Constitutive relation Conservation of mass Fluid-structure interaction: Darcy’s lawPoroelasticity in 1D Confined Compression(ε22 = ε33 = 0) Assumptions: Solid: Hookean (linear elastic, or viscoelastic), homogeneous, isotropic, incompressible Liquid: Inviscid (fluid friction negligible or μ=0, but there is solid-liquid friction), incompressible Constitutive relation Conservation of mass Conservation of momentum Fluid-structure interaction: Darcy’s law ()pHpGtot−=−+=1111112εελσ with 1111xu∂∂=ε (1) ()(sftotfsfvvAAvvUtuU −=−=+∂∂−=φ011) (2) 0111=∂∂xσ (3) 11xpkU∂∂−= (4) Diffusion equation For stress relaxation 21121xuHktu∂∂=∂∂ ()∑⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎠⎞⎜⎝⎛−⎟⎠⎞⎜⎝⎛−=ntLHknLxnnuLxutxu222101011expsin21,πππ Cell-Membrane Mechanics• Stretching • Bending • Shear • Composite Equation ()()ννενσσ−=Δ=Δ−=−===∫−1212100111122311EhKAAKAAEhEhhdxNeehh Typical Ke (N/m): Lipid bilayer ~ 0.1, red blood cell ~ 0.45, water /air ~ 0.07 Nrupture(N/m) ~ 0.04-0.06 2132213223223311)1(12 xuKxuvEhdxxMBhh∂∂=∂∂−−==∫−σ ()23112ν−=EhKB Typical KB (Nm): 10-18-10-19 hGKKhGhNss2221212121====εεσ Typical Ks (N/m): 10-6 (red blood cell) General: 03234=∇+∇− uNuKPB1D: 041342132=∂∂−∂∂+xuKxuNpB 2~NLKtensionbendingB Filipod Extension 3210~−NLKB RNRRNp2~)11(~21+Δ Neutrophil aspiration At max ∆p (stable shape) ccmRNPaNPP2;2==Δ+ If Pm=Pc ⎟⎟⎠⎞⎜⎜⎝⎛−=ΔcRaaNP 12* Cell Peeling κρ34322~~10~uLuKpNLKABB ρA = receptors/unit area h L0 M h σ21Rc, Pc 2a, ∆P RmPmκ = spring constant = bond stiffness Cytoskeletal Mechanics Structural elements: - actin: microfilaments - tubulin: microtubules - intermediate filaments Scaling & dimensional analysis Lx1~∂∂ ωτ~1~t∂∂ ∫fLfdx ~ ∫τffdt ~ Ignore first order dimensionless constants Governing equations Solid (Hookean) • Force balance (inertia neglected) • Constitutive equations AFxiijσσ==∂∂;0 pGGijijkkijijijkkij−+=+=εδλεσεδλεσ22 Note: δij=1 if i=j, and δij =0 if i≠j • Energy ∀∀⋅=∫2~21εεσEdU Experimental measurements • Cell indentation 32322222~~~~~~~~aaGaGEFUaGaaGaGaF⎟⎠⎞⎜⎝⎛∀δεεδδδεσ • Cell Squashing 212322232~~)(~~~RGFRaRRaaGFaGFUδδδδεδ⇒=−+ u3δ L G v 2 a F δ F δ 2a ν R G,• Magnetic twisting cytometry (relate torque to bead rotation angle to get G* = G’ + iG”) • Passive Particle Methods (particle tracking, Brownian motion) Microstructural models of the cell: • Cellular solids : 2ϕfnEE =ϕPEn~• Tensegrity: (P = press stress in tensile elements as ε→0) 255~ϕaKlEbpn• Biopolymer: at maximum cross link density ϕ23~alKlEbpn at low cross link