February 8, 2005 :

cypriotcamelUrban and Civil

Nov 29, 2013 (3 years and 6 months ago)

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February 8, 2005 :



Homework


Read Ch. 4,
5; Appendix


Biophysics
Review


Cell Types;


Mechanical
Testing


Percolation


Kinetics

Prokaryotes


Most have elevated osmotic pressure, I.e a few
tens of atmospheres.


Challenger Deep sea dropped to 10,896 m in last 8
million years (Deepest place in ocean)


Deficient in CaCO
3
at that depth


Foraminifera (shelled protists) quickly evolved
soft, non
-
calcareous shells


Likely are highly pressurized.



Biomembrane as an isotropic
material

Bilayer compression resistance, K
A
= 4
g

g= 0.04
J/M
2 (
g =
surface tension
)

Homogeneous lipid sheet:
Biomembrane

Stretching membrane thins it

exposing hydrophobic core to

Water. Rupture occurs at 2
-
10% area expansion, so say
lysis tension
~ 0.016 J/M
2
. For
a 1
m
m cell

:

P= 64,000

J/M
3

~ 0.6 atm. at

rupture.




=

=

A
m
K
T
A
A

10 atm = 10
6
J/m
3


Life @ 1,200 atmoshperes


How thick does the membrane need to be?


How thick does the wall need to be?


Compressibility properties:

p
V
A
V
V
p
V
A
d
K
K
E
K
K
d
K
K

=


=
m
m
3
/
9
/
4
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 when wall tension exceeds 5% of K
A
.
We can approximate K
A

by K
V
d, and for isotropic wall
material, Kv
~ E, so, assume a material E= 3 x 10
9

J/m
3

t
failure
= 0.05 K
A
= RP/2=0.05 E d


So to not fail,


d> RP/E


So for R = 0.5
m
M, P= 10 atm,



nM
x
x
d
3
5
10
3
10
10
5
.
0
9
6
6
=



Thick wall sphere

Thick walled sphere


Equilibrium


Pressure inside


Average stress in wall



Pressure from outside



Pressurized both sides



h
p
r
h
p
r
r
r
h
r
p
r
r
h
r
p
r
r
r
p
p
r
r
r
o
i
P
P
i
o
o
o
P
i
o
i
i
i
o
i
i
P
i
i
P
i
o
out
in
out
in
in
2
2
)
(
)
(
)
(
)
(
2
2
2
2
2
2
2
2

=


=

=

=
=








Alternate method (not too thick
wall)

max
2
2



E
h
Rp
h
Rp
fail
o
i
=

=
Laplace Law for Cylinder under pressure

Homework:

1.
Find wall stress for cylinder.

2.
Calculate stress for a foraminefora not

Assuming a thin wall.


Compression of a network


A cartesian lattice of fibers (a) subject
to compressive strain (b). The
boundary conditions are that the
angles between the segments arising
from each junction are fixed at 90

and the deformation consists in
movement of the junctions toward
each other along the three orthogonal
axes of the lattice, with no shear
deformation. The junctions, however,
are allowed to rotate as


compression progresses. This is
equivalent to the boundary conditions
in Feynman
et al
.
36
in which fiber
ends are fixed but direction changes
under compression.


Solution to
Donnan
Problem

mV
mM
A
A
solutes
solutes
mV
C
C
z
E
HCO
HCO
Cl
Cl
K
K
Na
Na
i
i
in
out
o
i
o
i
i
o
i
o
03
.
1
5
;
29
114
4
144
]
[
]
[
03
.
1
)
log(
58
144
150
]
[
]
[
]
[
]
[
]
[
]
[
]
[
]
[
3
3
=


=


=

=
=
=
=
=
=











A.
B.
C.
D.
Power from electrochemical
gradients (I.e batteries)

Driving Force determined by Nernst

Distributed Model

Electrical Model of Cell
Membrane

Inside
Outside
E Na
gNa
E K
g K

C m
)
(
m
x
x
x
x
x
V
E
N
p
g
I



=
Molecular model

Ca Wave in Oocyte

Mechanotransduction

What opens channels?

Types of mechanical analysis


Kinematics
-

just the connections


Statics
-

forces without motion


Dynamics
-

forces with motion



Rigid versus deformable body



FBDs

F
EL

F
BL
F
BR
F
ER

Loading Types


Tension
-

compression


Shear


Reaction


Traction


Friction


Bending


Uniaxial/bi
-
axial



Cytomechanical forces:


Gravitational:


Muscle contraction:


Contact:


Buoyant:


Hydraulic: (Static or dynamic)


Pneumatic



Cell Deformation and Stiffness


Most cells are constantly deformed
in vivo

by both internal and external forces.


Experimental deformations can be done by
poking, squishing, osmotic swelling,
electrical/magnetic fields, drugs, etc.


Cells have both area and shear stiffness,
mostly due to the cytoskeleton, although
lipids contribute some.


Material Parameters


Moduli: Young’s, area, shear, bending (flexural)


Stiff versus compliant


Strength versus weakness


Brittle versus ductile


Incompressible/Compressible


Failure


Ultimate tensile strength


Hardness: Moh’s scale


Comparative Mechanical
Properties

Strain


Steel





Wood


Bone



Cells

Steel Wood Bone






Cells

Cellular

‘pre
-
stress’

Comparative Stiffness
0.0002
0.007
14
21
210
1200
0.0001
0.01
1
100
10000
tissue
rubber
wood
bone
steel
diamond
Material
Modulus (GPa)
Elasticity


“ut tensio sic vis”


Young’s Modulus: Stress over strain


Shear Modulus: Related to Poisson


Comparative Strains


Comparative Stiffnesses




Y
0
0.7
X
1
2
For most engineering materials,

< 0.3
Materials with

= 0.5 are "Incompressible."
Some materials have

> 1
v =-(.7-1)/1 = 0.3
Poisson’s Effect

swelling

Incompressible

Means no volume change

Cauchy Strain

x
l
l
o

l
o

y
l
y
l
yo

l
yo
Poisson's Ratio


y

x

Elastic Behaviours



< 1

< 0

E =
/

K
A
= P/

A/A

Unixaxial stress Pressure


1 2

Applying forces (testing types)

Tension or
Compression
Uniaxial
Biaxial
Shear
Pressure
Bending
Twisting
Q: What are the relative resolutions?


Testing methods

AFM

Q? Why doesn’t the AFM needle poke right through?

Micropipet methods

Tension
Micropipette
Q
p
Q
p
Q
m
Mesangial
Cell

C
i

C
o
P
p
(K )
w
P
i
C =constant

i
A. Whole Patched Cell
Micropipette
Stretch
Mesangial
Cell

C (t)
i

C
o
P
i
Q
m
w
(K )
Stretch
Solutes
B. Isolated Cell
Solutes
Magnetic tweezers

Pulling on CSK
Wang et al, Science

Shear and compression

Example: Blood flow forces

Optical Tweezers


High resolution


Refractivity of bead


Trapping in the beam


Limited force

Optical Tweezer




Necturus

erythrocytes loaded with fluo
-
4 (10 µM) and exposed to UV
light emitted from a mercury vapor bulb and filtered through a FITC cube
(400
x). (A) Cells display little fluorescence under isosmotic
conditions (
n
=6). (B) Addition of A23187 (0.5 µM) to the
extracellular medium increased fluorescence under isosmotic
conditions (
n
=6). (C) Exposure to a hypotonic (0.5x) Ringer solution
increased fluorescence compared to basal conditions (
n
=6). (D) A
low Ca
2+

hypotonic Ringer solution (5 mM EGTA) did not display
the level of fluorescence normally observed following hypotonic
swelling (
n
=6).


Light et al.

Swelling

RBCs

Stimulation Protocols

Impulse Step Sinusoid Ramp
TIME
Magnitude
Figure
4.2 Modes (top) and timing protocols
(lower) of force application
Bone Loading Waveforms

Sickle Cell: A gel problem


Single point defect causes Hbs
-

a polymerizing
tendency in deoxygenated state


The stiff and deformed cells damage vessels


Main approaches:


1. Controlling kinetics of polymerization


2. Regulating stiffness (rheology) of sickle cells.

Thermal shape variations

Stiff

Flexible

(a)
-
(c) Serial images of a 23
m
m long
relatively stiff fiber. (b) and (c) are,
respectively, 21.9 and 41.4


seconds after (a). There is little visible
bending (see also
Figure 2(a))
,
consistent with a long persistence
length,
l
p
.
12.0 mm. (d)
-
(f) Serial
images of a 20
m
m long ¯flexible
fiber. (e) and (f) are, respectively,
51.8 and 60.8 seconds after (d).
There is marked bending and a short
persistence length,
l
p
.
0.28 mm (see
also
Figure 2(b))
. The fibers undergo
diffusional motion and hence are not
adhering to a glass surface, rather
are free in solution, a necessary
condition for using statistical
mechanics to obtain persistence
lengths. The width of each frame is
25
m
m.


Statistics of fluctuations in 1
dimension

Statistical Mechanics

2
2
=
=

=
R
R
kT
p
p
f
p




2
3
)
2
(
48
L
u
L
x
p
=

Dilute Semi
-
concentrated
Concentrated

Floppy

Chains











Rods

Isotropic Nematic

Harmonic motion (undamped)

0
)
(
)
(
)
(
2
2
0
=


=


=






x
x
x
k
mx
x
x
k
t
PAu
x
m

Gel motion follows simple rules

Model will predict dynamic and

Static equilibrium.

Natural Frequency

Damped Spring


Viscosity & Elasticity


A complex material can be modeled as a purely
viscous material combined with a purely elastic
material, thus mathematically separating the
viscosity of a material from its elasticity. A purely
viscous component is a Newtonian fluid
-

it has no
memory and no elasticity; it cannot deform as a
solid. Cells generally behave as solid
-
liquid
composites. V
-
E tools can quantify their
behaviour, since the models separate viscosity
from elasticity in a kind of finite element model.


Maxwell Model: Differential
method

)
(
1
1





















=

=

=
=
=

=



E
dt
d
E
dt
d
E
E
T
E
T
1/E


Maxwell model: Laplace Method

1/E


V

R
C
s
E
s
s
Z
s
E
s
Z
o
o








/
1
/
1
1
1
)
(

=
=
=

=
Viscosity: Pascal
-
sec

For a step input

Mechanical

Impedance.


Compliance


+

Slipperiness

t
=

/E

Gel Model


Make a complete model and label all
parameters


Describe the output, relating what happens
and why.


What is the time constant?


State the assumptions and simplifications

Mechanical Terms Review


Statics and dynamics


Kinematics and kinetics


Vector and scalars


Forces, resultants


Deformation


Classwork


Add damping to your model of cytogel


Describe how you can model thermal
fluctuations in cell diameter, and list all the
elements. List assumptions.


Write the model equation for the above.


Complete a simulink model of the above,
and do all labelling, including all parameter
values.