20.GEM GEM4 Summer School: Cell and Molecular Biomechanics ...€¦ · Continuum Modeling of the...

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

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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!