Raman, Trigueros et al · Raman, Trigueros et al on the other hand is the 2nd cosine coefficient...
Transcript of Raman, Trigueros et al · Raman, Trigueros et al on the other hand is the 2nd cosine coefficient...
SUPPLEMENTARY INFORMATIONDOI: 10.1038/NNANO.2011.186
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.
Raman, Trigueros et al
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
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Raman, Trigueros et al
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
Supplementary section page 3
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Raman, Trigueros et al
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)
Supplementary section page 4
© 2011 Macmillan Publishers Limited. All rights reserved.
Raman, Trigueros et al
tsF
Supplementary section page 5
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
© 2011 Macmillan Publishers Limited. All rights reserved.
Raman, Trigueros et al
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)
Supplementary section page 6
© 2011 Macmillan Publishers Limited. All rights reserved.
Raman, Trigueros et al
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
Supplementary section page 7
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Raman, Trigueros et al
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
Supplementary section page 8
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Raman, Trigueros et al
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 =
Supplementary section page 9
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Raman, Trigueros et al
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.
Supplementary section page 10
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Raman, Trigueros et al
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
Supplementary section page 11
© 2011 Macmillan Publishers Limited. All rights reserved.
Raman, Trigueros et al
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:
Supplementary section page 12
© 2011 Macmillan Publishers Limited. All rights reserved.
Raman, Trigueros et al
Supplementary section page 13
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
© 2011 Macmillan Publishers Limited. All rights reserved.
Raman, Trigueros et al
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)
Supplementary section page 14
© 2011 Macmillan Publishers Limited. All rights reserved.
Raman, Trigueros et al
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
Supplementary section page 15
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Raman, Trigueros et al
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
Supplementary section page 16
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Raman, Trigueros et al
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
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Raman, Trigueros et al
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.
Supplementary section page 18
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Raman, Trigueros et al
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
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Raman, Trigueros et al
morphology observed under AFM is due to the combined effects of buffer hypertonicity
and adhesion between RBC and the polylysine surface (e).
Supplementary section page 20
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Raman, Trigueros et al
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.
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Raman, Trigueros et al
[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.
Supplementary section page 22
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Raman, Trigueros et al
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.
Supplementary section page 23
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Supplementary section page 24
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.
© 2011 Macmillan Publishers Limited. All rights reserved.
Raman, Trigueros et al
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.
Supplementary section page 25
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Raman, Trigueros et al
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.
Supplementary section page 26
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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.
Supplementary section page 27
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Raman, Trigueros et al
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.
Supplementary section page 28
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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.
Supplementary section page 29
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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.
Supplementary section page 30
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Raman, Trigueros et al
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.
Supplementary section page 31
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Supplementary section page 32
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.
© 2011 Macmillan Publishers Limited. All rights reserved.