20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression...

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* 2kG (ω ) ≈ BT

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⎛ d ln ΔR (τ ) ⎞3π a ΔR (τ ) Γ ⎜1 + ⎟

⎝ d lnτ ⎠

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Three microstructural models for the cytoskeleton�

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Source: Stamenović, D., and Donald E. Ingber. "Models ofCytoskeletal Mechanics of Adherent Cells." Biomechanicsand Modeling in Mechanobiology 1, no. 1 (2002): 95-108.

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Cellular Solids Model�

Φ ��(������*�����0������,��������� �����0����1���"����������23�������������������

ε ��δ���� ������ε = � �

� ���� ��*��2��&���!�!�,��

�� ������������������ ������

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(Gibson & Ashby, 1988, Satcher & Dewey, 1997)

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Deformation

F/2 F/2L

L Compressionmembers

Tensionmembers

Tensegrity Model�δ

U ~ σ f 1ε f 1a2 dx +

0

L1

∫ σ f 2ε f 2a2 dx

0

L2

∫6��&����7�8�*�������������"�,�

Fδ ∼ La2 δL

⎛⎝⎜

⎞⎠⎟

2

2σ f 0 + E f( )

En ∼σ n

δ L∝ 2σ f 0 + E f( ) a

L⎛⎝⎜

⎞⎠⎟

2

∝ 2σ f 0 + E f( ) Φ ∝ 2σ n0 + E f Φ

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f

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D f ti:0��������

5

Deformation

F/2 F/2L

L Compressionmembers

Tensegrity Model

Tensionmembers

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=πσf0a

2/L2).

Where σf0 is the prestress in the individual tensile elements and ε0 is the initial strain in each.

See Stamenovic for a full derivation.

σEn = f 0Φ

3

1 + 4ε 0

1 + 12ε 0

© Springer-Verlag. All rights reserved. This content is excludedfrom our Creative Commons license. For more information,see http://ocw.mit.edu/help/faq-fair-use/.Source: Stamenović, D., and Donald E. Ingber. "Models ofCytoskeletal Mechanics of Adherent Cells." Biomechanics andModeling in Mechanobiology 1, no. 1 (2002): 95-108.

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For a single segment of polymer between cross-links (Isambert and Maggs, 1997, Maggs, 1999, Storm, et al., 2005)

lF = p

l 4

δεn =l

filaments FF ~

Kbδ

σ n ∼ ⋅area ξ 2

Low cross-link σEn = n

l~ pK

density b Φ

εn l 3a2

Maximum σcross-link En = n

l~ pKb Φ5 /2

density (l~ξ) εn a5

Biopolymer Models�

lp = persistence length l = distance between entanglements or cross-links ξ = filament spacing (pore size) εn = network strain En = network elastic modulus δ = change in distance between entanglements/cross-links Φ = solid fraction

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Tensegrity Predicts a linear dependence on prestress

Athermal

No ability to change �����-link density

No role for cross-link mechanics

Viscoelasticity?

Cellular Solids Filament bending stiffness dominates Maximal cross-link density Athermal No role for cross-link mechanics Viscoelasticity?

Biopolymer

G′ ~ 2σ n0 + E f Φ

G′ ~ E Φ2f

Thermal (WLC at high extensions) Viscoelastic. Captures ¾ power law at high frequency G′ ~ K 2

b Φ1 → K 2b Φ5 /2

Cross-link density and mechanics?

Scaling behaviors for the three models

© source unknown. All rights reserved.This content is excluded from our CreativeCommons license. For more information,see http://ocw.mit.edu/help/faq-fair-use/.

© Springer-Verlag. All rights reserved. This content isexcluded from our Creative Commons license. For moreinformation, see http://ocw.mit.edu/help/faq-fair-use/.Source: Stamenović, D., and Donald E. Ingber. "Modelsof Cytoskeletal Mechanics of Adherent Cells." Biomechanicsand Modeling in Mechanobiology 1, no. 1 (2002): 95-108.

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Computational Models of the Cytoskeleton

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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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11

A fatty acid molecule (left) and aggregation of phospholipids to form a cell membrane (below)

Attractive

B. Alberts et al. (2004)

: hydrophobic tails.

Repulsive: hydrophilic heads, ionic groups, steric effects.

© source unknown. All rights reserved. This contentis excluded from our CreativeCommons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.

12

D. L. Nelson and M. M. Cox (2005)

H. Lodish et al. (2004)

Figure 11-4 removed due to copyright restrictions.Source: Nelson, David L., et al. Lehninger Principles of Biochemistry. Macmillan, 2008.

Image of lipid bilayer removed due to copyright restrictions.

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Lipid bi-layer characteristics

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)

Figure 11-1 removed due to copyright restrictions.Source: Nelson, David L., et al. Lehninger Principles of Biochemistry. Macmillan, 2008.

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Uniform, disc-shaped normal erythrocyte

Red blood cells (erythrocytes)

Molecular Biology of the Cell, Bruce Alberts, Dennis Bray, Julian Lewis, Martin Raff, Keith Roberts, James D. Watson © 1994

Illustration courtesy of Blausen.com staff. "Blausen Gallery 2014". Wikiversity Journalof 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.

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White blood cells (leukocytes)

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Images removed due to copyright restrictions.

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Courtesy of Macmillan Publishers Limited. Used with permission.Source: Moeendarbary, Emad, et al. "The Cytoplasm of Living Cells Behavesas a Poroelastic Material." Nature Materials 12, no. 3 (2013): 253-61.

18

Force balance in the x3 direction N1(x1)

p(x1)V1(x1)

V1(x1 + dx1)

N1(x1 + dx )

pdx1 − V1(x1) − N1(x1)θ(x1) + V1(x1 + dx1) + N1(x1 + dx1)θ(x1 + dx1) = 0

M��.������+���������0�����������θ1�∂V

V 11(x1 + dx1) = V1(x1) +

∂x1

dx

p(x1) +∂V1

∂x1

+∂

∂x1

N1θ1( ) = p(x1) +∂V1

∂x1

+∂

∂x1

N1

∂u3

∂x1

���1�������<��1�����-�

⎡ ⎛ ⎞ ⎤⎢ ⎜ ⎟ ⎥ = 0⎣ ⎝ ⎠ ⎦

������

���V = ∫

hσ13dx30

19

Moment (torque) balance about the x2 axis

M1(x1)

V1(x1)

V1(x1 + dx1)

M1(x1 + dx )

dx1

∂M−M1(x1) + (M1 + 1

∂x1

dx1) − V1(x1 + dx1)dx1 = 0

∂M1

∂x1

= V1(x1)

M1(x1) = −Kb' ∂2u3

∂x12

p − Kb' ∂4u3

∂x14 + ∂

∂x1

N1

∂u3

∂x1

⎛⎝⎜

⎞⎠⎟

= 0

������

���

in 1D

20

=���!��� �����N�.���!�/����3������

Kb' ∂ 4 u3

∂ x14

⎛⎝⎜

⎞⎠⎟

− N∂ 2 u3

∂ x12

⎛⎝⎜

⎞⎠⎟

− p = 0

Kb' ∂ 4 u3

∂ x14

⎛⎝⎜

⎞⎠⎟

= p

Kb'

Nλ 2

�!���"�<����"�O!������0���������0�������� �����3��!��0�����0���1���"�������������������

<< 1

Bending stiffness Membrane tension

Bending

Tension

u = displacement p = pressure difference N = membrane tension

R = radius of curvature x = spatial coordinate λ = characteristic length

∝ Kb' u / λ 4

Nu / λ 2 ∝ Kb

Nλ 2 >> 1

p = −N∂ 2 u3

∂ x12

⎛⎝⎜

⎞⎠⎟

≅ N1

R⎛⎝⎜

⎞⎠⎟

21

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20.310J / 3.053J / 6.024J / 2.797J Molecular, Cellular, and Tissue BiomechanicsSpring 2015

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