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20.GEM GEM4 Summer School: Cell and Molecular Biomechanics in Medicine: Cancer
Summer 2007
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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
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
Mechanical Models for Living Cells
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
C.T. Lim et al., J Biomech , 39, 195–216, 2006
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 nucleu
Undergoes severe, reversible, large elastic deformation
Approx. 0.5 million circulations over 120 days
~ 3,000,000 red blood cells produced every second
Experimental Techniques
Courtesy of Subra Suresh. Used with permission.
G. Bao and S. Suresh, Nature Materials (2003)
Binding of silica
microbeads to cell
Video
Recorder
Bead adhered
to surface of
glass slide
Laser beam
trapping bead
Human red blood cell
Before
stretch
During
stretch
1.5 W diode
pumped
Nd:YAG
laser source
sample
CCD
Camera
Inverted
microscope
Dao, Lim and Suresh, J. Mech. Phys. Solids (2003)
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Lim et al., Acta Materialia (2004)
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
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
Spectrin molecules
Hereditary
blood cell
disorders:
Sickle-cell
disease
Figure by MIT OpenCourseWare.
Molecular structure of human RBC cytoskeleton
Malaria
B. Alberts et al. (1994)
Figure by MIT OpenCourseWare. After B. Alberts et al, 1994.
Spherocytosis, elliptocytosis, Asian ovalocytosis
(a)
Lipid bilayer
Cholesterol
Spectrin network
Structure of the cell membrane
(b)
Transmembrane
proteins
Spectrin Network + Lipid Membrane
Worm-like chain (WLC) model
with surface & bending energies
(c)
Effective Continuum
Membrane
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Dao, Lim & Suresh, J. Mech. Phys. Solids (2003)
In-plane shear modulus: m
Bending stiffness: k
h
Li, Dao, Lim & Suresh, Biophys J (2005)
Hyperelasticity Model


2
2
1
     3  m h        3
2
2
2
3
mh > 0
membrane
shear stress

m0
2
1
2
2
2
3
3
 1 2 3 = 1
3rd order hyperelasticity
mh = 0  neo-Hookean Model
m0
shear strain (2s)
Hyperelasticity Model

Neo-Hookean:
Incompressible Material:
Conserving Area:
m0
2
2
2
2
2
2
2






3

m





 1 2 3  h  1 2 3  3
 1 2 3 = 1
12 = 1 so that 3 = 1
Classical RBC Membrane Model
Response of a perfect spectrin network
y
(bx, by)
(bx-ax, by-ay)
Li
A1
A2
Aa
A0
x
A0
n
nb
(-ax, -ay)
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
(a)
 ab
(ax, ay)
A1
A2
(ax-bx, ay-by)
(-bx, -by)
(b)

f WLC (b)
f WLC (c)
1  f WLC (a)
 q 1


a
a

b
b

c
c

qC
A
ab 

a b
a b
a b
q

2 A  a
b
c

160
u0=8.0uN/m;
p = 8.5
nm; L0 =uh/u0=1/20
91 nm; Lmax = 238 nm
u0=7.3uN/m; uh/u0=1/30
p = 8.5
nm; L0 =uh/u0=1/90
87 nm; Lmax = 238 nm
u0=7.3uN/m;
p = 7.5
nm; LL0=91nm;
Lmax = 238 nm
p=8.5nm;
Lmax=238nm
0 = 75 nm;
L0=87nm;
m0 =p=8.5nm;
8.0 mN/m;
mh/m0 =Lmax=238nm
1/20
L0=75nm;
m0 =p=7.5nm;
7.3 mN/m;
mh/m0 =Lmax=238nm
1/30
m0 = 7.3 mN/m; mh/m0 = 1/90
140
s11
sxx (mN/m)
120
100
80
60
y
40
A0
A
x
20
0
0
0.5
1
1-1
1-1
1.5
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 Spectrin
Network and Human Erythrocyte." Mat Sci Eng C (2006): 1232-1244.
Constitutive Model: Prior literature
Constant Area Assumption: 12  1
Ts  2m s 
m
2
2
2



 1 2
m – membrane shear modulus
i – the principal stretches
Ts , s – membrane shear stress (force/length), shear strain
1
Ts  T1  T2 
2
E. Evans, Biophys. J., 1973
1
1 2
 s  1   2    1  22 
2
4
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-Rp) + spherical cap, thus
Deformed Area

 


  R2   Rp2  L  Rp 2 Rp  2 Rp2   R 2  Rp2  2LRp
Unreformed (Original) Area =
 R02
Z
Area Conservation gives
R  R  R  2LRp
2
0
2

2
p
.
Rp
Q
(1)
L
Rp
R
R0
Micropipette Aspiration
R R0
R 

R0
R
 
1
R

R
R0
Constitutive Law:
2
2
2
2
2

R

R

2
LR


R
m 2
m
R
m
R
p
p
Ts   R  2    02  2   
 2
2

2
2R
R0  2 
R
R  Rp2  2 LRp
Tz  pRp / 2  TR
pRp
m
dFzp
.
dFp
R  Rp
 2L 
2L

 1  ln 


Rp
 Rp 
p 
(1)
Fz= 2RpTz



Modeling Optical Tweezers Experiments
Binding of silica
microbeads to cell
Laser beam
trapping bead
Bead adhered to
surface of glass slide
Initial Model Setup
DT
dc
D0
DA
FEM Mesh
VRBC
VRBC
2
Before stretch
During stretch
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Spectrin network
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
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): 169-180.
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.
Dao, Lim & Suresh, J Mech Phys Solids (2003)
+ 0 experiments
-Po = 7.3 pN1m
R
4
#
.......Po = 5.3 pNlm
= 11.3 pNlm
--Po
- -p = 5.5 pN1m (const area)
*
*
+*
4
#
4
*
+
#
#
0
-
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. "NonlinearElastic 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)
1
tc = 0.19  0.06 s
(ave from 8 experiments)
tc = 0.25 s
tc = 0.13 s
exp(- t / t c)
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Dao, Lim and Suresh, J Mech Phys Solids, 2003
Maximum Principal Strain
0%
Experiment
20%
40%
Simulations with cytosol
60%
80% 100% 120%
Simulations without cytosol
0 pN
35 pN
68 pN
83 pN
(a)
(b)
(c)
(d)
(e)
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Dao, Lim and Suresh, J Mech Phys Solids, 2003
Experiments
Cell Diameter D = 8.5 mm
Cell Diameter D = 7.82 mm
Cell Diameter D = 7 mm
Diameter (mm)
18
16
14
12
10
ml = 3.3 mN/m
m0 = 7.3 mN/m
Axial diameter
8
6
4
Transverse diameter
2
0
00
50
25 150 200
50 250 300
75 350 100
100
400 450
Stretching Force (pN)
Dao, Lim and Suresh, J Mech Phys Solids, 2003
Experiments
dc = 2.0 mm
dc = 1.5 mm
dc = 1.0 mm
20
18
Diameter (m m)
16
ml = 3.3 mN/m
m0 = 7.3 mN/m
Axial diameter
14
12
10
8
6
4
Transverse diameter
2
0
00
50
100
25
150
200
50
250
300
75
350
400
100
450
Stretching Force (pN)
Dao, Lim and Suresh, J Mech Phys Solids, 2003
Experiments
Biconcave Model
Spherical Model
18
Diameter (mm)
16
14
ml = 3.3 mN/m
m0 = 7.3 mN/m
Axial diameter
12
10
8
6
4
Transverse diameter
2
0
00
50
100
400 450
25 150 200
50 250 300
75 350 100
Stretching Force (pN)
Dao, Lim and Suresh, J Mech Phys Solids, 2003
Scaling Functions of Optical Tweezers Expts
Binding of silica
microbeads to cell
Laser beam
trapping bead
Bead adhered to
surface of glass slide
Initial Model Setup
DT
dc
D0
DA
FEM Mesh
VRBC
VRBC
2
Before stretch
During stretch
Courtesy Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
Spectrin network
Scaling Functions of Optical Tweezers Expts
Considering the small influence of bending modulus, OT force
F  F  DA , m0 , mh , dc , D0 
Dimensional Analysis gives
F  m0
kBT
f WLC ( L)  
p
 DA D0 mh 
F
 
,
,

m0 D0
 D0 dc m0 
m0  p
Fp
 1

1
  x

2
 4(1  x) 4

L
x
 [01)
Lmax
Dao, Li & Suresh, Mat Sci Eng C (2006)
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): 1232-1244.
2um/7.82um tr
1.5um/7.82um tr
1um/7.82um tr
0.5um/7.82 tr
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): 1232-1244.
1.5um/7.82um tr 0
1.5um/7.82um tr 1/90
1.5um/7.82um tr 1/30
1.5um/7.82um tr 1/20
1.5um/7.82um tr 1/10
Start
Read D0, dc, obtain D0/dc
Select a mh/m0 ratio
Obtain B and C in Eq. (50)
Iterate to
optimize
mh/m0 ratio
Transform experimental data in terms of
1
C
 1  DA  
F
xr   
 1  , y f 
D0

 B  D0
Fit y f  m0 xr to obtain m0
Obtain optimized m0 and mh/m0
Stop
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 Cells
Harboring 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 and
Human 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
Computational cell mechanics
• Nonlinear deformation
• Continuum modeling framework of human RBC under
optical tweezers stretching developed, which significantly
facilitates mechanical property extraction in experiments
Thank you!