Measurements and Models of Cytoskeletal Rheology … · APS-DFD November, 2006 Measurements and...
Transcript of Measurements and Models of Cytoskeletal Rheology … · APS-DFD November, 2006 Measurements and...
APS-DFD
November, 2006
Measurements and Models of
Cytoskeletal Rheology
Roger D. Kamm
MITDepartment of Mechanical Engineering
Department of Biological Engineering
Acknowledgements: TaeYoon Kim, Hyunsuk Lee, Belinda Yap,Jeff Hsu, Jorge Ferrer, Seok Chung
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The Neutrophil -- naturally-occurring
deformations
(micrograph courtesy of C. Dongand R. Skalak)
- spherical- mean diameter = 6.8 µm
SEM
Cross-section
- multi-lobed nucleus- granules/cytoskeleton- actin-rich cortical region
(micrograph courtesy of C.Dong and R. Skalak)
Bathe et al., 2002
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Cells have a
multi-component
fibrous, gel-like
matrix.
Cytoskeletal composition/structure
Ingber, Scientific
American
Hartwick,http://expmed.bwh.harvard.edu
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Actin -- One of the primary structural
components of cells
Biochemistry, Wiley 1995
• 40kDa (375 AA residues)
• G-actin to F-actin : Nucleation (lag phase), elongation (growth phase),
steady state (equilibrium phase)
• Semi-flexible filament (neither flexible nor stiff) lp lc
• Viscoelastic network
• Monomer size << mesh size lp
• Self assembly with polarity
• Binding to a variety of ABPs (Actin Binding Proteins)
• Actin polymerization plays a key role in fundamental cellular processes
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G*= G + i G =
˜ T (t)˜ (t)
is a geometry-dependent prefactor determined
from finite element analysis
G*= G0
0
x 1
1 + i ( ) (2 x)cos2
(x 1)
+ i µ
is the Gamma-function; G0, 0 and x are
adjustable parameters
Consistent with models for soft, glassy materials
Magnetic Twisting Cytometry (MTC)
Cells exhibit a power-law rheology,
reminiscent of a soft , glassy
material.
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Different behaviors for different types of
measurement
Hoffman, et al., PNAS, 2006
But values for G range from 10’s of Pa to several
kPa, depending on the method of measurement.
Data appear to fall
along two characteris-
tic slopes, one for
methods that
emphasize cortical
structure, and another
for internal, cytoskele-
tal structure.
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Reconstituted actin gel rheology
Hypotheses for breakdown of
the cytoskeleton due to
mechanical deformation
1) Sudden
depolymerization of
F-actin
2) Rupture of actin
cross-links between
F-actin filaments
Incremental shear moduli exhibit strain hardening up to a
point, followed by a catastrophic drop in stiffness.
Tseng et. al. (2004) J Biol Chem; 279: 1819-1826
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November, 2006
a b c d
e f g h
Cytoskeletal shear modulus
computed from the Brownian
fluctuations of granules
As a function of time after enteringchannel, until protrusion
In adherent, migratingneutrophils
In non-adherent,neutrophils
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Entrance Time -- Passive Behavior
0.01
0.1
1
10
100
0.1 1 10 100
Pressure (mm H2O)
En
tra
nce
tim
e (
s)
DL, 37deg
DS, 37deg
DL, 23deg
Entrance time : interval
between leading edge
touching channel
entrance and trailing
edge clearing channel
entrance after
deformation
Entrance time appears
primarily influenced by
the magnitude of
mechanical stimulation;
less so by biological
activity
Yap & Kamm, APS, 2005
To what extent can the
cytoskeleton be treated as
a cross-linked actin gel?
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Viscoelastic moduli of neutrophils following deformation
0
50
100
150
200
250
300
0 10 15
(N=16)
20
(N=37)
25
(N=12)
30
(N=20)
Time (s)
G' (
dyn
/cm
2 )
DS,10mmH2O,37deg (n=4)
Control
0
50
100
150
200
250
300
0 10 15
(N=9)
20
(N=27)
25 30
Time (s)
G' (
dyn
/cm
2)
DL,1mmH2O,37deg (n=5)
Control
*
**
**
** ** **
†††
0
100
200
300
400
500
600
0 10 15
(N=9)
20
(N=27)
25 30
Time (s)
G"
(dyn/c
m2)
DL,1mmH2O,37deg (n=5)
Control
0
100
200
300
400
500
600
0 10 15
(N=16)
20
(N=37)
25
(N=12)
30
(N=20)
Time (s)
G"
(dyn/c
m2)
DS,10mmH2O,37deg (n=4)
Control
**
**
*
†
††
G G”
Moduli fall immediately upon deformation, and recover on a time
scale of about one minute
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0
50
100
150
200
250
300
0 10 15
(N=16)
20
(N=37)
25
(N=12)
30
(N=20)
Time (s)
G' (
dyn
/cm
2 )
DS,10mmH2O,37deg (n=4)
Control
Storage modulus drops
immediately, then recovers on a
time scale of 30-60s.
A similar time-course is
observed for F-actin content
following passage through
pores.
Dynamic shearmodulus and F-actinfollow similar patterns
Post-deformation
Resting
neutrophils
Actin depolymerization appears to
explain part, but perhaps not all, of
the drop in modulus
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Similar behavior -- abrupt fall, then
recovery of stiffness -- has been
observed in other cell types
Cytoskeletal fluidization is a common
attribute of cells
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Cytoskeletal remodeling
Forces are transmitted via theCSK and actin cross-links
Bonds may rupture andreform, leading to remodeling
Signaling pathways mightinitiate depolymerization
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Some outstanding questions
What is the physical basis for soft, glassy rheology?
How can strain stiffening in reconstituted actin gels bereconciled with fluidization in cells under abrupt strains?
Are the differences in behavior seen with differentmeasurement methods real, and how can they bereconciled?
Are the different power law behaviors really due to cortex vs.cytoskeleton? If so, why?
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• Estimate viscoelasticity of the actin cytoskeleton using
microbead tracking rheology, both active and passive.
• Investigate the effect of ACPs on viscoelasticity
Cytoskeleton Growth
Microbead Rheology
• Investigate the effects and roles of the properties of actin
cross-linking proteins (ACPs) on cytoskeletal structure
Modeling Objectives
Simulate actin cytoskeletal growth, rheology, and force-induced changes in biochemical activity using moleculardynamics, Brownian dynamics, and continuum models.
Relevant recent works: Storm, et al., Nature (London), 2005
Hoffman, et al., http://arxiv.org/pdf/physics/0504051
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Cytoskeletal Modeling
2
2( )i j
i j
d r drm f d t
dt dt= +
Equation of motion: Langevin equation
** * * * * *
*( ) ( )i j i j
i j i j
drf d t dr f d t dt
dt= + = +
Integration Method: Euler forward (dropping inertia)
Drag computed (initially)ignoring hydrodynamicinteractions
Interaction forces: Scalings:
ULJ
(r) =4
r
12
r
6
+ r 21/6
0 r > 21/6
U
Bend=
1
2 Bend 0( )2
U
Spring=
1
2k
s(r r
0)
U
Torsion=
1
2 Torsion 0( )2
Truncated Lennard-
Jones
Bending stiffness
Extensional
stiffness
Torsional stiffness
t*=
t2 / k
BT
Length scaled with
actin radius, energy
with kBT
2
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Cross-linking
Bundling cross-linkers (e.g., fimbrin,fascin, scruin, -actinin)
Network cross-linkers (e.g., filamin)
r
ACP= 0.75,
1= ,
2=
2,
3= 0
r
ACP= 1.5,
1= 1.158 (rad),
2=
2,
3=
2
Unbinding obeys Bellʼsequation and is force-dependent.
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Monomers : 8,000, ACPs : 1,000
Vol. % = 0.82% (+ACP ~0.1%), R=0.125
Conc = 150 M, Domain = 560 nm
Comparison of cross-linked and bundled structures(varied by changing the torsional stiffness and orientation of actin binding sites)
Cross-linked Bundled Mixed
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Computed 3D images
Computed Network Structures
“Confocal” images
1.12 µm x 1.12 µm
90° cross-
links
“Bundling”
cross-links200nm
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Analysis of the Polymerized Structure
Monomers : 8000, ACPs : 173
Vol. % = 0.82% = 150 M
Nucleation rate = 1ⅹ10-8 s-1, Domain = 560 nm
Filament length = total contour length
Connectivity = number of filaments attached
via cross-links
Avg = 328, S.D. = 112
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Analysis of the Polymerized Structure
Monomers : 8000, ACPs : 173
Vol. % = 0.82% = 150 M
Nucleation rate = 1ⅹ10-8 s-1, Domain = 560 nm
Mesh size = distance between cross-links
Pore size = max diameter of a sphere that can fit
inside a pore
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F-actin networks formed by -actinin
R = 0.01 R = 0.1 R = 0.2 R = 0.5
10 m
Stress fibers form of increasing diameter as the amountof -actinin is increased
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F-actin networks formed by filamin
Mean ± SD
0.93 ± 0.80 m
10 m
Rf = 10-1 Rf = 10-2 Rf = 10-3 Rf = 10-4
Rf = 10-1 Rf = 10-2 Rf = 10-4
Mean ± SD
0.98 ±
0.56 m
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Passive microrheology
Monitor thermal fluctuations of
embedded microspheres to
estimate the frequency
dependent complex modulus
High spatial and temporal
resolution using quadrant
photo detector
Fluctuation-dissipation theoremFluctuation-dissipation theorem
Kramers-Kronig Kramers-Kronig relationrelation
Generalized Stokes-EinsteinGeneralized Stokes-Einstein
relationrelation
Thermally driven dynamicsThermally driven dynamics
of the of the microspheremicrosphere
TrappingTrapping
laserlaser
MicrosphereMicrosphere
F-F-actinactin matrix matrix
[ ]*
2
*
*
2
1/
( )(1/ ) 1 ( )
'( ) ( ) cos( ( ) / 2)
''( ) ( ) sin( ( ) / 2)
d ln ( )( )
d ln
B
t
k TG
a r
G G
G G
r t
t=
+
=
=
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Apply a driving force to an embedded microsphere and
monitor its trajectory
• Sinusoidal and stepwise
Bulk rheometer
• Parallel plate, AR2000, TA Instruments
• Gap: 120 m – 40mm flat plate
• Frequency sweep
Active microrheology
timetime
xxtr
ap
trap(( tt
))
timetime
xxtr
ap
trap(( tt
))
tinput
tinput
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Comparison at different filamin concentrations
– Active sinusoidal forcing
• One order magnitude difference of moduli as Rf increases10 fold
• G´a ~ Rf0.2 for 10 M actin, Rf = 0.01 and 0.1
• G´a of Rf = 0.1 is comparable to cellular rheology
● Rf = 0.1
● Rf = 0.01○ Rf = 0.1
○ Rf = 0.01
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Comparison at different filamin concentrations –
Passive measurement
• Longer plateau region in G´p of Rf = 0.01
• Similar values of G´p in low frequency region
● Rf = 0.1
● Rf = 0.01
○ Rf = 0.1
○ Rf = 0.01
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Width : 924 nm, 1331 Monomers
Bead : 224 nm, Conc ~ O(10 M)
ComputationG’
G’’
Los
s sh
ear
Mod
ulus
, G’’
[Pa]
Sample : 20 x 5 x 0.2 mm
Bead dia. : 1 m, Conc. = 10 M
Experiment
Comparison of complex moduli
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Summary and remaining questions
• Simulated networks exhibit structures and rheology
“similar” to those found in experiments in cross-linked
gels.
• Can a computational model be created that replicates
cellular measurements?
• What is the fundamental basis for cytoskeletal fluidization,
correct power-law scaling?
• What is the role of measurement method (1- vs. 2-particle
methods, active vs. passive, bead coating, particle size
vs. mesh size) in cell rheological measurements?
"If we knew what it was we were doing, it would notbe called research, would it?” - Albert Einstein
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November, 2006
AcknowledgementsStudents:
TaeYoon Kim
Hyungsuk Lee
Jorge Ferrer
Belinda Yap
Jeff Hsu
Postdocs:
Seok Chung
Faculty collaborators:
Matt Lang
Peter So
Douglas Lauffenburger
Subra Suresh
Funding agencies:
NHLBI
NIGMS