On Cell Mechanics

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Possible Application of Cell Mechanics

2

Understanding the architecture of the cytoskeleton or other cellular compartments

follow changes in the cell, including reactions to external stimuli

determine differences in cell state

with possible bio-medical or clinical implications

cell mechanics as a diagnostic tool

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Why measuring mechanics ?

Understanding the structure and function of the actin cytoskeleton or other molecular players in mechanics

insights in mechanically interesting processes like migration or cell division

using cell mechanics as a marker of cellular function

3

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Cell Mechanics as Diagnostic Tool

4

cancer cells

mechanical diseases of cellular compartments! nucleopathies, Alzheimer, etc.

reproducibility and reliability becomes a big issue in clinical applications

data should be comparable also for scientists

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

What is this talk about ?

5

motivation for producing reliable and reproducible quantities

some personal ideas on this topic

suggestions how to do get reproducible numbers

timeline of errors I did in the past

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

How to we get from a force curve to the Young's modulus ?

6

-800

-750

-700

-650

-600

-550

-500

defle

ctio

n [n

m]

-13-12-11-10-9z height[µm]

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

How to we get from a force curve to the Young's modulus ?

6

-800

-750

-700

-650

-600

-550

-500

defle

ctio

n [n

m]

-13-12-11-10-9z height[µm]

6050

4030

2010

0µm

6050403020100µm

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

log

of Y

oung

s's

mod

ulus

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

There are Several Issues

7

need for a mechanical model

is the model appropriate?

calibration issues: deflection, force constant

model parameters: tip shape, Poisson ratio, ...

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Let's start with the points, often regarded as trivial: calibration of the deflection signal

8

-800

-750

-700

-650

-600

-550

-500

-450de

flect

ion

[nm

]

-8.0-7.8-7.6-7.4-7.2-7.0-6.8-6.6-6.4z height[µm]

fit a line to the linear part of a force curve

from a distance everything looks fine

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Let's start with the points, often regarded as trivial: calibration of the deflection signal

8

fit a line to the linear part of a force curve

from a distance everything looks fine

-750

-700

-650

-600

-550

-500

defle

ctio

n [n

m]

-6.60-6.55-6.50-6.45-6.40z height[µm]

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

However, there is some variation in the slope of force curves on stiff samples

9

600

400

200

01.041.021.000.980.96

histogramm of more than 2000 force curves on the substrate

mean is 1.00

sdev is on the order of 0.02

there is a 2% error in calibration of the deflection signal

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

This looks acceptable when looking at the big picture

10

6050

4030

2010

0µm

6050403020100µm

600

400

200

01.00.80.60.40.2

Histogram of the slope of a complete force map including cell data and data on the support

Cell Petri dish

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

How well do we know the force constant?

11

The manufacturer wants to be on the safe side and gives large error margins

from. http://www.brukerafmprobes.com/Product.aspx?ProductID=3444

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Several Ways to calibrate the force constant

12

calculate from geometry

added mass method

calibrating with a standard spring

estimate by resonance frequency (Cleveland: k ∝ f3 )

thermal noise method

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Thermal noise is easy to use and implemented in many commercial instruments

13

2

3

4

5

678910

-12

2

3

4

5

6789

10-11

100 Hz 1 kHz 10 kHz 100 kHz 1 MHz

resonance frequency: 7.286 kHz

spring constant: 17 mN/m

Cleveland says: 12 mN/m

see for more background: http://www.physics.uwo.ca/~hutter/calibration/afmcal.html

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Thermal noise is easy to use and implemented in many commercial instruments

14

2

3

4

5

678910

-12

2

3

4

5

6789

10-11

100 Hz 1 kHz 10 kHz 100 kHz 1 MHz

needs calibration of deflection

relies on equipartition theorem

requires low DC noise

requires DC calibration of deflection still valid at higher frequencies

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

What's the bottom line?

15

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Why do we need models?

16

because contact area changes during indentation

because stress is transmitted in the material in a complicated way

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Sometimes simple Shell Theory works

F = 0.8*E * d2

R*δ

kN = 0.8*E *d 2

R17

1.2

1.0

0.8

0.6

0.4

0.2

0.0

defle

ctio

n [µ

m]

1086420-2z height [µm]

Sample Slope Stiffness[mN/m]

control 0.12 1.34 control 0.15 1.81 37 µg/ml DNA 0.23 2.94 110 µg/ml DNA 0.40 6.64

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

More often the Hertz model is employed

18

homogenous material

isotropic material

no adhesion

infinite thickness

small stresses (i.e. linear regime)

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Variants of the Hertz model for different geometries

19

Paraboloid on Flat: F = 43

E(1−ν 2 )

R *δ32

Cone on Flat: F = 2π

E(1−ν 2 )

tanα *δ 2

Pyramid on Flat:

From: Lin DC, Dimitriadis EK, Horkay F. , J Biomech Eng. 2007 Jun;129(3):430-40. "Robust strategies for automated AFM force curve analysis--I. Non-adhesive indentation of soft, inhomogeneous materials."

F = 2 E(1−ν 2 )

a*δCylinder on Flat:

F = 12

E(1−ν 2 )

tanα *δ 2

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

The Hertz model can work amazingly well,here: poly-acrylamid gels

20

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Or not, if the model is not appropriate: fitting pyramidal data with the sphere model

21

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

How well do we know the parameters needed as input for the Hertz model?

22

Poisson ratio! generally assumed to be 0.5

tip opening angle or radius! specs: ! ! 17,5˚ ±2.5˚, 25˚ ±2.5˚

! measured:! 19˚, 29˚

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

How well do we know the parameters needed as input for the Hertz model?

22

Poisson ratio! generally assumed to be 0.5

tip opening angle or radius! specs: ! ! 17,5˚ ±2.5˚, 25˚ ±2.5˚

! measured:! 19˚, 29˚

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Summary of Errors

23

calibration of deflection!! ! ! ! 2%

calibration of force constant of cantilever! ! 5 %

tip shape (pyramidal tips)! ! ! ! 15 %

negligible: calibration of piezo/distance sensor!! -

total error in slope:! ! ! ! ! 20 %

could translate in an error in E-modulus:! ! 50 %

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

The largest uncertainty is in the tip shape

24

characterize tip shape very accurately

possibly use beads, which are better defined

make only relative measurements

referred to a reference sample

or compare cells before/after drug treatment

or compare different cells, like healthy versus diseased

Possible work arounds

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

What is wrong/missing in Hertz models?

25

cells are not homogenous and isotropic

cells are not infinitely thick

AFM measurements may lead to high strains, so any material will act non-linear

cells are visco-elastic

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Cancer versus normal cells - AFM data

Normal Cells

S784

Cancerous Cells

S277

26

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Cancer versus normal cells - AFM data

Normal Cells

S784

Cancerous Cells

S277

100

80

60

40

20

0

-20

defl

ecti

on [

nm

]

-11.0-10.5-10.0-9.5-9.0-8.5-8.0

z height [µm]

Force Data

100

80

60

40

20

0

-20

defl

ecti

on [

nm

]

-12.0-11.5-11.0-10.5-10.0-9.5-9.0

z height [µm]

Force Data

26

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Cancer versus normal cells - AFM data

Normal Cells

S784

Cancerous Cells

S277

100

80

60

40

20

0

-20

defl

ecti

on [

nm

]

-11.0-10.5-10.0-9.5-9.0-8.5-8.0

z height [µm]

Elastic Modulus Retract: 5100 PaApproach: 3700 Pa

Force Data Hertz Fit to Force Data

100

80

60

40

20

0

-20

defl

ecti

on [

nm

]

-12.0-11.5-11.0-10.5-10.0-9.5-9.0

z height [µm]

Elastic Modulus Retract: 1600 PaApproach: 670 P

Force Data Hertz Fit to Force Data

26

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Viscoelasticty is often neglected

27

-500

-400

-300

-200

-100

LVD

T (n

m)

1.00.80.60.40.20.0Time (s)

200

100

0

-100

-200

Def

lect

ion

(nm

)

1.00.80.60.40.20.0Time(s)

150

100

50

0

Def

lect

ion

(nm

)

-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.5LVDT (µm)

Approach Retract

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

New Model to include bottom effect

28

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Pyramidal Tips overestimate Young's moduli

29

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Testsample Poly-Acrylamide Gels

30

very homogenous

stable over time

E-modulus depends critically on concentration of cross linkers

preparation is hard to reproduce exactly

10

8

6

4

2

0

1.00.80.60.40.20.0

10

8

6

4

2

0

0.3880.3860.3840.3820.3800.3780.3760.374

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Testsample Poly-Acrylamide Gels

31

14

12

10

8

6

4

2

0

31.030.530.029.529.028.528.0x10

3

10

8

6

4

2

0

0.3880.3860.3840.3820.3800.3780.3760.374

Slope:! ! ! 0.38 +/- 0.002 (0.5%)Young's Modulus:! 29.5kPa +/- 0.5 kHz (2%)

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Does it work between labs ?

32

45

40

35

30

25

20

Youn

g M

odul

us [k

Pa]

Gel06 Gel07

Pos1 Pos2 Pos3 Pos4 Pos1 Pos2 Pos3 Pos4Barcelona Barcelona

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Variability in different positions

33

40

30

20

10

0

36343230282624x10

3

Position 1 0.5 Hz Position 2 0.5 Hz Position 2 1 Hz Position 2 2 Hz Position 2 3 Hz

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

However, real data show some miracles

34

180

160

140

120

100

80

60

40

20

Youn

gs's

Mod

ulus

[kP

a]

321

MLCT old α=40˚ k = 17.7 pN/m Glasbead R=5µm k= 27.5 pN/m MLCT new α=19˚ k = 15.8 pN/m

Same gel

different tips

very different E-Moduli

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

However there are many issues to be addressed

35

experimental details! same tips, similar forces

calibration sample! which sample ?! does the recipe allow to make the sample everywhere ?! or do we need a supplier ?

data analysis issues! which part of the data is analyzed ?! how is the contact point determined ?! how is the deflection signal calibrated ?! how is the force constant calibrated ?! how are tip shape parameters defined ?

COST AFM4BioMedNano

BIOPHYSICSINSTITUTE

Manfred Radmacher

Acknowledgements

36

Barcelona! Daniel Navajas! Tomas Luque!Bremen! Jens Schäpe

Cracow! Malgorzata Lekka