20.GEM GEM4 Summer School: Cell and Molecular Biomechanics in Medicine: Cancer Summer 2007

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1 MIT OpenCourseWare 20.GEM GEM4 Summer School: Cell and Molecular Biomechanics in Medicine: Cancer Summer 2007 For information about citing these materials or our Terms of Use, visit:

2 Continuum Modeling of the Cell Ming Dao Dept of Materials Science & Engineering, MIT, Cambridge, MA, USA GEM4 Summer School June 25 July 6, 2007 NUS, Singapore

3 Motivation Why Single-Cell Mechanics? Living cells and molecules sense mechanical forces, converting them into biological responses. Biological and biochemical signals are known to influence the ability of the cells to sense, generate and bear mechanical forces Red blood cell: Blood flow in microcirculation is influenced by the deformability of red blood cell

4 C.T. Lim et al., J Biomech, 39, , 2006 Mechanical Models for Living Cells Courtesy Elsevier, Inc., Used with permission.

5 Continuum Modeling of Human Red Blood Cell Studying the deformation characteristics of healthy & diseased red blood cell

6 red blood cell (erythrocyte) Simple model cell without nucleu Undergoes severe, reversible, large elastic deformation Approx. 0.5 million circulations over 120 days ~ 3,000,000 red blood cells produced every second

7 Experimental Techniques Courtesy of Subra Suresh. Used with permission. G. Bao and S. Suresh, Nature Materials (2003)

8 Binding of silica microbeads to cell Bead adhered to surface of glass slide Video Recorder Before stretch Human red blood cell During stretch Laser beam trapping bead sample CCD Camera 1.5 W diode pumped Nd:YAG laser source Inverted microscope Dao, Lim and Suresh, J. Mech. Phys. Solids (2003) Courtesy Elsevier, Inc., Used with permission. Lim et al., Acta Materialia (2004)

9 Courtesy Elsevier, Inc., Used with permission. Dao, Lim and Suresh, J Mech Phys Solids, 2003 Mills et al., Mechanics and Chemistry of Biosystems, 2004

10 Previous Efforts & Current Focus Micropipette Aspiration E. A. Evans, Y.C. Fung, R. Skalak, Optical Tweezers S. Henon, J. Sleep, D.E. Discher, Our Focus: Optical Tweezers Experiment Larger force range: > 200 pn Full 3-D Whole Cell Modeling Spectrin-Level Modeling & Continuum Verification Finite Deformation Formulations

11 Hereditary blood cell disorders: Spectrin molecules Sickle-cell disease Figure by MIT OpenCourseWare. Molecular structure of human RBC cytoskeleton B. Alberts et al. (1994) Spherocytosis, elliptocytosis, Asian ovalocytosis Malaria Figure by MIT OpenCourseWare. After B. Alberts et al, 1994.

12 (a) Structure of the cell membrane Lipid bilayer Cholesterol Spectrin network Transmembrane proteins (b) Spectrin Network + Lipid Membrane Worm-like chain (WLC) model with surface & bending energies (c) Effective Continuum Membrane In-plane shear modulus: m Bending stiffness: k Courtesy Elsevier, Inc., Used with permission. Dao, Lim & Suresh, J. Mech. Phys. Solids (2003) Li, Dao, Lim & Suresh, Biophys J (2005) h

13 membrane shear stress Hyperelasticity Model m0 m h = 1 m h > 0 3 rd order hyperelasticity m h = 0 neo-hookean Model m 0 shear strain (2 s )

14 Hyperelasticity Model Neo-Hookean: m0 m h Incompressible Material: = 1 Conserving Area: 1 2 = 1 so that 3 = 1 Classical RBC Membrane Model

15 Response of a perfect spectrin network y (b x, b y ) (b x -a x, b y -a y ) L i A 2 A 1 (a x, a y ) A a A 0 x n A 0 n b (-a x, -a y ) A 1 A 2 (a x -b x, a y -b y ) Courtesy Elsevier, Inc., Used with permission. (a) (-b x, -b y ) (b) 1 f ( a) f ( b) f ( c) WLC WLC WLC q1 ab aa ab ba bb ca cb qcq A ab 2A a b c

16 s11 s xx (mn/m) y p = 8.5 u0=8.0un/m; nm; L uh/u0=1/20 u0=7.3un/m; 0 = 91 nm; L uh/u0=1/30 max = 238 nm p = 8.5 u0=7.3un/m; nm; L 0 = uh/u0=1/90 87 nm; L max = 238 nm p = 7.5 p=8.5nm; LL0=91nm; 0 = 75 nm; Lmax=238nm L = nm m 0 = p=8.5nm; 8.0 mn/m; L0=87nm; m h /m 0 = Lmax=238nm 1/20 m p=7.5nm; L0=75nm; Lmax=238nm 0 = 7.3 mn/m; m h /m 0 = 1/30 m 0 = 7.3 mn/m; m h /m 0 = 1/90 A 0 x A Li, Dao, Lim & Suresh, Biophys J (2005); Dao, Li & Suresh, Mat Sci Eng C (2006)

17 Image removed due to copyright restrictions. Please see Figure 4 in Dao, M., J. Li, and S. Suresh. "Molecularly Based Analysis of Deformation of Spectrin Network and Human Erythrocyte." Mat Sci Eng C (2006):

18 Constitutive Model: Prior literature Constant Area Assumption: T m 2m s s 1 2 m membrane shear modulus i the principal stretches T s, s membrane shear stress (force/length), shear strain 1 T T T 2 s 1 2 E. Evans, Biophys. J., s

19 Micropipette Aspiration The total area in the deformed configuration would be divided in three parts: outside the pipette + in the pipette below the cap (L-R p ) + spherical cap, thus Deformed Area R 2 R p L Rp 2 Rp 2 Rp R Rp 2LRp Unreformed (Original) Area = 2 R 0 Area Conservation gives Z. R R R 2LR p p L R p Q (1) R p R R 0

20 Micropipette Aspiration R R R 1 R 0 R0 R R R0 Constitutive Law: T R R 2LR R0 R p p R s m R m m R R0 2 R R Rp 2LRp T pr T z p /2 R R R prp 2L 2L 1 ln m R p R p p df p z p. df p (1) F z = 2R p T z

21 Modeling Optical Tweezers Experiments Binding of silica microbeads to cell Bead adhered to surface of glass slide Laser beam trapping bead Initial Model Setup D T d c D 0 V RBC D A FEM Mesh VRBC 2 Before stretch During stretch Courtesy Elsevier, Inc., Used with permission. Spectrin network

22 Computational Model One half of the human red blood cell stretched by optical tweezers: 3-D computer simulation for comparison with experiment Dao, Lim and Suresh, J Mech Phys Solids, 2003

23 maximum principal strain 0% 20% 40% 60% 80%100%120% experiment simulations 0 pn 67 pn 130 pn 193 pn (a) (b) (c) Courtesy of Tech Science Press. Used with permission. Source: Mills, J. P., L. Qie, M. Dao, C. T. Lim and S. Suresh. "Nonlinear Elastic and Viscoelastic Deformation of the Human Red Blood Cell with Optical Tweezers." Mechanics and Chemistry of Biosystems 1 (2004): Dao, Lim & Suresh, J Mech Phys Solids (2003)

24 Courtesy of Tech Science Press. Used with permission. Source: Mills, J. P., L. Qie, M. Dao, C. T. Lim and S. Suresh. "Nonlinear Elastic and Viscoelastic Deformation of the Human Red Blood Cell with Optical Tweezers." Mechanics and Chemistry of Biosystems 1 (2004): Dao, Lim & Suresh, J Mech Phys Solids (2003)

25 + - 0 experiments R 4 # Po = 7.3 pn1m * Po = 5.3 pnlm * * Po = 11.3 pnlm - # # # -p = 5.5 pn1m (const area) * Axial diameter I I I I I Transverse diameter Stretching Force (pn) Courtesy of Tech Science Press. Used with permission. Source: Mills, J. P., L. Qie, M. Dao, C. T. Lim and S. Suresh. "Nonlinear Elastic and Viscoelastic Deformation of the Human Red Blood Cell with Optical Tweezers." Mechanics and Chemistry of Biosystems 1 (2004): Dao, Lim & Suresh, J Mech Phys Solids (2003)

26 exp(- t / t c ) Dao, Lim and Suresh, J Mech Phys Solids, t c = s (ave from 8 experiments) t c = 0.25 s t c = 0.13 s Courtesy Elsevier, Inc., Used with permission. Time (s)

27 Maximum Principal Strain 0% 20% 40% 60% 80% 100% 120% Experiment Simulations with cytosol Simulations without cytosol 0 pn 35 pn 68 pn 83 pn (a) (b) (c) (d) (e) Courtesy Elsevier, Inc., Used with permission. Dao, Lim and Suresh, J Mech Phys Solids, 2003

28 Diameter (mm) Experiments Cell Diameter D = 8.5 mm Cell Diameter D = 7.82 mm Cell Diameter D = 7 mm Axial diameter Transverse diameter Stretching Force (pn) m l = 3.3 mn/m m 0 = 7.3 mn/m Dao, Lim and Suresh, J Mech Phys Solids, 2003

29 Diameter (mm) Experiments d c = 2.0 mm d c = 1.5 mm d c = 1.0 mm Axial diameter m l = 3.3 mn/m m 0 = 7.3 mn/m Transverse diameter Stretching Force (pn) Dao, Lim and Suresh, J Mech Phys Solids, 2003

30 Diameter (mm) Experiments Biconcave Model Spherical Model Axial diameter m l = 3.3 mn/m m 0 = 7.3 mn/m Transverse diameter Stretching Force (pn) Dao, Lim and Suresh, J Mech Phys Solids, 2003

31 Scaling Functions of Optical Tweezers Expts Binding of silica microbeads to cell Bead adhered to surface of glass slide Laser beam trapping bead Initial Model Setup D T d c D 0 V RBC D A FEM Mesh Before stretch VRBC 2 During stretch Courtesy Elsevier, Inc., Used with permission. Spectrin network

32 Scaling Functions of Optical Tweezers Expts Considering the small influence of bending modulus, OT force F F D, m, m, d, D A 0 h c 0 F DA D0 Dimensional Analysis gives,, m D D d mh m c 0 F m 0 m0 p F p kbt 1 1 L fwlc ( L) x x [0 1) 2 p 4(1 x) 4 Lmax Dao, Li & Suresh, Mat Sci Eng C (2006)

33 Image removed due to copyright restrictions. Please see Fig. 10(a) in Dao, Li, Suresh. "Molecularly Based Analysis of Deformation of Spectrin Network and Human Erythrocyte." Mat Sci Eng C (2006):

34 Image removed due to copyright restrictions. Please see Fig. 10(b) in Dao, Li, Suresh. "Molecularly Based Analysis of Deformation of Spectrin Network and Human Erythrocyte." Mat Sci Eng C (2006): um/7.82um tr 0

35 Start Read D 0, d c, obtain D 0 /d c Select a m h /m 0 ratio Obtain B and C in Eq. (50) Iterate to optimize m h /m 0 ratio Transform experimental data in terms of x r 1 C 1 D A F 1, yf B D0 D0 Fit yf m0 xr to obtain m 0 Obtain optimized m 0 and m h /m 0 Stop Dao, Li & Suresh, Mat Sci Eng C (2006)

36 Critical experiments on single-protein effects Courtesy of National Academy of Sciences, U. S. A. Used with permission. Source: Mills, et al. "Effect of Plasmodial RESA Protein on Deformability of Human Red Blood Cells Harboring Plasmodium Falciparum." PNAS 104 (2007): Copyright 2007 National Academy of Sciences, U.S.A.

37 Image removed due to copyright restrictions. Please see Fig. 9 in Dao, Li, Suresh. "Molecularly Based Analysis of Deformation of Spectrin Network and Human Erythrocyte." Mat Sci Eng C (2006):

38 Summary Continuum mechanics model is a useful tool in extracting mechanical properties of the cell (within certain limitations) Geometry irregularity Computational cell mechanics Nonlinear deformation Continuum modeling framework of human RBC under optical tweezers stretching developed, which significantly facilitates mechanical property extraction in experiments

39 Thank you!