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A multiscale model for red blood cell mechanics

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Abstract

The objective of this article is the derivation of a continuum model for mechanics of red blood cells via multiscale analysis. On the microscopic level, we consider realistic discrete models in terms of energy functionals defined on networks/lattices. Using concepts of Γ-convergence, convergence results as well as explicit homogenisation formulae are derived. Based on a characterisation via energy functionals, appropriate macroscopic stress–strain relationships (constitutive equations) can be determined. Further, mechanical moduli of the derived macroscopic continuum model are directly related to microscopic moduli. As a test case we consider optical tweezers experiments, one of the most common experiments to study mechanical properties of cells. Our simulations of the derived continuum model are based on finite element methods and account explicitly for membrane mechanics and its coupling with bulk mechanics. Since the discretisation of the continuum model can be chosen freely, rather than it is given by the topology of the microscopic cytoskeletal network, the approach allows a significant reduction of computational efforts. Our approach is highly flexible and can be generalised to many other cell models, also including biochemical control.

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References

  • Alicandro R, Cicalese M (2004) A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J Appl Math 36: 1–37

    Article  MATH  MathSciNet  Google Scholar 

  • Alt HW (2002) Lineare Funktionalanalysis, 4th edn. Springer, Berlin

    MATH  Google Scholar 

  • Becker R, Braack M, Dunne T, Meidner D, Richter T, Schmich M, Stricker T, Vexler B (2009) Gascoigne 3D—a finite element toolbox. http://www.gascoigne.uni-hd.de

  • Berezhnyy M, Berlyand L (2006) Continuum limit for three-dimensional mass-spring networks and discrete Korn’s inequality. J Mech Phys Solids 54: 635–669

    Article  MATH  MathSciNet  Google Scholar 

  • Boal D (2002) Mechanics of the cell. Cambridge University Press, Cambridge

    Google Scholar 

  • Boey SK, Boal DH, Discher DE (1998) Simulations of the erythrocyte cytoskeleton at large deformation. I. Microscopic models. Biophys J 75: 1573–1583

    Article  Google Scholar 

  • Bozic B, Svetina S, Zeksand B, Waugh RE (1992) Role of lamellar membrane structure in tether formation from bilayer vesicles. Biophys J 61: 963–973

    Article  Google Scholar 

  • Braides A (2001) From discrete to continuous variational problems: an introduction. Lecture notes, School on homogenization techniques and asymptotic methods for problems with multiple scales

  • Braides A (2002) Γ-convergence for beginners. Oxford University Press, Oxford

    Book  MATH  Google Scholar 

  • Braides A, Defranceschi (1998) Homogenization of multiple integrals. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Canham P (1970) The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J Theor Biol 26: 61–81

    Article  Google Scholar 

  • Clarenz U, Diewald U, Dziuk G, Rumpf M, Rusu R (2004) A finite element method for surface restoration with smooth boundary conditions. Comput Aided Geom Des 21: 427–445

    Article  MATH  MathSciNet  Google Scholar 

  • Dacorogna B (2004) Introduction to the calculus of variations. Imperial College Press, London

    MATH  Google Scholar 

  • Dal Maso G (1993) An Introduction to Γ-convergence. Birkhäuser, Boston

    Google Scholar 

  • Dao M, Lim C, Suresh S (2003) Mechanics of the human red blood cell deformed by optical tweezers. J Mech Phys Solids 51: 2259–2280

    Article  Google Scholar 

  • Dao M, Li J, Suresh S (2006) Molecularly based analysis of deformation of spectrin network and human erythrocyte. Math Sci Eng C 26: 1232–1244

    Article  Google Scholar 

  • den Otter WK, Briels WJ (2003) The bending rigidity of an amphiphilic bilayer from equilibrium and nonequilibrium molecular dynamics. J Chem Phys 118: 4712–4720

    Article  Google Scholar 

  • Discher DE, Boal DH, Boey SK (1998) Simulations of the erythrocyte cytoskeleton at large deformation. II. Micropipette aspiration. Biophys J 75: 1584–1597

    Article  Google Scholar 

  • Evans EA, Skalak R (1980) Mechanics and thermal dynamics of biomembranes. CRC Press, Boca Raton

    Google Scholar 

  • Friesecke G, Theil F (2002) Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice. J Nonlinear Sci 12: 445–478

    Article  MATH  MathSciNet  Google Scholar 

  • Gov NS, Safran SA (2005) Red blood cell membrane fluctuations and shape controlled by ATP-induced cytoskeletal defects. Biophys J 88: 1859–1874

    Article  Google Scholar 

  • Hansen JC, Skalak R, Chien S, Hoger A (1997) Influence of network topology on the elasticity of the red blood cell membrane skeleton. Biophys J 72: 2369–2381

    Article  Google Scholar 

  • Hartmann D (2007) Multiscale modelling, analysis, and simulation in mechanobiology. Doctor of Sciences, Department of Mathematics and Computer Sciences, University of Heidelberg

  • Heinrich V, Waugh RE (1996) A piconewton force transducer and its application to measurement of the bending stiffness of phospholipid membranes. Ann Biomed Eng 24: 595–605

    Article  Google Scholar 

  • Helfrich W (1973) Elastic properties of lipid bilayers: Theory and possible experiments. Z Naturforsch C 28: 693–703

    Google Scholar 

  • Hénon S, Lenormand G, Richert A, Gallet F (1999) A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers. Biophys J 76: 1145–1151

    Article  Google Scholar 

  • Hwangand WC, Waugh RE (1997) Energy of dissociation of lipid bilayer from the membrane skeleton of red blood cells. Biophys J 72: 2669–2678

    Article  Google Scholar 

  • Ingber DE (2003) Tensegrity I. Cell structure and hierarchical systems biology. J. Cell Sci. 116: 1157–1173

    Google Scholar 

  • Kuzman D, Svetina S, Waugh RE, Zeks B (2004) Elastic properties of the red blood cell membrane that determine echinocyte deformability. Eur Biophys J 33: 1–15

    Article  Google Scholar 

  • Lee JCM, Wong DT, Discher DE (1999) Direct measures of large, anisotropic strains in deformation of the erythrocyte cytoskeleton. Biophys J 77: 853–864

    Article  Google Scholar 

  • Lenormand G, Hénon S, Richert A, Simeon J, Gallet F (2001) Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton. Biophys J 81: 43–56

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Lim GHW, Wortis M, Mukhopadhyay R (2002) Stomatocyte discocyte echinocyte sequence of the human red blood cell: Evidence for the bilayer couple hypothesis from membrane mechanics. Proc Natl Acad Sci USA 99: 16,766–16,769

    Article  Google Scholar 

  • Liu S, Derick LH, Palek J (1987) Visualization of the hexagonal lattice in the erythrocyte membrane skeleton. J Cell Biol 104: 527–536

    Article  Google Scholar 

  • Marko J, Siggia ED (1995) Stretching DNA. Macromolecules 28: 8759–8770

    Article  Google Scholar 

  • Miao L, Seifert W, Wortis M, Döbereiner HG (1994) Nbudding transitions of fluid-bilayer vesicles: The effect of area-difference elasticity. Phys Rev E 49: 5389–5407

    Article  Google Scholar 

  • Mills JP, Qie L, Dao M, Lim CT, Suresh S (2004) Nonlinear elastic and viscoelastic deformation of the human red blood cell with optical tweezers. Mech Chem Biosys 1: 169–180

    Google Scholar 

  • Mohandas N, Evans E (1994) Mechanical properties of the red cell membrane in relation to molecular structure and genetic defects. Annu Rev Biophys Biom 23: 787–818

    Article  Google Scholar 

  • Mukhopadhyay R, Lim GHW, Wortis M (2002) Echinocyte shapes: bending, stretching, and shear determine spicule shape and spacing. Biophys J 82: 1756–1772

    Article  Google Scholar 

  • Noguchi H, Gompper G (2005) Shape transitions of fluid vesicles and red blood cells in capillary flows. Proc Natl Acad Sci USA 102: 14,159–14,164

    Article  Google Scholar 

  • Pozrikidis C (2003a) Numerical simulation of the flow-induced deformation of red blood cells. Ann Biomed Eng 31: 1194–1205

    Article  Google Scholar 

  • Pozrikidis C (2003b) Modeling and simulation of capsules and biological cells. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Rannacher R (2006) Numerische Mathematik 3: Numerische Methoden für Probleme der Kontinuumsmechanik. Lecture notes

  • Rawicz W, Olbrich KC, McIntosh T, Needham D, Evans E (2000) Effect of chain length and unsaturation on elasticity of lipid bilayers. Biophys J 79: 328–339

    Article  Google Scholar 

  • Rusu RE (2005) An algorithm for the elastic flow of surfaces. Interface. Free Bound. 7: 229–239

    Article  MATH  MathSciNet  Google Scholar 

  • Scheffer L, Bitler A, Ben-Jacob E, Korenstein R (2001) Atomic force pulling: Probing the local elasticity of the cell membrane. Eur Biophys J 30: 83–90

    Article  Google Scholar 

  • Schmidt B (2008) On the passage from atomic to continuum theory for thin films. Arch Ration Mech An 190: 1–55

    Article  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Skalak R, Tözeren A, Zarda PR, Chien S (1973) Strain energy function of red blood cell membranes. Biophys J 13: 245–264

    Article  Google Scholar 

  • Steltenkamp S, Müller MM, Deserno M, Hennesthal C, Steinem C, Janshoff A (2006) Mechanical properties of pore-spanning lipid bilayers probed by atomic force microscopy. Biophys J 91: 217–226

    Article  Google Scholar 

  • Thompson D (1917) On growth and form (Abr. ed. by J. T. Bonner, 1961). Cambridge University Press, Cambridge

    Google Scholar 

  • Willmore TJ (1993) Riemannian geometry. Clarendon Press, Oxford

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Center for Modelling and Simulation in the Biosciences (BIOMS), University of Heidelberg, BQ 00 21 BIOQUANT, Im Neuenheimer Feld 267, 69120, Heidelberg, Germany

    Dirk Hartmann

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  1. Dirk Hartmann
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Correspondence to Dirk Hartmann.

Additional information

This work was supported by the German Science Foundation through the International Graduate College 710: “Complex Processes: Modeling, Simulation and Optimization”.

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Hartmann, D. A multiscale model for red blood cell mechanics. Biomech Model Mechanobiol 9, 1–17 (2010). https://doi.org/10.1007/s10237-009-0154-5

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  • DOI: https://doi.org/10.1007/s10237-009-0154-5

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