi: 10.1016/S0006-3495(73)85983-1 CrossRefGoogle Scholar
  51. Spencer TJ, Hidalgo-Bastida LA, Cartmell SH, Halliday I, Care CM (2013) In silico multi-scale model of transport and dynamic seeding in a bone tissue engineering perfusion bioreactor. Biotechnol Bioeng 110:1221–1230. doi: 10.1002/bit.24777 CrossRefGoogle Scholar
  52. Stott SL et al (2010) Isolation of circulating tumor cells using a microvortex-generating herringbone-chip. Proc Nat Acad Sci 107:18392–18397. doi: 10.1073/pnas.1012539107
  53. Sun D-K, Bo Z (2015) Numerical simulation of hydrodynamic focusing of particles in straight channel flows with the immersed boundary-lattice Boltzmann method. Int J Heat Mass Transf 80:139–149. doi: 10.1016/j.ijheatmasstransfer.2014.08.070 CrossRefGoogle Scholar
  54. Succi S (2001) The lattice Boltzmann equation: for fluid dynamics and beyond. Oxford University Press, OxfordMATHGoogle Scholar
  55. Sui Y, Chew YT, Roy P, Low HT (2008) A hybrid method to study flow-induced deformation of three-dimensional capsules. J Comput Phys 227:6351–6371. doi: 10.1016/ MathSciNetCrossRefMATHGoogle Scholar
  56. Wang S et al (2011) Highly efficient capture of circulating tumor cells by using nanostructured silicon substrates with integrated chaotic micromixers. Angew Chem Int Ed 50:3084–3088. doi: 10.1002/anie.201005853 CrossRefGoogle Scholar
  57. Wu Z, Xu Z, Kim O, Alber M (2014) Three-dimensional multi-scale model of deformable platelets adhesion to vessel wall in blood flow. Philos Trans Ser A. doi: 10.1098/rsta.2013.0380 MATHGoogle Scholar
  58. Zhang X, Jones P, Haswell SJ (2008) Attachment and detachment of living cells on modified microchannel surfaces in a microfluidic-based lab-on-a-chip system. Chem Eng J 135:S82–S88. doi: 10.1016/j.cej.2007.07.054 CrossRefGoogle Scholar
  59. Zheng X, Cheung LS, Schroeder JA, Jiang L, Zohar Y (2011) Cell receptor and surface ligand density effects on dynamic states of adhering circulating tumor cells. Lab Chip 11:3431–3439. doi: 10.1039/c1lc20455f CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of Mechanical EngineeringXi’an Jiaotong UniversityXi’anChina
  2. 2.Department of Orthopaedic Oncology, Xi-Jing HospitalThe Fourth Military Medical UniversityXi’anChina

Personalised recommendations