Homogeneous??
Cells in 3D matrix
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MDA-MB-231 breast cancer cells migratng
inside a collagen gel.
•  Dense cortcal actn with myosin.
•  Cross-linkers more homogeneously
distributed
Rajagopalan, unpublished
1
Indentaton by a microsphere
Importance of the cortex
Courtesy of Macmillan Publishers Limited. Used with permission.
Source: Moeendarbary, Emad, et al. "The Cytoplasm of Living Cells Behaves
as a Poroelastic Material." 1DWXUH0DWHULDOV 12, no. 3 (2013): 253-61.
2
Force balance in
the x3 direction
V1 (x1 )
p(x1 )
N1 (x1 )
Width w in x2-directon
X1
X2
θ (x1 + dx1 )
X3
θ (x1 ) ≈ ∂u3 / ∂ x1
V1 (x1 + dx1 )
h
dx1
V = ∫ σ 13dx3
N1 (x1 + dx )
0
pdx1 − V1 (x1 ) − N1 (x1 )θ (x1 ) + V1 (x1 + dx1 ) + N1 (x1 + dx1 )θ (x1 + dx1 ) = 0
Use Taylor expansions for V1 and N1θ1
V1 (x1 + dx1 ) = V1 (x1 ) +
Combine and divide by dX1:
∂V1
dx
∂x1
∂V1
∂
∂V1
∂ ⎡ ⎛ ∂u3 ⎞ ⎤
+
p(x1 ) +
( N1θ1 ) = p(x1 ) + + ⎢ N1 ⎜ ⎟ ⎥ = 0
∂x1 ∂x1 ⎣ ⎝ ∂x1 ⎠ ⎦
∂x1 ∂x1
3
Moment (torque) balance about the x2 axis X1
X2
V1 (x1 )
M 1 (x1 )
X3
∂M 1
− M 1 (x1 ) + (M 1 +
dx1 ) − V1 (x1 + dx1 )dx1 = 0
∂x1
∂M 1
= V1 (x1 )
∂x1
2
' ∂ u3
M1 (x1 ) = − K b 2
∂x1
4
∂
u3
∂ ⎛ ∂u3 ⎞
'
p − Kb 4 +
N1
=0
⎜
⎟
∂x1 ∂x1 ⎝ ∂x1 ⎠
M 1 (x1 + dx )
V1 (x1 + dx1 )
dx1
in 1D
4
Full governing equatons for linear deformatons, and the reduced forms for bending
or tension dominance
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
R
Molecular, Cell Tissue
=Biomechanics
radius of curvature
K b'
<< 1
2
Nλ
⎛ ∂ 2 u3 ⎞
⎛ 1⎞
p = −N ⎜
≅
N
⎜⎝ ⎟⎠
R
⎝ ∂ x12 ⎟⎠
p = pressure difference
N = membrane tension x = spatial coordinate
λ = characteristic length 5
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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