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Outline
• Defocusing Microscopy: a full-field technique for phase retrieval in transparent objects (phase objects) to study living cells.
• Theoretical backgroung: Fourier Optics and propagation of the Angular Spectrum; Paraxial and Fresnel approximation.
• Test of the optical model of Defocusing Microscopy on artificial transparent objects.
Lecture 3Defocusing Microscopy: a new way of phase retrieval and 3D imaging of
transparent objects
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Motivation: Study of Adhered Macrophage Motility
Film accelerate
d 16x
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Phagocytosis of Leishmania amazonensis at 37oC
Film accelerated
16x
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Contrast Fluctuations - Macrophage
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Defocusing microscopy
Agero et al., PRE 67 (5), 051904 (2003) and Phys.Rev. Focus, May 21 (2003); Agero et al., Microsc. Res. Tech. 65, 159 (2004); Mesquita et al., APL (2006); Coelho-Neto et al., Biophysical J. (2006)
Infinity corrected microscope
f < 0 f = 0 f > 0
Adhered Macrophage
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Light electric field for a defocused microscope
dezEzqA qi .),(),(
Angular spectrum of the light electric field
qdezqAzE qi
.
2),(
)2(
1),(
Considering a single polarization, propagation along z>0 and the paraxial approximation q<<k
2
2
1
)0,(),( k
qikz
eqAzqA
Free propagation of the angular spectrum
2D Fourier transformjyix ˆˆ
jqiqq yxˆˆ
From Helmholtz equation
0.22
deEkE qi 0),(),( 22
2
2
zqAqk
z
zqA
kq propagating wave
kq evanescent wave
k
zqi
ikz eeqAzqA 2
2
)0,(),(
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Angular spectrum through a thin lens
deAik
fzqA
qk
fi
l
2)(2
02)(
2
)2(
1),(
1. From the object (z=0) to L1 (z=f1-∆f);
2. through L1
3. from L1 to L2 (distance d)
4. through L2
5. from L2 to the image plane I (distance f2 )
qdeeqABe
E qik
fqii
.2
)( 2
02)(
)2()(
00
212
1
2
2
102001 222
)(k
d
k
f
k
f
f
k
kf
fkfkfkdkkf
Electric field for the defocused microscope on the image plane
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Diffraction by a sinusoidal phase grating with spacing L
light
ffocal plane
)sin(0
)(0
)(00
000)(xqnhiknhiki eEeEeEE
ximqm
mm enhkJEE 0)()( 000
Lq
20
02
002
0 ,)(2)( mqqqnhkJEqA xy
m
mm
ximqk
pzmqim
mm
i eenhkJEBeEf
0
12
0
2
)()(
00)( )()(
Electric field for the defocused sinusoidal phase grating
mJ are Bessel functions of order m
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Contrast of a defocused phase grating considering only first order diffraction
2)(and1)(
thatsuch
1if
00100
0
nhknhkJnhkJ
nhk
)sin(1)( 02
)(
00)(
120
xqenhikEBeE k
pzqii
f
Defining contrast as
0
0)()(
I
IIC
with
20
20
2)()(
EBI
EI
)sin(2
sin2)(
)( 0
)(20
00
0 1xq
k
qnhk
I
IIC
pz f
NAL
n
NA
k
q
0min
maxmaxsin
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)sin(2
sin2)( 0
20
0 xqk
fqhnkxC
Sinusoidal Phase Grating with Spacing L
Lq
20
mL 65.1
-0,6
-0,4
-0,2
0
0,2
0,4
0,6
-150 -100 -50 0 50 100 150
f (m)
0
1 104
2 104
3 104
4 104
5 104
6 104
7 104
8 104
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
FFT_contraste_1.65 m
kz (m-1)
0.076 m-1
desf
= 13.16 m
22Ldef
Shifted Talbot Images
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Test of the relation
22L
def
-2
0
2
4
6
8
10
12
14
-0,5 0 0,5 1 1,5 2
L (m)
13.052.1 n
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Defocused contrast for a general transparent interface
).sin()(1
)(
qhS
h
k
fqqh
S
nkC
).sin(
2sin)(2)(
2
0
12
For2
k
fq
)()( 2
hfn
nC
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Light
Objective Focal Position
Z
fz
h(x,y)
Glass-slide
solution
f
Curved Thick Phase Object
)()( 2
hhfn
nC
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Polystyrene Spherical Cap
hhfn
nC 2)()(
DMimage
AFMimage
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Linear Defocusing Region
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AFM DM
R=5.12m04.059.0
1.08.4
3.0
n
mR
mf
AFM and DM Profiles
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n=0.61±0.01
Refractive Index Difference Obtained with DM
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NCBN
NCBAIN
0
0
CameraDage-MTI – 8 bts
NCC
N
77.0
1270
NCC
N
98.0
20000
CCD Calibration
CameraUniqVision – 12 bits
Power meter intensity (W)
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Fluctuating transparent interfaces and contrast correlation function
H
h),()(),( tuhtH
Time average
0),(and
)(),(
tu
htH
).sin(2
sin),(2),(
),(2
0
0
0
qk
fqtqH
S
n
I
ItItC
q
k
).sin(2
sin)(2
),(2
0
qk
fqqh
S
ntC
q
k
with ),(),(),( tCtCtC
and for a stationary process such thattqequtququ )(2
)(),()0,(
).cos(2
sin)()(2
),()0,0(2
222
)(0
qk
fqequ
S
ntCC tq
q
k
Space-time correlation function of contrast fluctuations
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Mean-square fluctuation of contrast
k
fqqu
S
nfC
q
k
2sin)(
)(2)(
2
222
02
and for the continuum case
k
fqfCCfd
knkqu
2
2
2
cos)()()(
)( 22
0
4
2220
2
2
202
)()(
for
2sin)(
2
)()(
2
2
2
unC
f
k
fqquqd
nfC
k
k
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Spacial power spectrum of fluctuations
Mean square contrast
fluctuation
Numerical example
.9.4 ;1 ;4680 ;212 ;6.7 ],m[ 1
)(2
042424
2μmR
π
)nk(mcmba
cbqaqqu
][)( 42mqu
1min 3 mq
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Constrast correlation function
).cos(
2sin)(
2
)(sin)(
)(2
),()0,0(
}{2
22
2
2
122
1
2
2)(2)(10
q
k
qpzequ
k
qpzequ
S
nk
tCC
ftqftq
q
Diffraction by two transparent interfaces
}
{
)(2
sin
2
)(sin
)(2)(
2
2
1
2
22
10
)(
)(
qsenk
qpz
k
qpz
S
nkC
f
f
q
qH
qH
Average contrast
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Two symmetric interfaces
Numerical example
.9.4 ;1 ;4680 ;212 ;6.7 ,1
)(2
041
2
2,1242,1 μmR
π
)nk(mcmba
cbqaqqu
Two asymmetric interfaces
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Summary
By using the propagation of the light angular spectrum we develop an optical model for a defocused bright-field microscope.
Transparent objects can be visualized in a defocused microscope, since defocusing introduces a phase difference between the diffracted and transmitted light, which is translated into contrast after interference in the image plane.
For small defocusing the average contrast of a surface is proportional to its curvature.
We were able to obtain theoretical expressions for the correlation functions for one and two fluctuating interfaces. In the next lecture we will see, by using these expressions, how to obtain elastic information from the interfaces of living cells.
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Lecture 4Application of defocusing microscopy to study living cell motility
Outline
Application of the expressions obtained in Lecture 3 for testing motility models of living cells.
Macrophages and phagocytocis: 3D imaging and study of fluctuations. Effects of nonequilibrium.
Red Blood Cell: 3D imaging and study of coupling between the spectrin cytoskeleton and lipid bilayer via flickering. Effects of nonequilibrium.
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ruffle
SRMF
Results – Curvature Fluctuations
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Cytoskeleton
Polimerized protein filaments
Actin filaments just below the plasmatic membrane
Alberts, et al Mol. Biol. Cell. 3rd Ed.Garland Pub. Inc. NY(1994)
Svitkina, Verkhovsky, MacQuade & Borisy J. Cell Biol. 139 (2), 397 (1997)
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-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
2 3 4 5 6 7 8 9
h (m
)x (m)
w
xxhxh 00 tanh1
2)(
-1.5
-1
-0.5
0
0.5
1
1.5
2 3 4 5 6 7 8 9
(
m-1
)
x (m)
Ruffle hyperbolic
Ruffles: curvature and thickness profiles
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-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8h
(m
)x (m)
2
20
20)( w
xx
ehxh
-2
-1.5
-1
-0.5
0
0.5
1
0 1 2 3 4 5 6 7 8
(
m-1
)
x (m)
Ruffle gaussian
Ruffles: curvature and thickness profiles
Measuring ruffle contrast as a function of defocusing we are able to obtain its refractive index.
(Coelho Neto, Biophys. J. 91, 2006)015.0049.0 n
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Spatial correlation function
-0.02
0
0.02
0.04
0.06
0.08
-0.5 0 0.5 1 1.5 2 2.5 3 3.5
sp
ati
al
curv
atu
re c
orr
ela
tio
n f
un
cti
on
(m
-2)
distance (m)
-0,02
0
0,02
0,04
0,06
0,08
0,1
-5 0 5 10 15 20 25
sp
ati
al
corr
ela
tio
n f
un
cti
on
(m
-2)
distance (m)
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For bone marrow macrophages (extracted from healthy mice) this relaxation time is
s)26(
-0,02
0
0,02
0,04
0,06
0,08
0,1
0 50 100 150 200 250
cu
rva
ture
tim
e c
orr
ela
tio
n f
un
cti
on
(m
-2)
time (s)
y = m1 * exp(-M0/m2)
ErrorValue
0,000749780,10153m1
0,0575,7053m2
NA0,00053441Chisq
NA0,99463R
Time correlation function
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Curvature probability distribution function
1
1
0.25
0.61
0.024
m
m
10
100
1000
104
105
106
107
-3 -2 -1 0 1 2 3
N
curvature (m-1)
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Before and after addition of 100nM of Cytochalasin-D
Ruffles are inhibited
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After addition of 100nM of Cytochalasin-D
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Results: 24-37oC
2
2.4
2.8
3.2
3.6
322 324 326 328 330 332 334 336 338
ln ()
1/T (10-5 K-1)
-0.5
0
0.5
1
1.5
2
322 324 326 328 330 332 334 336 338ln
(V
ruff
les)
1/T (10-5 K-1)
TkE Ba 233 TkE Ba 536
Coelho Neto et al., Exp. Cell Res. 303 (2), 207 (2005)
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Discussion of models
Model of cellular motility : Brownian Ratchet
Dynamics of actin polymerization(diffusion + polymerization) TkE Ba 31
x
Actin filament
actin-g
fD
membrane
2.7 nm
Peskin, Odell & Oster Biophys. J. 65, 316 (1993)
Mogilner & Oster Biophys. J. 71, 3030 (1996)
Mogilner & Oster Biophys. J. 84, 1591 (2003)
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Phagocytosis of Leishmania amazonensis at 37oC
Film accelerated
16x
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0
0.05
0.1
0.15
0.2
0.25
-50 0 50 100 150 200 250 300
me
an s
qu
are
cu
rva
ture
(m
-2)
time (s)
st f 60
Behavior of <2> near the phagossome
0
0.05
0.1
0.15
0.2
0.25
-50 0 50 100 150 200 250 300m
ean
sq
ua
re c
urv
atu
re
(m
-2)
time (s)
st f 120
Results: Phagocytosis at 37oC
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3.5
4
4.5
5
5.5
6
322 324 326 328 330 332 334 336 338
ln (
ph
ago
cyto
sis
tim
e)
1/T (10-5 K-1)
TkE Ba 438
Results: Phagocytosis from 24 to 37oC
Coelho Neto et al., Exp. Cell Res. 303 (2), 207 (2005)
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Protein-MembraneCoupling Model
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Theoretical model of
Experimental data ofCoelho Neto et al. Exp. Cell Res.
303, 207 (2005)
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Objective focal plane above the RBC middle plane
Objective focal plane below the RBC middle plane
Brochard – Lennon (1975), flickering due to thermal motion of surfaces
Red Blood Cell (RBC)
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Defocused Image of a Red Blood Cell (RBC)
-0.2
0
0.2
-10 0 10
Con
trast
position (m)
Mesquita, Agero, Mesquita, APL 88, 133901 (2006)
)(2
)()(2)( 2
212
hhh
fnC
0
0.4
0.8
0 2 4
n=0.0422
RB
C p
rofil
e (m
)
r (m)
n = 0.056
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Limite assintótico para grandes desfocalizações
22
)( 2
k
qpf
k
qpf
k
qpf 2
1
2
12 )(cos
2
1
2
1
2
)(sin
222
122
202
21
201
2 hknhknC
Para hemácias
nmmh
h
hnkC
24024.0
3.9056.02103
2
2
224
220
2
Para
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Membrane Free Energy Variation
2222
222uuu
kdAF C
Fourier decomposition and energy equipartition
24
2)(
qqkA
kTqu
C
Curvature energy for curved surfaces – Helfrich free-energy (Phys. Lett.1973)
21
20212
CCkCCCk
dAF CC
S. A. Safram, Statistical Thermodynamics of Surfaces,Interfaces, and Membranes, Addison-Wesley (1994).
u
Monge representation
21
21
0
.
and
CCK
CC
C
spontaneous curvature
main curvatures
Gaussian curvature
Membrane Elasticity and Fluctuations
Ck bending modulus
surface tension
confinement potential
Lipid bilayer Hydrophilic
Hydrophobic
water
water
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RBC Elastic Model of Auth, Safran, and Gov
d
-Brochard F. and Lennon J.F., J. Physique , 36, 1035 (1975);-Zilker A., Engelhardt H., and Sackmann E., J. Physique 48, 2139 (1987);-Evans E., Methods Enzymol. 173, 3 (1989);-Tuvia S., Levin S. and Korenstein R, Proc Natl. Acad. Science, 94, 5045 (1997);-Tuvia S., Levin S. Bither A. and Korenstein R., J. Cell Biol. 141, 1551 (1998); -Gov N., Zilman A.G. and Safran S., Physical Review Letters 90 (22), 228101 (2003); -Gov N. and Safran S., Biophys. J. 88 (22), 1859 (2005);-Auth T., Safran S. and Gov N., Physical Review E 76 , 051910 (2007).
B. Alberts et al.,”MolecularBiology of the Cell”, (2002)
spectrin
bilayer
Spectrin filaments
Actin nodes
Detachedfilaments
Cytoskeleton is modeled as a hexagonal network of entropic springs
ATP driven non-thermal effects
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24
2)(
qqk
kTqu
efc
ef
C
efef k
kT
16
9
)2(3
3 2 KHdAA
fC
tqequtququ )(2)(),()0,(
242
4
))(22)2exp(1)(2exp()( qqk
q
qdqdqdqdq efc
bathef TT
RBC Elastic Model of Auth, Safran, and Gov
cytoskeleton shear modulus
cytoplasm viscosity
bilayer curvature modulusck
efT effective temperature
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se
2211 )()()( hzhzn
nC ff
)(2
)( fn
nC
2
21
2 hh
sendo2
21 hhzf f
Reference System
Glass-slide
Symmetry plane
lightorigen
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TransformFourier
q
C
fnhh
2
10
)(
2
1)(
2
2
25.0
13.0
2
1
mf
mf
C
C
Results
X (m)
X (m) (m)
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Middle region of a RBC
Defocusing microscopy is able to provide quantitative data about the fluctuations of each interface of a RBC separately.
Contrast correlation between the same pixel after 33ms. The decay oflarge wavenumber fluctutations isevident in the figure.
Measurements of RBC Flickering with DM
G. Glionna et al. APL (2009)
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]3
16
92
sin
316
92
sin
[2
)(
224
222
124
2122
02
CC
efC
ef
q
CC
efC
ef
fqk
kTqk
qk
pkT
fqk
kTqk
qk
pkT
S
nkC
ff
f
zz
z
0.0002
0.00022
0.00024
0.00026
0.00028
0.0003
0.00032
0.00034
9 9.5 10 10.5 11 11.5 12
<
C2(z
f)>
zf (m)
With DM we measured
22
21
25.0
13.0
mf
mf
C
C
bathef
ef
ef
C
TT
bkg
mkT
kT
k
3.3
.10)03.040.1(
;10)4.02.9(
8.06.7
4
23
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510)2.08.1(
)121(
bkg
nmd
2 10-5
4 10-5
6 10-5
8 10-5
0,0001
0,00012
8,5 9 9,5 10 10,5 11 11,5 12 12,5
<(
C(0
,0)
C(0
,0.0
33s)
>
zf (m)
water
c qqkq
qdqdqdqdq
3
4
))(22)2exp(1)(2exp()( 2,1
242
2,1
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Summary • We developed an optical model of a Defocused Microscope, such that
height profile of phase objects can be reconstructed from their defocused images.
• With Defocusing Microscopy (DM), fluctuations on cell surfaces with nanometer height amplitude can be analyzed. By scanning the microscope objective focal plane position, one can selectively obtain information about fluctuations on different interfaces in a multilayer material. Fluctuation spatial power spectra of each interface can separately be obtained.
• We used DM to study flickering of red blood cells. We are able to test a recent elasticity model of RBC, obtain the effective lipid bilayer curvature modulus, cytoskeleton shear modulus, normalized by the effective temperature, and the average distance between the bilayer and cytoskeleton.
• Defocusing microscopy is a full-field technique for phase retrieval in phase objects, which can be implemented in any standard optical microscope.