Lecture 3
Defocusing Microscopy: a new way of phase retrieval and 3D imaging of
transparent objects
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.
Motivation: Study of Adhered Macrophage Motility
Film
accelerated
 16x
Phagocytosis of Leishmania amazonensis at 37oC
Film accelerated
 16x
Contrast Fluctuations - Macrophage
Defocusing microscopy
Adhered Macrophage
Df < 0
Df = 0
Df > 0
Infinity corrected microscope
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)
Light electric field for a defocused microscope
Angular spectrum of the light electric field
 
E ( , z) 

  x iˆ  y ˆj
2D Fourier transform
1
 
  
iq . 
dq
 A(q , z ) e
( 2 )
 
 
 

 iq . 
A ( q , z )   E ( , z ) e
d

q  q x iˆ  q y ˆj
2
Free propagation of the angular spectrum
From Helmholtz equation
 
 A(q , z )
2

  i q .  
2
d  0
  E k E e

2

 ikz
 
 
A ( q , z )  A ( q ,0 ) e
1
z
q
2
k
2
2

 k q
2
2

 
A(q , z )  0
qk
propagating wave
qk
evanescent wave
Considering a single polarization, propagation along z>0 and the paraxial approximation q<<k
2
q z
i


ikz
A ( q , z )  A ( q ,0 ) e e 2 k
Angular spectrum through a thin lens

Al ( q , z ) 
2 f
1
( 2 )
2
ik
A

0
( ) e
i
f
2k
  2
( q  )

d
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 )
Electric field for the defocused microscope on the image plane

E ( ) 
Be

i (  )
( 2 )
2


A0 ( q ) e
i D fq
2k
2
e
 
iq . 

dq


k f  k
2
 f
f
d 

 (  )  kf 1  k 0 d  k 0 f 2  k D f    0 1   2  1  2 

 kf 2
 f1  2 k 2 k 0 2 k 0 
Diffraction by a sinusoidal phase grating with spacing L
focalfplane
q0 

E0 ( )  E0 e

i (  )
 E0 e

E0 ( )  E0
light

A0 ( q )   2 

2
2
L

ik 0 D nh (  )
 E0 e
m  
 J m ( k 0 D nh ) e
imq 0 x
m  
m  
E0
ik 0 D nh sin( q 0 x )
 J m ( k 0 D nh ) 
m  
2
q y , q x  mq 0 
J m are Bessel functions of order m
Electric field for the defocused sinusoidal phase grating
2

E (  )  Be

i (  )
m  
E 0  J m ( k 0 D nh ) e
m  
i
( mq 0 ) ( z f  p1 )
2k
e
imq 0 x
Contrast of a defocused phase grating considering only first order diffraction
if k 0 D nh  1
such that
J 0 ( k 0 D nh )  1 and J 1 ( k 0 D nh ) 
k 0 D nh
2
q 0 ( z f  p1 )


i



i (  )
2k
E (  )  Be
E 0 1  ik 0 D nh e
sin( q 0 x ) 




2

sin  max 
L min 
q max
k

NA
n
Defining contrast as
0
NA

C ( ) 

I ( )  I0
I0

C ( ) 

I ( )  I0
I0


I ( )  E ( )
with
2
I0  B E0
 q 02 ( z f 
 2 k 0 D nh sin 
2k

2
p1 )
2

 sin( q 0 x )

Sinusoidal Phase Grating with Spacing L
0,6
 D fq 02
C ( x )  2 D nk 0 h sin 
 2k
q0 
2
 def 
L

 sin( q 0 x )

2L
0,4
0,2
2
0

L  1 . 65  m
-0,2
-0,4
Shifted Talbot Images
-0,6
-150
-100
-50
0
50
100
150
Df (m)
FFT_contraste_1.65 m
8 10
7 10
6 10
5 10
4 10
3 10
2 10
1 10
4
0.076 m-1
4
4

4
desf
= 13.16 m
4
4
4
4
0
0
0,1
0,2
0,3
0,4
0,5
-1
k (m )
z
0,6
0,7
0,8
Test of the relation
 def 
2L
2

n  1.52  0. 13
14
12
10
8
6
4
2
0
-2
-0,5
0
0,5
1
L (m)
1,5
2
Defocused contrast for a general transparent interface

h( ) 

1

 
 h ( q ) sin( q .  )
S q
C ( )  2k0
 2
q Df


 h ( q ) sin
 2k
S q

Dn
 q 2Df 
For 
  1
 2k 


 sin( q .  )


C ( )  
Dn
n

Df  h( )
2
Curved Thick Phase Object
Objective Focal Position
Glass-slide
Light
solution
zf
Z
h(x,y)
Df

C ( )  
Dn
n
D f

 h  h(  )
2
Polystyrene Spherical Cap

C ( )  
Dn
( D f  h ) h
2
n
DM
image
AFM
image
Linear Defocusing Region
AFM and DM Profiles
AFM
DM
D f  0 .3  m
R=5.12m
R  4 .8  0 .1  m
D n  0 . 59  0 . 04
Refractive Index Difference Obtained with DM
Dn=0.61±0.01
CCD Calibration
N  AI  B
Camera
Dage-MTI – 8 bts
 N0 
C N
C  

 N0  B 
Camera
UniqVision – 12 bits
N 0  127
N 0  2000
C  0 . 77 C N
C  0 . 98 C N
Power meter intensity (W)
Fluctuating transparent interfaces and contrast correlation function
h



H ( , t)  h( )  u ( , t)
Time average


 H ( , t)   h( )

and  u (  , t )   0
H

C ( , t) 

I ( , t)  I0
I0


2 Dnk 0
D fq


 H ( q , t ) sin
 2k
q
S
2


2 Dnk 0
D fq



 C ( , t)  
 h ( q ) sin
 2k
S q
2


 sin( q .  )



 sin( q .  )


with



DC (  , t )  C (  , t )  C (  , t ) 


 2   ( q ) t
 u ( q , 0 ) u ( q , t )  u ( q ) e
and for a stationary process such that
Space-time correlation function of contrast fluctuations
2(Dnk 0 )


D C ( 0 ,0 ) D C (  , t ) 
u
(
q
)

q
S
2
2
e

 ( q )t
2

D
fq
2
sin 
 2k


 
 cos( q .  )

Mean-square fluctuation of contrast
2(Dnk 0 )
2
DC (Df ) 
2
S
2
DC (Df ) 
(Dnk 0 )
2 2

u
(
q
)

2
q
2
2

D
fq
2
sin 
 2k

 D fq

 2
2
 d q  u ( q ) sin 
 2k
2



and for the continuum case



for D f  
2
2
2
DC ( )  (Dnk 0 )  u    D
 2
u (q ) 

2


 D fq 
d
D
f
D
C
(

)

D
C
(
D
f
)
cos



2
( D nk 0 ) k
 k 
4
2
2
2
Numerical example
u (q )
2

1
aq  bq  c
4
[  m ],
4
2
a  7 . 6 ; b  212  m
2
; c  4680  m
Mean square
contrast
fluctuation
4
;
( D nk 0 )
2
π
 1; R  4 . 9 μm .
Spacial power
spectrum of
fluctuations
q min  3  m
2
u (q ) [m ]
4
1
Diffraction by two transparent interfaces
Average contrast

C ( ) 
2 ( D nk 0 )
S
2
 {
q
  z f  p 2 q 2

H 2 ( q ) sin 
2k

 ( z f  p1 ) q 2

H 1 ( q ) sin 
2k





 
sen
(
q
  )}


Constrast correlation function

D C ( 0 ,0 ) D C (  , t ) 
2 ( D nk 0 )
S
2
 {
q

u1 (q )
2
e

 ( q )t
1
2

(
z

p
)
q
f
1
2
sin 
2k

2






z

p
q
 2  2 ( q )t
 
f
2
2

u
(
q
)
e
sin
}
cos(
q
. )



2
2k



Numerical example
u 1, 2 ( q ) 
1
aq  bq  c1, 2
4
2
,
a  7 . 6 ; b  212  m
2
; c1  4680  m
4
;
( D nk 0 )
π
2
 1; R  4 . 9 μm .
Two symmetric
interfaces
Two asymmetric
interfaces
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.
Lecture 4
Application 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.
Results – Curvature Fluctuations
ruffle
SRMF
Cytoskeleton
Polimerized protein filaments
Alberts, et al Mol. Biol. Cell. 3rd
Ed.Garland Pub. Inc. NY(1994)
Actin filaments just below the
plasmatic membrane
Svitkina, Verkhovsky, MacQuade &
Borisy J. Cell Biol. 139 (2), 397
(1997)
Ruffles: curvature and thickness profiles
1 .5
0 .6
1
0 .5
0 .4
h ( m )
-1
 ( m )
0 .5
0
-0 .5
0 .3
0 .2
0 .1
-1
0
-0 .1
-1 .5
2
3
4
5
6
x ( m )
Ruffle
hyperbolic
7
8
9
2
3
4
5
6
7
8
x ( m )
h0 
 x  x0
h( x) 
1  tanh 
2 
 w



9
Ruffles: curvature and thickness profiles
1
0 .3
0 .5
0 .25
0 .2
h ( m )
-1
 ( m )
0
-0 .5
-1
0 .15
0 .1
0 .05
-1 .5
0
-2
-0 .05
0
1
2
3
4
5
x ( m )
Ruffle
gaussian
6
7
8
0
1
2
3
4
5
6
7
8
x ( m )
  x  x0
h ( x )  h0 e
2w
2
2
Measuring ruffle contrast as a function of defocusing we are able to obtain its refractive index.
D n  0 . 049  0 . 015 (Coelho Neto, Biophys. J. 91, 2006)
Spatial correlation function
0,1
-2
s p a tia l co rr e la tion fu n c tio n ( m )
-2
s p a tia l cu rv a tu re c orre la tio n fu n c tio n ( m )
0.08
0.06
0.04
0.02
0
- 0.02
0,08
0,06
0,04
0,02
0
- 0,02
-0.5
0
0.5
1
1.5
2
d is ta n c e (m )
2.5
3
3.5
-5
0
5
10
d is ta n c e (m )
15
20
25
Time correlation function
y = m 1 * exp(-M 0/m 2)
-2
c u rv a tu re tim e co rre la tio n fu n c tio n ( m )
0,1
0,08
V alue
E rror
m1
0,10153
0,00074978
m2
5,7053
0,057
C his q
0,00053441
NA
R
0,99463
NA
For bone marrow
macrophages
(extracted from healthy
mice) this relaxation
time is
0,06
  (6  2) s
0,04
0,02
0
- 0,02
0
50
100
150
tim e (s )
200
250
Curvature probability distribution
function
10
10
N
10
10
7
6
5
  0 .2 5  m
1
4
  0 .6 1  m
1
  0 .0 2 4
1000
100
10
-3
-2
-1
0
1
2
-1
cu rv a tu re ( m )
3
Before and after addition of 100nM of Cytochalasin-D
Ruffles are inhibited
After addition of 100nM of Cytochalasin-D
Results: 24-37oC
2
3 .2
1 .5
ru ffles
2 .8
ln (V
ln ( )
)
3 .6
2 .4
1
0 .5
0
2
3 2 2 3 2 4 3 26 3 28 3 3 0 3 3 2 3 3 4 3 3 6 3 3 8
1 /T (1 0
-5
-1
K )
E a  33  2  k B T
-0 .5
3 2 2 3 2 4 3 2 6 3 28 3 30 3 3 2 3 3 4 3 3 6 3 3 8
1 /T (1 0
-5
-1
K )
E a  36  5  k B T
Coelho Neto et al., Exp. Cell Res. 303 (2), 207 (2005)
Discussion of models
Model of cellular motility : Brownian Ratchet
2

Actin filament

x
D
f
membrane
actin-g
  2.7 nm
Dynamics of actin polymerization
(diffusion + polymerization)
Peskin, Odell & Oster Biophys. J. 65, 316 (1993)
Mogilner & Oster Biophys. J. 71, 3030 (1996)
Mogilner & Oster Biophys. J. 84, 1591 (2003)
E a  31 k B T
Phagocytosis of Leishmania amazonensis at 37oC
Film accelerated
 16x
Results: Phagocytosis at 37oC
Behavior of <2> near the phagossome
0 .1
0 .05
0
-5 0
t f  60 s
0
50
100 150
tim e (s )
200
250
300
0 .1 5
-2
-2
0 .15
0 .2
( m )
m e a n s q u a re c u rv a tu re
0 .2 5
0 .2
( m )
m e a n s q u a re c u rv a tu re
0 .25
t f  120 s
0 .1
0 .0 5
0
-5 0
0
50
100 150
tim e (s )
200
250
300
Results: Phagocytosis from 24 to 37oC
ln (p h ag o c y to sis tim e)
6
5 .5
E a  38  4  k B T
5
4 .5
4
3 .5
3 22 3 2 4 3 2 6 3 2 8 3 30 3 32 3 3 4 3 3 6 3 3 8
1 /T (1 0
-5
-1
K )
Coelho Neto et al., Exp. Cell Res. 303 (2), 207 (2005)
Protein-Membrane
Coupling Model
Theoretical model of
Experimental data of
Coelho Neto et al. Exp. Cell Res.
303, 207 (2005)
Red Blood Cell (RBC)
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
Defocused Image of a Red Blood Cell (RBC)




h 2 (  )  h1 (  )  2

C (  )  2Dn Df 
 h2 (  )
2


C on trast
0 .2
0
-0 .2
R B C p ro file (  m )
-1 0
10
0 .8
DDn
n==
0 .04
22
0.056
0 .4
0
Mesquita, Agero, Mesquita, APL 88, 133901 (2006)
0
p o s itio n (m )
0
2
4
r (m )
Limite assintótico para grandes desfocalizações
 ( D f  p1 ) q 2  1 1
 ( D f  p1 ) q 2 
sin 
  2  2 cos 

2
k
k




2
 (Df  p )q 2 

  2 
2k


Para
 D C 2
  D n1 k 0  h1   D n 2 k 0  h 2 
2
Para hemácias
2
2
2
 D C 2
3  10
h
2
 D  1 2

 2  D nk 0  h
2
4
 D  2 2
2
 2 0 . 056  9 . 3  h
2
 0 . 024  m  24 nm
2
Membrane Elasticity and Fluctuations
Membrane Free Energy Variation
k
2
D F   dA  C  u
 2


2


u
2
2

S. A. Safram, Statistical Thermodynamics of Surfaces,
Interfaces, and Membranes, Addison-Wesley (1994).
Monge representation

 u 
2

2
u
water
kC
bending modulus

surface tension

confinement potential
Lipid bilayer
Hydrophilic
Fourier decomposition and energy equipartition
Hydrophobic
u (q )
2


kT
A kC q  q  
4
2

water
Curvature energy for curved surfaces – Helfrich free-energy (Phys. Lett.1973)
k

2
FC   dA  C C 1  C 2  C 0   k C 1C 2 
 2

C0
spontaneous curvature
C 1 and C 2
main curvatures
K  C 1 .C 2
Gaussian curvature
RBC Elastic Model of Auth, Safran, and Gov
-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);
Cytoskeleton is modeled as a
-Auth T., Safran S. and Gov N., Physical Review E 76 , 051910 (2007).
hexagonal network of entropic springs
spectrin
Spectrin filaments
Actin nodes
bilayer
B. Alberts et al.,”Molecular
Biology of the Cell”, (2002)
ATP driven non-thermal effects
d
Detached
filaments
RBC Elastic Model of Auth, Safran, and Gov
 2
u (q )

 ef 
k
kT ef
q   ef q  
4
c
2

9  kT ef
 cytoskeleton shear modulus
T ef  T bath
16  k C
  3 fC 
k c bilayer curvature modulus
3
A
u ( q ,0 )u ( q , t ) 
 dA ( 2 H
u (q )
2
2
e
 K)
T ef effective temperature
 ( q ) t
exp(  2 qd )(  1  exp( 2 qd )  2 qd  2 ( qd ) )
2
 (q ) 
4 q
k
q   ef q  
4
c
2


cytoplasm
viscosity
Reference System
origen
light
Symmetry plane
Glass-slide
C ( ) 
Dn
n
( z
f
 h1 ) 1  ( z f  h 2 ) 2
se  h1   h 2  
2
2
sendo D f  z f 
h1  h 2
2

C ( ) 
2Dn
n
Df  ( )
Results

h (  )  h0 
  Fourier
X (m)
X (m)
f C 1  0 . 13  m
2
f C 2  0 . 25  m
2




C
(

) 
1
 

2
2DnDf
 q

1
Transform
(m)
Measurements of RBC Flickering with DM
G. Glionna et al. APL (2009)
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 of
large wavenumber fluctutations is
evident in the figure.
 D C ( z f ) 
2
2  D nk 0 
S
2


z f  p1 2 
z f  p2 2 
2
2
kT
sin
q
kT
sin
q



 

ef
ef
2
k
2
k



 
 [

]

9

kT
9

kT
q

ef
ef
4
2
4
2
k
q

q

3

f
k
q

q

3

f
C
C1
C
C2 

16

k
16

k
C
C


0 .0 0 0 3 4
With DM we measured
0 .0 0 0 3 2
0 .0 0 0 2 8
kC
2
f
< D C (z )>
0 .0 0 0 3
f C 1  0 . 13  m
2
f C 2  0 . 25  m
2
 7 .6  0 .8
kT ef
0 .0 0 0 2 6

2
 ( 9 . 2  0 . 4 )  10  m ;
3
kT ef
0 .0 0 0 2 4
bkg  (1 . 40  0 . 03 )  10
0 .0 0 0 2 2
T ef  3 . 3 T bath
0 .0 0 0 2
9
9 .5
10
1 0 .5
z ( m )
f
11
1 1 .5
12
4
.
exp(  2 qd )(  1  exp( 2 qd )  2 qd  2 ( qd ) )
2
 1, 2 ( q ) 
4 q
k
q   q   1, 2
c
4
2

  3 water
0 ,0 0 0 1 2
< (D C (0 ,0 )D C (0 ,0 .0 3 3 s )>
0 ,0 0 0 1
8 10
6 10
-5
-5
d  ( 21  1) nm
4 10
2 10
bkg  (1 . 8  0 . 2 )  10
-5
-5
8 ,5
9
9 ,5
10
1 0 ,5
z ( m )
f
11
1 1 ,5
12
1 2 ,5
5
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.