JH>KHP&(<LAINAJHIM
1
'+%"6-($,B"',-"'
%.('C'))+(4"&.('D=
G * (ω ) ≈
2kBT
⎛ d ln ΔR 2 (τ ) ⎞
2
3π a ΔR (τ ) Γ ⎜ 1 +
⎟
d ln τ
⎠
⎝
(+.%
.';
5-()%,&"
.';
+%5
"1,"' )+.%,;
J
2
Three microstructural models for
the cytoskeleton
%%1%+,(%", © source unknown. All rights reserved.
This content is excluded from our Creative
Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
', +"-5
© Springer-Verlag. All rights reserved. This content is
excluded from our Creative Commons license. For more
information, see http://ocw.mit.edu/help/faq-fair-use/.
Source: Stamenović, D., and Donald E. Ingber. "Models of
Cytoskeletal Mechanics of Adherent Cells." Biomechanics
and Modeling in Mechanobiology 1, no. 1 (2002): 95-108.
"()(%5&+
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Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
3
Cellular Solids Model
(Gibson & Ashby,
1988, Satcher &
Dewey, 1997)
Φ (C,(%"+.('D
+(&'"' '%5,",
3!+
ε δ
ε = C'-3(+$&(1%1,D
ΦΦ
+"1,(%&'-,
(%"+.('<9
4
Tensegrity Model
L
δ
F/2
F/2
Tension
members
Deformation
D
f
ti
(+&.('
L
L1
Compression
members
L2
U ~ ∫ σ f 1ε f 1a dx + ∫ σ f 2ε f 2 a 2 dx
2
0
0
(+$('Q8C,-(+%,.'+ 5D
2
⎛δ ⎞
Fδ ∼ La 2 ⎜ ⎟ 2σ f 0 + E f
⎝ L⎠
(
)
σn
⎛ a⎞
En ∼
∝ ( 2σ f 0 + E f ) ⎜ ⎟
⎝ L⎠
δ L
(
2
)
∝ 2σ f 0 + E f Φ ∝ 2σ n0 + E f Φ
+,-+,,
('-+"1.('
"
%
"+&(1%1,
)''-
('-+"1.('
5
Tensegrity Model
F/2
L
F/2
Tension
members
Deformation
L
Compression
members
σ f 0 Φ 1+ 4 ε 0
En =
3 1+ 12ε 0
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from our Creative Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
Source: Stamenović, D., and Donald E. Ingber. "Models of
Cytoskeletal Mechanics of Adherent Cells." Biomechanics and
Modeling in Mechanobiology 1, no. 1 (2002): 95-108.
Where σf0 is the
prestress in the
individual tensile
elements and ε0 is the
initial strain in each.
See Stamenovic
for a full
derivation.
Or, in the limit of ε0 0,
Gn ~ σn0/3
where σn0 is the pre-stress
in the tensile elements per
unit total cross-sectional
area (σn0=πσf0a2/L2).
6
Biopolymer
Models
For a single segment of polymer
between cross-links (Isambert
and Maggs, 1997, Maggs, 1999,
Storm, et al., 2005)
lp = persistence length
l = distance between entanglements or
cross-links
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Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
ξ = filament spacing (pore size)
εn = network strain
En = network elastic modulus
δ = change in distance between
entanglements/cross-links
Φ = solid fraction
σn ∼ F ⋅
F=
εn =
lp
l4
K bδ
δ
l
filaments F
~ 2
ξ
area
Low cross-link
σ n l p Kb
E
=
Φ
~
density
n
ε n l 3a 2
Maximum
l p Kb
σ
cross-link
En = n ~ 5 Φ 5 /2
εn
a
density (l~ξ)
7
Scaling behaviors for the three models
Tensegrity
Predicts a linear dependence on prestress
G′ ~ 2σ n0 + E f Φ
Athermal
No ability to change +(,,-link density
No role for cross-link mechanics
Viscoelasticity?
Cellular Solids
© Springer-Verlag. All rights reserved. This content is
excluded from our Creative Commons license. For more
information, see http://ocw.mit.edu/help/faq-fair-use/.
Source: Stamenović, D., and Donald E. Ingber. "Models
of Cytoskeletal Mechanics of Adherent Cells." Biomechanics
and Modeling in Mechanobiology 1, no. 1 (2002): 95-108.
Filament bending stiffness dominates
Maximal cross-link density
Athermal
No role for cross-link mechanics
G′ ~ E f Φ 2
Viscoelasticity?
Biopolymer
Thermal (WLC at high extensions)
Viscoelastic. Captures ¾ power law at high
frequency
Cross-link density and mechanics?
G′ ~ K b2 Φ1 → K b2 Φ 5 /2
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This content is excluded from our Creative
Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
8
Computational Models of the Cytoskeleton
•  '"2"1%&('(&+,'+(,,B
%"'$"' )+(-"',,%,,&%"'-(
K'-3(+$
•  (-(+,'-(,"&1%-
-!-,(&5(,"'
•  -3(+$,+-!+&%%5.2<
1%%5K'4!""-&'5(-!
!+-+",.,(%%,'.'
%,
•  !'"%)+()+.,,1!,?<
@' '+-"'-+'%,-+,,
'+"%5(&)1->
© source unknown. All rights reserved. This contentis excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
"&<-%><((&)"(%<JHHP
9
$!(&&,, ,
•  %%,4!""-3$)(3+B%3+!(%( 5
•  "-!"'+,"' ,-+"'=%"'+<,-+"'B,.'"' <,-+"'
,('"' •  %1""6.('C+(%(+(,,B%"'$,;D
•  "&1%.(',,!(3-!--','-3(+$,+(',",-'-3"-!
&1!(-!(,+2,-.!2"(+
•  5-(,$%-%'-3(+$,!2-!+&%%5<'+(,,B%"'$
+1)-1+))+,-(&(+"&)(+-'--!'1'(%"' "'
'-3(+$1""6.(''+&(%"' •  (-(+.2"-5",," '"'--+&"''-('-3(+$
&(+)!(%( 5
•  (-(+,"'1)+,-+,,'-!+5.2%5('-+(%
5-(,$%-%,.',,
10
Membrane
mechanics:
Micropipette
Aspiration
Measurements suggest a
model consisting of a viscous
core and a membrane of
constant surface tension.
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Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
11
A fatty acid molecule (left)
and aggregation of phospholipids
to form a cell membrane (below)
© source unknown. All rights reserved. This contentis excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
Attractive: hydrophobic tails.
Repulsive: hydrophilic heads, ionic groups, steric
effects.
B. Alberts et al. (2004)
12
Figure 11-4 removed due to copyright restrictions.
Source: Nelson, David L., et al. Lehninger Principles of Biochemistry. Macmillan, 2008.
D. L. Nelson and M. M. Cox (2005)
Image of lipid bilayer removed due to copyright restrictions.
H. Lodish et al. (2004)
13
Lipid bi-layer
characteristics
Figure 11-1 removed due to copyright restrictions.
Source: Nelson, David L., et al. Lehninger Principles of Biochemistry. Macmillan, 2008.
Thickness ~ 5 nm
Normal (resting) tension ~ 0.01 mN/m
Maximum areal strain ~4%
Rupture tension ~10 mN/m
(Surface tension of water ~ 70 mN/m)
(1 mN/m = 1 dyn/cm)
D. L. Nelson and M. M. Cox (2005)
14
Red blood cells
(erythrocytes)
Uniform, disc-shaped normal
erythrocyte
Illustration courtesy of Blausen.com staff. "Blausen Gallery 2014". Wikiversity Journal
of Medicine. DOI:10.15347/wjm/2014.010. ISSN 20018762. CC license BY.
Images of red blood cell membrane removed due to copyright restrictions.
Source: Alberts, B., et al. Molecular Biology of the Cell. 4th ed. Garland Science, 2002.
Molecular Biology of the
Cell, Bruce Alberts, Dennis
Bray, Julian Lewis, Martin
Raff, Keith Roberts, James
D. Watson © 1994
15
White blood cells (leukocytes)
Images removed due to copyright restrictions.
''"' %-+('&"+( +)!,,!(3"' D'"'--E),,"2E'1-+()!"%,3"-!&'5&&+'(1,(%,'
&"+(2"%%"'D'1-+()!"%-!-!,'-+-3"-!LBM+"-('B-(",,(%235-!&&+''
%2-!1'+%5"' 5-(,$%-('"'-!4)(,(+.%+ "('>
!0)=AA&&,> +>1$>1A1%-5A+!(!&1-!>!-&%
16
(&( '(1,;;
%%,"'K&-+"4
Images removed due to copyright restrictions.
BBJKI+,-'+%%,&" +.' "',"(%% ' %>
•  ',(+.%.'3"-!&5(,"'>
•  +(,,B%"'$+,&(+!(&( '(1,%5
",-+"1-
# ()%'<1')1%",!
17
''-.('5&"+(,)!+
&)(+-'(-!(+-4
Courtesy of Macmillan Publishers Limited. Used with permission.
Source: Moeendarbary, Emad, et al. "The Cytoplasm of Living Cells Behaves
as a Poroelastic Material." Nature Materials 12, no. 3 (2013): 253-61.
18
Force balance in
the x3 direction
V1 (x1 )
p(x1 )
N1 (x1 )
N1 (x1 + dx )
h
V = ∫ σ 13dx3
V1 (x1 + dx1 )
0
pdx1 − V1 (x1 ) − N1 (x1 )θ (x1 ) + V1 (x1 + dx1 ) + N1 (x1 + dx1 )θ (x1 + dx1 ) = 0
,5%(+4)',"(',(+'θ1
V1 (x1 + dx1 ) = V1 (x1 ) +
(&"''"2"5=
∂V1
dx
∂x1
∂V1
∂
∂V1
∂ ⎡ ⎛ ∂u3 ⎞ ⎤
p(x1 ) +
+
( N1θ1 ) = p(x1 ) + + ⎢ N1 ⎜ ⎟ ⎥ = 0
∂x1 ∂x1
∂x1 ∂x1 ⎣ ⎝ ∂x1 ⎠ ⎦
19
Moment (torque) balance about the x2 axis
V1 (x1 )
M 1 (x1 )
−M 1 (x1 ) + (M 1 +
∂M 1
dx1 ) − V1 (x1 + dx1 )dx1 = 0
∂x1
∂M 1
= V1 (x1 )
∂x1
2
∂
u3
M1 (x1 ) = −K b'
∂x12
4
∂
u3
∂ ⎛ ∂u3 ⎞
'
p − Kb 4 +
N1
=0
⎜
⎟
∂x1 ∂x1 ⎝ ∂x1 ⎠
M 1 (x1 + dx )
V1 (x1 + dx1 )
dx1
in 1D
20
1%% (2+'"' *1.(',(+%"'+(+&.(',<'-!+1(+&,(+'"' (+-',"('(&"''
Bending stiffness
Membrane tension
4
2
⎛
⎛
⎞
∂
u
∂
u3 ⎞
'
3
Kb ⎜
−N⎜
−p=0
4 ⎟
2 ⎟
⎝ ∂ x1 ⎠
⎝ ∂ x1 ⎠
Bending K b' u / λ 4
Kb
∝
∝
>> 1
2
2
Nu / λ
Nλ
Tension
4
⎛
∂
u3 ⎞
'
Kb ⎜ 4 ⎟ = p
⎝ ∂ x1 ⎠
u = displacement
(%1%+<%%F",,1
p = pressure difference
R ="(&!'",
radius of curvature x = spatial coordinate
K b'
<< 1
2
Nλ
⎛ ∂ 2 u3 ⎞
⎛ 1⎞
p = −N ⎜
≅
N
⎜⎝ ⎟⎠
R
⎝ ∂ x12 ⎟⎠
N = membrane tension
λ = characteristic length
21
MIT OpenCourseWare
http://ocw.mit.edu
20.310J / 3.053J / 6.024J / 2.797J Molecular, Cellular, and Tissue Biomechanics
Spring 2015
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