A Viscoelastic Model to
Explore Force
Mechanotransduction within
the Focal Adhesion Complex
Ricardo R. Brau
Nicholas A. Marcantonio
BE. 400 Project
December 11, 2002
Overview
Motivation
Mathematical Model
Model Results
Proposed Experiments
Conclusion
Motivation: Signal Transduction or
Transmission
External forces Important in Development induction of biological
responses in cells (apoptosis, differentiation, etc.)
Interface between mechanics and biochemistry must be explored
Localized Model vs. Decentralized Model for Mechanotransduction
Want to use model to
examine localization of
deformation in FAC

Explore potential for signal
transduction due to molecular
deformation in FAC vs.
cytoskeleton
Krammer et al, 1999; Shafrir and Forgacs, 2001; Bao, 2002
Deformations of Single Molecules
Most experimentation performed on
rodlike molecules (DNA/RNA and titin),
molecular motors (kinesin and myosin),
and fibronectin
Kellermayer et al, 1999
The Focal Adhesion Site
Zamir and Geiger, 2001
Model System
Applied Force
FAC is highly dynamic, and not
well characterized
Want to start with simple
model
α
β
Membrane
α=alpha
P
T
Β=beta
P=paxillin
V
T=talin
A
V=vinculin
A=actin
Zamir and Geiger, 2001
Mechanical Deformations in
Biology
Model molecules as
viscoelastic elements
k
m
F

dx
T t   kx  
dt
 kt  
T
x  1  exp    
k
  
Bao, 2002
Assume m~0
 controls kinetics
k affects steady-state
and kinetics
Model Assumptions
Deformation, not relative molecular position,
determines signaling


Relative deformation: x
Strain: x/L
FAC structure is time-invariant
Signaling due to molecular deformation, not
disruptions in FAC structure


2o and 3o structures preserved
hinge motion between molecules not significant
Molecules can be represented as single
domains
Subbiah, 1996; Oberhauser et al, 1998; Idiris et al, 2000
Viscoelastic Model of the FAC
T(t)
T1
T2
x
α
k

k

kP
P
kT
T
β
P
T
V
V
kV
A
kA
A
Mathematical Model
6 unknowns and 6 equations
x  x  xP  xT  0
(1)
k  x   x  k  x    x  T t 
(2)
k  x   x  k P xP   P xP  0
(3)
k  x    x  k T xT  T xT  0
(4)
k V xV  V xV  T t 
k A x A  T t 
(5)
(6)
Mathematical Model
Solve numerically using MATLAB: ode23s
Initial condition: system is at rest
dX
1
 D F  SX 
dt
dx dx dxT dxP



 0 (1a)
dt
dt
dt
dt
F=Forcing Matrix; S=Spring Matrix; D=Damping Matrix
Parameter Estimation
Molecular breathing due to thermal forces:
~ 1pN
Domain unfolding: ~ 100 pN
Deformation length scales: 0.05-5 nm
Deformation time scales: ~10 nsec
Only actin well characterized as
mechanical element
Only molecular weight information
available for other molecules
Marszalek et al, 1999; Zhu et al, 2000; Craig et al, 2001; Bao, 2002
Parameter Estimation
Hydrodynamic radius: model molecules as
spheres
Since η only affects kinetics, assume
similar for all molecules: ~60 pN sec/m
k ~ .02 nN/nm for 172 kDa molecule,
consider proportional to molecular weight
Signaling occurs at arbitrary threshold of
10% strain
Fisher et al, 1999; Bao, 2002,
Parameter Estimation
Scenario 1
Scenario 2
Each protein has
Each protein has the
same spring constant
same length
Length is calculated
k is proportional to
as a function of MW
MW
(hydrodynamic radius)
 Based on k=.02 N/m
for MW of 172 kDa
Measure strain (x/L)
Measure strain (x/L)
 x will be the same for
each protein, but strain
will be different

L will be the same for
each protein, but x
will be different
•Perform simulations for static and dynamic loading
Model Results
Static Force: 4 pN (thermal force)
Model Results (cont.)
Model Results (cont.)
Model Results (cont.)
Model Results (cont.)
Experimentation/Model Validation
Determine crystal structures and binding
sites of FAC molecules
Determine mechanical properties of
molecules of interest



AFM
Optical Tweezers
Single Molecule Fluorescence
Mehta et al., 1999; Lang et al., 2002
Experimentation/Model Validation
Recreate FAC in vitro (within microfluidic
chambers) and subject to external
loadings
Extend model to analyze kinetically
varying FAC
Analyze FAC from a finite elements
perspective, taking into account individual
domain linkages, deformations, and
unfolding
Experimentation/Model Validation
Study effects of mechanical deformations
on catalytical activities of enzymes and
proteins present in FAC
T(t)
x(t)
Ligand
Product
Enzyme
Catalysis rate function
of deformation
k=k(x)
Model Analysis
Loading dependent response


Static loading
Dynamic loading
Complex transient response
Identification of resonant frequency
Deformation of proteins dictated by their
mechanical properties


Suggests that mechanochemical coupling occurs at
FAC
Supports localized signaling
Conclusions
Model needs to be complemented by
experimental data
Results provide insight into protein synthesis of
dynamically loaded cells in culture
Deformations predicted by model correlate well
with single molecule simulations and
experimental data

Does not rule out energy propagation
as described by tensegrity and
percolation models
Shafrir and Forgacs, 2001
Acknowledgements
Ali Khademhosseini
Doug Lauffenburger
Paul Matsudaira
BE.400 Class
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