Tuesday, March 01, 2005 • Geometric and Mechanical Properties • Mechanical Statics • Review-
Thick walled sphere • Equilibrium • Pressure inside • Average stress in wall • Pressure from outside • Pressurized both sides
pKa =2.7 pKa =9.9 +H3N-CH-COO- CH3 Charged polymers: Electromechanical ChemistryI.e. Alanine charge Aqueous charge
Shape : Oblate sphere Meridions Curvature Latitudes Losing volume, not gaining area;
Tension on membrane patch Fappl Ri f T f Rc Tension force pulling down: FT = 2 p Ri Tsin(f) Force Balance Fappl + FT = Pp Ri 2
Tangent-Curvature t1 t2 R(s)= position
Forces on Rods • Does compressive force play a role? • Failure mode is buckling-To analyze must consider geometry when it buckles- • (1) get m.o.I; • (2) general formula for moment in the rod. (3) moment as a fxn of applied F. • (4) relation between R of curvature and x, (5) simplify eqn.
Step (1) Moment of Inertia of c.s. For hollow cylinder, subtract the hollow portion.
Step (2) Bending a rod s+Ds s dA y R (at neutral surface) is assumed constant on the small segment.
Step (3) Moment due to appl F P P h(x) x P P
Step (4) Minus sign because Curvature is negative. From before: Note similarity to harmonic Motion : (5) Hmax occurs at Lc/2 and h(0) = h(Lc)= 0.
Use spring equation. Hmax occurs at Lc/2. h(0) = h(Lc)= 0. We can relate F to Lc by double differentiating h, and then comparing it to the previous formula for the moment. • Buckle force is independent of hmax . Rod will buckle when P> Pbuckle • Can a microtubule withstand typical forces in a cell?
Buckling of Rods with Different Fixations
Living cells are both affected by and dependent upon mechanical forces in their environment.Cells are specialized for life in their own particular environments, whose physical stress patterns become necessary for normal functioning of the cells. If the forces go outside the normal range, then the cells are likely to malfunction, possibly manifesting as a disease or disability.
Material efficiencyStrength/weight Square Bar Rod
Fiber orientation for strength A: Actin fibers in two C2C12 cells. B,C: C2C12 cell with a schematic representation of the actin cytoskeleton, which is predominantly orientated along the first principal axis of the cell. As a result of the actin fibers, deformation of the cell and its nucleus is restricted in this direction.
Cell Walls for strength How thick does wall need to be to withstand normal pressures inside a bacterium, I.e. 30-60 atm. ? Lets say lysis occurs @ 50% strain. We can approximate KA By KVd, and for isotropic wall material, Kv ~ E, so, tfailure= 0.5 KA= RP= 0.5 E d. So to not fail, d> 2RP/E . So for R = 0.5 mM, P= 1 atm,
Homogeneous rigid sheet: Biomembrane Stretching membrane thins it exposing hydrophobic core to Water. Rupture at 2-10% area Expansion, so say lysis tension ~ 0.2 J/M2. For a 5 mm cell , P= ~ 8000 J/M3 ~ 0.08 atm. at rupture. Bilayer compression resistance, KA = 4 g g= 0.04 J/M2
Comparative Forces • To pull a 5 mm cell at a speed of 1 m/sec: • F= 6phRv = 0.1 pN • Compare this with force to bend or buckle hair, 10 cm length, R = 0.05 mm: • 5 x 10 4 pN • or to move it 1 cm: • F = 3 kf z/L3 = 1.5 x 10 6 pN F
Comparative Forces • Adhesion force between proteins on cell and on matrix: tens of pN. • Spectrin spring constant = 1-2 x 10 –5 J/m2 soto stretch by 0.1 um takes 1 pN.
Properties of the CSK • A dynamic structure that changes both its properties and composition in response to mechanical perturbations.
Uni- and Bi-axial Stress and Strain Take the case of unconstrained isotropic object compressed in the y direction: Before strain After strain x
Note that for an elastic material the strain occurs almost instantaneously upon application of the stress. Also note that to maintain constant stress, sy , the applied force must be reduced if the face area increases, but this would be a negligible change for all practical situations. • The strain in the y direction is:
Now, Consider the case where the x direction is constrained from movement. I.e. transverse movement is resisted, making Thus the new stress in the y direction is the original unconstrained stress plus the stress caused by transverse constraint:
Y X Z 3-Dimensional stresses (stress tensor)
Blood Forces Y.C. Fung
y x Flow z Cell 1 cell 2 cell 3 Analyze a Small element of upper EC membrane : (Also a mult-part solution)
y x z Analysis of EC upper membrane Symmetrical (Fluid Mosaic)
y x z On surface facing blood On surface facing cytosol
y x z
y x z On surface facing blood Define We need membrane tension as f(t)
y x z (if Tx= 0 @ x=0)
Stress on cell from flow @ x = -L For t = 1 N/m2 , L= 10 mm, h = 10 nm