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

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

Page 1: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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Page 2: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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������ !���"����������

* 2kG (ω ) ≈ BT

22

⎛ d ln ΔR (τ ) ⎞3π a ΔR (τ ) Γ ⎜1 + ⎟

⎝ d lnτ ⎠

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Page 3: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

Three microstructural models for the cytoskeleton�

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

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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Page 4: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

Cellular Solids Model�

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

ε ��δ���� ������ε = � �

� ���� ��*��2��&���!�!�,��

�� ������������������ ������

�������!���0�4�������

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

%����0������ �5�

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Page 5: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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

9�������������1!���� ��1����!�!��

�����������1!����

f

��1 �

D f ti:0��������

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Page 6: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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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Page 7: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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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Page 8: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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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Page 9: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

Computational Models of the Cytoskeleton

•  ;��<�!�������������������'���&��"�����������0�����1����������:���2��&�

•  =����������1���������!�����3� �����0��������;;�

•  >�2��&������3����������< �0!�����:����+3�1���������0��3��3�������������0��������������"���

•  =�3����������������!�3����#? �#@����"�����������������������1�����������!���

A�� ������ �9B�%������/��� ������

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

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Page 10: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

.�&�3�������"��•  �����+3�1�����2�&���2�'��2��3���"��•  6��3���������"�������-������ �������'�� ���" ����������C���"�

•  ��!��$�����*�����0������'���&��,�•  %��!��������3�2��3���������2��&���������������2��3��!�3��0��3��1��<�������13�<����

•  �����&�������2��&��13�<���3������ ���������'���&��!��!������������1����������������3���!�0����"������2��&�D!��$��������������"�

•  =��������<�����������"��4���������������0���2��&�����3���"��

•  =��������!��������������3�1�����<����������������&������� ����

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Page 11: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

Membrane mechanics:

Micropipette Aspiration

Measurements suggest a model consisting of a viscous core and a membrane of constant surface tension.

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Page 12: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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.

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Page 13: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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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Page 14: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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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Page 15: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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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��<��3�!������"������&���������3�+�������������"������3G�-������"��!&�!����!�����3��3�!�3�3����

Images removed due to copyright restrictions.

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H���"��!�������������:������+�

�=:I'=/'����1������������������"����"�������������"��"���•  :����������������2��3�����������•  �����'���&�������3���"��!����

�����1!��� � �)�K�"������ �!��!1���3� �

� �� � � ��

Images removed due to copyright restrictions.

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;��������1�����������3��;����������0��3�����+�

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.

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Page 19: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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

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Page 20: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

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

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Page 21: 20.310J Molecular, Cellular, and Tissue Biomechanics ... · Deformation F/2 F/2 L L Compression members. Tensegrity Model. Tension members. Or, in the limit of ε. 0. 0, G. n ~ σ.

=���!��� �����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⎛⎝⎜

⎞⎠⎟

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