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8/2/2019 Tissue Mechanics Slides
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Molecular, Cellular & Tissue Biomechanics
Goal: Develop a fundamentalunderstanding ofbiomechanics over a wide range of length scales.
20.310, 2.793, 6.024Fall, 2006
20.310, 2.793, 6.024
Biomolecules and intermolecular forcesSingle molecule biopolymer mechanicsFormation and dissolution of bondsMotion at the molecular/macromolecularlevel
MOLECULAR MECHANICS
Structure/function/properties of the cell
BiomembranesThe cytoskeletonCell adhesion and aggregationCell migrationMechanotransduction
CELLULAR MECHANICS
TISSUE MECHANICSMolecular structure --> physical propertiesContinuum, elastic models (stress, strain,constitutive laws)ViscoelasticityPoroelasticityElectrochemical effects on tissue properties
Fall, 2006
1
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Some Learning Objectives
1. To understand the fundamental concepts of mechanicsand be able to apply them to simple problems in thedeformation of continuous media
2. To understand the underlying basis for the mechanicalproperties of molecules, cells and tissues
3. To be able to model biological materials using methodsappropriate over diverse length scales
4. To be familiar with the wide spectrum of measurementtechniques that are currently used to determinemechanical properties
5. To appreciate the close interconnections betweenmechanics and biology/chemistry of living systems
20.310, 2.793, 6.024Fall, 2006
Biomechanics of tissues
Mechanics BiologyI. Linear elastic behavior I. Biochemical and
molecular biology ofII. Viscoelasticity
ECM moleculesIII. Poroelasticity
A. CollagenIV. Electrochemical and superfamily
physicochemical propertiesB. Proteoglycan
superfamily
C. Other glycoproteinsII. Nanomolecular
structures tissue
III. Mechanobiology20.310, 2.793, 6.024
Fall, 2006
2
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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
Youngs modulus; Poissons ratio, shear modulus
linear, isotropic, elastic materials
20.310, 2.793, 6.024Fall, 2006
20.310, 2.793, 6.024Fall, 2006
Force balance in a single cellDesprat et al.,
Biophys J., 2005.
Figure by MIT OCW.
3
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Stress distributions along the basal surfaceof a resting cell
20.310, 2.793, 6.024Fall, 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
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Tissue properties: common simplifyingassumptions
Linear-- the elastic modulus is constant,indpendent of strain amplitude
Homogeneous -- the material is spatiallyuniform
Isotropic-- the material exhibits the same elasticproperties in all directions
Time-independent-- stresses and strains areuniquely related, independent of rate of strain
20.310, 2.793, 6.024Fall, 2006
Constitutive laws for a linear elastic,
isotropic material= )+2G1 11 ( 11+22+33 11 (11 = 11 22 +33)
E 22 = 11+22+33)+2G22(
1( ) = )+2G22 =
E22 11 +33 33 ( 11+22+33 331 12 =2G12
33 =
E33 11 +22) 13=2G13 (
12 (1+)12 23=2G2312 = =
2G E
13 (1+)13 2G E = = = =13
2G E 12(1+
)(12
)
23 (1+)23
23 = =
2G E E= Youngs modulus = Lame constant= Poissons ratio G= shear modulus
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20.310, 2.793, 6.024Fall, 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-ACTINTENDON WOOD
TUBULIN
CONTRACTED SKELETAL MUSCLE
ELASTIN
RELAXED SKELETAL MUSCLECOLLAGEN GELS
LUNG PARENCHYMA FIBROBLAST CELLS
ENDOTHELIAL CELLS NEUTROPHILS LYMPHOCYTES
Values of the elastic or Youngs modulus (E) forvarious materials
20.310, 2.793, 6.024Fall, 2006
Linear? Unidirectional tensile tests
Stress-strain behavior of a peptide hydrogel
Linear behavior up tofracture
Relatively lowtoughness due to smallfracture strain
a bneedle needle
peptide
matrix
peptide
matrix
plastic
mesh
6
Figure by MIT OCW. Adapted from Leon, et. al., 1998
Constant Es11
e11
00.005 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
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7
Linear? Elastin is one of
main structuralcomponents of tissue
Linear; little hysteresis.
Provides thestretchiness of tissues.
Combination of single-molecule characteristicsand microscale structure.
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Linear? ACL -- different strain rates
20.310, 2.793, 6.024Fall, 2006
Figure by MIT OCW.
Figure by MIT OCW.
Slow Medium Fast
Range of linearity
E = ds/de = 109Pa
2 4 6 8 10 120
80
60
40
20
Strain(%)
Strass(MPa)
Control
100
80
60
40
20
05 10 15 20
Specimen
fixed at zero
stretch in 10% formalin
ELASTIN
Lig. Nuchae denatured
% Strain = (L/L0)*100
Stress(kPa)
The stress-strain curve of elastin. Data from Fung and Sobin (1981).
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Linear and Isotropic? Right tibia
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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's
aorta and a high-power view of the dense network of longitudinally
oriented 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
Stress(MPa)
Strain (%)
RA-direction
Sample no.
Sample no.
BA-direction
6
6
1
1
3
2
2
4
4
5
5
Longitudinal
direction
Radial direction
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20.310, 2.793, 6.024Fall, 2006
Collagen fiber arrangement in skin and cornea withalternating directions
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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 parallel
collagen fibrils in a tendon and the
mesh work of fibrils in skin removed
due to copyright restrictions.
Figure by MIT OCW.
Tendon
Fascicle
Tendon Hierarchy
X r ay EM X ray EM X ray EM
SEM
EM SEM
OM
SEM
OM
Evidence:
X- ray
Reticular
membrane
Fascicular
membrane
Waveform or
crimp structureFibroblasts
Tropocollagen
35 staining
sites
640
periodicity
15 35o
Ao
A
o
A
o
Ao
A
o
A
100-200 500- 5000 50-300 100-500
Size Scale
SubfibrilMicrofibril Fibril
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20.310, 2.793, 6.024Fall, 2006
Histological cross-sectionof a diseased carotid artery
stained for smooth musclecells.
Elastic response
initially, then stiff,collagen response athigh degrees ofextension.
H = hypertensive
High wall stressleads to functional
remodeling.
Time-dependent? Peptide gels Pre- and Post-
100 M KCl1000 M KCl
Storage Modulus (G)
Loss Modulus (G)
Gelation. Properties can change with time.1%wt KFE12
20.310, 2.793, 6.024Fall, 2006
10
Figure by MIT OCW.
Figure by MIT OCW.
1000
100
10
1
1
0.1
0.01
Oscillatory Frequency (rad/sec)
5 10
Moduli(Pa)
1000
100
10
1
1
0.1
0.01
Oscillatory Frequency (rad/sec)
5 10
Moduli(Pa)
100 M
1000
Cl
Cl
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 frequency
on 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)
Relativeradius
10 90 170 250 3300.1
1
10
Incremental modulus:
E = d/d
Incrementalelasticmodulus(MPa)
Stiffening
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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 inlung parenchyma
Surface tension effects20.310, 2.793, 6.024Fall, 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, g = 0.8% )
Equilibrium G, MPa
q.G
200
150
100
50
5 10 15 20
Liquid Inflation
Liquid
Air
Effect of
surface
tension of
gas-liquid
interface Effect of
collagen at
high degrees
of extension
Nearly linear response
at small extensions
Air Inflation
Relaxation Pressure
LungVolume(ml)
Von Neergaard Experiments
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20.310, 2.793, 6.024Fall, 2006
Striated
Othercomplications:
Activecontraction
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, 6.024Fall, 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
0
5 10 15 20
20
25
25
30 35
50 75 100
100
125 150 175
40
60
80
(mm)
(%)
Muscle length
Tension(mN)
Caption:A,resting tension;B, maximumtension produced by optimumstimulus;C, activetension (=B -A);
D,potentialed tension produced by successivestimuli.
Theslightest resting tension was produced at75% ofthe in situ length (20 mm).*
*
*
*
*
Theoptimum length, atwhich theactive tension becomemaximum, is between 100%and 125%ofthein situ length.
Theactive tension declines almostsymmetrically on eitherside ofthe optimallength.
Themaximumn tension potentialed by thesuccessive stimutiwas attained at75% ofthe insitu length.
Macroscopic view
120
100
Tension(%o
fmaximum)
80
60
40
20
01.0 2.0 3.0 4.0
Striation spacing (mm)
Many features of the length-tension relation are simply explained by the
sliding-filament theory.
The peak of the curve consists of a plateau between sarcomere lengths
of 2.05 and 2.2 mm.
Microscopsarcomere
ic -level
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Typical elastic behavior for tissues containingcollagen and elastin
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20.310, 2.793, 6.024Fall, 2006
Collagen fibril formation
Figure by MIT OCW.
Figure by MIT OCW.
STRAIN, (L/L0)
TENSILESTRESS,
(
F/A)
Young's Modulus = /
Elastin
dominated
Failure
Linear Region
Toe Region
Collagen dominated
Synthesis of pro- chain
Self-assembly OfThree pro- chain
2
1
4
5
3
6
7 8
9
Hydroxylation Of
Selected Prolines
And Lysines
Glycosylation Of
Selected Hydroxylysines
Propeptide
3 pro- chain
Procollagen Triple-helix
Formation
H2N
H2N
OHOH
OH
OH
OH
OH
OH
OH
OHOH
OHOH
OH
OH
OH
Secretory vesicle
ER/Golgi compartment
Collagen molecule
Self-Assembly
Into Fibril
Aggregation Of
Collagen Fibrils To
Form A Collagen Fiber
Collagen fiber
Collagen fibril
0.5-3m
10-300nm
Procollagen molecule
Secretion Plasma membrane
OHOH
OH
OH
OH
OHOHOH
OH
OH
OH
OH OH OHCleavage ofpropeptides
The Intracellular and Extracellular Events Involved in the Formation of a Collagen Fibril
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Collagen -- single moleculecharacteristics
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Fall, 2006
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As of 2003, therewere 28 (!)different types ofcollagenidentified.
14
Figure by MIT OCW.
Figure by MIT OCW.
Major Collagen Molecules
Type Molecule composition Structural features Representative
tissues
Fibrillar collagens
[ 1(I)]2
[ 2(I)]
[ 1(IV)]2
[ 2(IV)]
[ 1(VI)][ 2(VI)]
[ 1(IX)][ 2(IX)][ 3(IX)]
[ 1(II)]3
[ 1(III)]3
[ 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 fibrils
with globular N-terminal
domain; often with type I
Lateral association with type
I; periodic globular domains
Lateral association with
type II; N-terminal globular
domain; bound glycosami-
noglycan
Two-dimensional network
Skin, tendon, bone,
ligaments, dentin,
interstitial tissues
Skin, muscle, blood
vessels
Similar to type I; also
cell cultures, fetal
tissues
Most interstitial
tissues
Cartilage, vitreous
humor;
All basal laminaes
I
II
III
V
VI
IX
IV
Sheet-forming collagens
Cartilage, vitreous
humor
Figure by MIT OCW.
14
12
10
8
6
4
2
0
0-0
50 100 150 200 250 300 350
The force-extension curve of a single
collagen II.
Cover glass
XY Stage
Trap Center
Laser light
Procollagen
Stretching a procollagen II molecule with optical tweezers.
Figure by MIT OCW.
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Fall, 2006
Type IX collagen decorated with type II collagen
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Fall, 2006
15
Figure by MIT OCW.
Figure by MIT OCW.
Type IX collagen
moleculeFibril of type II
collagen
Stretch Relax
Elastic Fiber
Single elastin
molecule
Cross-link
Short section
of a collagen
fibril
collagen molecule
300 X 1.5 nm
50 nm
1.5 nmCollagen triple
helix
Contrast between elastin and collagen
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Entropic elasticity
Proteoglycans (PGs) and
glycosaminoglycans (GAGs)
a) GLYCOSAMINOGLYCANS (GAGs) form gelsi) polysaccharide chains of disaccharide unitsii) too inflexible and highly charged to fold in a compact wayiii) strongly hydrophiliciv) form extended conformations and gelsv) osmotic swelling (charge repulsion)vi) usually make up less than 10% of ECM by weightvii) fill most of the ECM spaceviii) four main groups
a. hyaluronanb. chondroitin sulfate and dermatin sulfatec. heparin sulfate and heparind. keratin sulfate
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Fall, 2006
16
Figure by MIT OCW.
Stretch Relax
Elastic Fiber
Single elastin
molecule
Cross-link
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b) Proteoglycans (PGs)i) form large aggregatesii) aggrecan is a large proteoglycan in cartilageiii) decorin is secreted by fibroblastsiv) PGs have varying amounts of GAGs.v) PGs are very diverse in structure and contentvi) PGs and GAGs can also complex with collagenvii) secreted proteoglycans have multiple functionsviii) PG/GAGs have important roles in cell-cellsignaling
20.310, 2.793, 6.024Fall, 2006
20.310, 2.793, 6.024Fall, 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 Biol98 (1985): 277-282.
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18
Figure by MIT OCW.
Figure by MIT OCW.
100 nm
GAG
Core Protein
Polypeptide chain
Short, branched
oligosaccharide
side 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 are
drawn to scale. The core proteins of both aggrecan and decorin contain oligosaccharide chains as well
as the GAG chains, but these are not shown. Aggrecan typically consists of about 100 chondroitin sulfate
chains and about 30 keratan sulfate chains linked to a serine-rich core protein of almost 3000 amino
acids. Decorin "decorates" the surface of collagen fibrils, hence its name.
Aggrecan Aggregate
Core protein
Link proteins
Chondroitin sulfate
Hyaluronan
molecule
Keratin sulfate
1 mm
Schematic drawing of an aggrecan aggregate. It is composed of 100 aggrecan
monomers noncovalently bound to a single hyaluronan chain through two link
proteins that bind to both the core protein of the proteoglycan and to the
hyaluronan chain, thus stabilizing the aggregate.
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Atomic Force Microscopy of Aggrecan
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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 GAGChains
(Fetal Bovine)
Ave. GAG length: ~36 nmInter-GAG spacing: 4-5 nm
(L. Ng, C. Ortiz, A. Grodzinsky)
Hyaluronan Molecule
Link Protein N-terminal Hyaluronan-
binding domain
Keratan Sulfate
Chondroitin
Sulfate
Linking SugarsAggrecan Core Protein
30
20
10
0
-10
0.5 1 1.5 2 2.5
Single Hyaluronan Molecule
Force(pN)
Displacement (mm)
A typical result of the force-displacement relationship from
the single hyaluronan molecule measurement.
Figure by MIT OCW.
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Like charge repulsion accounts for a large fraction (~50%) ofthe stiffness in tissues with high GAG content.
These effects can be eliminated either by shielding withcounter-ions or neutralization by changing pH.
+ + + + + ++
+ + +
++
++ +
+++ + +
+ ++ ++ +
+ +20.310, 2.793, 6.024Fall, 2006
20.310, 2.793, 6.024Fall, 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
otherEquilibriumModulus,MPa
Ionic Strength, M
Equilibrium Modulus of Adult Bovine Articular Cartilage
in Different Ionic Strengths
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ADHESION PROTEINSa) fibronectin
i) principal adhesion protein of connective tissuesii) fibronectin is a dimeric glycoproteiniii) fibronectin interacts with other molecules
b) laminini) found in basal laminaeii) form mesh-like polymersiii) has various binding sitesiv) assembles networks of crosslinked proteins
c) integrinsi) cell surface receptor, for attachment of cells to ECMii) family of transmembrane proteinsiii) two subunits, alpha and betaiv) about 20 different integrinsv) binding sites for ECM components
vi) binding sites for the cytoskeleton and linkage to ECM
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21
Figure by MIT OCW.
Some common proteoglycans
Proteoglycan
Approximate
molecular weight
of core protein
Type of GAG
chains
Number of
GAG chainsLocation 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 surface
and matrix
Widespread
in connective
tissue
Basal
laminae
Secretory vesicles
in white blood
cells
Fibroblast and
epithelial cell
surface
Mechanical support;
forms large aggregates
with hyaluronan
Binds to type 1
collagen fibrils
and binds TGF -
Structural and
filtering function
in basal lamina
Helps to package
and store secretory
molecules
Binds TGF -
Cell adhesion;
binds FGF
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Fiall, 2006
20.310, 2.793, 6.024
Fall, 2006
The bundle thickness, mesh size, elastic modulus, and critical strainas a function of R at cA = 11.9 M
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
0
G0 (Pa)
D (nm)
(m)
Bundle thickness
Mesh size
Shear modulus, G
Strain (21) at
which material becomes nonlinear
Collagen binding
Cell binding
Heparin
C C
N
N NH2
Arg
Gly
Asp
HOOC
100 nm
The Structure of a Fibronectin Dimer
SSSS
binding
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Fall, 2006
The stress-strain relationships of F-actin networks formed with different FLNmutants
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
Stress(Pa)
Strain,g
S
tress,s
s
g
ds
dg
10-2 10-1 100
100
101
101
102
102
103
103
Prestress(Pa)
Differential
Stiffness (Pa)
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Focal adhesion
complex
24
Figure by MIT OCW.
Figure by MIT OCW.
CYTOSOL
{{Lipidbilayer
Cell coat
(glycocalyx)
Transmembrane
proteoglycanAdsorbed
glycoprotein
Transmembrane
glycoprotein
Glycolipid
Simplified diagram of the cell coat (glycocalyx)
Sugar residue=
Fibronectin Extracellular
matrix
Cell membrane
Cytoplasm
Integrin
Focal adhesion
Actin
-
Actinin Signals that controlcellular activities
Talin
VinculinVinculin
Zyxin
Talin
Paxillin
Tensin
Ras
SOS
Grb2
FAK
Srckinase
Paxillin p130
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Viscoelastic models
k
FF
k FF
1k1
k2FF
u
u
u2u1
a)
b)
c)
Maxwell
Voigt
Standard linearsolid
Stress relaxation
Creep
Steady-state response tooscillatory displacement
F(t)=ku0exp(t/)
u(t)=Fk 1exp t/( )
=/k
E*()=E'+iE"Concept of a complex
elastic or shear modulus
Poroelastic materials
Governing equations:
1. Constitutive law
ijtot =2Gij +ijij pij
11
x1
u1
U k p =
1D forms
2. Fluid-solid viscous interactions (DarcysLaw)
3. Conservation of mass
uU=(vf vs) =vrel vs =
t4. Conservation of momentum
=0
20.310, 2.793, 6.024
Fall, 2006
tot
11 = (2G+)11pU
1=kpx
1U
1= u1 +U0t
11=0
x1
2u
1 u1=Hk
t x1
25
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Poroelasticity -- confined compression
Impose displacements at boundaries:u1(x1, t=0) = 0 11u1(x1=L, t>0) = 0
u1
u1
u1(x1=0, t>0) = u0
2u
1 u1 x1=Hk
t x1
Characteristic time ~ L2/Hk x1
Solution (Fourier series)
tu1(x1, t)=u0 1
x1 Ansin
nx1expL L
n n
L2
=n 2
2Hkn
20.310, 2.793, 6.024Fall, 2006
Finite rates of protein folding and unfolding can
also contribute to time-dependent behavior AFM can be used to measure both
the elastic (k) and viscous (z) properties of a single molecule as a
function of extension Sequential unfolding of the
immunoglobulin (Ig) domains of titinduring oscillations to measureviscoelasticity
Siingle molecule elastic and viscousproperties appear to scale witheach other
Kawakami et al., BJ, 200620.310, 2.793, 6.024Fall, 2006
26
Figure by MIT OCW.
Extension (nm)
k(pN/nm)
Force(pN)
(
10-7k
g/sec)
20
20
40
40
60 80 100
0
0
2
4
6
50
100
150
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20.310, 2.793, 6.024Fall, 2006
Additional factors: Ionic strength
+2GFigure by MIT OCW.
1.2
1
0.8
0.6
0.4
0.2
0.001 0.01 0.1 1
Electrostatic: GAG
otherEquilibriumModulus,MPa
Ionic Strength, M
Equilibrium Modulus of Adult Bovine Articular Cartilage
in Different Ionic Strengths