Outline

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

Motivation: Study of Adhered Macrophage Motility

Film accelerate

d 16x

Phagocytosis of Leishmania amazonensis at 37oC

Film accelerated

16x

Contrast Fluctuations - Macrophage

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

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

21

)0,(),( kqikz

eqAzqA

Free propagation of the angular spectrum

2D Fourier transformjyix ˆˆ

jqiqq yxˆˆ

From Helmholtz equation

0.22

deEkE qi 0),(),( 222

2

zqAqkz

zqA

kq propagating wave

kq evanescent wave

kzqiikz eeqAzqA 2

2

)0,(),(

Angular spectrum through a thin lens

deAik

fzqAq

kfi

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 qikfqii

.2

)( 2

02)(

)2()(

00

212

1

2

2

102001 222

)(kd

kf

kf

fk

kffkfkfkdkkf

Electric field for the defocused microscope on the image plane

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 2

0

02

002

0 ,)(2)( mqqqnhkJEqA xy

m

mm

ximqkpzmq

im

mm

i eenhkJEBeEf

0

12

0

2)()(

00)( )()(

Electric field for the defocused sinusoidal phase grating

mJ are Bessel functions of order m

Contrast of a defocused phase grating considering only first order diffraction

2)(and1)(

thatsuch1if

00100

0

nhknhkJnhkJ

nhk

)sin(1)( 02

)(

00)(

120

xqenhikEBeE kpzq

iif

Defining contrast as

0

0)()(

III

C

with20

20

2)()(

EBI

EI

)sin(2

sin2)(

)( 0)(2

00

0

0 1 xqk

qnhk

III

Cpz f

NAL

nNA

kq

0min

maxmaxsin

)sin(2

sin2)( 0

20

0 xqk

fqhnkxC

Sinusoidal Phase Grating with Spacing L

Lq 2

0

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

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

Defocused contrast for a general transparent interface

).sin()(1

)(

qq

qhS

h

qq

kfq

qhS

nkC

).sin(

2sin)(2)(

2

0

12

For2

k

fq

)()( 2

hfnnC

Light

Objective Focal Position

Z

fz

h(x,y)

Glass-slidesolution

f

Curved Thick Phase Object

)()( 2

hhfnnC

Polystyrene Spherical Cap

hhfnnC 2)()(

DMimage

AFMimage

Linear Defocusing Region

AFM DM

R=5.12m04.059.0

1.08.43.0

nmR

mf

AFM and DM Profiles

n=0.61±0.01

Refractive Index Difference Obtained with DM

NCBN

NCBAIN

0

0

CameraDage-MTI – 8 bts

NCCN

77.01270

NCCN

98.020000

CCD Calibration

CameraUniqVision – 12 bits

Power meter intensity (W)

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

nI

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

Mean-square fluctuation of contrast

k

fqqu

S

nfC

q

k2

sin)()(2

)(2

222

02

and for the continuum case

kfq

fCCfdknk

qu2

2

2cos)()(

)()( 22

0

4

2220

2

2202

)()(

for

2sin)(

2)(

)(2

22

unC

f

k

fqquqd

nfC

k

k

Spacial power spectrum of fluctuations

Mean square contrast

fluctuation

Numerical example

.9.4 ;1 ;4680 ;212 ;6.7 ],m[ 1)(2

042424

2 μmRπ

)nk(mcmbacbqaq

qu

][)( 42 mqu

1min 3 mq

Constrast correlation function

).cos(

2sin)(

2

)(sin)(

)(2

),()0,0(

}{2

22

2

2

122

1

22)(2)(10

q

k

qpzequ

k

qpzequ

S

nk

tCC

ftqftq

q

Diffraction by two transparent interfaces

}

{

)(2

sin

2

)(sin

)(2)(

2

2

12

22

10

)(

)(

qsenk

qpz

k

qpz

S

nkC

f

f

q

qH

qH

Average contrast

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

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 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.

ruffle

SRMF

Results – Curvature Fluctuations

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)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

2 3 4 5 6 7 8 9h

(m

)x (m)

w

xxhxh 00 tanh12

)(

-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

-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

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

spat

ial c

urva

ture

cor

rela

tion

func

tion

(m

-2)

distance (m)

-0,02

0

0,02

0,04

0,06

0,08

0,1

-5 0 5 10 15 20 25

spat

ial c

orre

latio

n fu

nctio

n (

m-2

)

distance (m)

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

curv

atur

e tim

e co

rrel

atio

n fu

nctio

n (

m-2

)

time (s)

y = m1 * exp(-M0/m2)ErrorValue

0,000749780,10153m1 0,0575,7053m2

NA0,00053441ChisqNA0,99463R

Time correlation function

Curvature probability distribution function

1

1

0.25

0.610.024

m

m

10

100

1000

104

105

106

107

-3 -2 -1 0 1 2 3

N

curvature (m-1)

Before and after addition of 100nM of Cytochalasin-DRuffles are inhibited

After addition of 100nM of Cytochalasin-D

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

(Vru

ffles

)1/T (10-5 K-1)

TkE Ba 233 TkE Ba 536

Coelho Neto et al., Exp. Cell Res. 303 (2), 207 (2005)

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)

Phagocytosis of Leishmania amazonensis at 37oC

Film accelerated

16x

0

0.05

0.1

0.15

0.2

0.25

-50 0 50 100 150 200 250 300

mea

n sq

uare

cur

vatu

re

(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

squa

re c

urva

ture

(m

-2)

time (s)

st f 120

Results: Phagocytosis at 37oC

3.5

4

4.5

5

5.5

6

322 324 326 328 330 332 334 336 338

ln (p

hago

cyto

sis

time)

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)

Protein-MembraneCoupling Model

Theoretical model of

Experimental data ofCoelho Neto et al. Exp. Cell Res.

303, 207 (2005)

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)

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)( 2212

hhhfnC

0

0.4

0.8

0 2 4

n=0.0422

RB

C p

rofil

e (

m)

r (m)

n = 0.056

Limite assintótico para grandes desfocalizações

22

)( 2

k

qpf

k

qpf

k

qpf 2

1

2

12 )(cos

21

21

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

Membrane Free Energy Variation

2222

222uuukdAF C

Fourier decomposition and energy equipartition

24

2)(qqkA

kTquC

Curvature energy for curved surfaces – Helfrich free-energy (Phys. Lett.1973)

21

20212

CCkCCCkdAF CC

S. A. Safram, Statistical Thermodynamics of Surfaces,Interfaces, and Membranes, Addison-Wesley (1994).

u

Monge representation

21

21

0

.and

CCKCC

C

spontaneous curvature

main curvatures

Gaussian curvature

Membrane Elasticity and Fluctuations

Ck bending modulus

surface tension

confinement potential

Lipid bilayer Hydrophilic

Hydrophobic

water

water

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

24

2)(qqk

kTqu

efc

ef

C

efef k

kT

169

)2(33 2 KHdAA

fC

tqequtququ )(2)(),()0,(

242

4))(22)2exp(1)(2exp()( qqk

qqdqdqdqdq efc

bathef TT

RBC Elastic Model of Auth, Safran, and Gov

cytoskeleton shear modulus

cytoplasm viscosity

bilayer curvature modulusck

efT effective temperature

se

2211 )()()( hzhznnC ff

)(2)( fn

nC

22

12 hh

sendo2

21 hhzf f

Reference System

Glass-slide

Symmetry plane

lightorigen

TransformFourier

qC

fnhh

2

10

)(2

1)(

2

2

25.0

13.0

2

1

mf

mf

C

C

Results

X (m)

X (m) (m)

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)

]3

169

2sin

3169

2sin

[2)(

224

222

124

2122

02

CC

efC

ef

q

CC

efC

ef

fqk

kTqk

qkp

kT

fqk

kTqk

qk

pkT

SnkC

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

<C

2 (zf)>

zf (m)

With DM we measured

22

21

25.0

13.0

mf

mf

C

C

bathef

ef

ef

C

TTbkg

mkT

kTk

3.3.10)03.040.1(

;10)4.02.9(

8.06.7

4

23

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

34

))(22)2exp(1)(2exp()( 2,124

2

2,1

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.