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MIT OpenCourseWare http://ocw.mit.edu
20.GEM GEM4 Summer School: Cell and Molecular Biomechanics in Medicine: CancerSummer 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Continuum Modeling of the Cell
Ming Dao
Dept of Materials Science & Engineering, MIT, Cambridge, MA, USA
GEM4 Summer SchoolJune 25 – July 6, 2007
NUS, Singapore
• 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
Motivation – Why Single-Cell Mechanics?
C.T. Lim et al., J Biomech , 39, 195–216, 2006
Mechanical Models for Living Cells
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Continuum Modeling of Human Red Blood Cell
Studying the deformation characteristics of healthy & diseased red blood cell
red blood cell (erythrocyte)
“Simple” model cell without nucleus
Undergoes severe, reversible, large elastic deformation
Approx. 0.5 million circulations over 120 days
~ 3,000,000 red blood cells produced every second
G. Bao and S. Suresh, Nature Materials (2003)
Experimental Techniques
Courtesy of Subra Suresh. Used with permission.
Inverted
microscope
Before
stretchDuring
stretch
Human red blood cell
Bead adhered
to surface of
glass slide
Binding of silica
microbeads to cell
Laser beam
trapping bead
Dao, Lim and Suresh, J. Mech. Phys. Solids (2003)
Lim et al., Acta Materialia (2004)
1.5 W diode
pumped
Nd:YAG
laser source
CCD
Camerasample
Video
Recorder
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Dao, Lim and Suresh, J Mech Phys Solids, 2003 Mills et al., Mechanics and Chemistry of Biosystems, 2004
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
• 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
Previous Efforts & Current Focus
Molecular structure of human RBC cytoskeleton
Spherocytosis, elliptocytosis, Asian ovalocytosis
Hereditary
blood cell
disorders:
Sickle-cell
disease
Malaria
B. Alberts et al. (1994)
B. Alberts et al. (1994)
Malaria
Spectrin molecules
Figure by MIT OpenCourseWare.
Figure by MIT OpenCourseWare. After B. Alberts et al, 1994.
(c)Effective Continuum
MembraneIn-plane shear modulus: m
Bending stiffness: k
h
Worm-like chain (WLC) model
with surface & bending energies
Spectrin Network + Lipid Membrane(b)
Transmembrane
proteins
Spectrin network
Lipid bilayer
Structure of the cell membrane
Cholesterol
(a)
Dao, Lim & Suresh, J. Mech. Phys. Solids (2003) Li, Dao, Lim & Suresh, Biophys J (2005)Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
3
2 2 2 2 2 201 2 3 h 1 2 33 3
2
m m
Hyperelasticity Model
shear strain (2s)
mem
bra
ne
shear
stre
ss
m0
mh = 0 neo-Hookean Model
mh > 03rd order hyperelasticity
123 = 1
Hyperelasticity Model
3
2 2 2 2 2 201 2 3 h 1 2 33 3
2
m m Neo-Hookean:
Incompressible Material: 123 = 1
Conserving Area: 12 = 1 so that 3 = 1
Classical RBC Membrane Model
x
y
(ax, ay)
(bx, by)
(bx-ax, by-ay)
(-ax, -ay)
(-bx, -by)
(ax-bx, ay-by)
A0
A1A2
A0
A2A1
Li
Aa
nb
n
(a) (b)
Response of a perfect spectrin network
1WLC WLC WLC( ) ( ) ( )1
2
q
q
f a f b f ca a b b c c qC A
A a b cab a b a b a b ab
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
0
20
40
60
80
100
120
140
160
0 0.5 1 1.5
1-1
s11
u0=8.0uN/m; uh/u0=1/20u0=7.3uN/m; uh/u0=1/30u0=7.3uN/m; uh/u0=1/90p=8.5nm; L0=91nm; Lmax=238nmp=8.5nm; L0=87nm; Lmax=238nmp=7.5nm; L0=75nm; Lmax=238nm
1-1
sxx
(mN
/m)
p = 8.5 nm; L0 = 91 nm; Lmax = 238 nmp = 8.5 nm; L0 = 87 nm; Lmax = 238 nmp = 7.5 nm; L0 = 75 nm; Lmax = 238 nmm0 = 8.0 mN/m; mh/m0 = 1/20m0 = 7.3 mN/m; mh/m0 = 1/30m0 = 7.3 mN/m; mh/m0 = 1/90
x
yA0 A
Li, Dao, Lim & Suresh, Biophys J (2005); Dao, Li & Suresh, Mat Sci Eng C (2006)
Image removed due to copyright restrictions.Please see Figure 4 in Dao, M., J. Li, and S. Suresh. "Molecularly Based Analysis of Deformation of SpectrinNetwork and Human Erythrocyte." Mat Sci Eng C (2006): 1232-1244.
Constitutive Model: Prior literature
Constant Area Assumption:
m – membrane shear modulus
2 2
s s 1 222
Tm
m
s 1 2
1
2T T T 2 2
s 1 2 1 2
1 1
2 4
1 2 1
i – the principal stretchesTs , s – membrane shear stress (force/length), shear strain
E. Evans, Biophys. J., 1973
Micropipette Aspiration
L
Rp
Rp
Z
R
Q
L
Rp
Rp
Z
R
Q
2 2 2 2 2
p p p p p p2 2 2R R L R R R R R LR
2
0R
2 2 2
0 p p2R R R LR
The total area in the deformed configuration would be divided in three parts: outside the pipette + in the pipette below the cap (L-Rp) + spherical cap, thus
.
(1)
Deformed Area
Unreformed (Original) Area =
R0
Area Conservation gives
Micropipette Aspiration
.
(1)
0R
0
RR
R R
R 0
1 R
R
2 22 2 2
p p2 2 0s R 2 2 2 2 2
0 p p
2
2 2 2 2
R R LRR R RT
R R R R R LR
m m m
Fz= 2RpTz
dFp
p
dFzp
Fz= 2RpTz
dFp
p
dFzp
p
p p
2 21 ln
pR L L
R Rm
pz p R/ 2
R RT pR T
Constitutive Law:
Before stretch During stretch
Binding of silica
microbeads to cell
dcD0
Bead adhered to
surface of glass slide
Laser beam
trapping bead
DT
DA
VRBC
FEM Mesh
Spectrin network
Initial Model Setup
RBC
2
V
Modeling Optical Tweezers Experiments
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
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
0 pN
67 pN
193 pN
(a) (b) (c)
experiment simulations
0% 20% 40% 60% 80% 100% 120%
maximum principal strain
130 pN
Dao, Lim & Suresh, J Mech Phys Solids (2003)
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): 169-180.
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): 169-180.
Dao, Lim & Suresh, J Mech Phys Solids (2003)
+ 0 experiments R 4
# - Po = 7.3 pN1m + *
Po = 5.3 pNlm * . . . . . . . *
- = 11.3 pNlm 4
--Po # # - -p = 5.5 pN1m (const area) # + *
4 0
-
Axial diameter
Transverse diameter I I I I I
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): 169- 180.
Dao, Lim & Suresh, J Mech Phys Solids (2003)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Time (s)
exp
(- t /
tc)
tc = 0.19 0.06 s
(ave from 8 experiments)
tc = 0.25 s
tc = 0.13 s
Dao, Lim and Suresh, J Mech Phys Solids, 2003
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
0 pN
68 pN
83 pN
35 pN
0% 20% 40% 60% 80% 100% 120%
Maximum Principal Strain
(a) (b) (c) (d) (e)
Experiment Simulations with cytosol Simulations without cytosol
Dao, Lim and Suresh, J Mech Phys Solids, 2003Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
0
2
4
6
8
10
12
14
16
18
0 50 100 150 200 250 300 350 400 450
Stretching Force (pN)
Dia
mete
r ( m
m)
Axial diameter
Transverse diameter
Experiments
Cell Diameter D = 8.5 mm
Cell Diameter D = 7.82 mm
Cell Diameter D = 7 mm
ml = 3.3 mN/m
m0 = 7.3 mN/m
0 25 50 75 100
Dao, Lim and Suresh, J Mech Phys Solids, 2003
0
2
4
6
8
10
12
14
16
18
20
0 50 100 150 200 250 300 350 400 450
Stretching Force (pN)
Dia
me
ter
( mm
)
Axial diameter
Transverse diameter
Experiments
dc = 2.0 mm
dc = 1.5 mm
dc = 1.0 mm
ml = 3.3 mN/m
m0 = 7.3 mN/m
0 25 50 75 100
Dao, Lim and Suresh, J Mech Phys Solids, 2003
0
2
4
6
8
10
12
14
16
18
0 50 100 150 200 250 300 350 400 450
Stretching Force (pN)
Dia
mete
r ( m
m)
Axial diameter
Transverse diameter
Experiments
Biconcave Model
Spherical Model
ml = 3.3 mN/m
m0 = 7.3 mN/m
0 25 50 75 100
Dao, Lim and Suresh, J Mech Phys Solids, 2003
Before stretch During stretch
Binding of silica
microbeads to cell
dcD0
Bead adhered to
surface of glass slide
Laser beam
trapping bead
DT
DA
VRBC
FEM Mesh
Spectrin network
Initial Model Setup
RBC
2
V
Scaling Functions of Optical Tweezers Expts
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
BWLC 2
max
1 1( ) [0 1)
4(1 ) 4
k T Lf L x x
p x L
A 0 h c 0, , , , F F D d Dm m
Dimensional Analysis gives 0A h
0 0 0 c 0
, , DF D
D D d
m
m m
Considering the small influence of bending modulus, OT force
0F m F p
Scaling Functions of Optical Tweezers Expts
0 pm
Dao, Li & Suresh, Mat Sci Eng C (2006)
2um/7.82um tr1.5um/7.82um tr1um/7.82um tr0.5um/7.82 tr
Image removed due to copyright restrictions.Please see Fig. 10(a) in Dao, Li, Suresh. "Molecularly Based Analysis of Deformation of Spectrin Networkand Human Erythrocyte." Mat Sci Eng C (2006): 1232-1244.
1.5um/7.82um tr 01.5um/7.82um tr 1/901.5um/7.82um tr 1/301.5um/7.82um tr 1/201.5um/7.82um tr 1/10
Image removed due to copyright restrictions.Please see Fig. 10(b) in Dao, Li, Suresh. "Molecularly Based Analysis of Deformation of SpectrinNetwork and Human Erythrocyte." Mat Sci Eng C (2006): 1232-1244.
Start
Select a mh/m0 ratio
Read D0, dc, obtain D0/dc
Obtain B and C in Eq. (50)
Transform experimental data in terms of1
A
0 0
11 ,
C
r f
D Fx y
B D D
Fit to obtain m0
0f ry xm
Obtain optimized m0 and mh/m0
Stop
Iterate tooptimizemh/m0 ratio
Dao, Li & Suresh, Mat Sci Eng C (2006)
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 CellsHarboring Plasmodium Falciparum." PNAS 104 (2007): 9213-9217. Copyright 2007 National Academy of Sciences, U.S.A.
Image removed due to copyright restrictions.Please see Fig. 9 in Dao, Li, Suresh. "Molecularly Based Analysis of Deformation of Spectrin Network andHuman Erythrocyte." Mat Sci Eng C (2006): 1232-1244.
Summary
• Continuum mechanics model is a useful tool in extracting
mechanical properties of the cell (within certain limitations)
• Geometry irregularity
• Nonlinear deformation
• Continuum modeling framework of human RBC under
optical tweezers stretching developed, which significantly
facilitates mechanical property extraction in experiments
Computational cell mechanics
Thank you!