Molecular, Cellular & Tissue Biomechanicsdspace.mit.edu/bitstream/handle/1721.1/100966/2-797j...Molecular, Cellular & Tissue Biomechanics Goal: Develop a fundamental understanding

Molecular, Cellular & Tissue Biomechanics

Goal: Develop a fundamental understanding of

biomechanics over a wide range of length scales.

20.310, 2.793, 6.024

Fall, 2006

20.310, 2.793, 6.024

Biomolecules and intermolecular forces

Single molecule biopolymer mechanics

Formation and dissolution of bonds

Motion at the molecular/macromolecular

level

MOLECULAR MECHANICS

Structure/function/properties of the cell

Biomembranes The cytoskeleton

Cell adhesion and aggregation

Cell migration

Mechanotransduction

CELLULAR MECHANICS

TISSUE MECHANICS

Molecular structure --> physical properties

Continuum, elastic models (stress, strain,

constitutive laws)

Viscoelasticity

Poroelasticity Electrochemical effects on tissue properties

Fall, 2006

1

Some Learning Objectives

1. To understand the fundamental concepts of mechanics

and be able to apply them to simple problems in the

deformation of continuous media

2. To understand the underlying basis for the mechanical

properties of molecules, cells and tissues

3. To be able to model biological materials using methods

appropriate over diverse length scales

4. To be familiar with the wide spectrum of measurement

techniques that are currently used to determine

mechanical properties

5. To appreciate the close interconnections between

mechanics and biology/chemistry of living systems

20.310, 2.793, 6.024

Fall, 2006

Biomechanics of tissues

Mechanics Biology

I. Linear elastic behavior I. Biochemical and

molecular biology ofII. Viscoelasticity

ECM molecules

III. Poroelasticity A. Collagen

IV. Electrochemical and superfamilyphysicochemical properties

B. Proteoglycan

superfamily

C. Other glycoproteins

II. Nanomolecular

structures <--> tissue

III. Mechanobiology

20.310, 2.793, 6.024

Fall, 2006

2

Some preliminaries

Equilibrium -- balance of forces

concept of a stress tensor

Compatibility -- relations between displacements and

strains or deformation

normal strains; shear strains; strain tensor

Constitutive laws

stress -- strain

Young’s modulus; Poisson’s ratio, shear modulus

linear, isotropic, elastic materials

20.310, 2.793, 6.024

Fall, 2006

20.310, 2.793, 6.024

Fall, 2006

Force balance in a single cell Desprat et al., Biophys J., 2005.

Figure by MIT OCW.

3

Stress distributions along the basal surface

of a resting cell

20.310, 2.793, 6.024

Fall, 2006

Several material testing methods

Linear extension Linear shear

Biaxial tension

Cone-and-plate rheometer Confined compression

20.310, 2.793, 6.024

Fall, 2006

4

Graphical image of stress distributions on a cell surface removed due to copyright restrictions.

Hu, et al., AJP Cell, 2003

Tissue properties: common simplifying

assumptions

Linear -- the elastic modulus is constant,

indpendent of strain amplitude

Homogeneous -- the material is spatially

uniform

Isotropic -- the material exhibits the same elastic

properties in all directions

Time-independent -- stresses and strains are

uniquely related, independent of rate of strain

20.310, 2.793, 6.024

Fall, 2006

Constitutive laws for a linear elastic,

isotropic material

= � � ) + 2G�1 �11 ( 11 + �22 + �33 11 � (�11 = ��11 � � � 22 + � 33 )��E � 22 = � �11 + �22 + �33 ) + 2G�22( 1

( ) = � � ) + 2G��22 = E ��� 22 � � �11 + � 33 �� � 33 ( 11 + �22 + �33 33

1 �12 = 2G�12 �33 =

E �� 33 � � �11 + � 22 )�� �13 = 2G�13 � ( �12 (1 + � )�12 � 23 = 2G�23�12 = = 2G E �13 (1 + � )�13 2G� E�

� = = � = = 13 2G E 1 � 2� ��(1 + � )(1 � 2� )��� 23 (1 + � )� 23�23 = = 2G E E = Young’s modulus � = Lame’ constant

� = Poisson’s ratio G = shear modulus

20.310, 2.793, 6.024

Fall, 2006

5

20.310, 2.793, 6.024

Fall, 2006

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

1.E+10

1.E+12 DIAMOND

STEEL

BONE CONCRETE

SILK F-ACTIN TENDON WOOD

TUBULIN

CONTRACTED SKELETAL MUSCLE

ELASTIN

RELAXED SKELETAL MUSCLE COLLAGEN GELS

LUNG PARENCHYMA FIBROBLAST CELLS

ENDOTHELIAL CELLS NEUTROPHILS LYMPHOCYTES

Values of the elastic or Young’s modulus (E) for

various materials

20.310, 2.793, 6.024

Fall, 2006

Linear? Unidirectional tensile tests

Stress-strain behavior of a peptide hydrogel

Linear behavior up to fracture

Relatively low toughness due to small fracture strain

a bneedle needle

peptide matrix

peptide matrix

plastic mesh

6

Figure by MIT OCW. Adapted from Leon, et. al., 1998

Constant Eσ11

ε11

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

10

20

30

50

60

40

Stress(N/m 2)>

Strain

7

Linear? Elastin is one of

main structural

components of tissue

Linear; little hysteresis.

Provides the “stretchiness” of tissues.

Combination of single-molecule characteristics and microscale structure.

20.310, 2.793, 6.024

Fall, 2006

Linear? ACL -- different strain rates

20.310, 2.793, 6.024

Fall, 2006

Figure by MIT OCW.

Figure by MIT OCW.

Slow Medium Fast

Range of linearity

E = dσ/dε = 109Pa

2 4 6 8 10 120

80

60

40

20

Strain(%)

Stra

ss (

MPa

)

Control

100

80

60

40

20

05 10 15 20

Specimen fixed at zero stretch in 10% formalin

ELASTINLig. Nuchae denatured

% Strain = (∆L/L0)*100

Stre

ss (k

Pa)

The stress-strain curve of elastin. Data from Fung and Sobin (1981).

Linear and Isotropic? Right tibia

20.310, 2.793, 6.024Fall, 2006

20.310, 2.793, 6.024

Fall, 2006

Canine aorta showing elastic fiber content

8

Figure by MIT OCW.

Scanning electron micrographs showing a low-power view of dog'saorta and a high-power view of the dense network of longitudinallyoriented elastic fibers in the outer layer of the same blood vessel.Images removed due to copyright restrictions. See Haas, K. S., S. J. Phillips, A. J. Comerota, and J. W. White. Anat. Rec. 230 (1991): 86-96.

250

200

150

100

50

00

1 2 3 4

Stre

ss (M

Pa)

Strain (%)

RA-direction

Sample no.

Sample no.

BA-direction

6

6

1

1

3

2

2

4

4

55

Longitudinaldirection

Radial direction

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Fall, 2006

Collagen fiber arrangement in skin and cornea with

alternating directions

20.310, 2.793, 6.024

Fall, 2006

Homogeneous? Only rarely

9

Electron micrograph of a cross-section of tadpole skin. The arrangement of collagen fibrils is plywoodlike, with successive layers of fibrils laid down nearly at right angles to each other. Image removed due to copyright restrictions.

Electron micrographs of parallelcollagen fibrils in a tendon and themesh work of fibrils in skin removed

due to copyright restrictions.

Figure by MIT OCW.

Tendon

Fascicle

Tendon Hierarchy

X ray EM X ray EM X ray EMSEM

EM SEMOM

SEMOM

Evidence:

X- ray

Reticularmembrane

Fascicularmembrane

Waveform orcrimp structure

Fibroblasts

Tropocollagen

35 stainingsites

640 periodicity

15 35o

Ao

A

o

A

o

Ao

A

o

A

100-200 500-5000 50-300 µ 100-500 µ

Size Scale

SubfibrilMicrofibril Fibril

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Fall, 2006

Histological cross-section

of a diseased carotid artery

stained for smooth muscle

cells.

Elastic response

initially, then stiff,

collagen response at

high degrees of

extension.

H = hypertensive

High wall stress

leads to functional

remodeling.

Time-dependent? Peptide gels Pre- and Post-

100 µM KCl

1000 µM KCl

Storage Modulus (G’) Loss Modulus (G’’)

Gelation. Properties can change with time.

1%wt KFE12

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10

Figure by MIT OCW.

Figure by MIT OCW.

1000

100

10

1

1

0.1

0.01

Oscillatory Frequency (rad/sec)

5 10

Mod

uli (

Pa)

1000

100

10

1

1

0.1

0.01

Oscillatory Frequency (rad/sec)

5 10

Mod

uli (

Pa)

100 µM

1000 µΜΚCl

ΚCl

Storage Modulus (G')

Loss Modulus (G")

Primary data for gel formation of 1wt % KFE12 equilibrated with (a) 0.1mM NaCl and (b) 1.0 mM NaCl. The storage modulus (squares),G', and loss modulus (circles), G", are plotted against oscillatory frequencyon log-log scales.

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.00 100 200 300 400

Group NGroup H

Pressure (mm Hg) Pressure (mm Hg)

Relat

ive r

adiu

s

10 90 170 250 3300.1

1

10

Incremental modulus:E = dσ/dε

Incr

emen

tal el

astic

mod

ulus

(MPa

)

Stiffening

Shear Modulus of cartilage. Modulus can be

frequency-dependent (Dynamic @ 0.5Hz, 0.8% strain)

20.310, 2.793, 6.024

Fall, 2006

Other complications:

Surface tension in

lung parenchyma

Surface tension effects 20.310, 2.793, 6.024

Fall, 2006

11

Images removed due to copyright restrictions.

Figure by MIT OCW.

Figure by MIT OCW.

1

0.75

0.5

0.25

0

0.01 0.1 1

10

12.5

15

17.5

20

2.5

2

1.5

1

0.5

0

Dynamic G

Equilibrium: G

Ionic Concentration, M

Dynamic Phase

Dynamic G

Dynamic G

Magnitude, MPa

Phase Angle (degree)

(Mean

(Mean

+-

+- S.D, n= 4,

S.D, n= 3.6)

f =0.5Hz, γ = 0.8% )

Equilibrium G, MPaq.G

200

150

100

50

5 10 15 20

Liquid Inflation

Liq

uid

Air

Effect ofsurfacetension ofgas-liquidinterface Effect of

collagen athigh degreesof extension

Nearly linear response at small extensions

Air Inflation

Relaxation Pressure

Lun

g V

olum

e (m

l)

Von Neergaard Experiments

20.310, 2.793, 6.024

Fall, 2006

Striated

Other

complications:

Active

contraction

Underlying basis for mechanical properties

Extracellular matrix, cartilage, and tendon are largely comprised of:

collagen

elastin

proteoglycan

(role of constituent cells??)

20.310, 2.793, 6.024

Fall, 2006

12

Figure by MIT OCW.

Figure by MIT OCW.Figure by MIT OCW.

A

B

CD

A- restingB- max contractionC- active component

0

0

05 10 15 20

20

25

25

30 35

50 75 100

100

125 150 175

40

60

80

(mm)

(%)

Muscle length

Tens

ion

(mN

)

Caption:A, resting tension; B, maximum tension produced by optimum stimulus; C, active tension (= B - A);D, potentialed tension produced by successive stimuli.

The slightest resting tension was produced at 75% of the in situ length (20 mm). *

*

*

*

*

The optimum length, at which the active tension become maximum, is between 100% and 125% of the in situ length.

The active tension declines almost symmetrically on either side of the optimal length.

The maximumn tension potentialed by the successive stimuti was attained at 75% of the in situ length.

Macroscopic view

120

100

Tens

ion

(% o

f m

axim

um)

80

60

40

20

01.0 2.0 3.0 4.0

Striation spacing (µm)

Many features of the length-tension relation are simply explained by thesliding-filament theory.The peak of the curve consists of a plateau between sarcomere lengthsof 2.05 and 2.2 µm.

Microscopsarcomere

ic - level

Typical elastic behavior for tissues containing

collagen and elastin

20.310, 2.793, 6.024

Fall, 2006

13

20.310, 2.793, 6.024

Fall, 2006

Collagen fibril formation

Figure by MIT OCW.

Figure by MIT OCW.

STRAIN, ε (∆L/L0)

TE

NSI

LE

ST

RE

SS, σ

(F/A

)

Young's Modulus = σ / ε

σε

Elastin dominated

Failure

Linear Region

Toe Region

Collagen dominated

Synthesis of pro-α chain

Self-assembly Of Three pro-α chain

2

1

4

5

3

6

7 8

9

Hydroxylation Of Selected ProlinesAnd Lysines

Glycosylation Of Selected Hydroxylysines

Propeptide

3 pro-α chain

Procollagen Triple-helixFormation

H2N

H2N

OHOH

OH

OH

OH

OH

OH

OH

OHOH

OHOH OH

OH

OH

Secretory vesicle

ER/Golgi compartment

Collagen molecule

Self-AssemblyInto Fibril

Aggregation Of Collagen Fibrils To Form A Collagen Fiber

Collagen fiber

Collagen fibril

0.5-3µm

10-3

00 n

m

Procollagen molecule

Secretion Plasma membrane

OHOH

OH

OH

OH

OH OH OH

OH

OH

OH

OH OH OHCleavage ofpropeptides

The Intracellular and Extracellular Events Involved in the Formation of a Collagen Fibril

Collagen -- single molecule characteristics

20.310, 2.793, 6.024

Fall, 2006

20.310, 2.793, 6.024

Fall, 2006

As of 2003, there

were 28 (!)

different types of

collagen

identified.

14

Figure by MIT OCW.

Figure by MIT OCW.

Major Collagen Molecules

Type Molecule composition Structural features Representativetissues

Fibrillar collagens

[α 1(I)]2 [α 2(I)]

[α 1(IV)]2 [α 2(IV)]

[α 1(VI)][α 2(VI)]

[α 1(IX)][α 2(IX)][α 3(IX)]

[α 1(II)]3

[α 1(III)]3

[α 1(V)]3

Fibril-associated collagens

300-nm-long fibrils

300-nm-long fibrils

300-nm-long fibrils;often with type I

390-nm-long fibrilswith globular N-terminaldomain; often with type I

Lateral association with typeI; periodic globular domains

Lateral association withtype II; N-terminal globulardomain; bound glycosami-noglycan

Two-dimensional network

Skin, tendon, bone,ligaments, dentin,interstitial tissues

Skin, muscle, bloodvessels

Similar to type I; alsocell cultures, fetaltissues

Most interstitialtissues

Cartilage, vitreoushumor;

All basal laminaes

I

II

III

V

VI

IX

IV

Sheet-forming collagens

Cartilage, vitreoushumor

Figure by MIT OCW.

1412

10

86420

0-0

50 100 150 200 250 300 350

The force-extension curve of a singlecollagen II.

Cover glass

XY Stage

Trap Center

Laser light

Procollagen

Stretching a procollagen II molecule with optical tweezers.

Figure by MIT OCW.

20.310, 2.793, 6.024

Fall, 2006

Type IX collagen decorated with type II collagen

20.310, 2.793, 6.024

Fall, 2006

15

Figure by MIT OCW.

Figure by MIT OCW.

Type IX collagenmolecule Fibril of type II

collagen

Stretch Relax

Elastic Fiber

Single elastinmolecule

Cross-link

Short sectionof a collagen

fibril

collagen molecule300 X 1.5 nm

50 nm

1.5 nmCollagen triple

helix

Contrast between elastin and collagen

20.310, 2.793, 6.024

Fall, 2006

Entropic elasticity

Proteoglycans (PGs) and

glycosaminoglycans (GAGs)

a) GLYCOSAMINOGLYCANS (GAGs) form gels

i) polysaccharide chains of disaccharide units

ii) too inflexible and highly charged to fold in a compact way

iii) strongly hydrophilic

iv) form extended conformations and gels

v) osmotic swelling (charge repulsion)

vi) usually make up less than 10% of ECM by weight

vii) fill most of the ECM space

viii) four main groups

a. hyaluronan

b. chondroitin sulfate and dermatin sulfate

c. heparin sulfate and heparin

d. keratin sulfate

20.310, 2.793, 6.024

Fall, 2006

16

Figure by MIT OCW.

Stretch Relax

Elastic Fiber

Single elastinmolecule

Cross-link

b) Proteoglycans (PGs) i) form large aggregates

ii) aggrecan is a large proteoglycan in cartilage

iii) decorin is secreted by fibroblasts

iv) PGs have varying amounts of GAGs.

v) PGs are very diverse in structure and content

vi) PGs and GAGs can also complex with collagen

vii) secreted proteoglycans have multiple functions

viii) PG/GAGs have important roles in cell-cell

signaling

20.310, 2.793, 6.024

Fall, 2006

20.310, 2.793, 6.024

Fall, 2006

17

Electron micrograph of proteoglycans in the extracellular matrix of rat cartillage removed due to copyright restrictions. See Hunziker, E. B., and R. K. Schenk. J Cell Biol 98 (1985): 277-282.

20.310, 2.793, 6.024

Fall, 2006

20.310, 2.793, 6.024

Fall, 2006

18

Figure by MIT OCW.

Figure by MIT OCW.

100 nm

GAG

Core Protein

Polypeptide chain

Short, branchedoligosaccharideside chain

Decorin(MW - 40,000)

Aggrecan(MW - 3 x 106)

Ribonuclease(MW - 15,000)

Examples of a large (aggrecan) and a small (decorin) proteoglycan found in the extracellular matrix.They are compared to a typical secreted glycoprotein molecule (pancreatic ribonuclease B). All aredrawn to scale. The core proteins of both aggrecan and decorin contain oligosaccharide chains as wellas the GAG chains, but these are not shown. Aggrecan typically consists of about 100 chondroitin sulfatechains and about 30 keratan sulfate chains linked to a serine-rich core protein of almost 3000 aminoacids. Decorin "decorates" the surface of collagen fibrils, hence its name.

Aggrecan Aggregate

Core protein

Link proteins

Chondroitin sulfate

Hyaluronanmolecule

Keratin sulfate

1 µm

Schematic drawing of an aggrecan aggregate. It is composed of 100 aggrecanmonomers noncovalently bound to a single hyaluronan chain through two linkproteins that bind to both the core protein of the proteoglycan and to thehyaluronan chain, thus stabilizing the aggregate.

20.310, 2.793, 6.024

Fall, 2006

Atomic Force Microscopy of Aggrecan

20.310, 2.793, 6.024

Fall, 2006

19

Figure by MIT OCW.

Electron micrograph of an aggrecan aggregate removed due to copyright restrictions.

Images removed due to copyright restrictions.

Aggrecan Chondroitin sulfate GAG

Chains

(Fetal Bovine)

Ave. GAG length: ~36 nm

Inter-GAG spacing: 4-5 nm

(L. Ng, C. Ortiz, A. Grodzinsky)

Hyaluronan Molecule

Link Protein N-terminal Hyaluronan-binding domain

Keratan Sulfate

ChondroitinSulfate

Linking SugarsAggrecan Core Protein

30

20

10

0

-10

0.5 1 1.5 2 2.5

Single Hyaluronan Molecule

Forc

e (p

N)

Displacement (µm)

A typical result of the force-displacement relationship fromthe single hyaluronan molecule measurement.

Figure by MIT OCW.

Like charge repulsion accounts for a large fraction (~50%) of the stiffness in tissues with high GAG content.

These effects can be eliminated either by shielding with counter-ions or neutralization by changing pH.

+ + + + + ++

+ + +

+ +

+ + +

++ + + + + ++ + + ++ +

20.310, 2.793, 6.024

Fall, 2006

20.310, 2.793, 6.024

Fall, 2006

Confined compression experiments

20

Figure by MIT OCW.

1.2

1

0.8

0.6

0.4

0.2

0.001 0.01 0.1 1

Electrostatic: GAG

otherEqui

libriu

m M

odul

us, M

Pa

Ionic Strength, M

Equilibrium Modulus of Adult Bovine Articular Cartilage in Different Ionic Strengths

20.310, 2.793, 6.024

Fall, 2006

ADHESION PROTEINS a) fibronectin

i) principal adhesion protein of connective tissues

ii) fibronectin is a dimeric glycoprotein

iii) fibronectin interacts with other molecules

b) laminin

i) found in basal laminae

ii) form mesh-like polymers

iii) has various binding sites

iv) assembles networks of crosslinked proteins

c) integrins

i) cell surface receptor, for attachment of cells to ECM

ii) family of transmembrane proteins

iii) two subunits, alpha and beta

iv) about 20 different integrins

v) binding sites for ECM components

vi) binding sites for the cytoskeleton and linkage to ECM

20.310, 2.793, 6.024

Fall, 2006

21

Figure by MIT OCW.

Some common proteoglycans

ProteoglycanApproximatemolecular weightof core protein

Type of GAGchains

Number ofGAG chains Location Functions

Aggrecan

Betaglycan

Decorin

Perlecan

Serglycin

Syndecan-1

210,000

36,000

40,000

600,000

20,000

32,000

Chondroitin

Chondroitin

Chondroitin

Chondroitin

Chondroitin

Sulfate +

Sulfate +

Sulfate/

Sulfate/

Sulfate/

KeratanSulfate

Dermatan

Dermatan

Dermatan

Sulfate

Sulfate

Sulfate

Sulfate

Heparan

Heparan

Sulfate

~130

1

1

2-15

10-15

1-3

Cartilage

Cell surfaceand matrix

Widespreadin connectivetissue

Basallaminae

Secretory vesiclesin white bloodcells

Fibroblast andepithelial cellsurface

Mechanical support;forms large aggregateswith hyaluronan

Binds to type 1collagen fibrilsand binds TGF - β

Structural andfiltering functionin basal lamina

Helps to packageand store secretorymolecules

Binds TGF - β

Cell adhesion;binds FGF

Fiall, 2006

20.310, 2.793, 6.024

Fall, 2006

The bundle thickness, mesh size, elastic modulus, and critical strain as a function of R at cA = 11.9 µM

22

Figure by MIT OCW.

Figure by MIT OCW. Adapted from Shin, J. H. et al. Proc Natl Acad Sci USA 101 (2004): 9636-9641.

R0.2

R0.3

R2

I/R0.6

100

100

100

10-1

10-1

10-1

101

101

102

102

100

10-210-2

R

γ0

G0 (Pa)

D (nm)

ξ (µm)

Bundle thickness

Mesh size

Shear modulus, G

Strain (ε21) at which materialbecomes nonlinear

Collagen binding

Cell bindingHeparin

C C

N

N NH2

Arg

Gly

Asp

HOOC100 nm

The Structure of a Fibronectin Dimer

SSSS

binding

20.310, 2.793, 6.024

Fall, 2006

Fall, 2006

The stress-strain relationships of F-actin networks formed with different FLN mutants

We apply a prestress to the network (Inset, single-headed filled arrow) and measure the deformation (Inset, dashed arrow) in response to an additional oscillatory stress (Inset, double-headed filled arrow)

23

Figure by MIT OCW. Adapted from Gardel, M. L. et. al. Proc Natl Acad Sci USA 103

(2006): 1762-1767.

Figure by MIT OCW. Adapted from Gardel, M. L. et. al. Proc Natl Acad Sci USA 103

(2006): 1762-1767.

FLNa

FLNa

FLNa h(-)

FLNa h(-)

FLNb h(-)

FLNb h(-)

FLNb

FLNb

0.0 0.2 0.4 0.6 0.8 1.0 1.2

12

10

8

6

4

2

0

Strain

Stre

ss (P

a)

Strain,γ

Stre

ss, σ σ

γ

dσ

dγ

10-2 10-1 100

100

101

101

102

102

103

103

Prestress(Pa)

Differential Stiffness (Pa)

20.310, 2.793, 6.024

Fall, 2006

20.310, 2.793, 6.024

Fall, 2006

Focal adhesion

complex

24

Figure by MIT OCW.

Figure by MIT OCW.

CYTOSOL

{{Lipid

bilayer

Cell coat(glycocalyx)

Transmembrane proteoglycan Adsorbed

glycoproteinTransmembrane glycoprotein

Glycolipid

Simplified diagram of the cell coat (glycocalyx)

Sugar residue=

Fibronectin Extracellularmatrix

Cell membrane

Cytoplasm

Integrin

Focal adhesion

Actin

α-Actinin Signals that control

cellular activities

α β

Talin

VinculinVinculin

Zyxin

Talin

Paxillin

Tensin

Ras

SOS

Grb2

FAKSrckinase

Paxillin p130

20.310, 2.793, 6.024

Fall, 2006

Viscoelastic models

µ k

FF

µ

k FF

µ1 k1

k2 FF

u

u

u2u1

a)

b)

c)

Maxwell

Voigt

Standard linear

solid

Stress relaxation

Creep

Steady-state response to

oscillatory displacement

F(t) = ku0 exp(�t / � )

u(t) = F k 1 � exp �t / �( )�� ��

� = µ / k

E* (� ) = E '+ iE"Concept of a complex

elastic or shear modulus

Poroelastic materials

Governing equations:

1. Constitutive law

� ij tot = 2G� ij + �� ij� ij � p� ij

�11

x1

u1

�

U k p� �=

���

1D forms 2. Fluid-solid viscous interactions (Darcy’s

Law)�

3. Conservation of mass �

� �u U = � (vf � vs ) = �vrel vs =

�t 4. Conservation of momentum

� �� = 0

20.310, 2.793, 6.024

Fall, 2006

tot �11 = (2G + �)�11 � p

U1 = �k �p �x1

U1 = ��u1

+ U0 �t

��11 = 0

�x1

2�u1 � u1

= Hk �t �x1

25

Poroelasticity -- confined compression

Impose displacements at boundaries:

u1(x1, t=0) = 0 �11

u1(x1=L, t>0) = 0

u1

u1

u1(x1=0, t>0) = u0

2�u1 � u1 x1

= Hk �t �x1

Characteristic time ~ L2/Hk x1

�

Solution (Fourier series)

�� ��

t �u1(x1,t) = u0 1 �

x1 � An sin n� x1 exp

L Ln n

�

L2

� = n 2

� ��

2 Hkn �

20.310, 2.793, 6.024

Fall, 2006

�� � ��

Finite rates of protein folding and unfolding can

also contribute to time-dependent behavior

• AFM can be used to measure boththe elastic (k) and viscous (z)properties of a single molecule as a

function of extension

• Sequential unfolding of theimmunoglobulin (Ig) domains of titinduring oscillations to measureviscoelasticity

• Siingle molecule elastic and viscousproperties appear to scale witheach other

Kawakami et al., BJ, 2006 20.310, 2.793, 6.024

Fall, 2006

26

Figure by MIT OCW.

Extension (nm)

k (p

N/n

m)

Forc

e (p

N)

ζ (1

0-7 k

g/se

c)

20

20

40

40

60 80 100

0

0

2

4

6

50

100

150

20.310, 2.793, 6.024

Fall, 2006

Additional factors: Ionic strength

� + 2G

27

Figure by MIT OCW.

1.2

1

0.8

0.6

0.4

0.2

0.001 0.01 0.1 1

Electrostatic: GAG

otherEqui

libriu

m M

odul

us, M

Pa

Ionic Strength, M

Equilibrium Modulus of Adult Bovine Articular Cartilage in Different Ionic Strengths