Isotropic: same in all directions
Neurofilaments
Cross-linked
In frog axon
Linker proteins for actin
Particle Tracking in fibroblasts
Lecture 5
Tracking particles: Regional
stiffness
• #3- Think of a balloon with stiff meridional
bands- networks can stretch more easily
along the axis with less stiff ropes.
• #4 hoop stress versus axial stress
Cylindrical Stresses
rP
 
d sh
rP
x 
2d sh
X Y Z
Buckling & Bending
20 cm
10 cm
P1 
 EI
2
L2
Tension Field Theory
Tx   xx h
so
Membrane
Flow
Cell 1 cell 2 cell 3
Tx
L
 xx   
h
h
Coupling:
Mechano-/Biochemical-/Cellular-
Inside a Blood Vessel
Endothelial cells with
Nucleus bulging out
Blood flow
10 microns
Cells- fluid or solid?
• Micropipet aspiration comparison between
ECs and chondrocytes
• Comparison between EC cell & nucleus
• Stiffness following spreading or adapting
to flow.
• ECs in flow will minimize force on nucleus
• Enucleus = 9 Ecytoplasm
Applying global strains to
Round
Spread
Nucleus
1
Nucleus
F  deflection * rigidity
Compression & relaxation done
quickly to measure passive props
while avoiding adaptation.
No hysteresis or plastic behaviour
seen in spread cells and nuclei.
1. Caille, N: J. Biomech, 2002
• Material properties, not inhomogeneity, explains
The non-linear behaviour
Slow cell squishing
FEM of compression
Bone Adaptation
• Most bones experience 1000’s of loads
daily
• Bone cells must detect mech signals in situ
and adjust bone architecture appropriately.
• Sensor cells: Osteocytes; Effector cells:
Osteoblasts, osteoclasts
• Signalling molecules: PGs, NO
• Responses: bone formation/resorption
• Bending forces not only cause
deformation of osteocytes, but
generate pressure gradients that
drive fluid flow through the
canalicular spaces. Bending causes
compressive stress on one side of
the bone and tensile stresses on
the other. This leads to a pressure
gradient in the interstitial fluids that
drives fluid flow from regions of
compression to tension. Fluid flows
through the canaliculae and across
the osteocytes, providing nutrients
and causing flow-related shear
stresses on the cell membranes.
The fluid flow also creates an
electric potential called a streaming
potential
• Strain detected by mechanoreceptors or by CAMs.
G protein in membrane causes Ca and other 2nd
messengers.
• osteocytes (Oc) and bone lining cells (BLC) detect
mechanical signals and communicate those signals
to the bone surface. Soluble mediators, which include
• prostaglandins (PGs) and nitric oxide (NO), are
released and cause the recruitment and/or
differentiation of osteoblasts (Ob) from proliferating
and nonproliferating osteoprogenitor cells.
• The error function, i.e., the daily loading stimulus (S)
• minus the normal loading pattern (F; So), drives bone
adaptation. Abnormally low values of the error
function cause increased osteoclast activity on bone
remodeling surfaces, while abnormally high values
cause increased osteoblast activity on bone modeling
surfaces
k
S   log( 1  N j ) E j
j 1
• Rats jumping various of numbers of times per day
showed that five jumps per day were sufficient to
increase bone mass, but increasing numbers of
jumps gave diminishing returns with respect to bone
mass. These data very closely fit the mathematical
relationship proposed in Eq. 1
• G proteind mechanochemical signal transducer
• Focal adhesions by
Integrin and associated
proteins.
Load type affects adaptation
• Long bones are loaded mostly in bending
• Strain @ neutral axis is small, and
increases away from axis
• Loading that changes the neutral axis,
changes bone formation 1
• 1. Turner, CH: J. Orthop. Sci, 1998.
• MC3T3-E1 osteoblasts subjected to fluid shear (12dynes/cm2)
for 60min undergo dramatic reorganization of the actin
• cytoskeleton. A Control cells not subjected to flow have poorly
organized stress fibers labeled with Texas red-phalloidin.
• B Cells subjected to fluid flow for 60 min develop prominent
stress fibers labeled with Texas red-phalloidin that are aligned
roughly parallel to each other. C and D Control cells not
subjected to fluid shear which have poorly organized stress
fibers
Adaptation Cascade
• Transduction … Biochemical…
transmission….effector cell…..tissue
• Ion channels….Ca++,NOS, COX, PGs, G
protein….Obs, Ocs…..trabeculae
• It is an error driven feedback system
• Driven more by infrequent abnormal strains
than by normal strains encountered during
predominant activity1
• 1. Layton, LE: The success and failure of the adaptive response to
functional loading-bearing in averting bone fracture; Bone:13:1992
Quantifying bone adaptation
k
S   log( 1  N j ) E j
j 1
n
E    i fi
i 1
S  stimulus; N # daily .loading .cycles
( for.each.loading .type)
E  strain.stimulus;   peak . principal .strain
f  fourier .component
Bone Loading Waveforms
Resonant Stimuli for Bone
• Loading frequencies near 20 Hz
• Vibration 1
• Error Driven
dm
 K (S L  S0 )
dt
1. Rubin, C.
Mechano - regulation
• Growth, proliferation, protein synthesis,
gene expression, homeostasis.
• Transduction process- how?
• Single cells do not provide enough material.
• MTC can perturb ~ 30,000 cells and is
limited.
• MTS is more versatile- more cells, longer
periods, varied waveforms..
Markov Chains
• A dynamic model describing random
movement over time of some activity
• Future state can be predicted based on
current probability and the transition matrix
Sliding filamentds
Dynamic equilibrium
d 2x
m 2  2 PAu(t )  k ( x  x0 )
dt
C  2 PA
x'  x  x0
2
d x'
m 2  Cu(t )  kx'
dt
d 2 x'
m 2  kx'  Cu(t )
dt
Sliding Filament Model

m x  kx  F (u )

u  v0  x
Ratchet
For A-M, vo = 0.5 um/s
Harmonic motion (undamped)
Gel motion follows simple rules
Model will predict dynamic and
Static equilibrium.

m x  2 PAu(t )  k ( x  x0 )

mx   k ( x )

x  2 x  0
Natural Frequency
Transition Probabilities
Tomorrow’s
Game Outcome
Today’s Game Outcome
Win
Lose
Win
3/4
1/2
Lose
1/4
1/2
Sum
1
1
Need a P for
Today’s game
Grades Transition Matrix
This Semester
Grade
Tendencies
Good
Bad
Good
3/4
1/2
Bad
1/4
1/2
Sum
1
1
To predict future:
Start with now:
What are the grade
probabilities for this
semester?
Markov Chain
1/4
Win
Lose
3/4
1/2
Pi 1  APi
1/2
 a11 a12   3 / 4 1 / 2 
  

A  
 a21 a22   1 / 4 1 / 2 
3 / 4

Pi  
1/ 4 
Pwin,i 1  3 / 4  3 / 4  1 / 2  1 / 4  11 / 16
Plose,i 1  1 / 4  3 / 4  1 / 2  3 / 4  5 / 16
Intial Probability
Set independently
Computing Markov Chains
% A is the transition probability
A= [.75 .5
.25 .5]
% P is starting Probability
P=[.1
.9]
for i = 1:20
P(:,i+1)=A*P(:,i)
end
Control System, I.e. climate
control
Set Point
Perturbation
Output
Sensor
-
-
Plant
-
Error
Feedback
y ( s)
G ( s) 
;
u (s)
Finding G

x(t )  Ax(t )  Bu (t )
y (t )  Cx(t )
sx ( s)  Ax( s)  Bu ( s)
( sI  A) x( s)  Bu ( s )
x( s )  ( sI  A) 1 Bu ( s )
y ( s )  C ( sI  A) 1 Bu ( s )
G ( s)  C ( sI  A) 1 B
G ( s)  cT ( sI  A) 1 B
Temperature Control
Example Control System
1/s
+
X2
1/s
X1
1
-
3
2
u

X (t )  AX (t )  BU (t )
y (t )  CX (t )
1
0
A

 2  3
0 
B 
1 
1 
C 
0 
Homework
• 1. Assuming the buckling force calculated in #6,
compare the energy required to bend the
microtubule as in #5. (State assumptions).
• 2. Find evidence (for or against) that the tension
field theory applies to endothelial cell regulation.
• 3. Make a model of bone adaptation. What kind of
function fits the data?
• 4. Make a model of A-M sliding filaments.
• 5. Based on bending forces of microtubules,
calculate how many would be present in the EC, in
the experiments shown (make simplifying
Bibliography
• 1. Hamill OP, Martinac B. Molecular basis of mechanotransduction
in living cells. Physiol Rev 81 2001; (2):685-740.
• 2. Lang F, Busch, GL, Ritter M, Volkl H, Waldegger S, Gulbins E,
Haussinger D. Functional significance of cell volume regulatory
mechanisms. Physiol. Rev 1998; 78:247-273.
• 3. Zhu C, Bao G, Wang N. Cell mechanics: Mechanical response,
cell adhesion, and molecular deformation. Annu Rev Biomed Eng
2000; 2:189-226.
• 4. Turner CH. Mechanical transduction mechanisms in bone. J
Bone Miner Res 2000; 15 (4):105.
5. Tavi P, Laine M, Weckstrom M, Ruskoaho H. Cardiac
mechanotransduction: from sensing to disease and treatment. Trends in
Pharmacological Sciences 2001; 22 (5):254-260.
Bibliography
•
•
•
•
•
•
6.
Craelius W. Stretch activation of rat cardiac myocytes. Experimental
Physiology 1993; 78 (3):411-423.
7.
Ingber DE and Folkman J. How does extracellular matrix control
capillary morphogenesis? Cell 1989; 58:803-805.
8.
Craelius, W, Huang, CJ, Palant, CE, Guber H: “Mechanotransduction of
swelling by rat mesangial cells,” Mechanotransduction 2000, Engineering and
Biological Materials and Structures, ENPC, France, 13-20, 2000.
9.
Craelius, W, Huang, CJ, Guber, H, Palant, CE: “Rheological behaviour
of rat mesangial cells during swelling in vitro,” Biorheology 35:397-405,
1998.
10. Pedersen SF, Hoffmann EK, Mills JW. The cytoskeleton and cell volume
regulation. Comp Biochem Phys A 2001; 130 (3):385-399, Sp Iss SI.
11. Lange K. Regulation of cell volume via microvillar ion channels. J Cell
Phys 2000; 185 (1): 21-35.