Tuesday, March 01, 2005

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Tuesday, March 01, 2005

• Geometric and Mechanical Properties
• Mechanical Statics
• Review-

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Thick walled sphere

• Equilibrium
• Pressure inside
• Average stress in wall
• Pressure from outside
• Pressurized both sides

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Charged polymers Electromechanical Chemistry
I.e. Alanine charge
Aqueous charge
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Shape Oblate sphere
Meridions
Curvature
Latitudes
Losing volume, not gaining area
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Slow cell squishing
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Curvature
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Membrane Tension
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Tension on membrane patch
Fappl
Ri
f
T
f
Rc
Tension force pulling down
FT 2 p Ri T sin(f)
Force Balance
Fappl FT Pp Ri 2
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Tangent-Curvature
t1
t2
R(s) position
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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.

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Step (1) Moment of Inertia of c.s.
For hollow cylinder, subtract the hollow portion.
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Step (2) Bending a rod
sDs
s
dA
y
R (at neutral surface) is assumed constant
on the small segment.
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Step (2) reiteration(Landau Lifschitz, 1986 ,
Theory of Elasticity)
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Step (2) continued Integrate
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Step (3) Moment due to appl F
P
P
h(x)
x
P P
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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.
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Step (5) Differentiate h twice
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• 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 Pgt Pbuckle
• Can a microtubule withstand typical forces in a
  cell?

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Buckling of Rods with Different Fixations
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Buckling of cell without reinforcement
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• 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.

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Material efficiencyStrength/weight
Square Bar
Rod
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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.
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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 _at_ 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, dgt 2RP/E . So for R 0.5 mM,
P 1 atm,
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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
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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
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Comparative Forces

• Adhesion force between proteins on cell and on
  matrix tens of pN.
• Spectrin spring constant 1-2 x 10 5 J/m2 so to
  stretch by 0.1 um takes 1 pN.

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Properties of the CSK

• A dynamic structure that changes both its
  properties and composition in response to
  mechanical perturbations.

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Pulling on CSK
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Uni- and Bi-axial Stress and Strain   Take the
case of unconstrained isotropic object compressed
in the y direction  
Before strain
After strain
 
x

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• 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

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• Because the transverse direction is
  unconstrained
• and,

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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
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Solving for ey we have the biaxial strain
equation  
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3-Dimensional stresses (stress tensor)
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Stress components _at_ Equilibrium
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Blood Forces
Y.C. Fung
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Analyze a Small element of upper EC membrane
(Also a mult-part solution)
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Analysis of EC upper membrane
Symmetrical
(Fluid Mosaic)
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On surface facing blood
On surface facing cytosol
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On surface facing blood
Define
We need membrane tension as f(t)
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(if Tx 0 _at_ x0)
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Stress on cell from flow
_at_ x -L
For t 1 N/m2 , L 10 mm, h 10 nm
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Shear stress from flow in a pipe
P1 P2
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Fluid Pressure is omnidirectional
Hence P1P2P3P4P5 P
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Two State Transitions
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Entropic springs
4-segment chain configurations
RNA
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22 nM
Sudden extensions of 22 nM (unfolding) when
forces above 14 pN are applied
tension
Small ree Many Config- urations
Large ree Few Configurations

Applying a tension to the zero ree state reduces
possible configurations to 10. S drops from
ln(16) to ln (10). Hence tension translates to
loss of entropy.
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Rate Constants
Kopen 0.9 sec-1 Kfold 8.5 sec-1
Kopen 7 sec-1 Kfold 1.5 sec-1
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RNA unfolding
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Coding of Probability
Integral pulse frequency modulation
Probability Pulse frequency and width
Modulation
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Pulse Width Modulator
Inputs
Leaky integrator
Thresholder
Pulses out
Reset
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Mechanical Models
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Voigt solution
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Classwork

• Make a simulink model of the RNA unfolding
  kinetics. Your model should be well documented,
  according to the following guidelines
• All parameter boxes should be labeled
• Document boxes should be included to describe
  operations
• Internal parameters, such as initial conditions,
  should be specified
• Sub-systems should be used so that the entire
  model can be fit onto 1 page and each sub-system
  can be printed separately, with documentation.
• A separate description of the system and all
  formulae should be made.
• Outputs should be the predicted, as well as
  measured probabilities
• A reasonable noise level should be placed in the
  model

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Control System, I.e. climate control
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Temperature Control
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G(s)
Y(s)
U(s)
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• Patterns on silicon with fibronectin.
• Cells grown on small pattern Apoptosis
• On a line they differentiate
• On a large surface, they grow.

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Mechanical Terms Review

• Statics and dynamics
• Kinematics and kinetics
• Vector and scalars
• Forces, resultants
• Deformation

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Homework

• Using the data shown in Figure previous, and the
  ground free energy, Fo 79 kT, graph the
  unfolding and folding probabilities, using Excel
  or other program. Put actual data points for the
  selected forces on your theoretical curve.

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Tensiometry
Plates coated with poly-HEMA Compression of
cells reduces the load measured by the balance by
an equivalent amount
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Liquid behaviour Surface tensions of embryonic
tissue
Liquid Properties
NR L H Ep M