Transcript Third Class Wednesday, January 28, 2004

Third Class
Tuesday February 2, 2005
• Previous: Cell Biophysics
•
Protein charge
•
Electroosmosis
•
Swelling & Stretching
•
Equilibrium
•
Gel
•
Model
Diffusion (Correction)
m
C
( ) x1   DA( ) x1
t
x
m
C
( ) x2   DA( ) x2
t
x
m
C
C
( ) x  DA[( ) x2  ( ) x1 ]
t
x
x
C
C
 C
( ) x2  ( ) x1  ( )x
x
x
x x
AxC
2
(
) x  DA 2 Cx
t
x
Corrections from
last time
m  AxC
m
C
 Ax
t
t
Types of transport
• Diffusion:
– Seconds response, random
targeting, gradient driven
• Microtubule tracking
– Milliseconds, point-to-point
target, energy driven
• Actin/myosin scaffolding
– Seconds to minutes, gradient
targeting, energy driven
• CSK Ringing
– milliseconds, global or
channeled, pre-stress driven
Intelligent Gels
Ca++ or Mg ++ can do the same and are reversible
Gelation: Phenomena
• by Non-polar solvent
• Electrical conductivity
and elastic modulus
are zero below
threshold, then
suddenly
•In solution, a measure of polymerization
is viscosity.
Tissue morphogenesis by gel
Electric gel motor
Chemical versus mechanical
signalling
Redundancy=reliability
Ties can break without
Losing NY-LA cnxn until
p< pcrit
When [branching polymers]
>
[
]crit
gel
Sequential images (left to right) from computer simulations of multimodular tensegrities (A,C)
or from time-lapse video recording of living cells (B,D)
Prestressed
fabric
Add
Trypsin
Ingber, D. E. J Cell Sci 2003;116:1157-1173
Rhodamine-phalloidin staining of filamentous actin in
HUVECs in the absence (A) and presence (B) of cytochalasin
D (0.1 µg/ml for 30 min). Note that in B, although some
RGD-coated beads were still stained with rhodaminephalloidin, the integrity of the actin CSK was disrupted, and
most actin bundles disappeared. Scale bar = 20 µm.
Tensegrity versus Percolation
• Geodesic dome follows
strict rules, fixed tie
lengths, fixed tension, and
collapses when one is
broken.
• Perc model is random in
orientation, tie length, and
tension.
• Properties: Redundancy
(reliability), fractal, signal
channeling, ie analogy to
prey falling on spider
web; adaptibility.
Three sequential fluorescent images from a time-lapse recording of the same cell expressing
GFP-tubulin showing buckling of a microtubule (arrowhead) as it polymerizes (from left to
right) and impinges end-on on the cell cortex at the top of the view [reproduced with
permission from the National Academy of Sciences (Wang et al., 2001)]
Ingber, D. E. J Cell Sci 2003;116:1157-1173
What determines connections?
• What happens if the
chain density is so low
that a connected lattice
doesn't exist?
• Consider the twodimensional square
lattice, on which
bonds have been
placed randomly
p low
p higher
Percolation (gelation)
• Consider a model: site percolation on a square
lattice. This uses an L x L square matrix of 1s and
0s, called the site matrix. A 1 represents an
occupied site and a 0 an empty site. The sites are
occupied randomly with some site occupation
probability, p. A cluster is a set of occupied sites
all of which are connected either horizontally or
vertically, i.e. an occupied site belongs to a cluster
if a member of the cluster is either above, below,
left, or right of it. A spanning cluster has an
element in both the top and bottom rows of the site
matrix.
Bond/site percolation
Probability of connection
• how does the probability of a site matrix having a
spanning cluster depend on the site occupation
probability p? Site percolation is an example of a
critical phenomenon. There is a special value of
p, called the critical site occupation probability
pc, such that for p < pc spanning clusters never
occur. While for p > pc they always occur. The
case p=pc is called a critical point.
Critical point: Phase transition
• At critical points a qualitative transition in the
behaviour of a system occurs: typically between
ordered and disordered states. In gelation, the
transition is from no spanning clusters to always
spanning clusters. Such qualitative changes are
known as phase transitions. Other examples are
the freezing of water and demagnetisation of a
ferromagnet at the Curie temperature. Strictly,
critical points only exist for infinite systems. For
finite systems, like we will investigate using the
computer, the transition is not sharp but smeared
out over a parameter range.
Critical probability
• for infinite systems, a connecting path
appears at the connectivity percolation
• threshold pC
 pC = 0.5 for a square lattice in 2D and pC
~ 0.35 for a triangular lattice in 2D
Direct approach
• One approach to finding
the critical site occupation
probability is to look at
every possible LxL matrix.
Since there are L2 matrix
sites and each can have
two values the total
number of site matricies is
2L2. Here is how this
number grows with L:
L
1
2
3
4
5
6
All combinations of matrices
L
2L2
1
2
3
4
5
6
2
16
512
65,536
33,554,432
6.87 x 1010
Monte Carlo short-cut
• A powerful method for dealing with the problem is
the Monte Carlo method. This involves random
sampling of the entire ensemble to obtain a
(hopefully) representative sample. Deductions are
then made from this sample and assumed to hold
for the entire ensemble.
• We will apply the Monte Carlo method to our
problem by generating small samples of random
matricies. We use these to estimate the critical site
occupation probability and a critical exponent.
Stress and Laplace Laws
Rectangular bar
F
F

A
Cells and balloons
F
Membrane Tension
R
3. Patch in x-z plane
dx
P
1. Hemis phere
Tm dy
Tm dy
P
Tm
Tm
R

T
Tm
Tm
dy
dx
P
Tdy
d
2. Patch
T
Tdy d
Tdy
4. V ertic al Res ultant
Thin walled sphere
Wall stress in a thick sphere
Ro
• To find equilibrium
forces:
 SFup = SFdown
h
P
Ri


Ri
P
Ro
h
.
Feedback Control of Volume
(V  V ) *Co  V * Ci
Anucleated cells
• RBCs-
• Platelets
• Fibroblasts
• Endothelial
Myocytes
Transformation
• Benign tumors grow rapidly, but respond normally
to ECM.
• Malignant cells have mutant actin, disorganized
CSK. They lose contact inhibition and invade
ECM, and climb over other cells.
• Cell shape affects malignancy, I.e. imposing a
spherical shape on melanoma cells makes them
more metastatic.
Polymerization rate
• definitions: ends of filament are not equivalent; n = number of
monomers in a single filament; t = time; [M] = concentration of free
monomer in solution
• capture rate of monomers by a single filament is proportional to the
number of monomers available for capture
• dn/dt = +kon [M] (capture) (1)
 kon = capture rate constant, with units of [concentration•time]-1
 release rate does not depend on [M]
• dn/dt = -koff (release)
Linkers
Kinetics
dn/dt = +kon [M]
1 dx'
C
 x'  u (t )
k dt
k
Laplace  transform
s
C
X ' ( s)  X ' ( s)  U ( s)
k
k
Transform pairs
f(t)
F(s)
Impulse (t)
1
Step
1/s
e-at
1/(s+a)
1
[e a1t  e a2t ]
a2  a
1
( s  a1 )(s  a 2 )
Transform back
C/k
X ' (s) 
U (s)
s / k 1
C
X ' (s) 
U (s)
sk
Step input
X ' (s) 
C
1
( )
(s  k ) s
C
C

[e 0t  e kt ]
(s  0)(s  k )
k 0
If non-zero initial conditions then add
:

x(0 )e
 kt
Cell spreading: polymerization
Vp = VL : n = Jp/rn
Jp is # of actin per second per area
and rn is # actin per volume at interface
Rules for analog simulation
• 1. Write diff-Q with highest order on left
• 2. Assume you know it, and integrate to get
x. (Check integration limits)
• 3. Perform required operations on lower
derivatives.
• 4. Check and simulate.
• 5. Scale magnitudes and timing.
Simulink Major Blocks
Block Name
Important Sources
Constant
Sine Wave
Pulse Generator
Signal Generator
Step
White Noise
Important Sinks
Graph
Scope
Kitchen
Important Operators
Discrete
Integrator 1/s
Description
Injects an input to the attached block
Injects a constant value
Injects a sine wave
Injects a rectangular pulse
Injects waves of selected forms
Injects one rectangular edge
Injects random noise
Plots output, axes, and permits printing
Same as oscilloscope (WYSIWYG)
(Not available in this version)
Cell inside is a gel
:
a state of matter produced by
electro-osmosis due to charged
polymers : water pressure inside.
Cartilage
H20
Swelling pressure = osmotic pressureelastic (compressive) pressure
Polymer-polymer
Intra-polymer
osmosis
Factors
Ca++, pH
0  2PA  k ( x  x0 )
Balance of forces in cytogel
F osmo
F elastic
Network condensed by shielding charge or reducing
Energy - I.e. divalent cations, Acetone, low temp.
 A
K  x  x o
Dynamic equilibrium
d 2x
m 2  2 PAu(t )  k ( x  x0 )
dt
C  2 PA
x'  x  x0
d 2 x'
m 2  Cu (t )  kx'
dt
d 2 x'
m 2  kx'  Cu (t )
dt
Mechanical Models




Voigt Model
I
R
sC
since d dt
and I = (1/R) V +
C dV/dt
t hen each of t he t op t erm s are equivalent t o t he bot t om
t erms. T h en t he im pedance analysis pro ceeds:
Voigt solution
1
1
1
Z R
s C
C
Z
s

Io 1
1



V ( s ) I ( s ) Z( s )
s C ( s 0)  s
Laplac e domain

1

1
e at1  e a2t = bi-exponential decay
a 2  a1
t
Time domain
V( t ) I o R 1 e


1

o
E
t
 1
e



E
Non-zero I.Cs.
dv

C
dt
v
( R)
I( t)
R earranging
Laplac e
( s 
1 ) V ( s )
 vdot( t )
R earranging
 v( 0 ) I ( s ) R
I ( s ) R
V( s )
s  1
Inv ers e
v( t ) I o R 1 e
v( t ) I ( t ) R
v( 0 )


s  1
Strain
t
t


v o e
 ( t)
o
E
t
 1 e
t
 o e

Cell crawling