raman, trigueros et al · raman, trigueros et al on the other hand is the 2nd cosine coefficient...

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NATURE NANOTECHNOLOGY | www.nature.com/naturenanotechnology 1

Raman, Trigueros et al

Mapping nanomechanical properties of live cells using multi-harmonic atomic force microscopy

A.Raman, S. Trigueros A. Cartagena, A.P. Z. Stevenson, M. Susilo, E. Nauman, and S. Antoranz Contera

In this supplementary material we provide many details to support the main text. In particular, we described in the following sections:

A. Speed and spatial resolution of the proposed method

B. Near-surface hydrodynamic corrections

C. Physics of 0th harmonic image formation

D. Extraction of local material properties from 0th, 1st, and 2nd harmonic observables

E. Comparison of mechanical properties of cells extracted using quasi-static curves

and the multi-harmonic method

F. Additional images

G. Additional data on red blood cells sample preparation

A. Speed and spatial resolution of the proposed method

Table 1 in the main text clearly shows that imaging throughput of our proposed method

in mapping local mechanical properties of cells represents ~10-1000 times improvement

in imaging throughput compared to the standard force-volume method. The high

resolution of the method can also be seen by examining the details of , say, the A0 map

on the rat tail fibroblast in Fig S1 (60 by 60 micron size) taken with 256 by 256 pixels in

~15 minutes. Even at such a large scan size, many cytoskeletal details are clearly

distinguished. As another example in Fig. S2 we show two different live fibroblast cells

captured in a 60 micron by 60 micron image (256 by 256 pixels) which clearly resolve

cytoskeletal details such as actin bundles in the A0 maps.

Supplementary section page 1© 2011 Macmillan Publishers Limited. All rights reserved.


B. Near-surface hydrodynamic corrections

Before proceeding to the theory of physics of image formation and the extraction

of local properties using the multi-harmonic variables, it is important to highlight an

experimental observation using the cantilevers described in the methods section.

Conventional theory generally assumes that once the cantilever has been tuned to

resonance far from the sample with amplitude and phase lag 1farA 1 2farπφ = then the

cantilever oscillation amplitude and phase change only due to tip-sample interaction

forces. In reality for many cantilevers, especially those with short tips such as SiN

probes, the hydrodynamic loading changes both the natural frequency and the damping

of the cantilever as it comes closer to the sampleS1. As a consequence the theories of

image formation and material property reconstruction require two important

considerations. First they needs to account for the difference (often significant) due to

viscous hydrodynamics, in the resonant response of the cantilever when located far and

near the sample. Secondly, the dynamics of the oscillating cantilever interacting with the

sample surface needs to be studied.

Far from the sample, the natural frequency and quality factor of the fundamental

eigenmode of the cantilever can be easily measured using a thermal tune in commercial

AFM systems, and are denoted as farω and respectively. The magnetic excitation at

a frequency

farQ

drω must be tuned to exact resonance with dr farω ω= with a steady state

amplitude so that the tip motion in the driven eigenmode, the phase lag of tip 1A far ( )rfaq t

Supplementary section page 2

© 2011 Macmillan Publishers Limited. All rights reserved.


oscillation relative to drive 1farφ , and the magnitude of the magnetic driving force are

given by:

magF

11 1 1( ) sin( ), ,

2far

far far dr tω φ= − far far magfar

kAq t A F

Qπφ = = (S1)

where is the equivalent stiffness of the first eigenmode for which standard calibration

methods exist.

k

When this resonant cantilever is brought within imaging distance (when the tip is

located <50 nm from surface) to the sample it is well known that the natural frequency

and Q-factor decrease significantly to near drω ω< and respectively due to

hydrodynamic squeeze film that develops S1 between the cantilever and the sample

surface. As a consequence of this important effect, the amplitude and phase of tip

motion change and the tip motion just before engaging the sample now becomes

nearQ

1( ) sin( )near near dr nearq t A t 1ω φ= − (S2)

with and 1near farA A< 1 1 2nearπφ > . nearω and can be measured by measuring the

thermal spectrum after withdrawing the cantilever from the sample by a small distance

(<50nm). Alternately,

nearQ

nearω and can also be estimated by knowledge of the

observables

nearQ

, 1far 1 1 ,far rQ A 1near, , far , neaA2farπω φ φ= using the forced steady state response of

an oscillator as follows. When near the sample, we have the following relations from

simple forced vibration steady state response theory




11 222 2

11

1 ; tan( )

11

; / 2

dr

near nearnearnear

mag drdr dr

nearnear near near

farmag far

far

QkAF

Q

kAFQ

ωω

φωω ω

ωω ω

φ π

⎛ ⎞⎜ ⎟⎝ ⎠= =

⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎜ ⎟− ⎜ ⎟⎜ ⎟− +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

= = (S3)

where is the magnitude of the magnetic excitation force. Using (S3) it can be easily

shown that the readily observable quantities

magF

φ1 1,near nearA are related to ωnear and as

follows:

nearQ

21

1 11 1

11 1

1 1

1 cos( ) cos( )

sin( ) sin( )

far magdrnear near

near near far near

far magdrnear near

near near near far near

A FA Q kA

A FQ A Q kA

ωφ φ

ω

ωφ φ

ω

⎛ ⎞− = =⎜ ⎟

⎝ ⎠

= =

. (S4)

The key point is that both the amplitude and phase of the cantilever change as it is

brought from far to within imaging distance of the sample, however this change is due to

viscous hydrodynamic effects and must be separated systematically from the amplitude

and phase changes that occur due to tip-sample interactions which are discussed now.

Consider the equation governing tip motion when it interacts with the

sample:

( )q t

2

, ,

sin( ) ( , )1

( , ) ( ) ( , )

mag dr ts

near near near

ts ts CONS ts DISS

F t F Z qq q qQ k

F Z q q F Z q F Z q q

ωω ω

+ ++ + =

+ = + + +

q

(S5)




tsF


is the tip sample-interaction force which is assumed to decompose additively into a

conservative (tip-sample position dependent) and a dissipative (tip velocity

dependent) component . is the magnitude of the magnetic excitation force as

derived in Eq. (S1). Z is the difference between cantilever position and the sample, also

known as the Z-piezo displacement (See Fig. S3).

,ts CONSF

,ts DISSF magF

Let the steady state motion of the tip interacting with the sample comprise of only

the 0th, 1st and 2nd harmonics so that the tip displacement and velocity are

0 1 1 2 2 0 1 2 1 2

1 2 1 2

( ) ( ) sin( ) sin(2 ) sin( ) sin(2 2 ) ( ) cos( ) 2 cos(2 2 )

dr dr

dr dr

q t q t A A t A t A A Aq t A A

ω φ ω φ θ θ φω θ ω θ φ φ

= = + − + − = + + + −

= + + −

φ

(S6)

assuming that these are the dominant harmonics that govern the motion, a fact that is

readily observable from experiments in liquids.

In order to calculate the 0th, 1st and 2nd Fourier components of in Eq. (S5) we

proceed as follows. Rewriting the interaction force as the sum of purely conservative

and non-conservative (in other words dissipative) tip-sample forces, and substituting

into it the assumed harmonic motion Eq. (S6) we find:

tsF

0 1 1 2 2, , , , ,( , ) cos( ) sin( ) cos(2 ) sin(2 )ts ts CONS ts DISS ts CONS ts CONS ts DISSF Z q q F F F F Fθ θ θ+ = + + + + θ (S7)

In the intermittent contact regime and while oscillating in permanent contact, it can be

shown that , ( )ts CONSF θ is symmetric about 3 / 2θ π= while , ( )ts DISSF θ is antisymmetric

about 3 / 2θ π= . As a result while is the 1st Fourier sine coefficient, the F1,ts CONSF 2

,ts CONS



on the other hand is the 2nd cosine coefficient since sin( )θ and cos(2 )θ are both

symmetric about 3 / 2θ π= .

Substituting Eqs. (S6, S7) into (S5a) and balancing separately the constant, and

the sine and cosine harmonic terms in the equation, readily leads to the following

results that link the Fourier components of the interaction forces to the observables

(cantilever amplitudes, phase etc). The ith Fourier coefficient of the conservative

interaction force is called the ith harmonic virial and the ith Fourier coefficient of the

dissipative interaction force is called the ith harmonic dissipation:

0, 0

2

co

sin(

cos(

mag

mag

kA

F

F

kA

φ

21, 1 1

1, 1 1

22, 1 2 1 22

,

( )

( ) s( ) 1

( ) )

4 2( ) 2 ) 1 s )

( )

ts CONS

drts CONS

near

drts DISS

near near

dr drts CONS

near near near

ts D

a F

b F kA

c F kAQ

d FQ

e F

ωφ

ω

ωω

ω ω in(2φ φ φ φω ω

⎛ ⎞⎛ ⎞⎜ ⎟+ − ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

⎡ ⎤⎛ ⎞= − − −⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

=

= −

= −

−

22

2 1 2 22

4 2) 1 cos( )dr drISS

near near nearQω ω

φ φω ω

⎡ ⎤⎛ ⎞= − − + −⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦12φ φsin(2kA

(S8)

Eliminating 1φ among Eqs.(S8b,c) we reach the classical amplitude reduction equation

of amplitude modulated-AFM:

21

1 ,

1,

ts CONS

ts DISS

F

F

⎛ ⎞⎜ ⎟− −⎜ ⎟⎝ ⎠

1 12 222

2

1 1, ,2

1 1

/, tan( )

1 ;

rmag

drdr dr

eff earnear near eff

ts CONS ts DISSdr dreff

near eff near near

kAF k

A

Q

F FkA Q Q kA

φω ωω

ω ω

ω ωω

ω ω

⎛ ⎞⎜ ⎟⎝ ⎠

= =⎛ ⎞ ⎛ ⎞ −

− +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= − = −

1

1 dr

nea

near n

kAQ

ωω

ωω (S9)




where effω is the non-dimensional natural frequency of the cantilever modified by

conservative tip-sample interactions and is the effective Q-factor of the cantilever

modified by dissipative tip-sample interactions. Thus amplitude reduction occurs due to

both conservative and dissipative tip-sample interactions while the phase reflects the

ratio of conservative and dissipative interactions.

effQ

Recall that when triangular cantilevers with short tips are used, the near-surface

hydrodynamic effects imply that near farω ω< and near farQ Q< so that and 1near farA A< 1

1 1 / 2near farφ φ π> = . Eq. (S4) connects the quantities ,near Qnearω to the observables

1 1,near nearA φ . Utilizing Eqs. (S3, 4) in Eq. (S8) we get the following expressions for the

0th, 1st and 2nd harmonics virials and dissipation in terms of the observables:

0, 0

1 1 1, 1 1

1

1 1 1, 1 1

1

2 1 1, 2 1 2 1

1 1

( )

( ) cos( ) cos( )

( ) sin( ) sin( )

4 2( ) cos(2 ) cos( ) 3 sin(

ts CONS

farts CONS near

far near

farts DISS near

far near

ts CONS nearnear near

a F kA

kA Ab FQ A

kA Ac FQ A

A Ad F kAA A

φ φ

φ φ

φ φ φ φ

=

⎛ ⎞= − +⎜ ⎟

⎝ ⎠⎛ ⎞

= − +⎜ ⎟⎝ ⎠

⎛ ⎞= − − −⎜ ⎟

⎝ ⎠1 1

2 1 1, 2 1 2 1 1 1 2

1 1

)sin(2 )

4 2( ) sin(2 ) cos( ) 3 sin( )cos(2 )

near

ts DISS near nearnear near

A Ae F kAA A

2φ φ

φ φ φ φ φ φ

⎡ ⎤−⎢ ⎥

⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞

= − − + −⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ (S10)

C. Physics of 0th harmonic image formation

The 0th harmonic refers to the DC signal of the of the vibrating cantilever

generated due to cycle averaged tip-sample interaction forces which is, in general,

different from the deflection curve for an unexcited cantilever in a quasi-static force




distance curve. For instance, a strong A0 signal is created in liquid in a high

concentration buffer (so that long range electrostatic forces are screened) when the

excited cantilever placed a certain height above the sample intermittently taps on the

sample while the same cantilever at the same height from the sample but not vibrating

would not deflect appreciably. Once the cantilever is firmly pushed into permanent

contact with the sample and if the vibration amplitudes are small compared to the

indentation (as in the case of soft live cells) then the 0th harmonic of the vibrating

cantilever equal the static bending of the unexcited cantilever at the same position.

However this equivalence is generally not true. Broadly speaking, the generation of the

0th harmonic is a direct result of nonlinearity of interaction. There are many fields in

physics where the AC excitation of a nonlinear system generates a 0th harmonic

(thermal expansion, acoustic streaming) as well as a 2nd harmonic.

From Eqs. (S9) and (S8a), the physics of image formation in the topography and

0th harmonic channels becomes clear. In amplitude modulated AFM, since the

amplitude is regulated by changing , a topography image of a live cell in tapping

mode in liquids consists of those values of Z that render a constant amplitude over

the scan area, reflecting the combination of tip-sample conservative and dissipative

interactions that reduce the amplitude. However since a live cell is so much softer than

the microcantilever, the Z-piezo actuator has to push down significantly on a cell to

reduce its amplitude to the setpoint amplitude. Thus the topography image of a live cell

in liquids using amplitude modulated AFM is not its “real” topography, and the perceived

cell “height” is significantly reduced since the Z-piezo has to push the cantilever into the

cell significantly to reduce the amplitude to the setpoint value. Thus the material

1A Z

1A

1A




contrast channels such as and 0A 1φ are not taken at a constant (or approximately)

cantilever height Z over the sample, as is the case with stiff or moderately stiff samples

(modulus>10MPa). So while on moderately or very stiff materials (glass, mica, purple

membrane) in liquids where the measured topography is close to the real one, the 0th

harmonic measures local conservative interactions, on very soft materials such as live

cells, the dominant contribution to the 0th harmonic map arises from the fact that the

measured topography itself depends strongly on local conservative and dissipative

properties, and this effect dominates the 0th harmonic map. As a consequence for soft

materials such as live cells (modulus 1-1000kPa), the best interpretation of the 0th

harmonic maps is simply that it is a measure of the average force needed to be applied

to the cantilever in order to reduce its amplitude to the setpoint amplitude.

To understand this better we have performed simulations of AFM microcantilever

dynamics on cells in liquids. We have used VEDA 2.0 - the virtual environment for

dynamic AFM, developed by the lead author’s group and available online on

www.nanohub.org. These simulation tools have been validated against experimental

data for tip simulations in liquid environments as described on the manual available

online. The simulations use single or multi-mode (or degree of freedom) cantilever

models with correct effective stiffness and mass parameters and, and use

experimentally measured Q-factors to account for hydrodynamics. In particular we have

used the following simulation parameters for a single mode cantilever model for a

magnetically excited Olympus TR400 cantilever with SiN tip:

1 1 10.08 / , 5 95%, 2 (8000) / , 1.720 , 260 , .25, 0.3

spfar far dr far far

tip tip tip sample

k N m A nm A rad s QR nm E GPa

ω ω πν ν

= = = = =

= = = =

, /0

A =



http://www.nanohub.org/


where Etip and νtip , νsample are respectively the elastic modulus of the tip and Poisson’s

ratio of the tip and the sample. The tip-sample interaction model is Hertz contact with

Kelvin-Voigt viscoelastic dissipation, so that the sample properties are represented by

Esample (kPa) and the sample viscosity by the viscosity parameter μsample (Pa-s). In these

simulations, we varied Esample from 50kPa to 500 MPa, and μsample from 0.001 Pa-s to 1

Pa-s, and calculate the mean deflection A0 at the Z distances where the amplitude

reduces to the 95% amplitude set point. The results are shown in Fig. S4.

Fig. S4 shows clearly that on harder (Esample >1MPa), A0 is strongly correlated to

local sample elasticity or local conservative interactions as the Eq. 8(a) would suggest.

However for softer, more viscous samples (Esample <500kPa), A0 becomes much more

sensitive to local dissipative interactions, μsample. The reason is not that Eq. (S8a) is

incorrect; on the contrary the simulations show that Eq. (S6) is an excellent

approximation of tip motion. A0 does measure local conservative interactions; however

for soft materials the Z distance at which the setpoint amplitude is achieved does

depend on a combination of local viscosity (dissipative properties) and elasticity

(conservative properties). As a result for soft materials such as cells in liquids, contrasts

in A0 appear both because the Z height depends strongly on the local viscoelasticity of

the cell, and also because the local conservative properties themselves change.




q

D. Extraction of local material properties from 0th, 1st, and 2nd harmonic

observables

We now present in further detail the propsoed method to quantify the local

mechanical properties by combining the 0th, 1st and 2nd harmonic data on live cells.

First, let us annotate the dynamic tip indentation into the sample as

( ) ( )t Zδ = − + (S11)

and the average tip indentation as

0 (Z A0 )δ = − + . (S12)

Next in recognition of the experimental observation that the tip oscillation is much

smaller compared to the net average indentation 0δ on live cells we describe the

interaction forces as a Taylor series in 0δ δ− to 2nd order to be consistent with the 2nd

harmonic description of the tip motion:

( )0

2 30 0 0 0

1( ) ( ) ( ) ( ) ,2

samplets ts sample sample

kF F k c O

δ δ

3δ δ δ δ δ δ δ δ δδ

=

∂= + − + − + + −

∂ (S13)

where (N/m) and (N-s/m) respectively are the conservative force gradient

(stiffness) and damping at that particular indentation value.

samplek samplec

0

,sample

sample

kk δ

δ δδ

=

∂=

∂is the

2nd gradient of the interaction force with respect toδ and is a measure of the

nonlinearity of interaction forces in a cycle of oscillation. , samplek ,samplek δ and

typically change depending on the mean force applied. This Taylor series expansion

samplec

0tsF




is only valid when the tip oscillation amplitude is small compared to the length scale of

the interaction forces, or in this case the mean indentation 0δ .

It is interesting to note that the quadratic terms in (S13) do not contain a

dissipative term. The reason for this is as follows – an interaction force term of the type

2δ would actually not be a truly dissipative force since it would act opposite the tip

velocity δ only for half the oscillation cycle, for the other half of the oscillation cycle it

would actually act in the direction of tip velocity δ . The first truly dissipative nonlinear

term in the Taylor series expansion (S13) would be a cubic term and is therefore not

included in the present analysis which only includes those nonlinear terms that influence

the 0th, 1st, and 2nd harmonics.

Substituting (S6) in (S11, S12) and substituting the resulting expression in (S13)

we evaluate the Fourier coefficients of the interaction force in terms of the local

properties:

0, 0

1, 1

1, 1

2 2, 2 1 2 2 1 2

2, 2 1 2 2

, 1

( ) ( )

( )

( )1( ) sin(2 ) 2 cos(24

( ) cos(2 ) 2 sin(2

ts CONS ts

ts CONS sample

ts DISS sample dr

ts CONS sample sample dr sample

ts DISS sample sample dr

a F F

b F k A

c F c A

d F k A c A k A

e F k A c A

δ

δ

ω

φ φ ω φ

φ φ ω φ

=

= −

= −

= − − − − −

= − − + −

(S14)

1

)φ

φ2 )

Finally we combine Eqs (S10) and (S14) to yield the expressions that connect

the 0th, 1st, and 2nd harmonic observables to the local material properties, and take into

account the near-surface hydrodynamic corrections required when these data are

acquired with triangular cantilevers with short tips in liquids:





0 0

1 11 1 1

1

1 11 1 1

1

22 1 2 2 1 2 , 1

( ) ( )

( ) cos( ) cos( )

( ) sin( ) sin( )

1( ) sin(2 ) 2 cos(2 )4

ts

farsample near

far near

farsample dr near

far near

sample sample dr sample

a F kA

kA Ab k AQ A

kA Ac c AQ A

d k A c A k A

kA

δ

δ

φ φ

ω φ φ

φ φ ω φ φ

=

⎛ ⎞− = − +⎜ ⎟

⎝ ⎠⎛ ⎞

− = − +⎜ ⎟⎝ ⎠

− − − − −

= 1 12 1 2 1 1 1 2

1 1

2 1 2 2 1 2

1 12 1 2 1 1 1

1 1

4 2cos(2 ) cos( ) 3 sin( )sin(2 )

( ) cos(2 ) 2 sin(2 )

4 2sin(2 ) cos( ) 3 sin( )cos(2

near nearnear near

sample sample dr

near nearnear near

A AA A

e k A c A

A AkAA A

φ φ φ φ φ φ

φ φ ω φ φ

φ φ φ φ φ

⎡ ⎤⎛ ⎞− − −⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦− − + −

⎛ ⎞= − − +⎜ ⎟

⎝ ⎠2 )φ

⎡ ⎤−⎢ ⎥

⎢ ⎥⎣ ⎦

−

(S15)

Eqs (S15b) and (S15c) can be used to make maps of and , while Eq (S15d)

can be used to extract the 2nd order force gradient (or equivalently the stiffness gradient)

samplek samplec

,samplek δ . Eq. S15e arising from is superfluous since the expansion (S13) did not

contain any nonlinear (quadratic) dissipative term as explained earlier. In the absence of

such a term Eq. S14e effectively fixes the phase of the second harmonic

2,ts DISSF

2φ based on

the linear stiffness and damping properties of the sample.

Thus the observables and can be used to determine the effective

sample stiffness and damping at a mean indentation, can be used to determine

the 2nd order conservative force gradient (or stiffness gradient), while the observable

measures the force needed to maintain a constant amplitude reduction due to

local stiffness and damping. The fact that at each point on the image we know the mean

force applied and can extract the effective sample stiffness and damping allows us to

estimate quantitatively the local mechanical properties such as the local elastic modulus

1,ts CONSF 1

,ts DISSF

2,ts CONSF

0,ts CONSF



so long as an analytical tip-sample interaction model is prescribed. For example, in the

case of a Hertz contact model with viscoelasticity:

3/24 * ( ) ,30,

tsF E R Z q when Z q

otherwise

= − − +

=

0<

(S16)

where *2 2

( ) ( )(1 ) (1 ) (1 )

storage losssample sample dr sample dr

sample sample sample

E E EE

ω ων ν ν

= = +− − − 2i is the complex effective sample

modulus (since the tip elastic modulus (SiN) is 5-6 orders of magnitude larger than that

of a live cell and thus can be considered essentially rigid) consisting of a storage and a

loss modulus representing the linear viscoelasticity of the sample evaluated at an

average indentation depth below the cell surface. Using the same small oscillation

assumptions as above, we find

δν

ω δν

=−

=−

1/202

1/202

2 ( )(1 )

2 ( )(1 )

storagesample

samplesample

losssample

sample drsample

Ek R

Ec R

(S17)

From (S16) and (S17) it is easy to find that:

( )

( )

2/32

00 ,

3/21/3 1

,1/32 0

1 ,

1/3 1,

12 01 ,

(1 )34

1 4(1 ) 2 3

1 4(1 ) 2 3

samplets CONSstorage

sample

storagesample ts CONS

sample ts CONS

losssample ts DISS

sample ts CONS

FR E

E FR

A F

E FR

A F

νδ

ν

ν

⎛ ⎞−= ⎜ ⎟

⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠

⎝ ⎠

⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠

3/2

/3

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

(S18)




These formulas clearly link the observables in a tapping mode scan , and

to quantitative local mechanical properties such as local and and

the mean indentation

1,ts CONSF

storagesample

1,ts DISSF

losssampleE0

,ts CONSF E

δ0 at which these are evaluated in the image. Clearly under the

basic assumptions of the theory presented here, the 2nd order force gradient ,samplek δ is

not required to determine and in the simple Hertz contact model. However

as other tip-sample models with additional unknown parameters are needed,

storagesampleE loss

sampleE

,samplek δ can

provide the necessary additional equation to solve for such a constitutive parameter.

So far we have only provided the Hertz contact model as an example, however it

should be clear that any contact mechanics model of an indentor on an

elastic/viscoelastic medium (such as in the Oliver-Pharr indentation model) can be used

since the only requirement for the method is the prescription of local stiffness, stiffness

gradient and damping coefficient, in terms of local constitutive properties.

In the next section we compare the elastic moduli and local stiffness obtained

while using the above multi-harmonic method and from the conventional pointwise

force-distance force spectroscopy.

E. Comparison of mechanical properties extracted from quasi-static F-Z curves

and the multi-harmonic method

In order to compare the mechanical properties extracted using the new multi-

harmonic method with those extracted using conventional F-Z curves, we made a

careful comparison of the two methods for fibroblasts and for bacterial cells. F-Z curves

during approach and retraction were repeated many times at slow speeds near the




center of the bacteria or cell at a point indicated by a cross in the Figs. S5 and S6.

These data were converted into Force indentation curves and either a linear stiffness

model or a Hertz contact model with tip radius of 45 nm and sample Poisson’s ratio of

0.3 were used to fit the measured curve. The Z position at contact is solved as a part of

the fitting process as described earlier in Radmacher et alS2.

Strictly speaking, the new multi-harmonic method for extracting local elastic

properties is so different from the conventional method using F-Z curves that comparing

the extracted values using the two methods is not really justifiable. For example in the

case of F-Z curves, one matches the entire force-distance curve to a model starting

from the first point of contact, while in the new method one tracks the local effective

force gradients at a specific mean indentation value which changes from point to point

on the image. Secondly the effective properties using the new method correspond to

viscoelastic properties measured at much higher frequencies than the conventional F-Z

curves which are performed at much smaller frequencies. Because viscoelastic

properties of biomaterials are strongly frequency dependent, it is only natural that the

values extracted using the new multi-harmonic method be different from those of the

quasi-static method. Nonetheless it is instructive to examine these differences for the

bacterial and fibroblast samples.

In Fig. S5b, for an E. Coli bacterium, the local stiffnesses are typically in the range

0.1-0.2 N/m. These maps reflect the internal turgor pressure in these cells as well as

local mechanical properties of the peptidoglycan network S3. The F-Z curves are

repeated on the equator of this bacterium (Fig. S5c) and the results are converted to

local elastic stiffness, revealing stiffness in the range 0.015-0.025 N/m, nearly an order




of magnitude softer than those obtained using the multi-harmonic method. This

surprising result is resolved by taking into account the strongly viscoelastic response of

the bacterial cell wallS4 which stiffen substantial when probed at higher frequencies. In

this case the cantilever is driven at ~12kHz at its resonance.

In Fig. S6, we consider data collected using a different set of fibroblast cells than

those presented in the main text. These cells were imaged towards the end of the

experimental period when many cells die. As can be seen the local elastic modulus

maps taken at a drive frequency of ~8kHz show values in the range 50-200kPa.

When compared with values extracted from repeated F-Z curves, we find a value

near the center of the cell in the range 80-100kPa, which is nearly half of what is

indicated at the same point on the cell using the new method.

storagesampleE

storagesampleE

In this case the extracted value from F-Z curves is itself quite high compared

to prior works on fibroblasts (Table S1), and by itself is somewhat lower than the value

extracted from the new method. That these modulus values are stiffer than those

indicated by prior AFM work on fibroblasts can be understood by the fact that the cell

properties vary a lot depending on their stage in the life cycle. These cells in particular

are older and closer to death since they were imaged towards the end of the

experimental period. Nonetheless we find consistently that the using the new

method typically is larger than that from quasi-static F-Z curves due to the natural

frequency dependent viscoelasticity of such samples.

storagesampleE

storagesampleE




Table S1. Summary of prior work using different experimental modalities to measure

the local elasticity of rat fibroblasts.

Experimental Modality Estimated modulus References

Rotation of ferromagnetic beads bound to cell membrane

0.1 - 1 kPa Eckes et al., 1998 S5

Quasi-static AFM indentation 3 – 30 kPa Rotsch, et al., 1999 S2 Thermal excitation of fluorescent microspheres

0.2 – 0.3 kPa Kole et al., 2005 S6

Quasi-static AFM indentation 5-30 kPa Solon et al., 2007 S7 Magnetic tweezers 1-10 kPa Klemm et al. 2010 S8 Force Mapping mode 1-150 kPa Haga et al. 2000 S9

It is also instructive to compare the elastic moduli reported on live fibroblasts

using AFM and other methods such as torsion of magnetic beads that probe the local

membrane properties (Table S1). It is clear that methods that locally probe the

mechanics of the membrane report much lower elastic moduli than AFM based methods

that are based on nanoindentation normal to the surface. This suggests that the cells

exhibit fundamentally different material properties across hierarchical length scales.

F. Additional Figures

In Fig. S7 we provide images taken with TR800 cantilevers in the acoustic mode

instead of the magnetically excited levers discussed in the main text, showing that the

multi-harmonics can easily be observed using the acoustic mode excitation also.

However the conversion of these maps into quantitative maps of local stiffness or

damping cannot be achieved since equations (S3, S4) do not hold for acoustically

excited cantilevers due to spurious resonances that change the shape (transfer

function) of the cantilever resonance S10.




In Figs. S8 and S9 we provide further material maps of E. Coli taken with Lorentz

force excited TR800 cantilevers to show that the 0th, 1st and 2nd harmonic channels do

indeed pick out distinct contrasts in local material properties on E.coli bacteria. First the

local material properties (stiffness, damping, 2nd order force gradients) on these

samples are similar to those presented in Fig. 2 in the main text. Moreover one can

clearly see the influence of the moving flagella of the bacteria in Fig. S9. In both Figs.

S8 and S9 one generally sees material property contrasts far from edges so that tip

convolution effects are not likely to play a role in these contrasts.

G. Additional data on red blood cells sample preparation

The characteristic biconcave morphology of live red blood cells (RBC) is highly

sensitive to the imaging buffer used. In this work we conducted AMPLITUDE

MODULATED AFM on live cells in phosphate buffered saline (PBS, 1x), and found RBC

to be semi-spherically shaped (Fig. 4 in text). This morphology has previously been

reported under similar conditionsS11, and we confirm this effect by optically imaging

fresh RBC in a range of PBS buffer concentrations and in Fetal Bovine Serum (Sigma-

Aldrich, Dorset, UK) as a physiological control (Fig. S10). RBC in serum exhibited the

expected biconcave shape (a), while cells in PBS 0.5x were swollen (b), due to the

hypotonic conditions. Cells in PBS 1x (c) exhibited a predominantly crenate morphology

as a result of hypertonic conditions, however cells with a biconcave morphology can

also be observed. PBS 10x (d) induced a higher level of hypertonicity with all cells

severely crenate, expected given the much higher salt concentration. Imaging in PBS 1x

therefore precluded the biconcave morphology in the majority of cells. We suggest the




morphology observed under AFM is due to the combined effects of buffer hypertonicity

and adhesion between RBC and the polylysine surface (e).




References for supplementary material

[S1] X. Xu, C. Carrasco, P. J. de Pablo, J. Gomez-Herrero, A. Raman, “Unmasking

imaging forces on soft biological samples in liquids: case study on viral capsids”,

95, 9520, Biophys. J., 2008.

[S2] C. Rotsch, K. Jacobsen, M. Radmacher, “Dimensional and mechanical dynamics of

active and stable edges in motile fibroblasts investigated by using atomic force

microscopy”, Proc. Natl. Acad. of Sci., 96(3), 921, 1999.

[S3] M. Arnoldi, M. Fritz, E. Bäuerlein, M. Radmacher, E. Sackmann, and A. Boulbitch,

“Bacterial turgor pressure can be measured by atomic force microscopy”, Phys.

Rev. E, 62(1), 1034, 2000.

[S4] V. Vadillo-Rodriguez, J. R. Dutcher, “Dynamic viscoelastic behavior of individual

gram negative bacterial cells”, Soft Matter, 5, 5012, 2009.

[S5] B Eckes, D Dogic, E Colucci-Guyon, N Wang, A Maniotis, D Ingber, A Merckling, F

Langa, M Aumailley, A Delouvee, V Koteliansky, C Babinet, and T Krieg,

“Impaired mechanical stability, migration and contractile capacity in vimentin-

deficient fibroblasts”, J. Cell Sci., 111, 1897, 1998.

[S6] T. P. Kole, Y. Tseng, I. Jiang, D. Wirtz, “Intracellular mechanics of migrating

fibroblasts”, Mol. Biol. Cell, 16(1), 328, 2005.

[S7] J. Solon, I. Levental, K. Sengupta, P.C. Georges, P. A., Janmey, “Fibroblast

adaptation and stiffness matching to soft elastic substrates”, Biophys. J., 93(12),

4453, 2007.




[S8] A. H. Klemm, S. Kienle, J. Rheinlander, T. E. Schaffer, W. H. Goldmann, “The

influence of Pyk2 on the mechanical properties in fibroblasts”, Biochemical and

Biophys. Res. Comm., 393 (4), 694, 2010.

[S9] H. Haga, S. Sasaki, K. Kawabata, E. Ito, T. Ushiki, T. Sambongi, “Elasticity

mapping of living fibroblast by AFM and immunofluorescence observation of the

cytoskeleton”, Ultramicroscopy, 82, 253-258, 2000.

[S10] X.Xu, A. Raman, “Comparative dynamics of magnetically, acoustically, and

brownian motion excited microcantilevers in liquid atomic force microscopy”,

J. App. Phys., 102(3), 2007.

[S11] S. Sen, S. Subramanian, D.E. Discher, “Indentation and adhesive probing of a cell

membrane with AFM: theoretical model and experiments”, Biophys. J., 89(5),

3203-13, 2005.




A0 A0

A0

Figure S1: The A0 map acquired on the rat fibroblast cell of Fig. 3 of the main text shown on a larger scale along with image zoom-ins clearly resolves the actin bundles and cytoskeletal features.





Figure S2: a, Topography images of two live rat tail fibroblast cells scanned with magnetically excited cantilever (Imaging conditions and parameters k=0.065N/m, wfar=7.4kHz, Qfar=1.7, Afar=22.5 nm, Anear=4.512 nm, setpoint ratio: 70%, scan rate: 0.25Hz/ line) show little contrast associated with the cytoskeleton. b, On the other hand A0 maps acquired simultaneously with the topography show clear contrasts in local material properties associated with the cytoskeleton. c, A zoom in of the material contrast maps clearly show the actin bundles and cytoskeletal features at high resolution. Length scale bar is 10 microns.



Figure S3. a, A schematic of an oscillating cantilever showing the key motion/displacement variables. the schematic emphasizes that while operating in liquids it becomes necessary to account for the average deflection of the tip A0, which is generally comparable to the setpoint amplitude of the drive harmonic A1. b, A schematic shows the time history of tip motion (in terms of θ ) to leading order, along with the conservative and non-conservative tip-sample interaction forces encountered by the tip during this motion.




Figure S4. a, A graph of A0 vs sample elastic modulus Esample (kPa) and the sample viscosity μsample (Pa-s) can be computed for a TR400 cantilever tapping on samples in liquid environments using the Virtual Environment for Dynamic AFM (VEDA 2.0 on www.nanohub.org) software. b, The same results can be shown as graphs of A0 vs μsample (Pa-s) for different Esample values and demonstrate that for moderate to high elastic stiffness samples A0 generally depends on local conservative (elastic stiffness) properties and is independent of viscosity, but for soft materials A0 also begins to depend on the local viscosity.



http://www.nanohub.org/


Figure S5. a, Topography image of an E. coli cell taken using magnetic (Lorentz force) excitation with a TR800 lever showing a well-defined bacterial cell. b, Image showing the corresponding map of local spring constant of sample derived from the A0, A1, φ maps as described in the text. c, A typical graph of quasi-static Force-Z response on the center of the E. coli cell as shown by the cross in a, d-e, Histograms showing the variations of local spring constant derived from multiple replicates of force-Z curves at a point indicated by “X” in a from both tip approach (or Z piezo extension) and tip retraction data.




Figure S6. a, Topography image of a rat fibroblast cell cell taken using magnetic (Lorentz force) excitation with a TR400 lever showing three different cells in buffer solution. b, Image of the local storage modulus of sample derived from the A0, A1, φ maps as described in the text. c, Quasi-static Force-Z curves on the center of a cell as shown by the cross in a. d-e, Histograms showing the variation of the local spring constant derived from multiple replicates of force-Z curves at a point indicated by “X” in a from both tip approach (or Z piezo extension) and tip retraction data.




Figure S7. a-e, Images of the topography, φ1, A0, A2, and φ2 of an E. coli cell on polylysine covered mica sample using piezo (acoustic) excitation on a TR800 cantilever clearly show heterogeneities in local mechanical properties. The operating conditions used are described in the methods section of the main text.




Figure S8. a, Topography image of an additional live E. coli cell scan using a magnetically excited Olympus TR800 cantilever (see materials and methods) showing a small bacterial cell. b-c, Maps of the multi-harmonic data (A0, φ1) acquired simultaneously clearly show heterogeneities not captured in the topography. d-f, Maps of mean indentation (nm), local dynamic stiffness (N/m), damping (N-s/m) can be extracted from the multi-harmonic variables using the theory described in the text. g-h, Maps of multi-harmonic data (A2, φ2) acquired with the topography show less contrast over the cell surface. i, These multi-harmonic observables are converted to a map of the local 2nd order force gradient (N/m2) using the theory described in the text. The scale bar represents 500nm, and this is a 256 by 256 pixel image taken in ~5 minutes.




Figure S9 a, Topography image of another live E. coli cell scan using a magnetically excited Olympus TR800 cantilever (see materials and methods) showing a well defined bacterial cell with a emergent flagella near the top of the cell. b-c, Maps of the multi-harmonic data (A0, φ1) acquired simultaneously clearly show heterogeneities not captured in the topography. d-f, Maps of mean indentation (nm), local dynamic stiffness (N/m), damping (N-s/m) can be extracted from the multi-harmonic variables using the theory described in the text. g-h, Maps of multi-harmonic data (A2, φ2) acquired with the topography show less contrast over the cell surface. i, These multi-harmonic observables are converted to a map of the local 2nd order force gradient (N/m2) using the theory described in the text. The scale bar represents 500nm, and this is a 256 by 256 pixel image taken in ~5 minutes.





Figure S10. a, Optical bright field image of red blood cells acquired with 100x oil immersion objective in Fetal Bovine Serum (Sigma-Aldrich, Dorset, UK) show the typical biconcave morphology of the cells. b-d, Dark field images with differential interference contrast (DIC) of the red blood cells in 0.5x PBS, and 1x PBS, and 10x PBS buffers show a different cell morphologies. e, A model of red blood cell morphology when landing on glass in 1x PBS solution. Scale bar: 15 μm.


raman, trigueros et al · raman, trigueros et al on the other hand is the 2nd cosine coefficient...

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