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Homogeneous?? Cells in 3D matrix )LJXUHVUHPRYHGGXHWRFRS\ULJKWUHVWULFWLRQV 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 • Cross-linkers more homogeneously distributed Rajagopalan, unpublished 1 Indentaton by a microsphere Importance of the cortex Courtesy of Macmillan Publishers Limited.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 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.